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OPTIMIZATION MODELS FOR CAPACITY PLANNING IN HEALTH CARE DELIVERY By CHIN-I LIN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2008 1
Transcript of OPTIMIZATION MODELS FOR CAPACITY PLANNING IN HEALTH … · 2013-05-31 · incorporate queueing...
OPTIMIZATION MODELS FOR CAPACITY PLANNING IN HEALTH CAREDELIVERY
By
CHIN-I LIN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2008
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c© 2008 Chin-I Lin
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To my family
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ACKNOWLEDGMENTS
I would like to thank all people who have helped and inspired me during my doctoral
study. I want to express my sincere gratitude to my dissertation advisor, Dr. Elif Akcalı,
for her guidance, insight and support during this research and study. I am also grateful
to my committee members, Dr. Farid AitSahlia, Dr. P. Oscar Boykin and Dr. Siriphong
Lawphongpanich, for their constructive suggestions and comments. I wish to extend my
warmest thanks to my mentor, Dr. Shangyao Yan, for leading me to the field of operations
research, his friendship and numerous fruitful discussions. My deepest gratitude goes to
to my parents, Zhe-Xiong and Fang-Xue, and my husband, Chung-Jui. Without their
understanding and encouragement, it would have been impossible for me to complete my
degree.
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TABLE OF CONTENTS
page
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
CHAPTER
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 AGGREGATE HOSPITAL BED CAPACITY PLANNING . . . . . . . . . . . . 13
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.1 Restricted Bed Capacity Planning Problem . . . . . . . . . . . . . . 192.3.2 Restricted Bed Capacity Planning Problem with Shuttering . . . . . 22
2.4 Illustration of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4.1 A Representative Decision-Making Scenario . . . . . . . . . . . . . . 252.4.2 Experiment 1 – An Application of the Model . . . . . . . . . . . . . 262.4.3 Experiment 2 – Assessing the Impact of Problem Parameters . . . . 29
2.5 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.6 Concluding Remarks and Future Research Directions . . . . . . . . . . . . 34
3 HEALTH CARE TEAM CAPACITY PLANNING . . . . . . . . . . . . . . . . 37
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.4 Queueing Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.1 Preemptive Case: Emergency Medicine Services . . . . . . . . . . . 483.4.2 Non-Preemptive Case: Outpatient Clinic Services . . . . . . . . . . 51
3.5 Computational Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.5.1 Computational Performance of the DBA Method . . . . . . . . . . . 573.5.2 Computational Performance of the HCTSCP Model . . . . . . . . . 62
3.6 Concluding Remarks and Future Research Directions . . . . . . . . . . . . 66
4 HOSPITAL BED ALLOCATION PROBLEM . . . . . . . . . . . . . . . . . . . 69
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.4 Solution Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
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4.4.1 Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.4.2 Greedy Randomized Adaptive Search Procedure . . . . . . . . . . . 804.4.3 Hybridization of GA & GRASP . . . . . . . . . . . . . . . . . . . . 81
4.5 Computational Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.5.1 Summary of Results Obtained by GA . . . . . . . . . . . . . . . . . 854.5.2 Summary of Results Obtained by GRASP . . . . . . . . . . . . . . 884.5.3 Summary of Results Obtained by HA . . . . . . . . . . . . . . . . . 88
4.6 Concluding Remarks and Future Research Directions . . . . . . . . . . . . 92
5 EMERGENCY ROOM SERVICES FACILITY LOCATION AND CAPACITYPLANNING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.4 Solution Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.4.1 Lagrangian Relaxation Approach . . . . . . . . . . . . . . . . . . . 1005.4.2 Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.4.3 Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.4.4 Lagrangian Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.5 Computational Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.5.1 Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.5.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.6 Concluding Remarks and Future Research Directions . . . . . . . . . . . . 111
6 CONCLUSIONS AND FUTURE RESEARCH DIRECTIONS . . . . . . . . . . 114
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
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LIST OF TABLES
Table page
2-1 Parameter settings for the base scenario, S1 . . . . . . . . . . . . . . . . . . . . 27
2-2 Scenario descriptions for experiment 1 . . . . . . . . . . . . . . . . . . . . . . . 27
2-3 Summary statistics for the RBCPwS problem’s solution time (in CPU seconds)as a function of initial effective capacity . . . . . . . . . . . . . . . . . . . . . . 31
2-4 Summary statistics for the RBCPwS problem’s solution as a function of capacitylevels and the length of the planning horizon . . . . . . . . . . . . . . . . . . . . 32
3-1 The possible transitions enter state (n1, n2,m) for the ED setting . . . . . . . . 49
3-2 Computational requirement of the DBA, AGE, and GE methods . . . . . . . . . 58
3-3 Relative and percentage error: preemptive case . . . . . . . . . . . . . . . . . . 60
3-4 Relative and percentage error: non-preemptive case . . . . . . . . . . . . . . . . 61
3-5 Team configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3-6 Parameter settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3-7 Impact of the fraction of class 1 patients on the similarity index . . . . . . . . . 65
3-8 Impact of unit patient delay cost on the similarity index . . . . . . . . . . . . . 65
3-9 Impact of maximum allowable average time in system on the similarity index . . 65
4-1 Near-optimal solutions obtained by using CPLEX . . . . . . . . . . . . . . . . . 86
4-2 Solutions obtained by GA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4-3 Solutions obtained by GRASP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4-4 Solutions obtained by HA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5-1 Experimental factor settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5-2 Experiment 1: effects of maximum number of facilities opened . . . . . . . . . . 109
5-3 Experiment 2: effects of capacity setting . . . . . . . . . . . . . . . . . . . . . . 110
5-4 Experiment 3: effects of diversion probability . . . . . . . . . . . . . . . . . . . 110
5-5 Experiment 4: effects of time value . . . . . . . . . . . . . . . . . . . . . . . . . 111
5-6 Performance of the heuristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
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LIST OF FIGURES
Figure page
2-1 Network flow representation for RBCP with c0=300, B=25, n=1, and T=4 . . . 21
2-2 Network flow representation for RBCPwS with c0=275, B=25, n=1, and T=4 . 24
2-3 Patient arrival rate for experiment 1 . . . . . . . . . . . . . . . . . . . . . . . . 28
2-4 Optimal capacity plans for experiment 1 . . . . . . . . . . . . . . . . . . . . . . 28
2-5 Number of nodes in the network as a function of initial effective bed capacity . . 30
3-1 An illustration of the network representation for HCTSCP . . . . . . . . . . . . 47
3-2 Two-dimensional CTMC approximation . . . . . . . . . . . . . . . . . . . . . . 52
4-1 Pseudo-code of the genetic algorithm . . . . . . . . . . . . . . . . . . . . . . . . 76
4-2 Pseudo-code of the population generating procedure . . . . . . . . . . . . . . . . 77
4-3 Pseudo-code of occupancy-driven allocation . . . . . . . . . . . . . . . . . . . . 77
4-4 Pseudo-code of random-rectified procedure . . . . . . . . . . . . . . . . . . . . . 78
4-5 Pseudo-code of crossover procedure . . . . . . . . . . . . . . . . . . . . . . . . . 79
4-6 Example of crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4-7 Pseudo-code of the mutation procedure . . . . . . . . . . . . . . . . . . . . . . . 80
4-8 Pseudo-code of GRASP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4-9 Pseudo-code of greedy randomized construction procedure . . . . . . . . . . . . 82
4-10 Pseudo-code of local search procedure . . . . . . . . . . . . . . . . . . . . . . . . 83
4-11 Pseudo-code of HA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4-12 Pseudo-code of elite set generation procedure . . . . . . . . . . . . . . . . . . . 84
4-13 The updating functions of α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5-1 Pseudo-code of the Lagrangian relaxation . . . . . . . . . . . . . . . . . . . . . 101
5-2 Pseudo-code of the feasible solution generation . . . . . . . . . . . . . . . . . . . 104
5-3 Pseudo-code of covering all demand nodes . . . . . . . . . . . . . . . . . . . . . 105
5-4 Pseudo-code of closing facilities . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5-5 Convergence speed of the modified LR . . . . . . . . . . . . . . . . . . . . . . . 109
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Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy
OPTIMIZATION MODELS FOR CAPACITY PLANNING IN HEALTH CAREDELIVERY
By
Chin-I Lin
May 2008
Chair: Elif AkcalıMajor: Industrial and Systems Engineering
Health care capacity planning is the art and science of predicting the quantity of
resources required to deliver health care service at specified levels of cost and quality.
Because of variability in the arrival of patients and in the delivery of health care services,
successfully meeting the demand for health care services is a daunting task that requires
an understanding of the inherent trade-off between its cost and quality of service.
In our work, we model the general health care systems as queueing stations and
incorporate queueing theory into an optimization framework. The queueing modeling
approach captures the stochastic nature of arrivals and service times that is typical in
health care systems. The optimization framework determines the minimum cost capacity
required to achieve a target level of customer service. The inclusions of queueing equations
and discrete capacity options result the capacity planning models in non-linear integer
programming formulations.
We develop effective solution algorithms to obtain high quality solutions particularly
for realistic-sized problems. For the analysis of underlying queuing systems, we either
use available results from the literature or develop approximations. For the solution of
optimization models, we employ network optimization, meta-heuristic, and Lagrangian
relaxation approaches to develop effective solution algorithms. We present results from
extensive computational experiments to demonstrate the computational efficiency and
effectiveness of the proposed solution approaches.
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CHAPTER 1INTRODUCTION
Capacity planning decisions are important to any industry, especially to health care
industry because not only it relates to the management of highly specialized and costly
resources (i.e., nurses, doctors, and advanced medical equipment), but also it makes a
difference between life and death in critical conditions. Health care capacity planning
involves predicting the quantity and particular attributes of resources required to deliver
health care service at specified levels of cost and quality.
According to the resource availabilities, capacity planning can be classified into three
levels including strategic, tactical and operational levels. In the strategic level, decision
makers focus on the long term capacity decisions such as locations of medical facilities or
sizes of medical facilities and workforce. The tactical level capacity plan concerns policies
which could improve the service performance, on the perspective of either health care
providers or receivers, by reallocating, expanding or downsizing the current resources.
In contrast, the capacity plan on the operational level concentrates on how to meet
demand by using the existing resources through appropriate methods such as scheduling or
overrun.
In our study, we focus on long-term level health care capacity planning decisions,
where the knowledge of the performance of the system at steady state is sufficient. In
addition, we model the health care service facilities as queueing systems to take the
stochastic patient arrivals and length of stay into account and measure the system
performance. The purpose of our study is to incorporate queueing models into optimization
framework to determine the optimal capacity level that minimizes cost while maintaining
a desired level of performance on patients’ service quality and financial and/or operational
restrictions.
In our work, we study four different, practically relevant, long-term capacity planning
problems. Chapter 2 introduces the aggregate hospital bed capacity planning (AHBCP)
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problem, in which we model the hospital as a G/G/c queueing system with a single bed
type and a single patient class. Using the approximation from Bitran and Tirupati [16, 17]
to estimate the patients’ expected waiting time, we determine the optimal bed capacity
plan over a finite planning horizon utilizing a network flow approach.
Chapter 3 presents the health care team capacity planning (HCTCP) problem,
which finds applications in hospital emergency room and outpatient clinical settings.
The underlying queueing system is more complex than the one used for AHBCP. In
particular, we consider a system with two classes of patients with different priorities and
two types of health care teams with patient class-dependent service rates. Moreover, there
is asymmetric substitutability between the teams, i.e., while one team can provide service
to both classes of patient, whereas the other team can serve only one class of patient. For
this queueing system, we develop an approximation approach to compute the average time
that a patient spends in the system for each patient class. Then, we integrate the results
from approximation method into an optimization model to make long-term health care
team capacity decisions.
Chapter 4 states the hospital bed allocation (HBA) problem, which is an extension the
AHBCP problem. After the aggregate bed capacity is specified, the next step involved is
concerned with the allocation of aggregate bed capacity among different medical care units
(MCUs). We model each MCU in a hospital as a M/M/c/c queueing system to estimate
the probability of rejection when there are c beds in the unit. We develop an optimization
model to allocate the aggregate bed capacity across different MCUs.
Chapter 5 details the emergency room services facility location and capacity planning
(ERSFLCP) problem, in which each facility is modeled as a M/M/c/c queueing system.
We construct a facility location model, which simultaneously determines the number
of facilities opened and their respective locations as well as the capacity levels of the
facilities so that the probability that all servers in a facility are busy does not exceed a
pre-determined level.
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Last, Chapter 6 provides a summary of the four problems we have investigated and
suggests future research directions.
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CHAPTER 2AGGREGATE HOSPITAL BED CAPACITY PLANNING
2.1 Introduction
Capacity planning is central to the pursuit of balancing the quality of health care
delivered with the cost of providing that care. Such planning involves predicting the
quantity and particular attributes of resources required to deliver health care service at
specified levels of cost and quality. In general, successful health care capacity planning
must address a variety of issues, including the duration of the planning horizon (i.e.,
operational, tactical, and strategic), the level of care provided (i.e., primary, secondary,
and tertiary), the type of care (i.e., inpatient and/or outpatient), the amount, capability,
cost, and types of available or desired resources (i.e., doctors, nurses, technicians, medical
and clinical support staff, facilities including buildings, rooms, beds, parking spaces,
medical diagnostic and monitoring equipment, or any other element that constitutes
an “input” to the delivery of health care) as well as the customer service metrics or
performance measures expected for the facility (e.g., patient length of stay, likelihood
of full capacity where all inpatient beds or examining rooms are occupied, utilization of
providers and facilities, and financial performance such as having expenses within or below
budget).
While capacity planning has challenged health care decision makers and researchers
for decades [90, 91, 101], there is a renewed sense of urgency to address this problem.
In addition to the perennial struggle between the continually increasing costs of highly
specialized and scarce inputs (i.e., skilled and flexible staff, advanced clinical and medical
technology and equipment, physical space and supplies) and declining government
and private reimbursements [89, 96], the demand for inpatient care has been growing
substantially. According to the American Hospital Association (AHA), while average
length of stay (ALOS) remained unchanged at 5.7 days, all community hospital volume
statistics increased from 2002 to 2003: inpatient admissions by 0.9% to 34.8 million,
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total hospital-based outpatient visits by 1.2% to 563.2 million, emergency department
visits by 1.0% to 111.1 million, adjusted average daily census (i.e., average number of
inpatients and outpatients receiving care per day) by 0.9% to 894,000, and average
inpatient occupancy rate increased by 1.9% to 66.8% [7]. However, the number of hospitals
of all types decreased by 30 to 5,764, there were 32 fewer community hospitals, and 8,000
fewer community hospital beds in 2003 [7].
In this paper, we focus on aggregate hospital bed capacity planning decisions. We
develop a model to simultaneously determine the timing and magnitude of changes in bed
capacity that minimizes capacity cost (including the cost of changing capacity as well as
the cost of operating capacity) while maintaining a desired level of facility performance
(e.g., limiting a patient’s expected delay before being admitted to a bed and keeping
expenses within budget) over a finite planning horizon. We divide the planning horizon
into discrete time periods of equal length, and assume that the system achieves steady
state in each of these intervals. This allows us to use queuing methodology to analyze
system performance, but this typically leads to nonlinear equations in our formulation. As
hospital bed capacity must be integer valued, our planning model is a large-scale nonlinear
integer optimization model that minimizes total cost while achieving a targeted level of
system performance. We show that some practical considerations lead to simplifications in
the model, which leads to a network flow formulation for the problem that can be solved
in polynomial time.
A variety of problems that arise in the context of transportation, finance, manufacturing,
and service systems can be modeled as network flow models [2]. A network is a collection
of (capacitated or uncapacitated) nodes and (directed/undirected and capacitated/uncapacitated)
arcs, where the arcs link one node to another and carry flow from one node to another.
Well-known network flow models are the shortest path, maximum flow, and minimum
total cost flow formulations, for which efficient solution algorithms exist [2]. In our work,
we show that the capacity planning model we consider can be transformed into a shortest
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path model, where the objective is to find the path from the source node to the sink node
with the shortest length (i.e., the minimum cost bed capacity plan from the current period
to the final period of a given planning horizon).
The remainder of this paper is organized as follows. Section 2.2 provides a brief
overview of the history and current research in hospital bed planning. In Section 2.3, we
describe the system and give three models for planning hospital bed capacity. In Section
2.4, using data from a medium-sized medical center, we provide a computational study to
illustrate how the model formulations can be used and how changes in problem parameters
can affect our ability to obtain an optimal solution. Section 2.5, offers several practical
extensions of our model. Last, we give concluding remarks and discuss future research
directions in Section 2.6.
2.2 Literature Review
During the 1990s, many hospitals in the United States reported having too many
beds and were exploring strategies to reduce space [10, 15, 29, 45, 46, 50]. Less than a
decade later, in part due to renewed growth in demand for inpatient services [7, 31], most
hospitals are currently facing considerable space and resource restrictions forcing them
to contemplate expensive renovations and/or new construction projects to increase bed
capacity [11, 31, 57]. However, whether hospitals, in fact, need the additional capacity
appears to be unresolved [10, 45]. On one hand, increased inpatient admissions coupled
with fewer hospitals and fewer hospital beds would support the argument in favor of
capacity increases [7, 31]. Conversely, level or decreasing average length of stay and a
corresponding decrease in the average inpatient occupancy rate may imply that existing
capacity is sufficient [10]. Regardless, determining the optimal number and organization of
hospital beds continues to be a challenge.
The ability to anticipate bed demand and match it with the appropriate bed supply
is critical to effective bed planning. Health care decision makers know that both will
be influenced by a number of factors. Factors internal to the decision makers include
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containing the costs associated with operating, contracting, and expanding current bed
capacity, reducing bed assignment waiting, maintaining quality of care when patients
are placed in inappropriate units (e.g., an intensive care patient may have to be placed
in a cardiac unit), eliminating emergency department bottlenecks (i.e., keeping patients
in the emergency department after initial treatment due to unavailability of beds in
the appropriate care unit), and reducing the probability of diverting patients to other
hospitals due to lack of bed capacity [45, 46]. Externally, factors facing decision-makers
include atypical changes in community health (e.g., severe flu strains), annual holidays
(e.g., Thanksgiving), and the availability, size, and composition of appropriate medical
personnel. Historically, starting with the Hill–Burton Act of 1946, bed capacity planning
has tended to be based on target occupancy levels (TOLs) that are assumed to reflect
capacity levels that achieve an appropriate balance of cost, patient delays, and resource
utilization. TOLs are derived using analytic models of typical hospitals in different
categories and are based on acceptable patient delays for different services. However,
Green and Nguyen [46] use queuing models to investigate the relationship between
occupancy levels and delay, and concluded that using TOLs as the primary determinant of
bed capacity is inadequate and may lead to excessive delays for beds. In particular, a TOL
does not necessarily correspond to a desired service level, and there is a need to quantify
the desired service level and measure its cost implications accurately.
Ryan [95] provides a capacity expansion model with exponential demand and
continuous time intervals and continuous facility sizes. In the context of health care
planning, however, it is more realistic to model capacity expansion as the product of
limited, discrete choices as routine planning sessions (e.g., bimonthly or quarterly) where
capacity increases or decreases occur in some fixed bed amount such as a 20-bed unit.
Bretthauer and Cote [26] model a general health care delivery system as a network of
queuing stations and incorporate the queuing network into an optimization framework to
determine the optimal capacity levels subject to a specified level of system performance
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(e.g., average total time spent at the facility). They use an algorithm combining
branch-and-bound with outer approximation cutting plane method to solve the nonlinear
optimization problem with discrete variables, but a disadvantage of this algorithm is that
in the worst case the algorithm could require complete enumeration of all integer solutions,
leading to very large solution times.
2.3 Problem Formulation
In the bed capacity planning problem, we start with a planning horizon of length
T indexed by t=1, 2, ..., T . Let λt denote the aggregate patient arrival rate in period
t, 1/µ be the ALOS per patient, and the service rate per bed per day is given by µ or
1/ALOS and the service rate per bed over period t is µt. In practice, there are alternative
patient streams (including admissions from the emergency department, admissions from
referrals, and elective admissions) for each of which the typical length of stay may be
different. As the objective of our work is to provide an aggregate planning tool for bed
capacity management, we assume that the arrival rates for different patient streams can be
combined and a representative value for the average length of stay per patient (regardless
type of services required by the patient) can be determined. Note that while ALOS has
been relatively stable over time [7], the actual λt for a given facility will not be known
until the demand presents itself. Therefore, for the purposes of capacity planning, λt
can be forecasted by a seasonally adjusted trendline, for example [32]. Let αt denote the
maximum allowable expected delay for a patient before the patient is admitted to a bed in
period t. We note that the number of beds in the system in a given period can be limited
due to other resource limitations including as the physical size of the facility and/or the
amount and type of personnel available. Let c0 be the initial bed capacity in the hospital.
Last, there is a budget limit on the amount of monetary resources that can be allocated to
purchasing additional bed capacity denoted by γt.
We have three types of decision variables. Let xt be number of beds in period t. Let
x+t be the amount of increase in bed capacity at the beginning of period t, and x−t the
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amount of decrease in bed capacity at the beginning of period t. Let f(xt, λt, µt) denote
the expected patient waiting cost as a function of number of beds xt, patient arrival
rate λt, and average service rate µt in period t. Similarly, let g(xt−1, xt) denote the cost
of changing bed capacity from xt−1 to xt (i.e., the cost of increasing or decreasing the
existing bed capacity) in period t. Let h(xt) denote the cost of operating xt beds in period
t. Finally, the expected delay for a patient in period t is a function of number of beds
xt, patient arrival rate λt, and service rate per bed µt, denoted by w(xt, λt, µt). We can
formulate the aggregate hospital bed capacity planning (AHBCP) problem as a nonlinear
integer programming formulation as follows:
minT∑
t=1
f(xt, λt, µt) +T∑
t=1
g(xt−1, xt) +T∑
t=1
h(xt) (2–1)
subject to
w(xt, λt, µt) ≤ αt ∀ t (2–2)
x0 = c0 (2–3)
xt−1 + x+t − x−t = xt ∀ t (2–4)
g(xt−1, xt) ≤ γt ∀ t (2–5)
xt, x+t , x−t are discrete varibles ∀ t (2–6)
The objective function (2–1) minimizes the total cost of patient waiting, changing
the bed capacity, and operating the existing bed capacity. Constraint (2–2) imposes
a maximum allowable limit on the expected patient waiting. For example, in order to
quantify the expected delay for a patient to be admitted to a bed, we assume that the
hospital can be represented as a GI/G/s queueing system and use the expected waiting
time approximation provided by Bitran and Tirupati [16, 17] to calculate a patient’s
expected wait for a hospital bed. Constraint (2–3) sets the initial bed capacity while
constraint (2–4) is a flow balance equation stating that the number of beds available in a
period is equal to the number of beds available in the previous period plus the increase in
bed capacity minus the decrease in bed capacity. Constraint (2–5) is the budget constraint
18
that limits the amount of the funds allocated to changing capacity. Last, constraint (2–6)
ensures that the number of beds available and changes in bed capacity are integer valued.
2.3.1 Restricted Bed Capacity Planning Problem
It should be readily apparent that the number of integer variables associated with the
AHBCP problem could be quite large as there is no restriction on how many beds can be
added or removed from service. For example, community hospitals may have 500 or more
beds [7]. In practice, bed capacity is increased or decreased in batches, and is typically
changed in integer multiples of a base value, say, in multiples of 10 or 25 corresponding
to the size of a unit. As a result, there are only a limited number of choices for changing
capacity in each period. Therefore, constraints that capture the change in capacity can be
replaced by a set of discrete alternative constraints, requiring that only one alternative is
chosen in the solution for each period. Then, the original non-linear integer programming
problem becomes a nonlinear binary (i.e., zeroone) integer programming problem, which
we refer to as the restricted bed capacity planning (RBCP) problem.
In the RBCP problem, we are given a base value of B in multiples of which the bed
capacity can be increased or decreased and we let n be the number of possible distinct
levels of capacity increase or decrease. That is, given bed capacity c in period t, the bed
capacity in period t + 1 can be one of (c−nB)+, (c− (n− 1)B)+, ..., (c−B)+, c, c + B, ...,
c+(n− 1)B, c+nB, where (x)+ = max{0, x}. We assume that all acquired new additional
capacity is available and becomes effective capacity in the same period. Let z+it = 1 if the
available bed capacity is increased by iB at the beginning of period t for i=1, 2, ..., n; and
0 otherwise. Similarly, let z−it = 1 if the bed capacity is decreased by iB at the beginning
of period t for i=1, 2, ..., n; and 0 otherwise. We can now formulate the RBCP problem as
a nonlinear zeroone integer programming problem as follows:
minT∑
t=1
f(xt, λt, µt) +T∑
t=1
g(xt−1, xt) +T∑
t=1
h(xt) (2–7)
subject to
19
w(xt, λt, µt) ≤ αt ∀ t (2–8)
x0 = c0 (2–9)
xt−1 +n∑
i=1
iBz+it −
n∑i=1
iBz−it = xt ∀ t (2–10)
n∑i=1
z+it +
n∑i=1
z−it ≤ 1 ∀ t (2–11)
g(xt−1, xt) ≤ γt ∀ t (2–12)
xt ≥ 0 ∀ t (2–13)
z+it , z
−it ∈ {0, 1} ∀ i, t (2–14)
As in the AHBCP problem, objective function (2–7) minimizes the total cost of
patient delay, changing the bed capacity, and operating the existing bed capacity,
constraint (2–8) imposes a maximum allowable limit on the expected patient delay,
constraint (2–9) sets the initial bed capacity, and constraint (2–10) is a flow balance
equation. Constraint (2–11) ensures that only one choice for changing the capacity is
allowed in each period. Constraint (2–12) imposes the budget constraint on the amount
of money allocated to changing bed capacity. Constraints (2–13) and (2–14) ensure the
nonnegativity of the bed capacity level and capacity level selection decision variables,
respectively.
An attractive feature of the RBCP problem is that a network representation can be
developed. Consider a T -partite graph with T layers each representing a time period t=1,
2, ..., T in the planning horizon. Let (t, c) denote the system when there are c beds in
period t. Let C(c) be the set of reachable capacity levels in the next period if the capacity
in the current period is c, and we have C(c) = {(c − nB)+, (c − (n − 1)B)+, ..., (c −B)+, c, c + B, ..., c + (n − 1)B, c + nB}. Let St be the set of all capacity levels reachable
in period t from all capacity levels in period t − 1. Let ds be a superficial source node
connected only to node (0, x0) with zero arc length. Node (0, x0) represents the beginning
state where there are x0 beds in the hospital at time t=0. Let (0, x0) be connected to all
nodes (1,x’) for x’ ∈ C(x0). If w(x’, λ1, µ1) ≤ α1 (i.e., the patient waiting time constraint
is not violated) and g(x0, x’) ≤ γ1 (i.e., the budget constraint is not violated), then the
20
length of these arcs are given by f(x’, λ1, µ1) + g(x0, x’) + h(x’) (i.e., the expected patient
waiting cost with x’ beds in the system, total cost of changing the bed capacity from x0 to
x’, and cost of operating x’ beds). However, if either constraint is violated, then the length
of the corresponding arc is set to M , where M is a very large number. Similarly, let each
node (t, x) for x ∈ St and t=1, 2, ..., T -1 be connected to (t + 1, x’) for x’ ∈ C(x) with
length f(x’, λt+1, µt+1) + g(xt, x’) + h(x’) if w(x’, λt+1, µt+1) ≤ αt+1 and g(xt, x’) ≤ γt+1,
and M otherwise. Last, let each node (T, x) for x ∈ ST be connected to a superficial sink
node dt with an arc of zero length.
Figure 2-1 provides an example of the network representation for the RBCP problem
where c0=300, B=25, n=1, and T=4. In this figure, a path from the superficial source
node to the superficial sink node represents a plan for the bed capacity over the planning
horizon. The shortest path without containing any arc with cost M yields the capacity
plan with total minimum cost that obeys the patient waiting time and budget constraints
over the planning horizon. If no such path can be found (i.e., the shortest path contains at
least one arc with cost M), then the problem is infeasible and no capacity plan that obeys
the waiting time and budget constraints over the planning horizon can be found.
ds 0,300
1,325
1,300
1,275
2,350
2,325
2,300
3,375
3,350
3,325
2,275
2,250
3,300
3,275
3,250
4,400
4,375
4,350
4,325
4,300
4,275
4,250
4,225
4,200
3,225
dt
Figure 2-1. Network flow representation for RBCP with c0=300, B=25, n=1, and T=4
21
Recalling that there are n distinct levels to increase or decrease capacity, the general
network flow representation representation for the RBCP problem has 2nt + 1 nodes in
layer t for t=1, 2, ..., T . Therefore, there are a total of nT (T + 1) + T + 2 nodes (including
the superficial sink and source nodes) and the shortest path for the RBCP problem can be
found in O(n2T 4) time using Dijsktra’s algorithm [2].
2.3.2 Restricted Bed Capacity Planning Problem with Shuttering
In the RBCP problem, we assume that the cost of increasing or decreasing bed
capacity is uniform. In practice, however, decreasing bed capacity can be achieved by
shuttering existing bed capacity. That is, a hospital unit is closed and the personnel may
be reassigned to other units in the hospital or laid off, thereby, reducing the effective bed
capacity. On the other hand, increases in bed capacity can be accomplished two ways. If
the existing capacity is larger than the effective capacity implying that shuttered capacity
is available, then restoring a shuttered unit into operation by reallocating personnel to
this unit can increase bed capacity. However, if the existing capacity is equal to the
effective capacity implying that no shuttered capacity is available, then bed capacity
can only be increased through a capital investment to open a new unit and purchase
new beds. We can incorporate this practical concern into our formulation easily by
changing the definition of the objective function by keeping track of the effective and
existing bed capacity in the hospital. We now distinguish between two types of capacity
changes, where g(x0|x’, x|x’) is the cost of changing effective capacity from x0 to x’ via
shuttering and the existing bed capacity from x to x’ via acquiring additional capacity
where x’ ≥ max{x’, x}. As before, we assume that all acquired new additional capacity
becomes effective capacity in the same period. The formulation is still a nonlinear zeroone
integer programming problem, which we refer to as the restricted bed capacity planning
with shuttering (RBCPwS) problem.
As with the RBCP problem, a network representation can be developed for the
RBCPwS problem. Consider a T -partite graph with T layers each representing a time
22
period t=1, 2, ..., T in the planning horizon. Let (t, c|c) denote an effective capacity
of c and an existing capacity of c in time period t, and c ≤ c. Let C1(c|c) denote the
set of reachable capacity levels via shuttering only, C2(c|c) the set of reachable capacity
levels by acquiring new additional capacity and C(c|c) = C1(c|c) ∪ C2(c|c) the set of
all reachable capacity levels in the next period if the effective and existing bed capacity
in the current period are c and c, respectively. If we have c + nB ≤ c, then we have
C1(c|c) = {((c− nB)+|c), ..., ((c−B)+|c), (c|c), (c + B|c), ..., (c + nB|c)} and C2(c|c) = {∅}.Also, if we have c ≤ c ≤ c+nB, we have C1(c|c) = {((c−nB)+|c), ..., ((c−B)+|c), (c|c), (c+
B|c), ..., (c|c)} and C2(c|c) = {(c+B|c), ..., (c+nB|c)}. Again, let ds be a superficial source
node connected only to node (0, x0|x) with zero arc length. Node (0, x0|x) represents
the beginning state where there are x beds in the system and x0 in operating condition
at t=0. Let (0, x0|x) be connected to all nodes (1, x’|x) in C(x0|x). Provided both the
patient waiting time constraint and the budget constraint are not violated, then the
length of these arcs are given by f(x’, λ1, µ1) + g(x0|x, x’|x’) + h(x’) (i.e., the expected
patient waiting cost with x’ beds in the system, total cost of changing the effective bed
capacity from x0 to x’ via shuttering and the existing bed capacity from x to x’ via new
bed acquisition, and cost of operating x’ beds). If either of these constraints is violated,
then the length of the corresponding arc is set to M . Similarly, each node (t, x|x) for
(x|x) ∈ St and t=1, 2, ..., T -1 be connected to all nodes (t+1, x’|x’) in C(x|x’) with length
f(x’, λt+1, µt+1)+g(x|x, x’|x’)+h(x’) if w(x’, λt+1, µt+1) ≤ αt+1 and g(x|x, x’|x’) ≤ γt+1, and
M otherwise. Finally, let each node (T, x|x) for (x|x) ∈ St be connected to a superficial
sink node dt with length zero.
Figure 2-2 provides an example of the network representation for the RBCPwS
problem where c0=275, c0=300, B=25, n=1, and T=4. For ease of exposition, the thin
arcs represent opening, maintaining, or shuttering of existing capacity, whereas the thick
arcs represent the acquisition of new capacity. In this network, a path from the superficial
source node to the superficial sink node represents a plan for the bed capacity throughout
23
the planning horizon. As before, the shortest path without containing any arc with
cost M in the network yields the capacity plan with total minimum cost that obeys the
patient waiting time and budget constraints throughout the planning horizon by allowing
capacity changes via shuttering and/or acquiring additional capacity. If no such path can
be found (i.e., the shortest path contains at least one arc with cost M), the problem is
infeasible and no capacity plan that obeys the waiting time and budget constraints over
the planning horizon can be found.
ds 0,275,300
1,300,300
1,275,300
1,250,300
2,325,325
2,300,300
2,275,300
3,325,325
3,300,325
3,300,300
2,250,300
2,225,300
3,275,300
3,250,300
3,225,300
4,325,325
4,300,325
4,275,325
4,300,300
4,275,300
4,250,300
4,225,300
4,200,300
4,175,300
3,200,300
dt
3,350,350
4,375,375
4,350,350
4,325,350
Figure 2-2. Network flow representation for RBCPwS with c0=275, B=25, n=1, and T=4
Recalling the RBCP problem, we have specified the number of arcs and nodes in
the network to determine the time to obtain the optimal solution. However, for the
RBCPwS problem, since existing capacity can be increased further through capital
acquisition, the analysis becomes slightly more complicated and dependent on the initial
state (i.e., the amount of effective and existing bed capacity). If no additional capacity
24
has to be purchased throughout the planning horizon, then the RBCPwS and RBCP
networks are identical and the size of the RBCP network is a lower bound on the size
of the RBCPwS network. If additional capacity has to be purchased in a period, at
most (t − 1)n2 nodes can be added to the network in that period. Hence, if the initial
effective capacity is equal to the existing capacity, then there can be at most a total of
nT (T + 1) + T + 2 + T (T + 1)(T − 1)n2/6 nodes (including the superficial sink and source
nodes) in the network, and the shortest path can be found again in O(n4T 6) time using
Dijsktra’s algorithm [2].
2.4 Illustration of the Model
In this section, we illustrate the practical applicability and computational behavior of
our model through two experiments. In the first experiment, we illustrate how our model
can be used to develop bed capacity plans. In the second experiment, we quantify the time
(in CPU seconds) needed to obtain optimal solutions. In both experiments, we use the
RBCPwS formulation and its associated network model.
2.4.1 A Representative Decision-Making Scenario
To set the stage for the computational experiments that follow, we present a
representative decision-making scenario based upon a real-world application of our model
to a medium-sized, non-government, not-for-profit, general medical and surgical medical
center. Administration at this facility provided us with information about their facility,
capacity planning decision-making processes, and facility-specific data for bed size, bed
operating cost, bed acquisition cost, and quarterly patient demand. However, note that at
the request of the facility’s administration, the data presented here have been modified to
protect their identity, but are representative of similar-sized facilities.
This facility would like to determine an optimal bed capacity plan for the next eight
quarters, corresponding to its operational, budgetary, and strategic planning periods.
Because capacity planning may involve a substantial capital commitment, it is imperative
that any capacity expansion plan be carefully developed and justified based upon the
25
facility’s current and expected demand. The facility’s decision makers would like to
minimize the total capacity cost associated with the cost of changing capacity as well
as the cost of operating capacity while ensuring that the average time a patient should
wait for a bed does not exceed one hour (an internal benchmark for bed assignment).
At this facility, both existing and effective bed capacities are 350 beds, capacity change
can occur in increments of 10 beds, and there are two levels of capacity increase (i.e.,
initially, bed capacity can range from 330 to 370 beds, in 10 bed increments). Based
on information from the facility’s administration, it costs $2,000/day to operate an
effective bed, $2,500/day to either shutter an effective bed or reactivate a shuttered bed,
and $200,000/bed to expand bed capacity through capital investment. Last, because
of seasonal migration (or “snow birds”), demand at the facility can be highly variable
throughout the year, and we were provided with data and guidance on values related to
patient arrival rates and service times.
2.4.2 Experiment 1 – An Application of the Model
The intent of this experiment is to illustrate how our network flow model can be used
to make bed capacity decisions and generate a T -period capacity plan. Our base scenario
was described in Section 2.4.1, and we refer to it as S1. Table 2-1 lists the relevant
parameter settings for S1, and other experimental scenarios relative to this scenario are
given in Table 2-2.
At the outset, we provide an estimated range of demand for the facility over the
planning horizon. Normally, a single seasonally adjusted trendline would be computed
to forecast the patient arrival rate based on historic demand data. Instead, to illustrate
the extent of variation in demand, Figure 2-3 displays a set of simulated patient arrival
rates over the planning horizon based upon the scenarios given in Table 2-2. We note
that some of the parameter changes directly impact the patient arrival rate, and different
patient arrival rates are generated. In S1, S3, S4, S5, and S6, the changed parameters do
not impact the arrival rate function, so these scenarios have identical arrival rates. (Note
26
Table 2-1. Parameter settings for the base scenario, S1
Parameter ValueLength of the planning horizon T = 8 quarters, t= 1, 2, ..., 8
Forecasted demand per time period t λt = smod(t,4)u(a + bt) where si is aquarterly seasonal index (i.e., s1=0.8,s2=1.0, s3=1.2, and s4=1.0), u is auniformly distributed random number(i.e., u ∼ U [0.8, 1.2]), a=6,400, and b=128
Initial existing bed capacity c0 = 350Initial effective bed capacity c = 350Number of levels of capacity increase ordecrease
n = 2
Incremental amount of capacity change B = 10Cost to operate an effective bed $2,000/bedCost to shutter an effective bed $2,500/bedCost to reactivate a shuttered bed $2,500/bedCost to acquire a new bed $200,000/bed(i.e., expand capacity through capitalinvestment)Coefficient of variation for arrivals cat = 0.5Coefficient of variation for service cst = 0.5Maximum expected delay per patient at = 1 hourCost of waiting $300/hourService rate µt = 15.8 patients per bed
Table 2-2. Scenario descriptions for experiment 1
Scenario Description Parameter changeS0 Level demand b = 0S1 Base scenarioS2 Increased rate of demand b = 256S3 Higher demand variability cat = 2.0S4 Higher service variability cst = 2.0S5 Higher cost of waiting per patient $1,200/hourS6 Smaller maximum expected delay per patient αt = 0.25 of an hour
that with S3, higher variability in the arrival rate impacts the performance constraint for
average waiting time, not the arrival rate function.) For S0 and S2, the patient arrival rate
function has no trend and a higher trend compared to S1, respectively. Hence, arrival rates
generated for these scenarios are significantly different from each other and S1.
We have implemented our network flow approach using the C++ programming
language and solved for the scenarios using a personal computer with 3.0 GHz Pentium IV
27
3000
4000
5000
6000
7000
8000
9000
10000
11000
1 2 3 4 5 6 7 8
Patie
nt a
rriv
al r
ate
(pat
ient
s/qu
arte
r)
Time periods
S0S1, S3, S4, S5, S6S2
Figure 2-3. Patient arrival rate for experiment 1
processor and 512 MB RAM memory. We obtained the optimal solution for each scenario
and the results are depicted in Figure 2-4, where each line represents the optimal capacity
plan that corresponds to one of the seven scenarios.
0 1 2 3 4 5 6 7 8
S0 350 330 310 290 270 250 230 250 250
S1 350 330 310 320 300 280 290 300 310
S2 350 330 310 320 300 310 330 350 370
S3 350 330 310 320 300 280 300 310 320
S4 350 330 310 320 300 280 300 300 320
S5 350 330 310 330 310 290 300 310 320
S6 350 330 310 320 300 280 300 300 310
200
225
250
275
300
325
350
375
400
Num
ber
of b
eds
Time periods
S0
S1
S2
S3
S4
S5
S6
Figure 2-4. Optimal capacity plans for experiment 1
28
In considering Figure 2-4, we have the following observations. For S1, we first observe
a general reduction in the bed capacity, then a gradual increase near the end of the
planning horizon. The initial bed capacity seems to be higher than needed, and as a
result, the bed capacity is reduced to reduce total costs over the planning horizon while
maintaining the average waiting time constraint. Of course, when the demand increases
due to the underlying trend, the bed capacity is increased. When demand is level as in
S0, a lower envelope is formed relative to the base case (i.e., the bed capacity for S0 is less
than or equal to the base case). Similarly, with an increased rate of demand as in S2, an
upper envelope is formed relative to the base case. With increased variation as in S3 and
S4, the optimal capacity plans are similar to S1’s capacity plan but tend to require higher
capacity when the arrival rate is increasing. When the arrival rate increases in periods
6, 7, and 8, because the higher arrival variability and higher service variability affect the
average waiting time constraint, more capacity is required to keep from violating this
performance constraint. Likewise, with a higher cost of waiting per patient as in S5 or a
tighter average waiting time performance constraint as in S6, the optimal capacity plans
tend to require capacity slightly higher than the base case. Not surprisingly, the net result
of this experiment indicates that optimal bed plans are driven substantially by changes in
demand. While health care decision makers may not be able to affect overall demand for
their services, if they can reduce variability in arrivals [78] or are willing to tolerate a less
stringent performance constraint, less capacity will be required.
2.4.3 Experiment 2 – Assessing the Impact of Problem Parameters
As we have discussed earlier, an upper bound on the size of the network (i.e., number
of nodes in the network) representing a problem instance of RBCPwS can be characterized
in terms of the number of levels for capacity increase or decrease and the number of time
periods in the planning horizon. The ratio of the effective bed capacity to existing bed
capacity impacts the size of the network. The size of the network can also be used to
quantify the computing time required to obtain the optimal solution. The time required
29
to build the network and find the optimal solution may change as the number of levels
increases, the planning horizon length increases, or the ratio of effective to existing bed
capacity changes. In order to illustrate the change in computational time, this experiment
has two parts: 1) the impact of effective to existing bed capacity and 2) the impact of
changes to the number of levels of bed capacity and the length of the planning horizon.
In the first part of this experiment, we fix the number of levels to vary bed capacity
and the duration of the planning horizon in addition to some other problem parameters
constant and examine the impact of different ratios of existing to effective bed capacity.
Using the assumptions for the base case scenario, S1, from the previous experiment, we
consider ten different levels of the effective bed capacity in the interval [260, 350]. We
generated 30 random test instances for each of these levels and the summary results are
provided in Figure 2-5 and Table 2-3.
0
100
200
300
400
500
600
250 260 270 280 290 300 310 320 330 340 350 360
Num
ber
of n
odes
in t
he n
etw
ork
Initial number of effective beds
Figure 2-5. Number of nodes in the network as a function of initial effective bed capacity
In Figure 2-5 we depict the number of nodes in the network, and in Table 2-3 we
report the time to build the network and time to obtain the solution for each level of the
initial effective bed capacity. The number of nodes increases as the ratio of effective bed
capacity to existing bed capacity approaches one, and this behavior is clearly depicted
in Figure 2-5. However, in Table 2-3, we see that an increase in the size of the network
30
increases the time to build the network only slightly, and its impact on the time to obtain
the solution is almost negligible. Therefore, our solution method is robust to changes in
the problem size that are induced by the initial effective bed capacity.
Table 2-3. Summary statistics for the RBCPwS problem’s solution time (in CPU seconds)as a function of initial effective capacity
Initial level Time (in CPU seconds) to Total timeof effective Build the netowrk Obtain the solution (in CPU seconds)
bed capacity Min. Avg. Max. Min. Avg. Max. Min. Avg. Max.260 0.0 0.0 0.2 0.0 0.0 0.0 0.0 0.0 0.2270 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.1280 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.1290 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.1300 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.1310 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.1320 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.1330 0.0 0.1 0.1 0.0 0.0 0.0 0.0 0.1 0.1340 0.1 0.1 0.1 0.0 0.0 0.0 0.1 0.1 0.1350 0.1 0.1 0.1 0.0 0.0 0.0 0.1 0.1 0.1
For the second part of this experiment, we vary the number of levels to change bed
capacity as well as the duration of the planning horizon. We consider four different levels
to vary the bed capacity (i.e., n=2, 3, 4, 5) where capacity is increased in increments of
B=10, and three different time horizons (i.e., T=8, 12, 16) that correspond to two-, three-,
and four-year planning horizons. Therefore, we have 12 settings in total. For each setting,
we generated 30 random instances and for each of the instances, we also generated the
effective bed capacity as a fraction of the existing bed capacity. The summary results for
this experiment are provided in Table 2-4.
From Table 2-4, as the number of levels of capacity change and the length of the
planning horizon increases, the number of nodes in the network increases. The increase
in the number of nodes impacts the total time required to obtain the optimal solution.
However, a closer examination of the results reveals the increase in the number of nodes in
the network has a direct impact on the time required to build the network, and has almost
no impact on the time to obtain the solution. Only in the setting with the largest test
instances (i.e., n=5 and T=16) do we observe an increase in the time to obtain the
31
Tab
le2-
4.Sum
mar
yst
atis
tics
for
the
RB
CP
wS
pro
ble
m’s
solu
tion
asa
funct
ion
ofca
pac
ity
leve
lsan
dth
ele
ngt
hof
the
pla
nnin
ghor
izon
(n,T
)N
umbe
rof
Tim
e(i
nC
PU
seco
nds)
toTot
alti
me
(in
CP
Use
cond
s)N
odes
inth
ene
twor
kA
rcs
inth
ene
twor
kB
uild
the
netw
ork
Obt
ain
the
solu
tion
Avg
.(M
in.,
Max
.)A
vg.
(Min
.,M
ax.)
Avg
.(M
in.,
Max
.)A
vg.
(Min
.,M
ax.)
Avg
.(M
in.,
Max
.)(2
,8)
323.
7(1
94,49
0)1,
213.
9(7
37,1,
865)
0.04
(0.0
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32
optimal solution. Even in that case, the maximum solution time is still less than a few
seconds. Therefore, our solution method is robust to changes in the problem size induced
by the number of levels of capacity change and the duration of the planning horizon.
2.5 Extensions
In this section, we discuss several extensions to our model. These extensions may
arise out of practical considerations associated with how our model addresses facility
performance.
In our model, we treat the performance constraints as a hard constraint. That is,
if a particular capacity level violates the performance constraint, then a solution with
that particular capacity level is not feasible, and is dropped from further consideration.
However, the performance constraint can be modeled as a soft constraint where we can
deliberately allow the violation of the performance constraint while incurring a penalty
cost to be added to the objective function. We can justify this constraint by noting that
lags typically exist between capacity levels so there might be periods of time where the
facility is operating above its typical utilization and the capacity expansion cannot occur
quickly enough to allow the organization to react to the change in demand. To illustrate
how our model can be reformulated with the soft constraint, let vt be the amount of the
violation, st be the amount of slack in the performance constraint, and π(vt) be thepenalty
cost incurred for violating the performance constraint in period t. Then, considering the
RBCP problem, objective function (2–7) would be replaced with:
minT∑
t=1
f(xt, λt, µt) +T∑
t=1
g(xt−1, xt) +T∑
t=1
h(xt) +T∑
t=1
π(vt) (2–15)
Similarly, constraints (2–8) and (2–13) would be replaced with:
w(xt, λt, µt)− vt + st = αt ∀ t (2–16)
vt = max{w(xt, λt, µt)− αt, 0} ∀ t (2–17)
st = max{αt − w(xt, λt, µt), 0} ∀ t (2–18)
xt, vt, st ≥ 0 ∀ t (2–19)
33
It is easy to observe that this modified version of the RBCP problem can still
be formulated and solved as a network flow problem. The variables vt and st are
calculated by constraints (2–17) and (2–18), respectively, once w(xt, λt, µt) is known.
The only modification of the network is to include the cost associated with violating the
performance constraint.
When evaluating a hospital, we recognize that the average waiting time to be assigned
a bed or having expenses within budget are not the only metrics to assess facility
performance. Indeed, it may be necessary to include measures for facility utilization,
likelihood of patient diversion, and the like. Regardless, we note that more performance
constraints can easily be added to the formulations for the BCP, RBCP, and RBCPwS
problems. An increase in the number of performance constraints does not increase the
time to obtain the solution significantly, as there is only a need to take these additional
constraints into account in setting up the network and assigning a large arc cost in
case any of the constraints are violated. Therefore, our modeling approach is robust
and additional constraints can be considered without increasing the complexity of the
formulation significantly.
2.6 Concluding Remarks and Future Research Directions
We have presented a network flow approach to optimize bed capacity planning
decisions for hospitals. Our model incorporates the reasonable concerns associated
with determining hospital bed size, such as a finite planning horizon, an upper bound
on the average waiting time before a patient is admitted to a hospital bed, and a
budget constraint that limits the amount of money that can be allocated to changing
bed capacity. Further, our model accommodates capacity change through shuttering,
as well as expansion of bed capacity through new capital investment. Our series of
computational experiments illustrated both the ease of implementation of our model and
the sensitivity of the computational time required to obtain the optimal solution to several
34
problem parameters. We have also discussed extensions of our model in the form of soft
performance constraints and multiple performance constraints.
Our model is based on a generic view of a hospital where we have assumed that
the demand (i.e., patient arrivals) and service (i.e., beds) components are homogeneous.
From an aggregate planning perspective, such uniformity may be acceptable. However, in
order to apply this research to operational decision support for health care delivery, there
are additional avenues of research worth pursuing. First, if cost depends on all previous
stages, for example, the cost of maintaining the beds depends not only on the number
of beds but also the duration of the beds are in the system, then the number of vertices
in the network will be exponential with respect to T and the optimal solution to the
network will not be solved with a polynomial time algorithm. Consequently, alternative
model formulations and solution techniques to determine the optimal bed plan would
be necessary. Second, recognizing that hospital beds are not identical, facility capacity
could be separated to distinguish the various specialties, with specialty-specific demand
rates, lengths of stay, and costs. In determining the average waiting time associated
with being assigned a bed, we have used closed-form approximations to calculate this
statistic. Therefore, we are implicitly assuming that this general distribution accounts
for different types of patients that require different types of hospital-based health care.
This may not necessarily be the case, and should be investigated further. Third, our work
can be expanded to include multiple types of patients (e.g., electives, admissions coming
through the emergency department, and referrals from physicians). Also, in estimating
the cost of patient waiting, we assume that this cost is identical regardless of patient
type. Clearly, for example, there should be different waiting costs associated placing an
emergency department admission in an appropriate unit versus an inappropriate unit.
As such, representations of patient waiting cost need to be developed in the presence
of congested, heterogeneous resources. Fourth, the current form of our model does not
account for the potential time delay that may exist between the decision to expand
35
capacity and actually starting to use the new capacity. Our current model formulations
would have to be amended to include the length of delay relative to a capacity expansion
(e.g., if a capacity expansion requires k time periods and we need to use the capacity in
period t, the decision to expand should occur on or before period t − k) and reconciliation
of multiple capacity expansions over the planning horizon (e.g., if a capacity expansion
decision is made in period t, can the facility make another decision in subsequent periods
until t + k when the earlier decision comes into effect). However, because these capacity
expansion considerations would destroy the underlying polynomially bounded network
structure of the current model, other solution methodologies would have to be developed.
Last, as evidenced by the current nurse shortage [3] and the ongoing debate regarding
nurse-to-patient ratios [4], the ability to use physical capacity hinges upon the availability
of suitable medical personnel. A natural extension of our model would be to incorporate
workforce planning to simultaneously determine the quantity and composition of the
health care resources to construct a comprehensive capacity plan.
36
CHAPTER 3HEALTH CARE TEAM CAPACITY PLANNING
3.1 Introduction
With a size of $1.9 trillion and growth rate of 7.9 percent [102], the health care
industry accounts for the largest sector of the economy in the United States (US). Despite
advances in medical technology and, thereby, the increasing use of medical diagnostic,
monitoring, and treatment equipment, the health care industry is highly labor-intensive.
According to the US Department of Labor, the health care industry provided 13.5 million
jobs in 2004, out of which 13.1 million jobs are for wage and salary workers and about
411,000 are for the self-employed [107]. It follows that personnel wages and salaries
account for the largest portion of the total expenditures for any health care facility. For
instance, hospitals spend on average about 54 percent of all expenditures on wages and
salaries [86]. Hence, health care personnel planning, i.e., determining the appropriate mix
of health care personnel, needed to provide safe, effective, timely, and cost-efficient service
to patients [55], is an important problem.
In practice, both in inpatient and outpatient facilities, a health care team, comprised
of a group of health care personnel with different, and complementary, skill sets, provides
health care services to individual patients [34]. The members of the team (e.g. physicians,
physician’s assistants, and registered nurses) are responsible to perform a set of tasks
required for the diagnosis, monitoring, and treatment of the patients. Some additional
tasks may have to be performed by other personnel (e.g., laboratory technicians,
radiological technicians, and radiologists) who are not a part of the health care team
but provide assistance for diagnosis and treatment. In the delivery of services by health
care teams, the safety and effectiveness of the service is ensured by the appropriate
selection of the service capability of the team, whereas the timeliness and cost-effectiveness
of the service is ensured by the appropriate selection of the service capacity of the team.
The service capability of a team is characterized by the collection of the skills possessed
37
by each of the individual members of the team, whereas the service capacity of the team is
given by the total number of members included in the team.
In a health care facility, there are typically multiple types of health care teams each
with different capabilities. The patients that arrive to the facility are classified according
to their conditions (i.e., acuity levels) or medical requests. Based on this classification,
each patient is assigned to a health care team and admitted to an examination/treatment
(E/T) room that has the equipment necessary to provide the service needed by the
patient. As the set of tasks are shared among the members of the team and a typical
facility has multiple E/T rooms, a health care team usually serves multiple patients
simultaneously. For instance, while the team waits for the test results from the lab for
a patient, a registered nurse may be collecting specimens from another, a physician’s
assistant may be suturing a wound of another, and a physician accompanied by a
registered nurse may be discussing a treatment plan with another. In our work, we
consider two particular settings where health care services are provided by teams.
Shands at Alachua General Hospital in Gainesville, Florida is a community hospital
that provides emergency medicine services. triage nurse, who determines the acuity level
of the patient and identifies whether the patient requires immediate (i.e., emergency) or
delayed (i.e., urgent) care. The emergency care services are delivered by an emergency
care (EC) team, which is composed of physicians and registered nurses, and the urgent
care services are delivered by an urgent care (UC) team, which is also composed of
physicians and registered nurses. We note that the triage nurse does not belong to either
of these teams, but acts as a gatekeeper to route an arriving patient to either of the teams.
There are eight and two E/T rooms dedicated to the EC and UC teams, respectively.
Moreover, the EC team has the capability to attend to urgent care patients, but the
UC team does not have the capability to attend to emergency care patients. Finally,
emergency care patients have preemptive priority over urgent care patients. That is, if
an emergency care patient arrives to the ED, while all E/T rooms dedicated to the EC
38
team are occupied by other patients and one of them is an urgent care patient, then
the emergency care patient preempts the urgent care patient out of the room and the
emergency care patient is immediately admitted to the room.
The Women’s Clinic at the Student Health Care Center at the University of Florida
in Gainesville, Florida provides women’s health care services. The outpatient clinic
(OC) serves not only non-acute patients, i.e., those who need routine services, but treats
acute patients also. The services for the diagnosis and treatment of acute illnesses and
abnormalities are provided by a physician (P) team, which is composed of a physician and
a physician’s assistant. The routine clinical services are delivered by a nurse practitioner
(NP) team, which is composed of a nurse practitioner and a registered nurse. There are
four E/T rooms dedicated to the NP team and two rooms to the P team. Moreover, the
P team has the capability to attend to non-acute patients but the NP team does not have
the capability to attend to acute patients. Finally, acute patients have non-preemptive
priority over the non-acute patients. That is, if an acute patient arrives to the OC, while
all E/T rooms dedicated to the P team are occupied by other patients and one of them is
a non-acute patient, then the P team does not interrupt service to the non-acute patient
and the acute patient has to wait until an E/T room dedicated to the P team becomes
available.
In the settings we discussed above, the service capabilities of the health care teams
are fixed as dictated by the service needs of the patients, and personnel planning is mainly
concerned with determining the service capacity of the teams. In determining the service
capacity, however, administrators must take several additional facility capacity, budgetary,
and legislative constraints into account that limit the minimum and maximum total
number of each personnel type employed. For instance, budget constraints may limit the
total number of physicians employed, whereas legislated nurse-to-patient staffing ratios
may prescrobe a lower limit on the total number of registered nurses. Therefore, given the
lower and upper bounds on the total number of personnel with different skills that can be
39
employed, there is a finite number of team configurations that can be utilized by a health
care facility. For example, consider a health care team for which physicians and registered
nurses are required. Suppose that due to a budget constraint, the health care facility can
employ at most two physicians and four registered nurses at a time. Also, suppose that
due to a legislative constraint, the facility should employ at least one physician and two
registered nurses. Then, for this team, there are at most six feasible configurations, i.e.,
{(1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4)}, that administrators can choose from. Therefore, in
such settings, personnel planning is concerned with choosing an appropriate configuration
for each type of health care team.
In this paper, motivated by the practical settings we discussed above, we address
the long-term health care team service capacity planning (HCTSCP) problem in the
context of health care facilities where there are two patient classes and two types of teams.
We assume that the capability of each type of team is known, and the number of E/T
rooms allocated to each type of team is fixed over the planning horizon. Therefore, the
service capacity of a team can only be changed by modifying the configuration of the
team. Moreover, the service capacity of a team can be quantified by the service rate of
the E/T rooms dedicated to the team, which is the reciprocal of the average time that the
patients spend in the E/T rooms prior to discharge or transfer to another department.
We formulate a non-linear binary integer programming model to determine the service
capacity plan for the health care teams such that health care services are delivered in a
timely (i.e., the average time a patient from a particular class spends in the system does
not exceed a pre-specified threshold) and cost-effective (i.e., the total costs associated with
changing service capacity by hiring additional personnel, reassigning existing personnel
or laying off existing personnel and operating the service capacity are minimized) manner
over a given planning horizon considering some additional constraints.
To estimate the average time a patient from a particular class spends in the system,
we develop queuing models and decomposition based approximation (DBA) methods.
40
We note that in the practical settings we consider the steady state results obtained from
queuing analysis can be used in capacity planning, as both systems are highly utilized,
and, hence, the systems reach to steady state quickly in both of the settings. To model
these settings, we consider queueing systems where there are two classes of patients and
two types of teams. Each team has a set of of E/T rooms dedicated to it, and there is
asymmetric substitutability between the teams and their dedicated E/T rooms. That is,
class 1 patients (i.e., emergency care patients in the ED and acute patients in the OC) can
only be served in the E/T rooms dedicated to type 1 team (i.e., EC team in the ED and
P team in the OC), class 2 patients (i.e., urgent care patients in the ED and non-acute
patients in the OC) can be served in the E/T rooms dedicated to either type 1 or type
2 team. An arriving class 2 patient is admitted to a vacant type 1 E/T room only when
no class 1 patient waits in the system and no type 2 E/T room is available. In addition,
service rates are patient-class and team-type dependent, i.e., the time required to serve
a patient depends on both the class of the patient being served and the type of the team
that delivers the service. We consider both the preemptive and non-preemptive cases. Our
computational results illustrate that the DBA method and the capacity planning model
can effectively be used to make long-term personnel planning decisions.
The remainder of this paper is organized as follows. In Section 3.2, we review the
related literature. Section 3.3 presents our capacity planning model for HCTSCP and
show how it can be interpreted as a network flow model. We develop queueing models for
the ED and OC settings and present DBA methods to analyze these models in Section
3.4. In Section 3.5, we present results from our computational study that evaluates the
accuracy of the approximations from Section 3.4 and investigate the computational
performance of the formulation presented in Section 3.3. Section 3.6 includes a discussion
of the results and suggests future research directions.
41
3.2 Literature Review
Health care personnel capacity planning has been been studied for decades to address
long-term budgeting, medium-term staffing, and short-term scheduling decisions. A critical
examination of the analytical literature to date illustrates that a considerable amount of
effort has been allocated to the short-term personnel planning decisions. In particular,
the nurse rostering (also known as the nurse scheduling or the nurse staffing) problem
has been studied extensively, as nurse staffing costs account for a significant portion of
personnel costs in health care system [61]. Burke et al. [28] provide an excellent review of
the existing work in this area.
Analytical work that focuses on medium- to long-term personnel planning is relatively
limited in scope. Schneider and Kilpatrick [97] develop optimization models for personnel
planning in health care facilities. Kao and Tung [62] present a linear programming model
for the aggregate (nursing) workforce planning problem, which is later extended by Brusco
and Showalter [27] to account for the exogenous impact of nursing shortage. Kropp and
Carlson [72] propose the integrated use of optimization and simulation modeling. Existing
work that uses simulation modeling is reviewed in Jun et al. [58]. Although the earlier
work in this area primarily focuses on the perspective of the health care provider by
placing an emphasis on minimizing the cost of personnel resources or maximizing the
utilization of personnel resources, there is a need to take the patient’s perspective into
account by considering the timeliness (e.g., minimizing of the waiting time and/or time
in system) or the availability (e.g., minimizing the probability of finding all servers busy
and being diverted to another service provider). To this end, we focus on minimizing the
sum of capacity costs and cost of not providing service in a timely manner, while ensuring
that the average time in system for a patient does not exceed a pre-specified threshold.
To this end, we divide the planning horizon into discrete time periods of equal length and
assume that the system achieves steady state in each of these intervals. This allows us to
use queueing analysis to capture the stochastic behavior of the system and compute the
42
average time in system for the patients. Using the results of this analysis, we formulate
the HCTSCP problem as a mathematical programming model as in [5].
In the literature, there is a number of studies that investigate queueing systems
that are closely related to the setting we consider. Stanford and Grassmann [105] derive
the expected waiting time in a call center with unilingual and bilingual servers serving
majority- and minority-language-use customers. A minority-language-use customer can
only be served by a bilingual server, and the type of a customer is not be known prior to
the first service. The service rates for all server types are the same, i.e., independent of
the customer type. Green [44] and Shumsky [99] also investigate expected waiting time of
a system with two types of customers and limited- and general-use servers. Both of them
consider the case where the service rates depend on the type of the server. Hence, the
distinguishing characteristic of the system we investigate is the dependence of the service
time on customer class for one of the server types. This seemingly a simple attribute of
the system leads to significant modeling challenges.
In queueing theory, to obtain the stationary probabilities of a system, the major
approaches used include the matrix analytic methods [44, 60, 74, 75, 87, 105], power-series
algorithm (PSA) approach [19, 20, 54, 71], and DBA method [99]. The matrix analytic
methods formulate the system as a Markov chain to which the stationary probability π
has the matrix-geometric form πn = π0Rn, where rate matrix R can be obtained through
an iterative algorithm [87]. The PSA approach first represents the stationary probabilities
of a system as power-series expansions of the traffic intensity of the system, and then
recursively solves for the coefficients of the power-series expansions by using the set of
stationary equations. Kao and Wilson [64] compare the performance of the PSA and
matrix analytic methods with three iterative algorithms proposed in [60, 74, 75], and
conclude that the PSA performs extremely well in terms of computational speed, though
it may encounter difficulties in parameter settings which may lead to losses in accuracy.
Shumsky [99] proposes the DBA method. The DBA method first divides the state space
43
of the system into several regions, and then estimates the stationary probabilities in each
region by using simple approximate queueing models. Shumsky [99] illustrates that this
method generates performance measures rapidly with sufficient accuracy that can be used
in call center capacity decisions.In our queueing analysis, we also use this DBA method.
3.3 Problem Formulation
In the HCTSCP problem, we are given a planning horizon of length T , indexed by
t = 1, . . . , T . In each planning period, there is a limit on the funds that can be allocated
to changing service capacity of the teams in the facility, denoted by γt for t = 1, . . . , T .
There are two patient classes, indexed by i = 1, 2. For each patient class there is an
upper limit on the number of patients in the facility, denoted by bi, and let b be the
two-dimensional vector representing these limits. In addition, there is an upper bound
on the average amount of time that a class i patient spends in the facility, denoted by αi,
for i = 1, 2. The forecasted arrival rate of class i patients in period t is denoted by λit
for i = 1, 2 and t = 1, . . . , T , and let λt be a two-dimensional vector associated with the
arrival rates of patients in both classes in period t.
There are two types of health care teams, indexed by j = 1, 2, and two types of
E/T rooms. Team type 1 can provide service to both patient classes with different service
rates in its dedicated E/T rooms, type 1, while team type 2 can only provide service to
patient class 2 in its dedicated E/T rooms, type 2. Let rj denote the number of E/T
rooms allocated to team type j for j = 1, 2, and r be the associated two-dimensional
vector representing the room allocation. As we discussed earlier, the service capacity
of each type of team can only be changed by modifying the configuration of the team
and can be quantified by the service rates of the associated E/T rooms. Given the lower
and upper bounds on the number of personnel with different skills that can be employed
in the facility and the types of personnel that must be included in a particular type of
team, let Kj denote the total number of possible configurations for team type j, indexed
by k = 1, . . . , Kj. Suppose that it is possible for the administrators to estimate the
44
service rate θijk of type j E/T rooms when the associated team, i.e., team type j, has
configuration k, and the patient in the room belongs to class i for i = 1, 2, j = 1, 2 and
k = 1, . . . , Kj. We note that θ12k = 0 for k = 1, . . . , K2 since the class 1 patients cannot be
treated by the type 2 team or in the type 2 E/T rooms.
We have two sets of decision variables. Let xjkt take the value of one if for team type
j configuration k is selected in period t for j = 1, 2, k = 1, . . . , Kj and t = 1, . . . , T ,
and zero otherwise. Also, let xjt be a Kj-dimensional vector associated with the team
configuration decision for team type j in period t. Finally, let φijt denote the service rate
for class i patients who are treated in type j E/T rooms in period t for i = 1, 2, j = 1, 2
and t = 1, . . . , T , and φt be a 2 × 2 matrix associated with the service rates of the E/T
rooms in period t.
We let wi(φt,λt,b, r) and si(φt,λt,b, r) represent the waiting cost of class i patients
and the average time spent in the system by class i patients, respectively, as a function
of the service rate of the E/T rooms φt in period t, patient arrival rates λt in period t,
maximum number of patients allowed in the facility b and E/T room allocation r. Also,
we let cj(xjt−1,xjt) denote the cost of modifying the configuration of type j team from
period t − 1 to t and oj(xjt) denote the cost of employing the personnel necessary for the
chosen configuration for team type j in period t. Assuming that all acquired additional
personnel capacity is available and becomes effective in the same period, the HCTSCP
problem can be formulated as a non-linear binary integer programming problem as follows:
min2∑
i=1
T∑t=1
wi(φt, λt,b, r) +2∑
j=1
T∑t=1
cj(xjt−1,xjt) +2∑
j=1
T∑t=1
oj(xjt) (3–1)
subject to
Kj∑
k=1
xjkt = 1 ∀j, t (3–2)
φijt −Kj∑
k=1
θijkxjkt = 0 ∀i, j, t (3–3)
si(φt,λt,b, r) ≤ αi ∀i, t (3–4)
45
2∑j=1
cj(xjt−1,xjt) ≤ γt ∀t (3–5)
xjkt ∈ {0, 1} ∀j, k, t (3–6)
φijt ≥ 0 ∀i, j, t (3–7)
The objective function (3–1) minimizes the sum of the cost associated with the
time that the patients spend in the system, the cost of modifying team configurations to
change the service capacity of the teams, i.e., service rates of the associated E/T rooms,
and the cost of employing the personnel necessary for the selected team configuration.
Constraints (3–2) stipulate that one configuration must be selected for each team type in
each planning period. Constraints (3–3) assign the service rates for E/T rooms according
to the selected configurations for each team type with respect to different patient classes.
Note that φ12t = 0 for t = 1, . . . , T due to θ12k = 0 for k = 1, . . . , K2. Constraints (3–4)
impose upper limits on the average time that patients spend in the system for each patient
class. Constraints (3–5) limits the amount of funds that can be allocated to changing
team configuration in each planning period. Finally, constraints (3–6) and (3–7) ensure
the integrality of team configuration and non-negativity of service rate decision variables,
respectively.
HCTSCP is a difficult non-linear binary integer programming problem with non-linear
constraints. Note that, however, HCTSCP can be represented by a T + 1-partite network,
where each layer in the network represents a time period t = 0, . . . , T in the planning
horizon. Let (k1, k2, t) denote the facility when configuration kj for type j team is used
in period t. Layer t = 0 include a single node (k1, k2, 0), which denotes the initial
configurations for type 1 and type 2 teams. A superficial source node S is connected
to node (k1, k2, 0) only with zero arc cost. Each layer t = 1, . . . , T contains K1 × K2
nodes, each of which represents a feasible pair of team configurations for the two teams in
period t. The cost of the arcs connecting a node (k1, k2, t − 1) in period t − 1 to a node
(k1’, k2’, t) in period t is given by∑2
i=1 wi(φt,λt,b, r) +∑2
j=1 cj(xjt−1,xjt) +∑2
j=1 oj(xjt),
where xjkj ’t = 1 for j = 1, 2, for t = 1, . . . , T . However, for a given node (k1, k2, t), if
46
the upper limit on either the expected patients’ time in system for any patient class or
the cost of modifying the team configurations is violated, i.e., either constraints (3–4) or
constraints (3–5) is violated, then the cost of the incoming arcs to this node are set to
M , where M is a very large number. Finally, each node in layer t = T is connected to
a superficial sink node D only with zero arc cost. Figure 3-1 provides an example of the
network representation for HCTSCP for K1 = 3, K2 = 3, and T = 2. For each team type
(a1, a2) represents a configuration where the number of type ` personnel in the team is
a` for ` = 1, 2. In this figure, a path from the superficial source node to S the superficial
sink node D represents a team capacity plan over the planning horizon. The HCTSCP
problem finds a capacity plan with minimum cost, if the shortest path on this graph does
not contain any arc with cost M . Otherwise, the problem is infeasible. In the next section,
we explain how we obtain the average time in system for each patient class.
S (2,5), (1,2), 0
(2,4), (1,2), 1
(2,4), (2,2), 1
(2,5), (1,1), 1
D
(2,4), (1,1), 1
(2,5), (1,2), 1
(2,5), (2,2), 1
(3,4), (1,1), 1
(3,4), (1,2), 1
(3,4), (2,2), 1
... ...
(2,4), (1,2), 2
(2,4), (2,2), 2
(2,5), (1,1), 2
(2,4), (1,1), 2
(2,5), (1,2), 2
(2,5), (2,2), 2
(3,4), (1,1), 2
(3,4), (1,2), 2
(3,4), (2,2), 2
... ...
Figure 3-1. An illustration of the network representation for HCTSCP
3.4 Queueing Analysis
In the development and analysis of the underlying queueing models, the E/T rooms,
rather than health care teams, are viewed as the servers because a team may serve more
than one patient simultaneously, while each E/T room can be occupied by only one
patient at a time. The service of a patient is considered to begin once the patient enters
a server, i.e., an E/T room, even though all the members of the associated team may be
47
busy with other patients and the patient may have to wait in the room. To analyze the
time that each patient class spends in the system, we assume that the patient arrivals in
both classes are Poisson processes, and the service times for the patients are exponentially
distributed with patient-class- and team-type-dependent service rates. Given the service
rate of each E/T room type for each patient class, we can represent such a health care
system by a continuous time Markov chain (CTMC) model. We develop a DBA method
to estimate the average time that the class 1 and class 2 patients spend in a health care
facility for the preemptive and the non-preemptive cases.
3.4.1 Preemptive Case: Emergency Medicine Services
The evolution of the ED in a given period t of the planning horizon can be represented
by a CTMC. To simplify the notation, the time index t is ignored in this section. To
characterize the status of the system, we need to know not only the number of class i
patients in the system (health care facility), denoted by ni for i = 1, 2, but also the
number of class 2 patients treated in the type 1 E/T rooms, denoted by m. Therefore, we
use a triplet (n1, n2,m) to represent the state of the system, and the associated state space
is defined by
Sp = {(n1, n2,m) : 0 ≤ n1 ≤ b1, 0 ≤ n2 ≤ b2, (r1−n1)+∧(n2−r2)
+ ≤ m ≤ ((r1−n1)+∧n2)}.
Note that because class 1 patients have preemptive priority, if the number of class 1
patients is not less than the number of type 1 E/T rooms, there is no class 2 patient being
served in the type 1 E/T rooms, i.e., m = 0 when r1 ≤ n1.
The transition rate out of state (n1, n2,m) ∈ Sp is the sum of arrival and departure
rates of both patient classes, that is, λ1I(n1 < b1) + λ2I(n2 < b2) + [n1 ∧ (r1)]φ11 + mφ21 +
[(n2 −m) ∧ r2]φ22. The transitions entering state (n1, n2, m) are summarized in Table 3-1.
Using the lexicographical order sequence for the states (n1, n2), the infinitesimal
generator Q of the CTMC described above has the form of a nonhomogeneous quasi-
birth-death (QBD) process and has the stationary probability of matrix-product form
πn1 = πn1−1Rn1 , where vector πn1 represents the stationary probabilities of the states
48
Table 3-1. The possible transitions enter state (n1, n2,m) for the ED setting
Event Transition probability1) A class 1 patient arrives to the system λ1P (n1 − 1, n2,m)I(n1 > 0)2) A class 1 patient arrives to the system and
preempts a class 2 patient served in a type 1E/T room
λ1P (n1−1, n2,m+1)I(n1 > 0)I(n1 +m =r1)
3) A class 2 patient arrives to the system, andis admitted immediately to a type 2 E/Troom or joins the queue of class 2 patients
λ2P (n1, n2 − 1,m)I(n2 > m)
4) A class 2 patient arrives to the system, andis admitted immediately to a type 1 E/Troom
λ2P (n1, n2−1,m−1)I(m > 0)I(n1 +m ≤r1)I(n2 −m = r2)
5) A class-1-patient departure results in eitherfreeing a type 1 E/T room or starting theservice of a waiting class 1 patient
[(n1 +1)∧ r1]φ11P (n1 +1, n2,m)I(n1 < b1)
6) A class-1-patient departure results instarting the service of a waiting class 2patient
(n1 + 1)φ11P (n1 + 1, n2, m − 1)I(n1 <b1)I(m > 0)I(n1 + m = r1)I(n2 −m ≥ r2)
7) A class-2patient departure from a type 1E/T room results in starting the service of awaiting class 2 patient
mφ21P (n1, n2 + 1,m)I(n2 < b2)I(m >0)I(n1 + m = r1)I(n2 −m ≥ r2)
8) A class-2-patient departure from a type 2E/T room results in starting the service of awaiting class 2 patient
[(n2+1−m)∧r2)]φ22P (n1, n2+1,m)I(n2 <b2)
9) A class-2-patient departure from a type 1team results in freeing a type 1 server
(m + 1)φ21P (n1, n2 + 1,m + 1)I(n2 <b2)I(m < r1)
with first dimension equal to n1 [75]. Although the stationary probabilities of such a
process can be obtained through matrix-analytic methods, our goal is to develop fast
approximation methods to obtain the average time in system for each patient class with
sufficient accuracy. To estimate the stationary probabilities, we develop a DBA method
with the following steps: (1) decompose the state space into three submodels, for each of
which the stationary probabilities can be computed easily; and (2) combine the results of
the submodels using the linking probabilities to obtain the stationary probabilities for the
original CTMC and compute the average time in system for each patient class.
Submodels. The state space of the CTMC for the preemptive case can be
decomposed into three submodels. Let N1 and N2 be random variables that denote
the number of class 1 and class 2 patients in system, respectively. The first submodel
49
determines the stationary probabilities of the number of class 1 patients in the system, N1.
To determine the stationary probabilities of the number of class 2 patients in the system,
N2, the state space Sp is decomposed with respected to the first dimension by N1 > r1 and
N1 = n1 for n1 ≤ r1. This decomposition procedure forms the second and third submodels,
which are used to determine the stationary probabilities in regions N1 > r1 and N1 = n1
for n1 ≤ r1, respectively, i.e., the conditional probabilities P (N2 = n2|N1 > r1) and
P (N2 = n2|N1 = n1) for n1 ≤ r1. The details of the submodels are described as follows:
• Submodel 1: P (N1 = n1): The arrival and service rates of class 1 patients areindependent of the value of N2 because class 1 patients have preemptive priority overclass 2 patients. Therefore, the stationary probabilities P (N1 = n1) of the originalCTMC can be determined by using an M/M/r1/b1 queueing system with arrival rateλ1 and service rate φ11.
• Submodel 2: P (N2 = n2|N1 > r1): Given N1 > r1, all type 1 E/T rooms must beoccupied by class 1 patients and no class 2 patient is served in the type 1 E/T rooms.Therefore, the service rates of the class 2 patients are independent of the value of N1,and the stationary probabilities P (N2 = n2|N1 > r1) can be determined by using anM/M/r2/b2 queueing system with arrival rate λ2 and service rate φ22.
• Submodel 3: P (N2 = n2|N1 = n1) for n1 ≤ r1: If N1 = r1, all type 1 E/T roomsmust be occupied by class 1 patients or at least one type 1 E/T room is empty,but a waiting class 2 patient can be admitted to an empty type 1 E/T room oncea class 1 patient leaves an E/T room and no class 1 patient arrives to the system.Therefore, for N1 ≤ r1 the service rate of the system with n2 class 2 patients isthe sum of the service rates of all type 2 E/T rooms, the empty type 1 E/T room,and one possibly available type 1 E/T room which is occupied by a class 1 patient,i.e.,µn2 = (r2 ∧ n2)φ22 + [(n2 − r2)
+ ∧ (r1 − n1) + Pon1I(n2 > r2)]φ21, where Pon1 isapproximated by Pon1 = (n1φ11 + λ2)/(n1φ11 + λ1 + λ2).
Linking Probabilities. The unconditional probabilities P (N2 = n2) can be
represented by the conditional probabilities as follows:
P (N2 = n2) = P (N2 = n2|N1 > r1)P (N1 > r1) +
n1=r1∑n1=0
P (N2 = n2|N1 = n1)P (N1 = n1).
Note that the linking probabilities, P (N1 > r1) and P (N1 = n1) for n1 ≤ r1 can
be obtained from the results in submodel 1. If we know the stationary probabilities
P (N1 = n1) and P (N2 = n2), we can calculate the expected number of patients of each
50
class accordingly. Then, by applying Little’s formula, we can obtain the average time in
the system for each class 1 and class 2 patients.
3.4.2 Non-Preemptive Case: Outpatient Clinic Services
The evolution of the OC in a given period t of the planning horizon can also be
represented by a CTMC. As before, we again ignore the time index t to simplify the
notation. The status of the system is also characterized by the triplet (n1, n2,m) defined
for the ED. However, the state space associated with the OC setting is different from that
in the ED setting and is defined by
Snp = {(n1, n2,m) : 0 ≤ n1 ≤ b1, 0 ≤ n2 ≤ b2, (r1 − n1)+ ∧ (n2 − r2)
+ ≤ m ≤ (n2 ∧ r1)}.Note that if all type 2 E/T rooms are occupied, i.e., n2 ≥ r2, an arriving class 2 patient
can be served in an empty type 1 E/T room, i.e., r1 − n1 > 0. Thus, there is no state
(n1, n2,m) with m less than (r1 − n1)+ ∧ (n2 − r2)
+.
The transition rate out of state (n1, n2,m) ∈ Snp is the sum of arrival and departure
rates of both patient classes, that is, λ1I(n1 < b1) + λ2I(n2 < b2) + [n1 ∧ (r1 −m)]φ11 +
mφ21 + [(n2 −m) ∧ r2]φ22. The types of transitions entering state (n1, n2, m) are the same
as the ED setting, except that the second type of transition in Table 3-1 does not occur in
the OC setting because the class 1 patients only have non-preemptive priority over class 2
patients.
Similar to the preemptive case, the matrix Q of the CTMC for the non-preemptive
case has the form of a nonhomogeneous QBD process, for which we estimate the stationary
probabilities using the DBA method. The DBA method for the non-preemptive case is
slightly different than the one for the preemptive case and includes the following four
major steps: (1) approximate the three-dimensional CTMC by a two-dimensional one;
(2) decompose the state space into four submodels, for each of which the stationary
probabilities can be computed easily; (3) combine the results of the submodels to
obtain the stationary probabilities for the approximate two-dimensional CTMC and then
compute the expected number of patients of each class; and (4) represent the system by
51
Type 1 Team
Type 2 Team
Class 1 Patients
Class 2 Patients Subsystem 2
Subsystem 1
Figure 3-2. Two-dimensional CTMC approximation
two independent simple subsystems and use the performance measures of these simplified
subsystems to adjust the results obtained in step (3).
State-dimension Reduction. The dimension of state space can be reduced from
three to two. We can treat the OC as a combination of two interacting subsystems as
shown in Figure 3-2. Subsystem 1 contains the type 1 team, type 1 E/T rooms and the
patients that are served in the type 1 E/T rooms. Similarly, subsystem 2 contains the
type 2 team, type 2 E/T rooms and the patients that are served in the type 2 E/T rooms.
We then use a two-dimensional CTMC with state (z, n2’) to approximate the original
three-dimensional CTMC, where z is the total number of patients in subsystem 1 and
n2’ is the number of class 2 patients in subsystem 2. Note that z includes the number of
class 1 and class 2 patients in subsystem 1, and n2’ includes the number of class 2 patients
served in subsystem 2 only. The state space of the two-dimensional CTMC is defined by
S’np = {(z, n2’) : 0 ≤ z ≤ r1 + b1, 0 ≤ n2’ ≤ r2I(z < r1) + [b2 − (z − b1)+]I(z ≥ r1)}.
Since the OC has patient class-dependent service rates for type 1 team, we need to
know the number of patients in each class to calculate the service rate of subsystem 1.
Let Z be a random variable that denotes the total number of patients in subsystem 1, M
a random variable that denotes the number of class 2 patients in subsystem 1, and N2’ a
random variable that denotes the number of class 2 patients in subsystem 2. Similarly, let
Pc2 be the probability that a patient in subsystem 1 belongs to class 2. We first assume
that given Z = z, M follows a binomial distribution with parameters min(z, r1) and Pc2
Then, to approximate the expected number of class 2 patients in subsystem 1 given Z = z,
52
E[M |Z = z], we need to know whether there is a queue in subsystem 1 since it affects the
entries of class 2 patients to subsystem 1. Therefore, we have two cases to analyze:
• Case 1: For 0 ≤ z ≤ r1, no class 1 patient waits in the system, and the next patiententering subsystem 1 can be either a class 1 or a class 2 patient. An arriving class2 patient can enter subsystem 1 only when all type 2 E/T rooms are occupied bypatients and no other class 2 patients wait in queue, i.e., N2’ = r2. Thus, the totalarrival rate in this case can be approximated by λ1+λ2P (N2’ = r2|Z ≤ r1). Therefore,for 0 ≤ z ≤ r1, E[M |Z = z] can be approximated by
E[M |Z = z] = zPc2 , (3–8)
where Pc2 = [λ2P (N2’ = r2|Z ≤ r1)]/[λ1 + λ2P (N2’ = r2|Z ≤ r1)]. Note that theprobability P (N2’ = r2|Z ≤ r1) can be approximated by P (N2’ = r2|Z < r1) fromsubmodel 1 below.
• Case 2: For r1 < z ≤ r1 + b1, all type 1 E/T rooms are occupied by patients,and only class 1 patients can enter subsystem 1. Thus, the existing class 2 patientsin subsystem 1 must satisfy two conditions: (1) the service time of the existingclass 2 patients must be greater than the sum of the interarrival time of class 1patients in queue; and (2) the existing class 2 patients must have entered subsystem1 before there is a queue of class 1 patients. Therefore, for r1 < z ≤ r1 + b1, we canapproximate E[M |Z = z] by
E[M |Z = z] = max(r1Pc2 , r1Pc2 + (z − b1)(1− Pc2)), (3–9)
where Pc2 = e−φ21
z−r1λ1 [(λ2P (N2’ = r2|Z ≤ r1))/(λ1 + λ2P (N2’ = r2|Z ≤ r1))]
z−r1 . Notethat under some particular parameter settings, we could have E[M |Z = z] ≈ 0 andE[N1|Z = z] ≈ k because Pc2 ≈ 0 and N1 + M = Z, for example, when the value ofφ21 or z is large enough. However, the number of class 1 patients in the OC can be b1
at most. Therefore, for the state (z, n2’) with z > b1, E[M |Z = z] should be adjustedby (z − b1)(1− Pc2) as in equation (3–9).
Submodels. After approximating the OC by a two-dimensional CTMC, we apply
the DBA method to estimate the stationary probabilities of the system. The state space
S’np can be decomposed with respect to the first dimension by Z < r1 and Z ≥ r1 and
the second dimension by N2’ > r2 and N2’ ≤ r2. Then, the DBA method uses some
well-known queueing models to determine the stationary probabilities in each region, i.e.,
conditional probabilities P (N2’ = j|Z < r1), P (Z = z|N2’ > r2), P (N2’ = j|Z ≥ r1), and
P (Z = z|N2’ ≤ r2). The details of the submodels are described as follows:
53
• Submodel 1: P (N2’ = j|Z < r1): Given Z < r1, no class 2 patient waits insubsystem 2 because at least one type 1 E/T room is available in subsystem 1.Therefore, the value of N2’ can only range from 0 to r2, which means P (N2’ = j|Z <r1) = 0 for r2 < j ≤ b2, and the system can be approximated by an M/M/r2/r2 queuewith arrival rate λ2 and service rate φ22.
• Submodel 2: P (Z = z|N2’ > r2): Given N2’ > r2, all type 1 E/T rooms must beoccupied by patients in subsystem 1. Otherwise, the waiting class 2 patients wouldoverflow to subsystem 1 and be served in a type 1 E/T room. Therefore, the valueof z can only range from r1 to r1 + b1, which means P (Z = z|N2’ > r2) = 0 for0 ≤ Z < r1. This system can be viewed as an M/M/1/r1 + b1 queue with arrival rateλ1 and state-dependent service rate µz given by
µz = (r1 − E[M |Z = z])φ11 + E[M |Z = z]φ21, (3–10)
where the value of E[M |Z = z] follows from equation (3–9).
• Submodel 3: P (N2’ = j|Z ≥ r1): Given K ≥ r1, all type 1 E/T rooms areoccupied by patients. The behavior of subsystem 2 under this condition can bemodeled as an M/M/r2/b2 queueing system. For state j with j > r2, subsystems1 and 2 may interact. That is, a waiting class 2 patient in subsystem 2 is admittedto a type 1 E/T room if no class 1 patient waits in subsystem 1, i.e., Z = r1.Therefore, the service rate of state j with j > r2 is the sum of the service rates ofall type 2 servers and a possibly available type 1 server, i.e., µj = r2φ22 + Pon1µr1 ,where µr1 is computed from equation (3–10) and Pon1 is the probability that atype 1 server is available for a waiting class 2 patient, and can be approximated byPon1 = P (Z = r1|N2’ > r2)(µr1 + λ2)/(µr1 + λ2 + λ1).
• Submodel 4: P (Z = z|N2’ ≤ r2): Similar to Submodel 3, we model the behaviorof the subsystem 1 as an M/M/r1/r1 + b1 queueing system. For state z with z < r1,the subsystems 1 and 2 may interact. That is, an arriving class 2 patient overflowto subsystem 1 and receives service in an empty type 1 E/T room if all type 2 E/Trooms are occupied by patients and no class 2 patient waits in subsystem 2, i.e.,N2’ = r2. Thus, for z < r1, the arrival rate of state z is the sum of arrival ratesof class 1 and possible class 2 patients, i.e., λz = λ1 + Pon2λ2, where Pon2 is theprobability that an arriving class 2 patient receives service in a type 1 E/T room, i.e.,P (N2’ = r2|Z < r1) from submodel 1. The service rate µz can be computed by
µz = (min(z, r1)− E[M |Z = z]) φ11 + E[M |Z = z]φ21, (3–11)
where the value of E[M |Z = z] is computed from equation (3–8) for 0 ≤ z ≤ r1, andfrom equation (3–9) for r1 < z ≤ r1 + b1.
Linking Probabilities. The unconditional probabilities P (Z = z) and P (N2’ = j)
can be represented by the conditional probabilities as follows:
54
P (Z = z) = P (Z = z|N2’ ≤ r2)P (N2’ ≤ r2) + P (Z = z|N2’ > r2)P (N2’ > r2) andP (N2’ = j) = P (N2’ = j|Z < r1)P (Z < r1) + P (N2’ = j|Z ≥ r1)P (Z ≥ r1).
Note that the linking probabilities, P (N2’ ≤ r2) and P (Z < r1), can further be represented
by the conditional probabilities as follows:
P (N2’ ≤ r2) = P (N2’ ≤ r2|Z < r1)P (Z < r1) + P (N2’ ≤ r2|Z ≥ r1)P (Z ≥ r1) andP (Z < r1) = P (Z < r1|N2’ ≤ r2)P (N2’ ≤ r2) + P (Z < r1|N2’ > r2)P (N2’ > r2).
If we substitute P (Z ≥ r1) = 1− P (Z < r1) and P (N2’ > r2) = 1− P (N2’ ≤ r2) above, we
can solve the resulting equations to obtain:
P (N2’ ≤ r2) =P (N2’ ≤ r2|Z ≥ r1)
1− P (N2’ > r2|Z ≥ r1)P (Z < r1|N2’ ≤ r2)and
P (Z < r1) = P (Z < r1|N2’ ≤ r2)P (N2’ ≤ r2).
Therefore, after computing the conditional probabilities in all the four submodels, the
unconditional probabilities, P (Z = z) and P (N2’ = j), and the expected number of
patients in subsystem 1 and 2, E[Z] and E[N2’], respectively, can be computed. Then,
E[N1] and E[N2], can be computed using the following set of equations:
E[M ] =
r1+b1∑z=0
E[M |Z = z]P [Z = z], (3–12)
E[N1] = max{E[Z]− E[M ], 0}, and (3–13)
E[N2] = E[N2’] + E[M ]. (3–14)
Again, by applying Little’s formula, we obtain the average time in system for each class 1
and class 2 patients.
Bounds. To improve the performance of the proposed DBA method for the
non-preemptive case, we use some simple queueing systems to obtain bounds on E[Z],
E[N1], and E[N2], and then adjust the results from equations (3–12)-(3–14).
• Expected total number of patients in subsystem 1 (E[Z]): The lower andupper bounds on E[Z] are obtained by considering two M/M/r1/r1 + b1 queues withstate-dependent service rates from equation (3–11) with arrival rates λ1 and λ1 +λ2f2,respectively. Let f2 be the fraction of class 2 patients served in the type 1 E/T rooms,
55
which is approximated by
f2 ≈ P (N2’ = r2, Z < r1) + P (N2’ = r2, Z = r1)λ2
λ1 + λ2
≈ P (N2’ = r2|Z < r1)P (Z < r1) + P (N2’ = r2|Z ≥ r1)P (Z = r1)λ2
λ2 + λ1
. (3–15)
Note that the first term in equation (3–15) represents the probability that an arrivingclass 2 patient finds all type 2 E/T rooms occupied and overflows to subsystem 1immediately. The second term represents the probability that an arriving class 2patient waits in subsystem 2 until a type 1 E/T room becomes available. Let E[N1]be the upper bound of E[N1]. If E[Z] is greater than E[N1], then E[N1] and E[M ] inequations (3–12) and (3–13) are decreased proportionally, i.e.,
E[N1]new = E[N1]previousE[N1]
E[Z]and E[M ]new = E[M ]previous
E[N1]
E[Z].
Otherwise, i.e., if E[Z] is less than the upper bound of E[N1], E[N1] and E[M ] inequations (3–12) and (3–13) are increased proportionally.
• Expected number of class 1 patients in the system (E[N1]): The lower andupper bounds on E[N1] are obtained by considering two M/M/r1/b1 queueingsystems with service rates φ11 and with arrival rates λ1 and λ1 + λ2f2, respectively.
• Expected number of class 2 patients in the system (E[N2]) The lower andupper bounds on E[N2] are obtained by considering two multi-server queueingsystems. Let E[Y ] be the total number of patients in an M/M/r1 + r2/b2 queueingsystem with arrival rate λ1 + λ2 and service rate µ, where r1 + r2 ≤ b2. The servicerate µ is assumed to be µ = (λ∗1 + λ∗2)/(λ
∗1/φ11 + λ∗2/φ22), where λ∗i is the effective
arrival rate of class i patients and is approximated by
λ∗1 = λ1(1−b1+r1∑
z=b1
P (Z = z)) and λ∗2 = λ2(1− P (N ’2 = b2)).
The resulting E[Y ] is a lower bound on the total number of patients in the OCbecause class 1 patients can be served in the type 2 E/T room here. Thus, the lowerbound on E[N2] can be computed by E[Y ] − E[N1]. Last, the upper bound on E[N2]is obtained by considering an M/M/r2/b2 queueing system with arrival rate λ2 andservice rate φ22.
3.5 Computational Study
In this section, we present results from our computational study, where we first
investigate the efficiency and the accuracy of the DBA method. We then use the DBA
56
method in conjunction with the HCTSCP formulation to test the efficiency of our capacity
planning model in making long-term personnel planning decisions.
3.5.1 Computational Performance of the DBA Method
To assess the computational performance of the DBA method, we compare the
approximate results obtained by the DBA method with the exact results obtained by
solving the steady state equations using Gaussian Elimination (GE). In preliminary
experiments we observe that GE may require significant computational effort as the size
of the model increases. Therefore, we consider an adaptation of Gaussian Elimination
(AGE) proposed by Thorson [106]. AGE algorithm avoids unnecessary row operations
by considering the fact that Q is a banded matrix, containing a large number of zero
elements. The DBA, GE, and AGE methods are implemented using C++ programming
language, and the numerical results reported are obtained using a personal computer with
a 3.0 GHz Pentium IV processor and 1 GB RAM memory.
Our parameter choices for our computational study are based on the data collected
from a participating ED. In the base case, there are eight and two rooms allocated to type
1 and type 2 teams, respectively, i.e., (r1, r2) = (8,2); service rates for class 1 and class 2
patients in type 1 E/T room are 0.25 and 0.80 patients/hour, respectively, i.e., (φ11, φ21)
= (0.25, 0.80); service rate for class 2 patients in type 2 E/T room are 0.75 patients/hour,
i.e., φ22 = 0.75; and the maximum number of patients from each class allowed in system
are 20 and 20, respectively, i.e., (b1, b2) = (20,20). We note that in practice, an ED is
required by law to admit all the patients that request emergency medicine services.
Therefore, essentially, the waiting room capacity is infinite, and no arriving patient is
denied of entry to the system because of lack of waiting room capacity. However, when
there is not enough service capacity, then an arriving patient can be diverted to a sister
hospital. Therefore, for modeling purposes, we include a limit on the number of patients of
each type in the system.
57
In our study, we consider three experimental factors including the E/T room
allocation, service capacity, and system utilization. In the first and second experiments,
we test the impact of E/T room allocation and service rate on the accuracy of DBA,
respectively. We consider r1 ∈ {2, 4, 8} and r2 ∈ {2, 4, 8} for E/T room allocation
in the first experiment and (φ11, φ21) ∈ {(0.20, 0.70), (0.25, 0.80), (0.30, 0.90)} and
φ22 ∈ {0.65, 0.75, 0.85} for service rates in the second experiment. Let ρ denote the system
utilization. For each scenario, we use ρ ∈ {0.6, 0.7, 0.8, 0.9} to generate four instances with
different pairs of patient arrival rates, i.e., (λ1, λ2), using λ1 = r1φ11ρ and λ2 = r2φ22ρ.
In Table 3-2, we report the size of the CTMC as well as the CPU time (in seconds)
required to obtain the solution using the DBA, AGE, and GE methods. In the preemptive
case, the size of the CTMC model grows if r1 or r2 increases. In addition, an increase
in r1 has a more significant effect than that in r2. Similar behavior can be observed
in the non-preemptive case, however, the impact of increasing r1 on problem size in
non-preemptive case is more significant than that in the preemptive case. Furthermore,
under the same E/T room allocation, the problem size of non-preemptive case is at least
twice larger than that of preemptive case for the tested scenarios in Table 3-2. Our results
show that the time to obtain the approximate results using the DBA method is negligible
and AGE is considerably more effective than GE in obtaining the exact solution.
Table 3-2. Computational requirement of the DBA, AGE, and GE methods
Preemptive Non-Preemptiver1 r2 State Count DBA AGE GE State Count DBA AGE GE2 2 447 0.0 0.0 0.5 2707 0.0 0.3 10.82 4 453 0.0 0.0 0.6 2713 0.0 0.3 10.92 8 465 0.0 0.0 0.6 2725 0.0 0.3 11.24 2 461 0.0 0.0 0.6 4225 0.0 0.7 36.64 4 481 0.0 0.0 0.7 4245 0.0 0.7 37.94 8 521 0.0 0.0 0.9 4285 0.0 0.7 39.88 2 513 0.0 0.0 0.8 6609 0.0 1.9 116.18 2 585 0.0 0.0 1.2 6681 0.0 2.0 129.28 2 729 0.0 0.0 2.3 6825 0.0 2.1 161.0Unit of running time: Second
58
In Tables 3-3 and 3-4, we report the percentage error associated with the expected
time in system for the two patient classes obtained by the DBA method (when compared
to the exact solution obtained by the AGE method) in the preemptive and non-preemptive
case, respectively. Table 3-3 shows clearly that the average time in system for class 1
patients can be correctly computed by the DBA method because submodel 1 in Section
3.4.1 captures the exact behavior of class 1 patients in the original CTMC. For class 2
patients, the first experiment shows that for the instances with the same levels of r1 and
ρ, the absolute percentage error (APE) tends to decrease as r2 increases. For example,
for the instances with r1 = 2 and ρ = 0.6, APE decreases as r2 increases. In addition,
for the same levels of r1/r2 and ρ, APE tends to decrease as r1 or r2 increases as system
utilization is low, such as the instances with r1/r2=1 and ρ=0.6, APE decreases as r1 (or
r2) increases. In other words, for a given setting of r1 or r1/r2 and ρ, DBA performs better
in the problems with larger r2. In the second experiment, 9 scenarios of service capacity
are tested. For the same level of (φ11, φ21), APE tends to increase as φ22 increases for the
instances with ρ=0.6, which is opposite to the results of the instances with ρ=0.9, where
APE tends to decrease as φ22 increases. Last, both experiments show that DBA tends to
underestimate class 2 patients’ expected time in the system when the system utilization
is low, i.e., ρ = 0.6, while overestimate class 2 patients’ expected time in the system when
the system utilization is high, i.e., ρ = 0.9.
Table 3-4 shows the results for non-preemptive case. In the first experiment, we
observe that for the instances with the same levels of r1 and ρ, the APE for class 2
patients tends to decrease as r2 increases, which is the same as the preemptive case. In the
second experiment, for the same levels of (φ11, φ21) and ρ, APE tends to increase as φ22
increases for both class 1 and class 2 patients. In addition, DBA tends to underestimate
class 1 patients’ expected time in system while it tends to overestimate class 2 patients’
expected time in system. For class 1 patients, APE is less than 5 percent for all tested
instances, i.e., DBA yields a more reliable estimate for class 1 patients’ time in system.
59
Tab
le3-
3.R
elat
ive
and
per
centa
geer
ror:
pre
empti
veca
se
Exp
erim
ent
1:E
ffect
sof
E/T
room
allo
cati
on,(φ
11,φ
21,φ
22)=
(0.2
5,0.
8,0.
75)
ρ=
0.6
ρ=
0.7
ρ=
0.8
ρ=
0.9
r 1r 2
Cla
ss1
Cla
ss2
Cla
ss1
Cla
ss2
Cla
ss1
Cla
ss2
Cla
ss1
Cla
ss2
22
0.0(
0.0
%)
-5.3
(-6.
1%
)0.
0(0.
0%
)-5
.3(-
4.4
%)
0.0(
0.0
%)
-0.1
(0.0
%)
0.0(
0.0
%)
10.1
(3.1
%)
40.
0(0.
0%
)-5
.2(-
3.4
%)
0.0(
0.0
%)
-8.2
(-4.
1%
)0.
0(0.
0%
)-1
0.0(
-3.6
%)
0.0(
0.0
%)
-9.5
(-2.
3%
)8
0.0(
0.0
%)
-3.1
(-1.
1%
)0.
0(0.
0%
)-6
.6(-
1.8
%)
0.0(
0.0
%)
-10.
8(-2
.4%
)0.
0(0.
0%
)-1
2.5(
-2.2
%)
42
0.0(
0.0
%)
-3.2
(-4.
0%
)0.
0(0.
0%
)-1
.2(-
1.1
%)
0.0(
0.0
%)
9.1(
5.6
%)
0.0(
0.0
%)
29.9
(10.
6%
)4
0.0(
0.0
%)
-3.8
(-2.
5%
)0.
0(0.
0%
)-5
.1(-
2.7
%)
0.0(
0.0
%)
-3.6
(-1.
4%
)0.
0(0.
0%
)1.
3(0.
3%
)8
0.0(
0.0
%)
-2.5
(-0.
9%
)0.
0(0.
0%
)-5
.3(-
1.5
%)
0.0(
0.0
%)
-8.5
(-2.
0%
)0.
0(0.
0%
)-1
0.1(
-1.8
%)
82
0.0(
0.0
%)
-2.3
(-3.
1%
)0.
0(0.
0%
)0.
6(0.
6%
)0.
0(0.
0%
)14
.9(1
1.0
%)
0.0(
0.0
%)
51.5
(22.
9%
)4
0.0(
0.0
%)
-2.7
(-1.
9%
)0.
0(0.
0%
)-2
.9(-
1.6
%)
0.0(
0.0
%)
2.2(
0.9
%)
0.0(
0.0
%)
15.4
(4.5
%)
80.
0(0.
0%
)-1
.9(-
0.7
%)
0.0(
0.0
%)
-3.9
(-1.
1%
)0.
0(0.
0%
)-5
.6(-
1.3
%)
0.0(
0.0
%)
-5.9
(-1.
1%
)A
vera
ge0.
0(0.
0%
)3.
3(2.
6%
)0.
0(0.
0%
)4.
4(2.
1%
)0.
0(0.
0%
)7.
2(3.
1%
)0.
0(0.
0%
)16
.2(5
.4%
)E
xper
imen
t2:
Effe
cts
ofse
rvic
eca
paci
ty,(r
1,r
2)=
(8,2
)ρ
=0.
6ρ
=0.
7ρ
=0.
8ρ
=0.
9(φ
11,φ
21)
φ22
Cla
ss1
Cla
ss2
Cla
ss1
Cla
ss2
Cla
ss1
Cla
ss2
Cla
ss1
Cla
ss2
(0.2
0,0.
70)
0.65
0.0(
0.0
%)
-2.4
(-3.
2%
)0.
0(0.
0%
)0.
3(0.
4%
)0.
0(0.
0%
)14
.1(1
0.3
%)
0.0(
0.0
%)
49.3
(21.
7%
)0.
750.
0(0.
0%
)-3
.6(-
4.7
%)
0.0(
0.0
%)
-1.6
(-1.
6%
)0.
0(0.
0%
)10
.5(7
.3%
)0.
0(0.
0%
)41
.4(1
7.5
%)
0.85
0.0(
0.0
%)
-4.9
(-6.
2%
)0.
0(0.
0%
)-3
.5(-
3.4
%)
0.0(
0.0
%)
7.1(
4.8
%)
0.0(
0.0
%)
34.6
(14.
1%
)(0
.25,
0.80
)0.
650.
0(0.
0%
)-1
.2(-
1.7
%)
0.0(
0.0
%)
2.3(
2.5
%)
0.0(
0.0
%)
18.4
(14.
1%
)0.
0(0.
0%
)59
.4(2
7.6
%)
0.75
0.0(
0.0
%)
-2.3
(-3.
1%
)0.
0(0.
0%
)0.
6(0.
6%
)0.
0(0.
0%
)14
.9(1
1.0
%)
0.0(
0.0
%)
51.5
(22.
9%
)0.
850.
0(0.
0%
)-3
.4(-
4.4
%)
0.0(
0.0
%)
-1.1
(-1.
1%
)0.
0(0.
0%
)11
.7(8
.3%
)0.
0(0.
0%
)44
.5(1
9.0
%)
(0.3
0,0.
90)
0.65
0.0(
0.0
%)
-0.4
(-0.
6%
)0.
0(0.
0%
)3.
9(4.
3%
)0.
0(0.
0%
)22
.0(1
7.5
%)
0.0(
0.0
%)
68.0
(33.
2%
)0.
750.
0(0.
0%
)-1
.3(-
1.8
%)
0.0(
0.0
%)
2.3(
2.5
%)
0.0(
0.0
%)
18.6
(14.
2%
)0.
0(0.
0%
)60
.0(2
7.9
%)
0.85
0.0(
0.0
%)
-2.3
(-3.
0%
)0.
0(0.
0%
)0.
8(0.
8%
)0.
0(0.
0%
)15
.5(1
1.4
%)
0.0(
0.0
%)
53.0
(23.
7%
)A
vera
ge0.
0(0.
0%
)2.
4(3.
2%
)0.
0(0.
0%
)1.
8(1.
9%
)0.
0(0.
0%
)14
.8(1
1.0
%)
0.0(
0.0
%)
51.3
(23.
1%
)U
nit:
min
.(%
)
60
Tab
le3-
4.R
elat
ive
and
per
centa
geer
ror:
non
-pre
empti
veca
se
Exp
erim
ent
1:E
ffect
sof
E/T
room
allo
cati
on,(φ
11,φ
21,φ
22)=
(0.2
5,0.
8,0.
75)
ρ=
0.6
ρ=
0.7
ρ=
0.8
ρ=
0.9
r 1r 2
Cla
ss1
Cla
ss2
Cla
ss1
Cla
ss2
Cla
ss1
Cla
ss2
Cla
ss1
Cla
ss2
22
-3.2
(-2.
8%
)1.
1(1.
3%
)-5
.6(-
3.3
%)
-4.5
(-3.
7%
)-6
.2(-
2.3
%)
-14.
8(-8
.1%
)13
.7(3
.3%
)-2
3.6(
-7.4
%)
4-3
.1(-
2.7
%)
2.3(
1.5
%)
-5.9
(-3.
5%
)-2
.6(-
1.3
%)
-9.1
(-3.
5%
)-1
0.4(
-3.8
%)
9.9(
2.4
%)
-14.
2(-3
.4%
)8
-2.3
(-2.
0%
)3.
8(1.
3%
)-5
.2(-
3.1
%)
0.2(
0.1
%)
-9.4
(-3.
6%
)-4
.5(-
1.0
%)
5.9(
1.4
%)
-4.6
(-0.
8%
)4
2-2
.7(-
1.6
%)
9.5(
12.0
%)
-5.5
(-2.
4%
)4.
1(3.
9%
)-9
.5(-
2.9
%)
-10.
9(-7
.0%
)-1
2.9(
-2.7
%)
-32.
6(-1
2.0
%)
4-2
.8(-
1.6
%)
10.8
(7.2
%)
-6.1
(-2.
6%
)6.
0(3.
2%
)-1
0.9(
-3.3
%)
-8.8
(-3.
5%
)-1
5.0(
-3.1
%)
-27.
0(-7
.1%
)8
-2.1
(-1.
2%
)9.
1(3.
1%
)-5
.6(-
2.4
%)
9.1(
2.6
%)
-11.
2(-3
.4%
)-1
.5(-
0.3
%)
-16.
0(-3
.3%
)-1
1.1(
-2.0
%)
82
-1.6
(-0.
5%
)19
.6(2
6.6
%)
-4.2
(-1.
1%
)19
.0(2
0.4
%)
-8.3
(-1.
7%
)8.
5(6.
6%
)-1
2.4(
-2.0
%)
-13.
0(-6
.1%
)4
-1.7
(-0.
6%
)21
.4(1
4.7
%)
-4.8
(-1.
3%
)22
.5(1
2.7
%)
-10.
0(-2
.1%
)11
.0(4
.8%
)-1
5.2(
-2.5
%)
-13.
7(-4
.1%
)8
-1.4
(-0.
5%
)11
.5(4
.0%
)-4
.7(-
1.3
%)
25.2
(7.3
%)
-10.
8(-2
.3%
)17
.5(4
.2%
)-1
7.0(
-2.8
%)
-1.4
(-0.
3%
)A
vera
ge2.
3(1.
5%
)9.
9(8.
0%
)5.
3(2.
3%
)10
.3(6
.1%
)9.
5(2.
8%
)9.
8(4.
4%
)13
.1(2
.6%
)15
.7(4
.8%
)E
xper
imen
t2:
Effe
cts
ofse
rvic
eca
paci
ty,(r
1,r
2)=
(8,2
)ρ
=0.
6ρ
=0.
7ρ
=0.
8ρ
=0.
9(φ
11,φ
21)
φ22
Cla
ss1
Cla
ss2
Cla
ss1
Cla
ss2
Cla
ss1
Cla
ss2
Cla
ss1
Cla
ss2
(0.2
0,0.
70)
0.65
-1.5
(-0.
5%
)20
.7(2
8.1
%)
-3.9
(-1.
0%
)19
.8(2
1.2
%)
-7.7
(-1.
6%
)7.
4(5.
7%
)-1
1.5(
-1.9
%)
-19.
5(-9
.1%
)0.
75-1
.8(-
0.6
%)
22.7
(29.
7%
)-4
.4(-
1.2
%)
21.7
(22.
3%
)-8
.5(-
1.8
%)
8.1(
5.9
%)
-12.
5(-2
.1%
)-2
2.5(
-10.
0%
)0.
85-2
.0(-
0.6
%)
24.4
(31.
0%
)-4
.8(-
1.3
%)
23.4
(23.
2%
)-9
.3(-
2.0
%)
8.8(
6.2
%)
-13.
5(-2
.2%
)-2
5.1(
-10.
8%
)(0
.25,
0.80
)0.
65-1
.4(-
0.5
%)
17.8
(24.
9%
)-3
.7(-
1.0
%)
17.2
(19.
2%
)-7
.4(-
1.6
%)
7.9(
6.4
%)
-11.
3(-1
.9%
)-1
0.6(
-5.3
%)
0.75
-1.6
(-0.
5%
)19
.6(2
6.6
%)
-4.2
(-1.
1%
)19
.0(2
0.4
%)
-8.3
(-1.
7%
)8.
5(6.
6%
)-1
2.4(
-2.0
%)
-13.
0(-6
.1%
)0.
85-1
.8(-
0.6
%)
21.3
(28.
0%
)-4
.6(-
1.2
%)
20.6
(21.
4%
)-9
.1(-
1.9
%)
9.1(
6.8
%)
-13.
4(-2
.2%
)-1
4.8(
-6.7
%)
(0.3
0,0.
90)
0.65
-1.4
(-0.
4%
)15
.4(2
2.1
%)
-3.5
(-0.
9%
)15
.1(1
7.4
%)
-7.1
(-1.
5%
)7.
9(6.
7%
)-1
1.0(
-1.8
%)
-4.0
(-2.
1%
)0.
75-1
.5(-
0.5
%)
17.1
(23.
8%
)-4
.0(-
1.1
%)
16.7
(18.
6%
)-7
.9(-
1.7
%)
8.6(
6.9
%)
-12.
1(-2
.0%
)-6
.2(-
3.1
%)
0.85
-1.7
(-0.
6%
)18
.7(2
5.3
%)
-4.4
(-1.
2%
)18
.3(1
9.6
%)
-8.7
(-1.
8%
)9.
1(7.
1%
)-1
3.1(
-2.1
%)
-8.2
(-3.
9%
)A
vera
ge1.
6(0.
5%
)19
.8(2
6.6
%)
4.2(
1.1
%)
19.1
(20.
4%
)8.
2(1.
7%
)8.
4(6.
5%
)12
.3(2
.0%
)13
.8(6
.4%
)U
nit:
min
.(%
)
61
3.5.2 Computational Performance of the HCTSCP Model
To illustrate the effectiveness of the HCTSCP model used in conjunction with the
DBA method in making long-term personnel planning decisions, we compare the capacity
plans generated by the HCTSCP model using the approximate and exact average time in
system for different patient classes obtained by the DBA and AGE methods, respectively.
We also study the sensitivity of the results obtained by the HCTSCP model to several
problem parameters.
We consider the personnel planning problem over a three-year planning horizon where
the unit planning period corresponds to a quarter of a year, i.e., T = 12. We assume that,
for all t, the initial service rates for class 1 and class 2 patients are (φ110, φ210, φ220) =
(0.25, 0.80, 0.75) patient/hour, the allowable maximum numbers of patients in system
are (b1, b2) = (20,20) patients, the E/T room allocation is (r1, r2) = (8,2), and the upper
bound on the amount of time that a patient spends in the system are (α1, α2) = (4.75,
4.00) hours. We consider a case where there are two types of personnel included in each
team, and there are six and four feasible configurations for type 1 and type 2 teams,
respectively. Table 3-5 shows the number of different types of personnel with different
skill sets included in each configuration and the corresponding service capacity as well as
operating cost for each team. Without loss of generality, we assume that patient waiting
costs increase linearly with patients’ time in system, unit delay cost for each patient class
(UP1t, UP2t) are set to ($400,$100) /hour per patient, and the funds that can be allocated
to change the service capacity of the teams in the facility γt= $17,000 for all t. Finally, we
assume that personnel hiring or termination costs are zero.
An examination of emergency medicine practices show that patient arrivals to the
ED exhibit seasonality. We generate the total quarterly arrival rate of patients using a
seasonally adjusted trend line, represented by a function of the form λt = δmod(t,4)u(λ0 + bt)
where δi is the quarterly seasonality factor for season i (where we have the following
estimates for the seasonality factors δ1 = 0.8, δ2 = 1.0, δ3 = 1.2, and δ4 = 1.0), u is
62
Table 3-5. Team configurations
Type 1 Team Type 2 Teamk1 a1 a2 φ11 φ21 Cost k2 a1 a2 φ22 Cost1 2 4 0.23 0.75 56,000 1 1 1 0.60 11,5002 2 5 0.25 0.80 63,000 2 1 2 0.75 16,5003 2 6 0.28 0.85 70,000 3 2 2 1.00 23,0004 3 4 0.35 1.00 70,000 4 2 3 1.10 28,0005 3 5 0.40 1.10 77,0006 3 6 0.45 1.20 84,000Unit of φij: patients/hour. Unit of cost: $/quarter
a uniformly distributed random number (where we have u ∼ U [0.8, 1.2]), λ0 = 2, b
= 0.04, and t = 1, . . . , 12. Note that we implicitly assume that the patient demand
increases linearly by 2 percent every quarter. We assume that the fraction of class 1
patient is fit = 0.85 for all t, i.e., λ1t = f1tλt and λ2t = (1 − f1t)λt. We note that our
parameter choices are mainly according to the characteristics of the data collected from
the ED, which is represented by the preemptive case. In order to eliminate the effects that
may be due to a specific health care facility, we use the same set of parameters for the
non-preemptive case also.
In our study, we considered three experimental factors including the fraction of class
1 patients, f1t, the unit patient treatment cost for each patient class, (UP1t, UP2t), and
the maximum allowable average time in system for each patient class, (α1, α2). Each
parameter is tested at three levels as listed in Table 3-6. For each experimental setting, we
generated 25 random instances of the patient arrival streams over the planning horizon.
We solved all the instances for each of the settings by using the HCTSCP model with the
AGE method and with the DBA method to compare the performance of the two methods.
Table 3-6. Parameter settings
Parameters Level 1 Level 2 Level 3Experiment 1 f1t(%) 80 85 90Experiment 2 UP1t ($/hour/patient) 200 400 800
UP2t($/hour/patient) 50 100 200Experiment 3 α1 (hours) 4.50 4.75 5.00
α2 (hours) 3.50 4.00 4.50
63
As explained in Section 3.3, we build a network to represent the HCTSCP problem.
To compute the patient delay cost for each arc, we obtain average time in system for each
patient class approximately (exactly) using the DBA (AGE) method. We note that we
do not use the GE method in this experiment, since the AGE method is shown to be
considerably more efficient in Section 3.5.1. After we construct the network, we find the
shortest path using Dijkstra’s algorithm [2]. The results show that the network for the
12-period HCTSCP problem with six and four feasible configurations for type 1 and type
2 teams, respectively, contains 290 nodes and 6,384 arcs. The average time required to
obtain the optimal capacity plan is 11.0 seconds for preemptive case and 549.8 seconds for
non-preemptive case if we use the AGE method in building the network for an instance of
the HCTSCP problem. In contrast, the DBA method requires 0.03 seconds, on average,
for both the preemptive and non-preemptive cases. Therefore, using the DBA method
in building the network for an instance of the HCTSCP problems is considerably more
efficient than using the AGE method.
In order to measure the accuracy of the DBA method, we compare the team capacity
plans generated by the two approaches based on a similarity index. Specifically, we count
the number of periods where team configurations chosen are the same in both approaches
and then divide this counting by 12 for each care team type to determine the value of
the similarity index. The results of our comparison for each experimental factors are
summarized in Tables 3-7, 3-8, and 3-9. Cells range from 0 percent (absolutely different)
to 100 percent (perfectly similar). For instance, if a cell has a value of 75 percent, then
among 12 periods, the HCTSCP model with DBA gives the same results in 8 of them as
the HSCTSCP model with GE.
In Table 3-7, we observe that the HCTSCP model with DBA works very well in
the preemptive case. However, in the non-preemptive case, as the fraction of class 1
patients increases, the performance of the HCTSCP model with DBA deteriorates as it
overestimates the required service capacity of type 2 team by choosing the team
64
Tab
le3-
7.Im
pac
tof
the
frac
tion
ofcl
ass
1pat
ients
onth
esi
milar
ity
index
Pre
empti
veN
on-p
reem
pti
vef 1
tT
ype
1T
ype
2T
ype
1T
ype
280
%10
0.0%
99.0
%10
0.0%
88.0
%85
%10
0.0%
98.3
%10
0.0%
84.7
%90
%10
0.0%
98.0
%10
0.0%
82.7
%
Tab
le3-
8.Im
pac
tof
unit
pat
ient
del
ayco
ston
the
sim
ilar
ity
index
Pre
empti
veN
on-p
reem
pti
veU
P1t=
50U
P2t=
100
UP
2t=
200
UP
2t=
50U
P2t=
100
UP
2t=
200
UP
1t
Type
1T
ype
2T
ype
1T
ype
2T
ype
1T
ype
2T
ype
1T
ype
2T
ype
1T
ype
2T
ype
1T
ype
220
010
0.0%
97.0
%10
0.0%
99.7
%10
0.0%
99.7
%10
0.0%
73.3
%10
0.0%
84.0
%10
0.0%
83.7
%40
010
0.0%
97.0
%10
0.0%
99.7
%10
0.0%
99.7
%10
0.0%
74.3
%10
0.0%
84.7
%10
0.0%
84.3
%80
010
0.0%
97.0
%10
0.0%
99.7
%10
0.0%
99.7
%10
0.0%
86.0
%10
0.0%
86.7
%10
0.0%
86.7
%U
nit
ofU
P1tan
dU
P2t:
$/h
our
per
pat
ient.
Tab
le3-
9.Im
pac
tof
max
imum
allo
wab
leav
erag
eti
me
insy
stem
onth
esi
milar
ity
index
Pre
empti
veN
on-p
reem
pti
veα
2=
2.5
α2=
3.0
α2=
3.5
α2=
2.5
α2=
3.0
α2=
3.5
α1
Type
1T
ype
2T
ype
1T
ype
2T
ype
1T
ype
2T
ype
1T
ype
2T
ype
1T
ype
2T
ype
1T
ype
26
100.
0%10
0.0%
100.
0%10
0.0%
100.
0%10
0.0%
100.
0%10
0.0%
100.
0%10
0.0%
100.
0%10
0.0%
710
0.0%
100.
0%10
0.0%
100.
0%10
0.0%
100.
0%10
0.0%
100.
0%10
0.0%
100.
0%10
0.0%
100.
0%8
100.
0%10
0.0%
100.
0%10
0.0%
100.
0%10
0.0%
100.
0%10
0.0%
100.
0%10
0.0%
100.
0%10
0.0%
Unit
ofα
1an
dα
2:
hou
rper
pat
ient.
65
configuration with a higher service rate. This can be attributed to our earlier observation
that the DBA method overestimates the average time in system for class 2 patients.
Table 3-8 shows that the unit patient delay cost does not impact the accuracy of the
HCTSCP model in the preemptive case. But in the non-preemptive case, its accuracy
in determining the team configuration of type 2 team goes down, as unit class 1 patient
delay cost or unit class 2 patient delay cost decreases, and it tends to overestimate the the
required service capacity of the type 2 team by 1 level. Table 3-9 shows that the maximum
allowable average time in system does not materially impact the accuracy of the HCTSCP
model in either the preemptive case or the non-preemptive one. In summary, the HCTSCP
model with DBA is efficient and accurate in solving the capacity planning problems,
particularly in the preemptive case, e.g., the ED application. For the non-preemptive case,
e.g., the OC application, its accuracy in type 2 care team requirements is not as precise.
3.6 Concluding Remarks and Future Research Directions
In this paper, motivated by the emergency medicine services at a community hospital
and specialty services at an outpatient clinic, we examined the health care team service
capacity planning problem. These health care systems provide services to two major
patient classes using two types of health care teams and two types of E/T rooms. While
class 1 patients are served by the type 1 team in the dedicated E/T rooms only, class 2
patients can be served by either the type 1 or the type 2 team in the E/T rooms dedicated
to the corresponding team. However, the type 1 team delivers service to a class 2 patient
only if there are no class 1 patients waiting in the system and all type 2 E/T rooms are
occupied by other class 2 patients. The service capacity of each team is measured by the
service rates of the dedicated E/T rooms for each patient class. Although this setting is
similar to some other studies in the literature, the distinguishing characteristic of our work
is that we have patient-class- and team-type-dependent service rates.
In our work, we first considered the case where the class 1 patients can preempt class
2 patients served in type 1 E/T rooms, which is typical in emergency rooms in hospitals.
66
We also analyzed the non-preemptive case, common in outpatient clinic settings. We
developed queueing models for both cases, and developed approximation procedures
to estimate the average time that each patient class spends in the system. Through
an extensive computational study, we illustrated that our DBA method provides the
performance measures of interest efficiently with sufficient accuracy. Using the results
of the queueing analysis, we then developed a non-linear binary integer programming
model to determine the minimal cost capacity plan of health care teams that a health
care facility should employ over a planning horizon to deliver service for the patients
while ensuring that the average time that each patient class spends in the system does not
exceed certain values. Our computational study showed that our approximation approach
provides sufficiently accurate results that can be used in practice to make long-term health
care team service capacity planning decisions.
We note that in our queueing analysis, we assumed exponentially distributed
interarrival and service times to preserve analytical tractability. Extending our work
to consider general arrival and service processes is a potential area for future research.
Moreover, in our study we analyzed systems where patients are categorized into two
classes. Although this classification is widely used in the health care industry, some clinics
and hospitals further classify each of the classes into two or more subclasses [100]. The
presence of multiple subclasses in each patient class magnifies the problem size rapidly and
cannot be solved through general numerical methods (e.g., the GE or the AGE method).
Therefore, developing approximation procedures for such settings would be of theoretical
and practical interest. In our work, we focused on the long-term capacity planning of
the health care teams, treating the E/T rooms as the servers and assuming that the
service rates of the rooms for different patient classes are given, and room allocation is
fixed over the planning horizon. For more detailed analysis, the set of tasks required for
the diagnosis, monitoring, and treatment of a patient can be modeled using a queueing
67
network, and the service rates of the E/T rooms under different room allocations and
team configurations can be further investigated.
68
CHAPTER 4HOSPITAL BED ALLOCATION PROBLEM
4.1 Introduction
In this chapter, we introduce the hospital bed allocation (HBA) problem. The HBA
problem is an extension of the AHBCP problem in Chapter 2, which is concerned with
determining the optimal aggregate bed capacity plan over a finite planning horizon. After
the aggregate bed capacity is specified, the next step involved is concerned with the
allocation of aggregate bed capacity among different medical care units (MCUs) (e.g.,
neurosurgery, oncology, pediatrics, etc.).
Ineffective allocation of existing bed capacity among different medical service units
can lead to service quality problems for the patients along with operational and/or
financial inefficiencies for the hospitals. An arriving patient may be declined or placed
on hold by an MCU, if there is no bed available to accommodate the patient in the unit.
In this case, the patient may be subject to same health risks due to the necessity to
find an alternative health care provider or wait until a bed becomes available. From the
hospital’s perspective, the potential revenue is either lost or deferred, which may lead to
some financial inefficiencies. Similarly, an arriving patient may be accepted for treatment
but can be accommodated in another MCU (e.g., an arriving obstetrics patient can be
boarded in neurosurgery). In this case, the patient may be subject to same unnecessary
health risks due to the necessity to be boarded together with patients with more different,
possibly more serious, health conditions. From the hospital’s perspective, the potential
revenue is collected but the operational costs may increase, as the patient may be boarded
in an MCU where the service resources are more expensive (e.g., neurosurgery nurses
may have additional qualifications than obstetrics nurses and the medical equipment in a
neurosurgery department are typically more expensive than the ones in obstetrics).
To effectively utilize existing bed capacity, hospital administration could choose from
among a number of alternative planning strategies to to find an allocation of existing
69
aggregate bed capacity among different MCUs. In particular, there are four practical
planning strategies, including
• the expected bed occupancy is balanced across entire hospital,
• the expected net profit of the hospital is maximized,
• the occurrence of bed shortages is minimized, or
• the number of patients rejected is minimized.
In this work, we focus on the first strategy and develop a mathematical programming
formulation to address this problem. We also develop effective solution approaches to
obtain high quality solutions particularly for large-sized, realistic test instances.
The remainder of this chapter is organized as follows. In Section 4.2, we review the
related literature. Section 4.3 presents mathematical programming formulation for HBA.
We develop three heuristic solution approaches in Section 4.4. In Section 4.5, we present
results from our computational study that evaluates the computational performance of the
approaches developed in Section 4.4. Section 4.6 includes a discussion of the results and
suggests future research directions.
4.2 Literature Review
Most health care managers apply relatively simple approaches, such as the use of
target occupancy level with average length of stay, to forecast bed capacity required for a
hospital or an MCU. Yet, the failure to adequately consider the uncertainties associated
with patient arrivals and time needed to treat patients by using such simple approaches
may result in bed capacity configurations where a large portion of patients may have to
be turned away [46]. To take the stochastic nature of health care systems into account,
researchers utilize queuing and simulation models in determine the appropriate bed
capacity configuration.
The application of queueing theory allows for the evaluation of the expected
(long-run) performance measure of a system by solving the associated set of flow balance
equations. Mackay and Lee [80] evaluate the choice of models for forecasting bed capacity
and suggest to use compartmental flow model, which models patient flow through a
70
hospital as flow through a sequence of compartments. Patients with short length of stay
may leave the system after visiting the first compartment, otherwise, patients move
to the next compartment until their length of stay are reached. The benefit of using
compartmental flow model is that it can capture the variation in bed occupancy without
using a sophisticated method.
Gorunescu et al. [43] model a department of geriatric medicine as an M/M/c/K
queueing system to investigate the interrelationships between admission rates, length
of stay, number of allocated beds, and probability that an arriving patient is denied
admission. Kao and Tung [63] present an approach for allocating beds to care units in
a hospital to minimize the expected patient overflow, i.e., the rate at which patients are
denied admission due to inavailability of bed capacity. They model each service as an
M/G/∞ queueing system and use normal approximation in computing number of patient
overflows. The bed allocation problem is solved in two stages. The first stage distributes
the majority of beds such that no gross imbalances in bed utilization among all care
units are observed and a pre-specified fraction of patients can stay in the units designated
for the unit. The second stage uses marginal analysis to allocate the remaining beds to
minimize the expected total patient overflow.
Discrete-event simulation is useful for the analysis of systems with complex behavior,
including health care systems. Discrete-event simulation has been widely applied in
health care services [58] to study the interrelationships between admission rates, hospital
occupancy, and several different policies for allocating beds to MCUs. Harrison et al.
[52] construct a simulation model where patients’ stay in hospitals are classified into
three stages, which represent different phases of care provided. The output obtained from
the model matches the mean and the variability associated with actual bed occupancy
data. The model is used to identify daily occupancy distributions, study trade-offs
between overflow and bed capacity levels, and investigate the effects of various changes.
Akkerman and Knip [6] use Markov chain approach to specify the number of beds needed
71
for two hospital wards, and then utilize discrete-event-simulation to obtain detailed
information on expected bed occupancy and patient rejection levels. Masterson et al. [84]
use discrete-event simulation to investigate the interrelationships between bed occupancy,
average number of patient deferred, and different bed allocation and operating policies.
Harper [51] develops a simulation model for the planning and management of hospital
beds, operating theaters, and workforce needs. The model captures the complexity
of health care systems by incorporating the variability for each patient group such as
monthly, daily, and hourly demand as well as the distributions of length of stay and
operation times. Kim et al. [68] analyze an intensive care unit with 14 beds and develop
a simulation model to evaluate different bed-reservation schemes to reduce the number of
cancelled surgeries.
Some papers consider the hierarchical relation between care units. For example, after
a mother-to-be delivers her child in the labor and delivery unit, she should be moved to
the postpartum unit for recovery. When the capacity downstream is insufficient, patients
are forced to stay at the current care units with typically more expensive equipment
blocking the capacity at these upstream care units. To take the interactions among care
units in a hospital into account, Cochran and Bharti [30] first apply queueing network
methodology (without blocking) to find a balanced bed allocation, which is obtained
through trial-and-error work. Then, they use simulation analysis to estimate the blocking
behavior and patient sojourn times. Galvao et al. [39, 40] apply a three-level hierarchical,
capacitated model to determine the capacity required in perinatal health care facilities,
which is categorized into three levels: basic units, maternity homes, and neonatal clinics
where intensive care unit for babies is available.
In our work, we utilize a different approach by integrating results from queueing
theory into an optimization framework. Specifically, we model each MCU in a hospital
as an M/M/c/c queueing system to estimate the probability of rejection when there are
c beds in the unit. We then develop an optimization model to allocate the aggregate bed
72
capacity across different MCUs. The purpose of this work is to develop efficient solution
approaches to solve this problem.
4.3 Problem Formulation
We now begin the mathematical formulation with the objective of balancing bed
occupancy throughout the hospital. We consider a hospital with D MCUs and B beds
available. For each service there are lower and upper limits on the number of beds
allocated, denoted by li and ui, respectively. In addition, there is a lower bound on the
bed occupancy for each MCU, denoted by γ. To analyze the bed occupancy of each MCU,
denoted by ρi, we assume that the patient arrivals of each MCU are Poisson processes
and the lengths of stay at each MCU are exponentially distributed. We assume that an
arriving patient to MCU i is rejected, if all beds designated for the MCU i are occupied.
Let pi denote the probability of rejecting an arriving patient of MCU i, and xi be the
number of beds allocated to MCU i. Let ρ be the bed occupancy of the entire hospital, λi
be the patient arrival rate and 1/µi be the average lengths of stay at MCU i. Let xi be the
decision variables of HBA problem, i.e., the number of bed allocated at MCU i. We then
formulate the HBA problem as a nonlinear integer programming formulation as follows:
minD∑
i=1
|ρi − ρ| (4–1)
subject to
D∑i=1
xi = B (4–2)
pi =(λi/µi)
xi
xi!/
xi∑m=0
(λi/µi)m
m!∀ i (4–3)
ρi =(1− pi)λi
xiµi
∀ i (4–4)
ρ =D∑
i=1
ρixi
B(4–5)
li ≤ xi ≤ ui ∀ i (4–6)
γ ≤ ρi ≤ 1 ∀ i (4–7)
xi ∈ Z∗ ∀ i (4–8)
73
The objective function (4–1) minimizes the total deviation of the bed occupancy for
each of the MCUs from the overall average bed occupancy for the hospital. Constraint
(4–2) limits the total number of beds allocated among the MCUs to the number of total
beds available in the hospital. Constraint set (4–3) represents the probability of rejecting
patients arriving to service i as a function of arrival rate λi, service rate µi and bed
capacity xi for service i. Constraint set (4–4) represents the bed occupancy of the service i
as a function of effective arrival rate (1− pi)λi, service rate µi and bed capacity xi for each
service i. Constraint (4–5) specifies the overall average bed occupancy for the hospital.
Constraint sets (4–6) and (4–7) impose the lower and upper bounds on the bed capacity
and the bed occupancy for each service i, respectively. Finally, constraint (4–8) ensures
that the decision variables are non-negative integers.
HBA is a difficult nonlinear binary integer programming problem with nonlinear
constraints. Note that constraint set (4–3) involves the factorial function on xi and the
summation of the term (λi/µi)m
m!from m = 1 up to m = xi, which increases the complexity
of the problem significantly. Since decision variables xi’s assume discrete values, we
can introduce the splitting variable yij, where yij takes value of one if xi = bij and zero
otherwise, where bij ∈ {li, li + 1, ..., ui − 1, ui}. Let ρij and pij represent the bed occupancy
and probability of rejecting an arriving patient of MCU i, respectively, when bed capacity
of MCU i equals bi.
As a result, we can obtain an equivalent linear binary integer programming problem
that can be stated as follows:
minD∑
i=1
ui−li∑j=1
δijyij (4–9)
subject to
D∑i=1
ui−li∑j=1
bijyij = B (4–10)
74
pij =(λi/µi)
bij
bij!/
bij∑m=0
(λi/µi)m
m!∀ i, j (4–11)
ρij =(1− pij)λi
bijµi
∀ i, j (4–12)
ρ =D∑
i=1
ui−li∑j=1
ρijbijyij
B(4–13)
γ ≤ ρij ≤ 1 ∀ i, j (4–14)
ui−li∑j=1
yij = 1 ∀ i (4–15)
δij ≥ ρij − ρ ∀ i, j (4–16)
δij ≥ ρ− ρij ∀ i, j (4–17)
yij ∈ {0, 1} ∀ i, j (4–18)
bij ∈ {li, li + 1, ..., ui − 1, ui} ∀ i, j (4–19)
We have four sets of additional constraints. Constraint set (4–15) ensures that
only one splitting variable takes the value of one. Constraint sets (4–16) and (4–17)
compute the deviation of the bed occupancy for each service i from the overall average bed
occupancy for the hospital. Constraint set (4–18) ensures that the decision variables take
binary values. Finally, constraint set (4–19) provides information of the discrete options of
xij.
4.4 Solution Algorithms
The HBA problem with variable splitting formulation can be solved by commercial
MILP solvers, however, it takes time to obtain the optimal solutions of real-size problems.
This section describes three solution approaches we develop for the HBA problem, which
includes genetic algorithm (GA), greedy randomized adaptive search procedure (GRASP)
and a hybridization of GA and GRASP.
4.4.1 Genetic Algorithm
GA constructs a population of solutions and generates new generations of solutions
mimicking the behavior of population genetics [41, 53]. In particular, two members of the
current population of solutions are chosen randomly and used to generate new offspring
75
which are then retained for the next generation of solutions if they qualify. Figure 4-1
depicts the pseudo-code of the GA. A preprocessing step ensures that a test instance is
feasible by verifying the validity of the following inequalities:
D∑i=1
li ≤ B (4–20)
D∑i=1
ui ≥ B (4–21)
ρ(li) ≥ γ ∀i (4–22)
where ρ(li) represents the bed occupancy with bed capacity li of service i.
procedure GeneticAlgorithm(instance)
1 Read and preprocess the input data;2 Solution* ← 0;3 Population ← GeneratePopulations(PopulationSize);4 NumGenNotImprove ← 0;5 for m = 1 to MaxNumGenerations do6 Children ← ProduceOffspring(Population, NumGenNotImprove);7 Population ← UpdatePopulation(Population, Children);8 Solution* ← UpdateSolution(Solution*, Children, NumGenNotImprove);9 end for
end GeneticAlgorithm.
Figure 4-1. Pseudo-code of the genetic algorithm
GA starts by generating a set of distinct feasible solutions to form an initial
population of solutions. These solutions are generated through an occupancy-driven
approach, and then adjusted to feasible by a random-rectified approach (see Figure 4-2).
The occupancy-driven approach first treats each MCU as an M/M/c queueing
system, and then allocates a number of beds to the MCU such that the bed occupancy is
close to γ (see Figure 4-3). As it can be observed from constraint set (4–3), specifying the
probability of rejecting a patient from an MCU that is modeled as an M/M/c/c queueing
system requires the knowledge of the number of beds allocated to that particular MCU.
Since our objective is to generate some solutions for the initial population of GA, rather
76
procedure GeneratePopulations(PopulationSize)
1 Population ← ∅;2 while NumInSet (Populaiton) < PopulationSize do3 x ← OccupancyDrivenAllocation(λ, µ, γ);4 x ← RandomRectified( x, l, u, γ);5 if x /∈ Population then6 Population ← Population ∪ x;7 end if8 end while9 return (Population);
end GeneratePopulations.
Figure 4-2. Pseudo-code of the population generating procedure
than spending time to find the corresponding bed capacity for an M/M/c/c queueing
system, we model the MCU as an M/M/c queueing system to initialize the bed capacity.
procedure OccupancyDrivenAllocation(λ, µ, γ)
1 x ← 0;2 for i = 1 to D do3 xi ← λi
γµi;
4 end for5 return (x);
end OccupancyDrivenAllocation.
Figure 4-3. Pseudo-code of occupancy-driven allocation
The solutions generated from the occupancy-driven approach may not satisfy
constraint sets (4–2), (4–6) and (4–7), i.e., constraint sets associated with the total
number of beds available, lower and upper bounds and bed occupancy (of an M/M/c/c
queueing system). We use a random-rectified procedure to fine-tune the initial bed
allocation. Figure 4-4 shows the pseudo-code of the random-rectified procedure. We
first revise the bed capacity of each MCU such that the constraints (4–6) and (4–7) are
satisfied. Then, when the total number of beds allocated is greater (less) than B, an MCU
is chosen randomly and its bed capacity is decreased (increased) by one, if this operation
does not violate constraint (4–6). The step of choosing an MCU randomly and adjusting
its capacity accordingly is repeated until the total number of beds allocated is equal to the
overall bed capacity available, B.
77
procedure RandomRectified( x, l, u, γ)
1 for i = 1 to D do2 while ρ(xi) < γ do3 xi ← xi − 1;4 end while5 xi ← min{xi, ui};6 xi ← max{xi, li};7 end for8 while NumBedAllocated(x) < B do9 i ← RandomlySelectInteger(D);
10 xi ← min{xi + 1, ui};11 end while12 while NumBedAllocated(x) > B do13 i ← RandomlySelectInteger(D);14 xi ← max{xi − 1, li};15 end while16 return (x);
end RandomRectified.
Figure 4-4. Pseudo-code of random-rectified procedure
After the initial population is formed, GA starts to produce offspring. Two types of
genetic operators are considered in this work; one is simple single-point crossover, and
the other is mutation. In simple single-point crossover (see Figure 4-5), we first randomly
select two parents from the population, and then swap bed capacity between the randomly
selected dD/2e MCUs. That is, if the MCU i is selected, the bed capacity of MCU i of
the first parent’s chromosome is swapped with the bed capacity of MCU i of the second
parent’s chromosome. The resulting solutions are the offspring solutions produced by the
selected parent solutions. Figure 4-6 shows an example with four MCUs and 100 beds,
where the second and the third MCUs are chosen to swap their bed capacity. Note that
the crossover may produced an offspring which does not have a chromosome specifying a
feasible solution. In this case, the random-rectified approach is employed to adjust the bed
capacity allocation.
The second genetic operator, mutation, is invoked if the current best solution is not
improved throughout the evolution of a pre-specified number of consecutive generations.
The mutation employs the occupancy-driven approach with the average occupancy
78
procedure Crossover(Population)
1 x ← RandomlySelectElement(Population);2 y ← RandomlySelectElement(Population);3 while y=x do4 y ← RandomlySelectElement(Population);5 end while6 SwapSet ← ∅;7 while NumInSet(SwapSet) 6= dD/2e do8 i ← RandomlySelectInteger(D);9 if i /∈ SwapSet then
10 SwapSet ← SwapSet ∪ i;11 end if12 end while13 for i = 1 to D do14 if i ∈ SwapSet then15 temp ← xi;16 xi ← yi;17 yi ← temp;18 end if19 end for20 x ← RandomRectified(x);21 y ← RandomRectified(y);22 if f(x) < f(y) then23 return (x);24 else25 return (y);26 end if
end Crossover.
Figure 4-5. Pseudo-code of crossover procedure
(10, 20, 30, 40) (10, 16, 24, 40) ↕ ↕ �
(38, 16, 24, 22) (38, 20, 30, 22)
Figure 4-6. Example of crossover
of the current best solution to produce an offspring. Again, this mutated offspring is
adjusted through the random-rectified procedure to ensure feasibility. Then, the value of
NumGenNotImprove is set to zero.
If the offspring produced from either crossover or mutation procedure does not exist
in population and is better than the current worst solution in population, then the current
79
procedure Mutation(x*)
1 ρ* ← ρ(x*);2 x ← OccupancyDrivenAllocation(λ, µ, ρ*);3 x ← RandomRectified( x, l, u, γ);4 NumGenNotImprove← 0;5 return (x);
end Mutation.
Figure 4-7. Pseudo-code of the mutation procedure
worst solution in the population is replaced with the offspring to form a new generation
with the other existing solutions. Otherwise, i.e., either the offspring exists in population
already or the offspring is not better than the current worst solution in the population,
this offspring is ignored and the next offspring is produced using the procedure described
above. Note that the current best solution is also updated if the new offspring has a better
objective function value than that of the current best.
4.4.2 Greedy Randomized Adaptive Search Procedure
Greedy randomized adaptive search procedure is a multi-start approach [36] that is
widely used for combinatorial optimization problems. Each GRASP iteration consists of
two phases: construction and local search. Figure 4-8 depicts the pseudo-code for GRASP.
The construction phase creates a feasible solution, whose neighborhood is explored by
the local search phase to find a locally optimal solution. The algorithm stops after a
pre-specified number of iterations is executed, which is denoted by MaxNumIterations in
Figure 4-8.
procedure GRASP(instance)
1 Read and preprocess the input data;2 Solution* ← 0;3 for n = 1 to MaxNumIterations do4 Solution ← GreedyRandomizedConstruction(α);5 Solution ← LocalSearch(Solution);6 Solution* ← UpdateSolution(Solution, Solution*);7 end for
end GRASP.
Figure 4-8. Pseudo-code of GRASP
80
The construction phase builds a feasible solution in a greedy manner. First, each
MCU is allocated an initial bed capacity at its lower bound to form an incomplete
solution. Then, one bed is added to an MCU if this capacity expansion does not destroy
the upper bound constraint and the increment on the objective value is at an acceptable
level. The pseudo-code of the construction procedure is illustrated in Figure 4-9, where
the ej in line 22 represents a zero vector except the jth element equals one. Lines 15
through 20 in Figure 4-9 build a restricted candidate list (RCL), which records the set
of MCUs for which adding one more bed to the MCU does not violate the feasibility and
has the potential to improve the objective function value at an acceptable level. The
threshold of increment on the objective function value is controlled by the parameter
α ∈ {0, 1} on line 17 in Figure 4-9. An MCU is included in RCL, if adding one more bed
to the unit is feasible, and the increment on the objective function value is not greater
than δmin + α(δmax − δmin), where δmin and δmax represent the minimum and maximum
increments on objective function value after incorporating one more bed to current
solution, respectively. Note that the lower the α is, the greedier the procedure is. A bed is
added to an MCU selected randomly from RCL, until the total number of beds allocated
equals B.
During the local search phase, the neighborhood of the feasible solution created in the
construction phase is fully investigated to find the local optimum. A neighbor solution is
produced by swapping one bed from one MCU to another, if this swap does not destroy
the feasibility of the solution. The pseudo-code of the local search procedure is illustrated
in Figure 4-10. Here, we adopt the best-improving strategy that is we evaluate all feasible
neighbors and choose the neighbor that improves the objective function value the most.
4.4.3 Hybridization of GA & GRASP
We develop a hybridization of GA and GA. The pseudo-code for the hybrid approach
(HA) is given in Figure 4-11. On each iteration of the HA, a set of elite solutions is
generated, in contrast to a single initial solution used by GRASP. Then, an offspring is
81
procedures GreedyRandomizedConstruction(α)
1 x ← 0;2 for i = 1 to D do3 xi = li;4 end for5 while NumBedAllocated(x)6= B do6 δmin ← +∞;7 δmax ← −∞;8 for i = 1 to D do9 if xi + 1 ≤ ui then
10 δi ← f(x+ei)− f(x);11 δmin ← min{δmin, δi};12 δmax ← max{δmax, δi};13 end if14 end for15 RCL = ∅;16 for i = 1 to D do17 if xi + 1 ≤ ui and δi ≤ δmin + α(δmax − δmin) then18 RCL ← RCL ∪ i;19 end if20 end for21 j ← RandomlySelectElement(RCL);22 x ← x + ej;23 end while24 return (x);
end GreedyRandomizedConstruction.
Figure 4-9. Pseudo-code of greedy randomized construction procedure
produced using the crossover procedure of GA and improved by the local search procedure
of GRASP. The previous two steps are repeated, until a pre-specified number of iterations
are completed. The characteristic of the HA is that on each iteration the threshold α for
the greedy randomized construction phase is updated by a decreasing function p(n), i.e.,
α decreases as the number of iterations, n, increases. In this way, the HA can test more
than one setting of α and reduce the probability of converging to a local optimum solution
prematurely.
To form an elite set of solutions, we first use the GreedyRandomizedConstruction
procedure of GRASP to generate NumGRASP distinct solutions, and then choose the best
ESize of them, where NumGRASP > ESize. The pseudo-code is presented in Figure 4-12.
82
procedure LocalSearch(x)
1 x*←x;2 f*← f(x);3 for i = 1 to D do4 for j = 1 to D do5 if i 6= j, xi − 1 ≥ li, xi + 1 ≤ ui and ρ(xi + 1) ≥ γ then6 if f(x−ei + ej) < f* then7 x*←x−ei + ej;8 f*← f(x−ei + ej);9 end if
10 end if11 end for12 end for13 return (x*);
end LocalSearch.
Figure 4-10. Pseudo-code of local search procedure
procedure Hybrid(instance)
1 Read and preprocess the input data;2 Solution* ← 0;3 for n = 1 to MaxNumIterations do4 α ← p(n);5 EliteSet ← GenerateEliteSet(α, Solution*);6 for m = 1 to MaxNumGenerations do7 Solution ← Crossover(EliteSet);8 Solution ← LocalSearch(Solution);9 Solution*← UpdateSolution(Solution, Solution*);
10 end for11 end for
end Hybrid.
Figure 4-11. Pseudo-code of HA
4.5 Computational Study
In this section, we present results from our computational study, where we investigate
the computational efficiency of the proposed solution approaches. Specifically, we measure
the efficiency of the approach using the CPU time needed to obtain the solution and
relative error in the objective function value, given by
fA − f ∗
f ∗× 100%
83
procedure GenerateEliteSet(α, Solution*)
1 EliteSet ← ∅;2 for j = 1 to NumGRASP do3 x ← GreedyRandomizedConstruction(α);4 while x∈ EliteSet do5 x ← GreedyRandomizedConstruction(α);6 end while7 y ← the worst solution in EliteSet;8 if f(x) < f(y) then9 EliteSet ← EliteSet \ y ∪ x;
10 end if11 Solution*← UpdateSolution(x, Solution*);12 end for13 return (EliteSet);
end GenerateEliteSet.
Figure 4-12. Pseudo-code of elite set generation procedure
where fA denotes the objective function value obtained by using approach A, where
A ∈ {GA, GRASP, HA}.In our study, we consider three settings each with different instance sizes to evaluate
the impact of the size of the instance on the performance of the proposed solution
approaches. The problem size is varied with the number of total beds available in the
hospital, B, and number of MCUs in the hospital, D. For the total number of beds and
the number MCUs available in the hospital, we consider three levels that correspond to
small, medium, and large sized hospitals. In particular, we consider hospitals with 750,
1,000 or 1,250 beds and 30, 40 or 50 MCUs. For each setting, we generate 30 random
instances. Each random instance is obtained by generating parameters that correspond
to patient arrival and service rates along with lower and upper bounds on the number of
beds available in each MCU. Specifically, each parameter is obtained by using the formula:
(mean value)∗u, where u is drawn from the distribution U [0.4, 1.6]. The mean values of
the random parameters are listed as follows:
1. Mean arrival rate (λi): 10 persons/per unit time for each MCU i;
2. Mean service rate (µi): 0.5 persons/per unit time for each MCU i;
3. Mean lower bound (li): 10 beds for each MCU i; and
84
4. Mean upper bound (ui): 80 beds for each MCU i.
Last, the minimum bed occupancy, γ, is 70% for all MCU all problems. The
experiment is implemented on a workstation with two Pentium 4 3.2 GHz processor
and 6 GB of memory.
To obtain a near-optimal solution as a comparison basis, we use CPLEX to obtain
the optimal solutions for the test instances. For each test instance, CPLEX is stopped
if the relative stopping tolerance of 0.01% is satisfied, or CPU time of 3,600 seconds is
used. Table 4-1 reports the results obtained by CPLEX. As we expect, the number of
instances for which CPLEX can determine and verify the optimal solution within one
hour of CPU time decreases as the problem size increases. Moreover, for the instances
with small problem size, CPLEX takes more than 1,800 seconds on average to solve one
instance.
4.5.1 Summary of Results Obtained by GA
GA is executed for 200D generations and performs a mutation when best solution
is not improved for D generations, where D is the number of MCUs in consideration. To
evaluate the impact of the population size, we consider three levels of D, 2D and 3D for
each setting. Table 4-2 reports the results of GA, where the information is classified into
two layers. The first layer distinguishes the instances with different sizes, and the second
specifies the parameter setting of GA, including number of generations and population
size. Note that if the relative error of an instance is less than zero, it is replaced by zero
when computing the average relative error of the corresponding instance.
In general, GA performs quite well in terms of average relative error and CPU time.
For any of the 90 instances, GA spends less than 2 seconds to find a solution and the
average relative error is less than 2%. Furthermore, for more than one third of the test
instances, GA finds solutions better than CPLEX. For example, for the fourth large-sized
instance, GA finds a solution with an objective function value which is about 5.6% lower
than that of the solution found by CPLEX.
85
Table 4-1. Near-optimal solutions obtained by using CPLEX
Problem size Small Medium Large(D, B) (30, 750) (40, 1000) (50, 1250)Instance Objective Time (sec.) Objective Time (sec.) Objective Time (sec.)
1 0.1964 37 0.2691 318 0.5298 > 3,6002 0.6132 4 0.4122 > 3,600 1.0745 > 3,6003 0.2574 > 3,600 0.6417 > 3,600 0.5083 > 3,6004 0.2869 1,972 0.3864 > 3,600 0.7238 > 3,6005 0.4828 > 3,600 0.4141 > 3,600 1.0353 > 3,6006 0.4588 21 0.7636 > 3,600 0.2228 > 3,6007 0.2817 1,369 0.3180 > 3,600 0.5841 > 3,6008 0.2484 709 0.5691 > 3,600 0.4073 > 3,6009 0.1351 2,111 0.4530 > 3,600 0.6679 > 3,60010 0.4558 110 0.3737 > 3,600 1.1034 > 3,60011 0.3447 1,277 0.3347 > 3,600 0.5721 > 3,60012 0.4320 47 0.4972 > 3,600 0.6960 > 3,60013 0.2872 > 3,600 0.6381 > 3,600 0.4622 > 3,60014 0.4589 4 0.7216 > 3,600 0.7431 > 3,60015 0.5883 > 3,600 0.6496 > 3,600 0.4568 > 3,60016 0.2848 375 0.8077 > 3,600 0.9320 > 3,60017 0.2751 > 3,600 0.5183 > 3,600 0.5609 > 3,60018 0.1942 > 3,600 0.2022 543 0.5702 > 3,60019 0.3408 > 3,600 0.6919 29 0.7131 > 3,60020 0.2600 8 0.4640 > 3,600 0.6928 > 3,60021 0.3262 > 3,600 0.6155 > 3,600 1.0242 > 3,60022 0.4457 > 3,600 0.5205 1,727 0.6448 > 3,60023 0.1813 8 0.4363 > 3,600 0.4886 > 3,60024 0.3991 > 3,600 0.1789 > 3,600 0.7533 > 3,60025 0.6148 > 3,600 0.3602 > 3,600 0.7442 > 3,60026 0.3832 > 3,600 0.7231 > 3,600 0.4640 > 3,60027 0.8989 12 0.7193 > 3,600 0.5765 > 3,60028 0.3392 730 0.5233 > 3,600 0.7700 > 3,60029 0.1377 1,770 0.6959 > 3,600 0.4138 > 3,60030 0.4448 13 0.4833 577 1.0881 > 3,600
No. of instancessolved by CPLEX 18 5 0
The parameter of population size does not appear to have significant impacts on
relative errors and CPU times. For example, for medium-sized test instances, when the
population size grows from 30 to 90, the average relative error changes from 1.18% to
1.55% and the average CPU time slightly increases from 1.01 seconds to 1.05 seconds.
Moreover, about 80% of the 30 instances obtain the relative errors at the same level with
respect to three different sizes of population.
86
Tab
le4-
2.Sol
uti
ons
obta
ined
by
GA
Pro
ble
mSm
all
Med
ium
Larg
esi
zeN
o.of
6000
6000
6000
8000
8000
8000
10000
10000
10000
gen
erati
ons
Popula
tion
30
60
90
40
80
120
50
100
150
size
Err
.T
ime
Err
.T
ime
Err
.T
ime
Err
.T
ime
Err
.T
ime
Err
.T
ime
Err
.T
ime
Err
.T
ime
Err
.T
ime
Inst
ance
(%)
(sec
.)(%
)(s
ec.)
(%)
(sec
.)(%
)(s
ec.)
(%)
(sec
.)(%
)(s
ec.)
(%)
(sec
.)(%
)(s
ec.)
(%)
(sec
.)1
0.0
00.5
60.0
00.5
70.0
00.5
80.0
01.0
00.0
01.0
00.0
01.0
26.2
61.5
66.2
61.6
06.2
61.6
82
9.2
80.5
89.2
80.6
09.2
80.6
23.8
61.0
13.8
61.0
43.8
61.0
6-2
.10
1.6
0-2
.10
1.5
9-2
.10
1.6
73
0.0
00.5
70.0
00.5
80.0
00.6
00.0
01.0
30.0
01.0
40.0
01.0
60.0
01.5
60.0
01.5
90.0
01.6
74
0.0
00.5
60.0
00.5
70.0
00.5
71.7
41.0
01.7
41.0
31.7
41.0
4-5
.58
1.5
2-5
.58
1.5
6-5
.58
1.6
45
0.0
00.6
00.0
00.6
10.0
00.6
10.0
00.9
80.0
00.9
90.0
01.0
10.6
91.5
60.6
91.5
80.6
91.6
56
0.0
00.5
50.0
00.5
60.0
00.5
73.0
91.0
43.2
91.0
73.3
11.0
90.0
01.5
80.0
01.6
20.0
01.7
07
0.0
00.5
70.0
00.5
80.0
00.5
90.0
00.9
90.0
01.0
10.0
01.0
20.0
01.5
90.0
01.6
40.0
61.7
38
0.0
00.5
50.0
00.5
60.0
00.5
70.0
01.0
31.4
11.0
60.0
01.0
90.0
01.5
80.0
01.6
20.0
01.7
09
0.0
00.5
70.0
00.5
70.0
00.5
80.0
01.0
30.0
01.0
30.0
01.0
40.8
81.6
01.2
81.6
91.8
71.7
710
0.0
00.5
60.0
00.5
80.0
00.5
80.4
40.9
90.4
41.0
20.4
41.0
47.0
61.6
07.1
31.6
68.2
01.7
411
0.0
00.5
70.0
00.5
80.0
00.5
90.0
01.0
20.0
01.0
40.0
01.0
60.0
01.5
60.0
01.6
10.0
01.6
812
0.0
00.5
70.0
00.5
90.0
00.6
20.0
01.0
00.0
61.0
20.0
61.0
35.7
81.5
85.7
81.6
05.7
81.6
613
0.0
00.5
60.0
00.5
60.0
00.5
70.9
00.9
90.9
01.0
00.9
01.0
20.0
01.5
70.0
01.6
00.0
01.6
814
3.4
50.5
53.4
50.5
63.4
50.5
79.4
50.9
99.4
51.0
09.4
51.0
2-0
.11
1.5
8-0
.11
1.6
2-0
.11
1.7
115
0.0
00.5
70.0
00.5
80.0
00.5
80.0
01.0
10.0
01.0
20.0
01.0
40.0
01.5
40.0
01.5
50.0
01.6
316
0.0
00.5
60.0
00.5
70.0
00.5
90.0
01.0
00.0
01.0
30.0
01.0
41.8
21.5
62.0
01.5
92.3
51.6
717
0.0
00.5
70.0
00.5
70.0
00.5
80.0
01.0
10.0
01.0
30.0
01.0
51.0
91.6
01.0
91.6
41.0
91.7
318
0.0
00.5
80.0
00.6
10.0
00.6
20.0
01.0
40.0
01.0
50.0
01.0
80.5
91.6
00.5
91.6
60.4
11.7
619
0.0
00.5
70.0
00.5
70.0
00.5
83.4
31.0
42.4
91.0
84.6
81.1
10.6
61.5
80.6
61.6
10.6
71.7
020
0.0
00.6
00.0
00.6
20.0
00.6
40.0
01.0
19.3
41.0
29.3
41.0
50.9
11.5
81.0
41.6
14.3
41.6
921
0.0
00.5
70.0
00.5
70.0
00.5
90.0
01.0
20.0
01.0
50.0
01.0
70.2
71.5
50.2
71.5
90.0
01.6
822
2.0
60.5
60.0
00.5
80.0
00.5
90.0
00.9
70.0
00.9
80.0
01.0
00.3
21.6
00.3
21.6
40.3
21.7
323
0.0
10.5
60.0
10.5
60.0
10.5
60.0
01.0
30.0
01.0
60.0
01.0
91.0
11.6
11.0
81.6
61.1
01.7
724
4.8
30.5
54.8
30.5
64.8
30.5
80.0
01.0
10.0
01.0
10.0
01.0
33.4
01.5
23.4
01.5
43.4
01.6
225
2.8
20.5
73.4
80.5
83.5
50.5
90.0
01.0
10.0
01.0
40.0
01.0
60.0
01.5
30.0
01.5
51.1
01.6
226
0.0
00.5
70.0
00.5
80.0
00.5
97.2
01.0
37.2
01.0
57.2
11.0
8-2
.24
1.5
5-2
.24
1.5
8-2
.24
1.6
527
0.0
00.5
60.0
00.5
70.0
00.5
8-2
.06
1.0
0-2
.06
1.0
2-2
.06
1.0
5-0
.62
1.5
61.0
01.6
01.9
61.6
928
0.0
00.5
80.0
00.5
90.0
00.6
00.0
01.0
20.0
01.0
50.0
01.0
8-3
.76
1.5
6-3
.69
1.5
8-2
.31
1.6
729
0.0
00.5
80.0
00.6
00.0
00.6
25.4
01.0
05.4
01.0
25.4
01.0
40.0
01.5
30.0
01.5
70.0
01.6
330
4.9
50.5
95.0
30.6
05.2
20.6
20.0
01.0
20.0
01.0
40.0
01.0
72.4
51.6
03.0
41.6
42.5
01.7
3M
in.
0.0
00.5
50.0
00.5
60.0
00.5
6-2
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0.9
7-2
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0.9
10.5
70.8
70.5
80.8
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51.1
11.5
71.1
91.6
11.4
01.6
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ax.
9.2
80.6
09.2
80.6
29.2
80.6
49.4
51.0
49.4
51.0
89.4
51.1
17.0
61.6
17.1
31.6
98.2
01.7
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o.of
13
13
13
11
10
11
12
10
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87
4.5.2 Summary of Results Obtained by GRASP
GRASP is executed for 20D starts (iterations). The threshold parameter, α, of
construction phase is examined at 0.25, 0.30 and 0.35 three levels. Table 4-3 displays
the results of GRASP. As is obvious to see that GRASP takes CPU times longer than
GA to obtain solutions. GRASP spends 3.1 seconds in average to obtain a solution of
the problem with small size, in contrast to GA’s 1.6 seconds with respect to the problem
with large size. In general, the accuracy of GRASP and GA does not appear significantly
different, if we compare the best results from each of them. For example, in the problem
with medium size, the relative errors of GRASP (with α = 0.25) are 1.37% and 8.24% in
average and maximum, respectively, and GA (with population size = 40) are 1.18% and
9.45%, respectively. Furthermore, among 30 instance GRASP (with α = 0.25) finds results
better than CPLEX for 9 instances, compared to 11 instances of GA (with population size
= 40).
Table 4-3 presents the tendency that the lower α is, the lower the average relative
error is. However, a low value of α has the drawback that it may lead the solution to a
local optimal solution and result in a large error. For example, the instance 20 in problem
with medium size has relative error 8.24% when α is 0.25. In contrast, the instance has
less relative error, 3.77%, when α is increased to 0.35.
In GA and GRASP, we do observe that some instances have relative error greater
than 8%, such as the instance 14 of the problem with medium size in Table 4-2, and the
instance 20 of the problem with medium size in Table 4-3. To reduce the errors of those
instances, we combines these two algorithms to develop a hybridization.
4.5.3 Summary of Results Obtained by HA
HA is run for bD/10c iterations, in which 200D generations are produced. In the
beginning of each iteration, we generate 2D distinct solutions and pick the best D of these
solutions to form an elite set. The added feature of HA is that α decreases as the number
of iterations increases. To evaluate the effects of α on the proposed approach, we consider
88
Tab
le4-
3.Sol
uti
ons
obta
ined
by
GR
ASP
Pro
ble
mSm
all
Med
ium
Larg
esi
zeN
o.
of
600
600
600
800
800
800
1000
1000
1000
start
sα
0.2
50.3
00.3
50.2
50.3
00.3
50.2
50.3
00.3
5E
rr.
Tim
eE
rr.
Tim
eE
rr.
Tim
eE
rr.
Tim
eE
rr.
Tim
eE
rr.
Tim
eE
rr.
Tim
eE
rr.
Tim
eE
rr.
Tim
eIn
stance
(%)
(sec
.)(%
)(s
ec.)
(%)
(sec
.)(%
)(s
ec.)
(%)
(sec
.)(%
)(s
ec.)
(%)
(sec
.)(%
)(s
ec.)
(%)
(sec
.)1
0.0
03.0
00.0
02.9
60.0
02.9
40.0
08.9
00.0
08.8
40.0
09.3
96.3
322.7
86.7
622.5
18.5
624.1
92
0.9
83.2
42.8
13.1
82.8
13.1
54.3
69.1
04.5
78.9
57.9
19.2
1-2
.10
23.7
2-2
.10
23.5
4-2
.10
24.0
03
0.0
02.9
20.0
02.9
00.0
02.8
80.0
09.8
30.0
09.7
50.0
910.3
90.0
023.3
10.0
622.9
81.7
324.5
54
0.0
03.2
20.0
03.2
10.0
03.2
02.7
69.5
35.2
79.5
25.2
710.0
3-5
.58
23.5
0-5
.58
23.3
4-5
.58
24.8
45
0.0
03.2
00.0
03.1
41.0
83.1
30.0
09.2
10.0
09.1
30.0
09.3
60.4
724.3
60.9
524.2
11.2
725.0
26
-0.7
83.2
7-0
.78
3.2
3-0
.78
3.2
22.8
89.6
53.3
19.5
411.1
910.0
20.0
022.3
00.0
022.1
83.0
623.7
17
0.0
03.0
90.0
03.0
50.0
03.0
50.0
09.8
40.0
09.7
60.0
010.3
40.4
922.4
40.9
422.1
02.5
223.9
38
0.0
03.1
30.0
03.0
80.0
03.0
60.0
09.3
50.0
09.2
00.1
49.4
20.0
024.0
20.1
723.8
51.4
825.3
19
0.0
03.5
40.0
03.5
20.0
03.5
11.1
49.6
31.4
89.5
32.3
210.1
92.6
822.4
73.0
221.9
85.0
422.5
910
0.0
02.9
30.0
02.9
30.0
02.9
30.4
49.7
20.4
49.6
40.4
410.1
16.0
822.7
78.6
522.4
419.5
923.9
511
0.0
03.0
30.0
02.9
90.0
02.9
70.0
09.9
20.0
09.8
50.0
010.3
60.0
024.2
10.0
023.9
00.0
025.4
312
0.0
03.0
90.0
93.0
61.1
13.0
30.0
69.5
80.0
69.3
90.0
69.7
85.7
823.7
55.7
823.5
25.8
723.9
313
0.0
03.1
70.0
03.1
30.0
03.1
30.9
09.7
10.9
09.6
60.9
010.1
50.0
023.8
70.0
023.6
60.0
024.8
014
-0.1
43.3
9-0
.14
3.3
5-0
.14
3.3
27.5
29.6
07.8
89.5
89.0
010.2
50.5
023.2
54.8
123.0
317.5
824.2
115
0.0
03.1
20.0
03.1
00.0
03.1
10.1
89.9
70.1
89.8
00.5
710.1
20.0
023.5
50.0
023.4
40.0
024.8
716
0.0
03.2
50.0
03.2
12.0
83.2
00.4
29.4
50.0
99.3
22.3
89.6
33.2
023.9
13.9
023.6
63.8
625.1
417
0.0
03.4
50.0
03.4
10.0
03.3
60.0
09.8
40.0
09.7
50.7
110.0
30.5
123.9
72.1
423.8
54.8
325.0
618
0.0
03.4
10.0
03.3
73.8
83.3
30.0
09.1
70.0
09.0
10.0
09.5
50.0
923.4
41.3
023.1
94.8
824.7
119
0.0
02.9
90.0
02.9
50.0
02.9
41.7
38.9
62.6
68.8
64.9
29.1
70.6
621.9
70.7
521.7
81.4
923.2
120
0.0
03.2
20.0
03.1
60.0
03.1
38.2
49.3
66.0
19.1
93.7
79.5
04.4
323.1
44.4
622.8
68.0
823.2
021
0.0
03.3
30.0
03.3
30.0
03.2
90.0
09.5
20.4
49.3
718.1
99.9
1-1
.48
23.2
1-1
.42
22.9
27.3
724.8
322
0.5
03.1
00.4
03.0
60.0
03.0
4-1
.27
9.5
4-1
.27
9.4
6-1
.14
9.7
41.0
821.5
20.9
721.0
85.5
422.3
323
0.0
13.1
70.0
13.1
70.0
13.1
40.0
39.3
90.3
99.2
72.2
39.5
21.2
823.2
32.5
322.9
511.6
024.4
224
4.8
33.1
94.8
33.1
44.8
33.1
30.0
010.0
10.0
09.8
70.0
010.5
13.2
724.7
43.2
824.5
43.3
026.1
625
5.5
13.2
06.1
73.1
95.8
33.1
70.0
09.6
80.0
09.5
30.0
09.8
16.2
622.8
82.7
722.5
36.5
424.1
326
0.0
03.2
20.0
03.1
80.0
03.1
74.5
69.4
94.5
99.4
09.3
99.6
7-0
.50
23.2
9-0
.50
23.1
6-0
.50
24.5
727
-7.0
43.3
1-7
.04
3.2
6-7
.04
3.2
5-2
.06
9.5
3-2
.06
9.4
2-2
.06
9.8
82.6
323.5
71.2
423.2
81.3
724.5
328
0.0
03.1
34.0
03.0
611.9
53.0
30.0
59.1
10.6
69.0
02.1
59.2
98.3
823.4
98.0
623.0
88.8
424.0
529
0.0
03.0
10.0
02.9
80.0
02.9
35.6
89.7
75.5
29.7
06.2
310.2
90.0
024.0
00.0
023.9
20.0
025.3
630
3.7
42.9
51.1
12.9
33.9
82.9
10.0
09.5
00.2
59.3
46.9
19.8
81.9
822.7
92.0
122.2
55.6
924.0
7M
in.
-7.0
42.9
2-7
.04
2.9
0-7
.04
2.8
8-2
.06
8.9
0-2
.06
8.8
4-2
.06
9.1
7-5
.58
21.5
2-5
.58
21.0
8-5
.58
22.3
3A
vg.
0.5
23.1
80.6
53.1
41.2
53.1
21.3
79.5
31.4
99.4
23.1
69.8
51.8
723.3
22.1
523.0
64.6
724.3
7M
ax.
5.5
13.5
46.1
73.5
211.9
53.5
18.2
410.0
17.8
89.8
718.1
910.5
18.3
824.7
48.6
524.5
419.5
926.1
6N
o.
of
15
13
12
98
68
75
inst
ance
sG
RA
SP
finds
bet
ter
soln
.th
an
CP
LE
X
89
three updating functions of α as follows:
Constant: p1(n) = 0.3,
Linear: p2(n) = 1.0− 0.7n
N, and
Nonlinear: p3(n) = 1.0− 0.7
√n
N
where N equals bD/10c and is the total number of iterations.
To provide a benchmark, we include the constant function p1(n) in our experiment,
where α is set to 0.3 on each iteration n for n = 1,2,. . . , bD/10c. Using function p2(n), we
linearly decrease α to 0.3 as the number of iterations increase. Using function p3(n), we
decrease α to 0.3 in a non-linear fashion over the iterations. Note that α values generated
by function p3(n) is no larger than those generated by function p2(n) for each n as shown
in Figure 4-13.
0.00.20.40.60.81.01 2 3 4 5α n
p1(n)p2(n)p3(n)Figure 4-13. The updating functions of α
The results are shown in Table 4-4. In each test instance, the first column reports the
relative errors and CPU times obtained by using the updating function p1(n), and so on.
For the medium-sized test instances, the average (maximum) relative error decreases from
1.28% (9.34%) to 0.35% (4.56%), when p2(n) is used instead of p1(n) . The function p3(n),
which has smaller α than p2(n) at each iteration n, does not appear to make HA perform
better than the HA with p2(n). For example, the average relative error for large-sized
instances increases from 0.35% to 0.44%, if the p2(n) is replaced by p3(n). This is different
90
Tab
le4-
4.Sol
uti
ons
obta
ined
by
HA
Pro
ble
mSm
all
Med
ium
Larg
esi
zeN
o.
of
33
34
44
55
5it
erati
ons
No.
of
6000
6000
6000
8000
8000
8000
10000
10000
10000
gen
erati
ons
αp1(n
)p2(n
)p3(n
)p1(n
)p2(n
)p3(n
)p1(n
)p2(n
)p3(n
)fu
nct
ion
ESize
30
30
30
40
40
40
50
50
50
Err
.T
ime
Err
.T
ime
Err
.T
ime
Err
.T
ime
Err
.T
ime
Err
.T
ime
Err
.T
ime
Err
.T
ime
Err
.T
ime
Inst
ance
(%)
(sec
.)(%
)(s
ec.)
(%)
(sec
.)(%
)(s
ec.)
(%)
(sec
.)(%
)(s
ec.)
(%)
(sec
.)(%
)(s
ec.)
(%)
(sec
.)1
0.0
07.9
40.0
07.9
30.0
07.9
20.0
027.6
50.0
027.6
60.0
027.3
56.2
671.1
10.0
071.8
60.0
072.5
12
2.8
17.6
50.0
07.7
10.0
07.6
63.8
625.3
10.0
025.8
00.0
026.1
8-2
.10
62.1
9-2
.10
64.4
9-2
.10
63.2
63
0.0
07.1
80.0
07.7
00.0
07.4
40.0
024.2
20.0
024.5
60.0
024.7
90.0
077.7
00.0
075.5
30.0
077.4
64
0.0
08.0
70.0
08.1
70.0
07.9
51.7
425.7
51.7
425.5
21.7
425.8
4-5
.58
74.4
1-5
.58
73.7
1-5
.58
74.2
95
0.0
07.2
90.0
07.5
00.0
07.3
50.0
025.7
60.0
023.7
10.0
025.2
60.6
966.5
40.6
965.0
90.6
965.6
96
-0.5
88.0
00.0
08.1
40.0
08.0
52.5
525.0
60.0
025.6
80.0
025.8
10.0
077.3
40.0
077.1
90.0
077.7
67
0.0
07.3
90.0
07.7
40.0
07.3
30.0
027.9
90.0
028.0
60.0
028.2
90.0
070.4
20.0
073.2
70.0
072.7
68
0.0
07.9
40.0
07.0
70.0
07.9
10.0
024.6
50.0
024.6
30.0
024.7
40.0
072.0
10.0
073.6
10.0
074.5
19
0.0
08.1
80.0
08.3
60.0
08.1
30.0
026.0
50.0
026.4
50.0
026.4
12.6
870.8
12.6
869.4
80.0
070.4
510
0.0
07.0
40.0
07.2
70.0
07.0
80.4
425.8
70.4
425.3
10.4
425.7
02.6
067.4
9-1
.35
67.5
91.5
267.8
311
0.0
07.8
00.0
07.9
30.0
07.7
50.0
026.4
90.0
026.4
20.0
026.6
80.0
075.0
10.0
074.5
40.0
075.3
012
0.0
07.7
10.0
07.9
00.0
07.8
20.0
626.6
30.0
026.5
20.0
626.8
05.7
876.0
6-1
.57
75.6
1-1
.57
77.0
113
0.0
08.2
20.0
08.2
60.0
08.2
20.9
026.9
10.9
026.8
00.9
026.8
10.0
072.4
90.0
071.2
60.0
072.3
014
-0.1
47.4
60.1
67.4
71.9
27.3
58.6
525.3
72.8
325.6
63.7
625.7
4-0
.11
69.4
4-0
.11
70.5
5-0
.11
70.0
615
0.0
07.3
30.0
07.7
10.0
07.4
00.0
027.1
90.0
027.2
00.0
027.5
30.0
078.0
20.0
077.3
20.0
078.3
016
0.0
07.6
00.0
07.7
70.0
07.6
30.0
023.7
70.0
023.3
7-0
.68
24.4
61.6
267.2
30.0
067.1
70.0
068.1
217
0.0
08.1
40.0
08.4
00.0
08.1
50.0
025.3
60.0
025.2
50.0
025.5
30.0
069.2
00.7
870.6
90.7
871.0
118
0.0
07.8
80.0
08.3
00.0
08.0
70.0
027.8
50.0
027.9
20.0
028.1
40.0
072.9
30.0
073.5
10.0
073.4
119
0.0
07.4
60.0
07.8
80.0
07.5
11.4
625.2
00.0
025.0
70.0
025.0
90.6
670.7
90.0
070.1
90.6
671.4
620
0.0
07.4
50.0
07.7
60.0
07.5
49.3
426.6
80.0
027.3
00.0
027.4
24.3
472.9
50.0
074.3
64.3
474.2
821
0.0
07.4
00.0
07.6
80.0
07.4
70.0
025.8
20.0
025.0
30.0
025.7
40.0
066.7
10.2
769.0
60.1
468.9
522
0.0
07.5
30.0
07.8
20.0
07.5
6-1
.05
26.0
80.0
025.7
0-0
.14
26.3
50.0
067.3
70.0
068.3
40.0
067.7
223
0.0
16.1
90.0
16.3
80.0
16.1
70.0
025.5
50.0
026.1
80.0
026.3
51.0
169.3
61.0
173.4
50.0
072.9
424
4.8
37.5
74.8
37.9
54.8
37.5
50.0
028.0
10.0
028.1
80.0
028.5
23.4
072.4
13.4
071.3
63.4
073.0
825
2.8
27.8
02.8
28.0
72.8
27.8
30.0
026.6
10.0
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00.0
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40.0
071.3
20.0
070.6
10.0
071.8
026
0.0
07.7
90.0
08.2
20.0
07.9
04.0
325.8
64.5
625.4
94.2
325.7
4-2
.24
78.8
6-2
.24
77.0
7-2
.24
78.6
227
-6.0
57.3
7-5
.91
7.3
2-6
.70
7.0
8-2
.06
25.5
2-2
.06
25.5
1-2
.06
25.7
8-1
.06
75.3
5-1
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75.1
8-1
.06
75.1
328
0.0
07.4
70.0
07.8
10.0
07.6
20.0
023.7
40.0
024.2
00.0
024.4
6-3
.76
70.7
9-3
.76
70.8
7-3
.76
71.5
329
0.0
07.9
20.0
08.2
30.0
07.9
75.4
026.8
40.0
027.1
72.0
627.2
60.0
078.7
60.0
078.2
40.0
079.1
230
3.8
97.3
30.0
07.6
90.0
07.6
20.0
024.8
00.0
025.4
00.0
025.3
50.0
069.4
40.0
068.7
50.0
069.2
2M
in.
-6.0
56.1
9-5
.91
6.3
8-6
.70
6.1
7-2
.06
23.7
4-2
.06
23.3
7-2
.06
24.4
6-5
.58
62.1
9-5
.58
64.4
9-5
.58
63.2
6A
vg.
0.4
87.6
00.2
67.8
00.3
27.6
31.2
825.9
50.3
525.9
50.4
426.2
30.9
771.8
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91
from the observation in GRASP, where lower α tends yield a lower average relative error.
This also supports the viewpoint mentioned in GRASP, i.e., with a low value of α, the
approach may converge to a local optimum prematurely.
HA outperforms both GA and GRASP in terms of solution quality. The relative error
is less than 0.5% in average and 5% in maximum for all problems whether p2(n) or p3(n) is
used. Furthermore, HA (with p2(n) or p3(n)) finds better solutions than CPLEX for about
half of the instances, compared to GA or GRASP’s one third. As to the computational
requirement, HA spends less than 80 seconds to obtain a solution for the large-sized
instances, which is larger than either GA or GRASP, but is still at an acceptable level
from a practical perspective. From these, we can conclude that HA is an effective and
efficient solution approach for the HBA problem.
4.6 Concluding Remarks and Future Research Directions
In this chapter, we focused on allocating the available aggregate hospital bed
capacity among MCUs to balance the bed occupancies among different units. To take
the uncertainties associated with health care systems into account, closed-form results
from queuing theory were incorporated into an optimization framework, which resulted
in large scale nonlinear integer programming formulations for the HBA problem. To
efficiently solve the problem, we proposed three solution approaches. Our computational
study showed that GA and GRASP were very efficient, both of which solved the problem
within 30 seconds with average and maximum errors less than 3% and 10%, respectively.
To decrease the maximum relative errors observed in GA or GRASP, we proposed a
hybridization of GA and GRASP, denoted by HA. HA outperformed either GA or GRASP
in terms of accuracy, whose relative error is <0.5% and <5 % in average and in maximum,
respectively, by spending <80 seconds in terms of CPU time. In summary, HA is an
efficient approach that finds good quality solutions for the HBA problem.
In our work, each MCU is modeled as an M/M/c/c queueing system. Future research
can consider different queueing system configurations with general arrival and service
92
processes or queueing networks. The design of the queueing network can model the
hierarchical relation between sub-units in each MCU, i.e., patients flow through units in
a specific order. We note that the mathematical expressions of stationary probabilities
of a queueing network would be more complicated than that of an M/M/c/c queueing
system. Specifically, the expressions of stationary probabilities of a queueing network have
a product form of capacity allocated in each station, which correspond to the the decision
variables of the HBA problem. Another practically relevant research direction is concerned
with the modeling of the blocking behavior between sub-units in each MCU. Note that
the blocking in a health care system is different from that in a manufacturing system,
where the work of a blocked part can not be started before entering a designated station.
However, the recovery process of a patient who is blocked from entering the bed in a
downstream unit does not stop. For example, a mother-to-be is typically allocated to a
bed in the labor and delivery unit first, and moved to a bed in postpartum unit to recover
from delivery. It may happen that the mother spends her entire recovery time in the labor
and delivery unit and is discharged from there directly. Because she fully recovers in the
labor and delivery unit before a bed becomes available in the postpartum unit.
93
CHAPTER 5EMERGENCY ROOM SERVICES FACILITY LOCATION AND CAPACITY
PLANNING
5.1 Introduction
This chapter introduces the emergency room services facility location and capacity
planning (ERSFLCP) problem. In traditional facility location models, facility location
and demand allocation decisions are made based on the objectives of minimizing the total
number of facilities opened or minimizing the total or weighted) distance traveled. This
approach ignores the fact that some demand may not be satisfied due to a shortage of
capacity or system congestion as the system operates in real-time.
In a health care service facility, when the emergency room (ER) is full or all intensive
care beds are occupied, hospitals send out divert status. When a hospital is on divert
status, incoming patients might be sent to hospitals which are farther away or kept at
the hospitals where they currently are that may not able to provide adequate service. To
a critical patient, the consequence of divert status can be the difference between life and
death.
The purpose of this work is to construct a facility location model, which simultaneously
determines the number of facilities opened and their respective locations as well as the
capacity levels of the facilities so that the probability that all servers in a facility are busy
does not exceed a pre-determined level. In other words, we want to locate ER services
on a network and determine their respective capacity levels such that the probability of
diverting patients is not larger than a particular threshold.
The remainder of this chapter is organized as follows. In Section 5.2, we review
the related literature. Section 5.3 presents mathematical programming formulation for
ERSFLCP. Section 5.4 details the Lagrangian relaxation algorithm that we propose for the
ERSFLCP problem. In Section 5.5, we present results from our computational study that
evaluates the computational performance of the Lagrangian relaxation algorithm. Section
5.6 includes a discussion of the results and suggests future research directions.
94
5.2 Literature Review
Traditional facility location models (such as the set covering model, the P -median
model, or the P -center model) focus on determining the location of facilities and the
allocation of demand to facilities with the objective of either minimizing the number of
facilities opened, minimizing the sum of fixed facility location and/or transportation costs,
or maximizing the demand covered. In practice, however, a customer may choose to go
to a facility different from the one identified by the optimization model for a number of
reasons, such as the designated facility experiences a temporary shutdown, is associated
with long queues and wait times.
There is a rich body of literature that develop robust facility location models under
uncertainty to hedge the randomness on costs, demands, and travel times utilizing the
robust or stochastic optimization approach. Snyder [103] presents a detailed review of on
stochastic and robust facility location models.
In the area of health care application, Beraldi et al. [12] investigate the problem
of characterizing the optimal locations of emergency medical service sites and numbers
of emergency vehicles required for each site. They consider the problem formulation
in a stochastic optimization setting, where the ability to cover the random requests of
emergency service at demand points is restricted by a set of probabilistic constraints.
Specifically, the probabilistic constraints ensure that the probability that the number
of vehicles located at a facility can cover the random service request is greater than a
prescribed probability value. Some papers consider the hierarchical relation between
facilities in a health care setting. For example, Koizumi et al. [70] classify mental health
care system into levels of extended acute hospitals, residential facilities and supported
housing, and patients flow through the levels according to their health conditions. Because
of the hierarchical structure of health care service, the capacity requirements of units in a
higher level of hierarchy usually correlate with the units in a lower level. Galvao et al. [39,
40] formulate a hierarchical location-allocation model to determine the capacity required
95
in perinatal health care facilities, which are categorized into three levels: basic units,
maternity homes and neonatal clinics. They develop a mixed integer linear programming
model which aims to determine the optimal locations of facilities for each level and
allocations of mothers-to-be to these locations for needed type of service.
Another branch of literature that is related to our work develops facility location
models which require backup (multiple) coverage for each demand point. Snyder and
Daskin [104] consider a P -median-based model that minimizes total costs associated with
a location-allocation plan and the expected failure cost. As each demand point is assigned
primary and backup facilities, the expected failure cost is quantified by the additional
transportation cost incurred to cover the demand by the back-up facility. The problem
is formulated as a 0-1 integer programming formulation and solved by the Lagrangian
relaxation algorithm. Jia et al. [56] present a detailed review of traditional facility location
models and propose a general facility location model suited for large-scale emergencies.
In their model, a demand point is considered to be covered if a pre-specified number of
facilities are assigned.
To address the issue of service quality, some papers incorporate queuing systems into
facility location models to consider the randomness on availability of servers and focus on
reducing the demand lost due to the shortage of capacity or system congestion. Marianov
and ReVelle [81] formulate a maximal availability location model, which uses a M/M/c/c
queueing system to model the server availability of a demand point. The model wants to
locate a set of ambulances such that the demand covered is maximized, where a demand
point is considered to be covered, if the probability that there is an ambulance nearby and
available is greater than a threshold. They show how to transform the nonlinear queueing
expression to an equivalent linear one, and solve the problem by using a commercial solver
(LINDO). Berman et al. [13] model each facility as an M/M/1/a queueing system, where
a is the maximum number of customers allowed in the facility, and consider a facility
location problem with the upper bounds on the amount of demand lost due to insufficient
96
coverage and system congestion. Berman et al. [14] also investigate the facility location
problem under the objective of maximizing captured demand. They, again, model each
facility as an M/M/1/a queueing system and assume that a customer is lost if the closest
facility and all other facilities that he/or she can reach are full. Both papers ([13, 14])
obtain the solutions, i.e., the location of the facilities, through heuristic approaches.
We note that the capacities of facilities to be opened has an impact on both the total
number and the locations of facilities, particularly when the congestion associated with
the potential facilities is taken into account. In what follows, we model each facility as an
M/M/c/c queueing system, where c is number of servers in the facility, which designates
the capacity of the facility. The goal of our model is to identify locations of facilities and
specify the capacity levels of the facilities simultaneously.
5.3 Problem Formulation
We consider a network G = (N,A), where N = {1, 2, . . . , N} is the set of nodes, and
A denotes the set of arcs. We assume that at each node i ∈ N the occurrence of patients
needing ER services is Poisson distributed with rate ni, and all patients at a node need to
be directed to one open ER service facility. We consider that an ER service facility can
be opened at any node j ∈ N, and at most p facilities can be opened for providing the
ER services to patients, which is similar to most of the facility location models. Let dij be
the distance (measured in time units) between nodes i and j for i, j ∈ N, and d be the
coverage range of an ER. The allocation parameter aij takes the value of one if dij ≤ d,
zero otherwise. Let t represents the monetary value for travel time for a critical patient.
Let fj be the fixed cost of opening a facility at node j ∈ N, and cjk be the operating
cost of the facility j with capacity level k ∈ K, where K = {1, 2, . . . , K} is the set of
capacity levels of an opened ER. In this problem, the capacity is measured as number of
beds in an ER. That is, if an ER is to be opened at capacity level k ∈ K, then there are
mk beds in the ER. We assume that the lengths of stay of a patient in the ER at node
j are exponentially distributed with rate µj, and each ER is modeled as an M/M/c/c
97
queueing system. In other words, when all beds in an ER are occupied, patients are
diverted (rejected) to other ERs. Let γ be the upper bound of probability of diverting
a patient from an ER, λj be the patient arrival rate to an ER service facility at node j,
and zj be the bed capacity of the ER service facility at node j. Note that zj = mk if the
facility at location j is chosen to be opened at capacity level k.
We have four sets of decision variables. The first set is patient allocation variable xij,
which takes the value of one if node i is served by the ER service facility at location j,
zero otherwise. The second set is ER location and capacity allocation variable yjk, which
takes the value of one if an ER service facility is placed at node j and operated at capacity
level k, zero otherwise. The last two sets are λj and zj, which are determined once the
values of xij and yjk are assigned.
The ERSFLCP problem can be formulated as a nonlinear integer programming
formulation as follows:
minN∑
i=1
N∑j=1
tnidijxij +N∑
j=1
K∑
k=1
(fj + cjk)yjk (5–1)
subject toN∑
j=1
aijxij = 1 ∀ i (5–2)
xij ≤K∑
k=1
yjk ∀ i, j (5–3)
N∑j=1
K∑
k=1
yjk ≤ p (5–4)
K∑
k=1
yjk ≤ 1 ∀ j (5–5)
N∑i=1
nixij = λj ∀ j (5–6)
K∑
k=1
mkyjk = zj ∀ j (5–7)
98
π(λj, µj, zj) ≤ γ
K∑
k=1
yjk + (1−K∑
k=1
yjk) ∀ j (5–8)
xij, yjk ∈ {0, 1} ∀ i, j, k (5–9)
zj ∈ Z∗ ∀ j (5–10)
where π(λj, µj, zj) is the probability of diverting patients from an ER service facility at
location j. The formulas for π(λj, µj, zj) of an M/M/c/c queueing system can be found
in any queueing book (e.g., Gross and Harris, 1998). Specifically, π(λj, µj, zj) can be
represented as a function of arrival rate λj, service rate µj, and bed capacity zj, that is
π(λj, µj, zj) =(λj/µj)
zj
zj!/
zj∑q=0
(λj/µj)q
q!. (5–11)
The objective function (5–1) minimizes the value of time of ER patients and the
costs of opening and operating ER service facilities. Constraint (5–2) stipulates that each
demand node must be covered by one facility. Constraint (5–3) restricts that demand
nodes can only be assigned to opened facilities. Constraint (5–4) imposes the upper bound
on the number of facilities opened. Constraint (5–5) states that an opened facility must be
associated with a single capacity level. Constraints (5–6) and (5–7) obtain the arrival rates
and capacity levels for all the facilities. Constraint (5–8) imposes the upper bound on the
probability of all beds being occupied for an opened facility. Finally, constraints (5–9) and
(5–10) ensure that decision variables are binary and nonnegative integers, respectively.
Note that the factorial terms in equation (5–11) make the ERSFLCP problem
intractable. To overcome this problem, we replace the constraint (5–8) by constraint
(5–12),
λj ≤ φjkyjk + Mj(1− yjk) ∀ j, k (5–12)
where φjk is the largest value which satisfies inequality (5–13) and equation (5–14).
π(φjk, µj,mk) =(φjk/µj)
mk
mk!/
mk∑q=0
(φjk/µj)q
q!≤ γ (5–13)
99
Mj =N∑
i=1
niaij (5–14)
As a result, we transform the previous nonlinear integer programming formulation to
a linear binary integer programming formulation as follows:
minN∑
i=1
N∑j=1
tnidijxij +N∑
j=1
K∑
k=1
(fj + cjk)yjk (5–15)
subject to
N∑j=1
aijxij = 1 ∀ i (5–16)
xij ≤K∑
k=1
yjk ∀ i, j (5–17)
N∑j=1
K∑
k=1
yjk ≤ p (5–18)
K∑
k=1
yjk ≤ 1 ∀ j (5–19)
N∑i=1
nixij ≤ φjkyjk + Mj(1− yjk) ∀ j, k (5–20)
xij, yjk ∈ {0, 1} ∀ i, j, k (5–21)
5.4 Solution Approach
The ERSFLCP problem can be solved by commercial MILP solvers, however, it takes
time to obtain the optimal solutions of problems with large size or tight constraints, for
example, large values of N or small values of p. Here, we develop a Lagrangian relaxation
(LR) approach to obtain solutions to the ERSFLCP problem.
5.4.1 Lagrangian Relaxation Approach
The general idea of Lagrangian relaxation is to move hard constraints into the
objective and penalize the objective if the constraints are violated. Figure 5-1 presents
the general procedure we use to obtain a solution to the ERSFLCP problem using a
Lagrangian relaxation approach. In Figure 5-1, n is the iteration counter. Also, UB∗ and
LB∗ denote the incumbent upper and lower bounds on the objective function value of
100
Lagrangian dual problem, respectively. Similarly, UBn and LBn denote the upper and
lower bounds on the objective function value of Lagrangian dual problem at iteration n,
respectively. u is the counter of number of iterations that UB∗ does not improved. Last,
nmax and umax are upper bounds of n and u. In the following subsections, we discuss the
elements of the iterative approach.
procedure LagrangianRelaxation()
1 n ← 1; u ← 1; UB∗ ←∞; LB∗ ← 0;2 Initialize Lagrangian multipliers;3 while n ≤ nmax and u ≤ umax do4 Formulate and solve the Lagrangian problem to obtain LBn;5 if LBn > LB∗ then6 LB∗ ← LBn;7 end if8 Obtain UBn using problem specific approach;9 if UBn < UB∗ then
10 UB∗ ← UBn;11 u ← 0;12 else13 u ← u + 1;14 end if15 Revise Lagrangian multipliers using subgradient optimization;16 n ← n + 1;17 end while
end LagrangianRelaxation.
Figure 5-1. Pseudo-code of the Lagrangian relaxation
5.4.2 Lower Bound
To obtain the lower bound of ERSFLCP, we relax the constraint sets (5–17) and
(5–20) with Lagrange multipliers α and β, respectively, where α and β are matrices with
sizes N × N and N × K, respectively. The relaxation yields the following Lagrangian
problem (ERSFLCP-LR):
minN∑
i=1
N∑j=1
(tnidij + αij + ni
∑
k
βjk)xij
+N∑
j=1
K∑
k=1
(fj + cjk −∑
i
αij − βjkφjk + βjkMj)yjk −N∑
j=1
K∑
k=1
βjkMj (5–22)
101
subject to
N∑j=1
aijxij = 1 ∀ i (5–23)
N∑j=1
K∑
k=1
yjk ≤ p (5–24)
K∑
k=1
yjk ≤ 1 ∀ j (5–25)
xij, yjk ∈ {0, 1} ∀ i, j, k (5–26)
Note that the Lagrangian problem can be separated into subproblems LX and LY, where
subproblem LX contains variables xij, and the subproblem LY contains variables yjk as
follows:
(LX) minN∑
i=1
N∑j=1
dijxij
subject toN∑
j=1
aijxij = 1 ∀ i
xij ∈ {0, 1} ∀ i, j
(LY) minN∑
j=1
K∑
k=1
cjkyjk
subject to
N∑j=1
K∑
k=1
yjk ≤ p
K∑
k=1
yjk ≤ 1 ∀ j
yjk ∈ {0, 1} ∀ j, k
where dij = tnidij + αij + ni
∑Kk=1 βjk and cjk = fj + cjk −
∑Ni=1 αij − βjkφjk + βjkMj.
Given a set of α and β, both subproblems LX and LY are easy to solve. For
subproblem LX, for each i ∈ N the decision variable xij∗ is set to one if dij∗ ≤ dij
for all j ∈ N. Similarly, subproblem LY can be solved according to each variable yjk’s
102
contribution to the objective function, while for each j ∈ N at most one yjk can be set to
one for all k ∈ K, and at most p of the yjk’s can be set to one for all j ∈ N and k ∈ K.
Note that if yjk = 1, then cjk ≤ 0 must hold.
5.4.3 Upper Bound
In the beginning of the algorithm, we first generate an upper bound by the heuristic
that will introduced later. Also, at each LR iteration, the heuristic is applied to improve
the lower bound solution from the Lagrangian subproblem, i.e., subproblems LX and LY,
to become a feasible solution. Figure 5-2 depicts the pseudo-code of the heuristic.
The heuristic starts by resetting the infeasible solution (X, Y) according to the
strategy selected randomly from following:
1. For each i, if xij > 0, then∑N
k=1 yjk > 0; and
2. If∑N
k=1 yjk = 0, then xij = 0 for all i ∈ N.
The first strategy resets facility variables, yjk, based on demand allocation, xij,
determined from subproblem LX. The facilities are ordered according to their patient
arrival rates, i.e., λj =∑
i nixij, in non-increasing ordern, and then the first p facilities
are set to open at capacity level one, i.e., yj1 = 1. The second strategy resets demand
allocation variables, xij, according to solution yjk form subproblem LY. That is, if a
facility j is not opened, then the demand nodes allocated to facility j are reset to not
covered, i.e., the corresponding variable xij’s are reset to 0 and re-allocate them to other
facilities using the next procedure.
Line 2 resets the capacity level k of the opened facilities such that the facilities is
opened at appropriate capacity level k, i.e.,∑
i nixij ≤ φjk. For an opened facility j, if
the demand allocated exceeds its largest capacity, i.e.,∑
i nixij > φjK , then an allocated
demand node i is selected randomly and its xij is reset to 0, until∑
i nixij ≤ φjk holds.
Next, the while loop in line 3 in Figure 5-2 ensures that all demand nodes are covered
and the number of facilities opened is less than p. The pseudo-code of covering all demand
nodes is presented in Figure 5-3. We randomly select a node u which has not covered by
103
procedure GenerateFeasibleSolution(X,Y)
1 ResetSolution(X,Y);2 Reset capacity level;3 while
∑i
∑j xij < N or
∑j
∑k yjk > p do
4 if∑
i
∑j xij < N then
5 CoverDemand(X,Y);6 end if7 if
∑j
∑k yjk > p then
8 CloseFacility(X,Y);9 end if
10 end while11 return (X,Y);
end GenerateFeasibleSolution.
Figure 5-2. Pseudo-code of the feasible solution generation
any facility, and generate a set OpenF which contains the facilities meeting the following
criteria:
1. The facility j is opened, i.e.,∑
k yjk > 0;
2. The facility j can covered the demand node u, i.e., auj = 1; and
3. After allocating node u to the facility j, the sum of arrival rates of the allocateddemand node does not exceed facility j’s maximum capacity, i.e.,
∑i nixij +nu ≤ φjK .
If the set OpenF is empty, then a set NotOpenF is generated. The set NotOpenF
includes the facilities which are not opened, i.e.,∑
k yjk = 0, and the node u is within
their coverage range, i.e., auj = 1 for all j ∈ N. Then, in the line 8 in Figure 5-3 we open
a facility j ∈ NotOpenF to cover node u. There are many strategies that we can apply
to choose which facility to open, such as, open the facility j ∈ NotOpenF which is the
closest one to node u, the one with the largest capacity, or the one with the smallest fix
cost. Here we apply the three strategies and find the corresponding facilities for each of
them. If these strategies yield different facility options, we randomly select one of them
to open. Once all demand nodes are covered by the opened facilities, the capacity level of
each opened facility is reset to appropriate level k ∈ K (line 14 in Figure 5-3) such that∑
i nixij ≤ φjk.
So far, we have ensured that all demand nodes are covered by facilities which are
opened, and the demand allocation does not exceed the maximum capacity of the opened
104
procedure CoverDemand(X,Y)
1 while∑
i
∑j xij < N do
2 u ← RandomlySelectElement(UncoveredNode);3 Generate set OpenedF = {j :
∑k yjk > 0, auj = 1,
∑i nixij + nu ≤ φjK , j ∈ N};
4 if OpenF 6= ∅ then5 j ← the closest facility j ∈ OpenF ;6 else7 Generate NotOpenedF = {j :
∑k yjk = 0, auj = 1, j ∈ N};
8 select j from the set NotOpenF ;9 NotOpenF ← NotOpenF \ j;
10 OpenF ← NotOpenF ∪ j;11 end if12 xij ← 1;13 end while14 Reset capacity level;15 return (X,Y);
end CoverDemand;
Figure 5-3. Pseudo-code of covering all demand nodes
facilities. The next thing to do is to inspect whether the number of facilities opened is no
larger than p. If the number of facilities opened is less than or equal to p, then a feasible
solution is generated. Otherwise, one of the opened facilities is selected randomly and
closed, until the number of facilities opened is no larger than p. In addition, the associated
demand nodes are reset to not covered, i.e., reset xij = 0, where facility j is chosen to
close.
procedure CloseFacility(X,Y)
1 Generate set OpenedF = {j :∑
k yjk > 0, j ∈ N};2 while
∑j
∑k yjk > p do
3 j ← RandomlySelectElement(OpenF );4 yjk ← 0, for all k ∈ K;5 for i = 1 to N do6 xij ← 0;7 end for8 OpenF ← OpenF \ j;9 end while
10 Reset X based on Y;11 return (X,Y);
end CloseFacility;
Figure 5-4. Pseudo-code of closing facilities
105
The procedures of CoverDemand and CloseFacility are repeated, until a feasible
solution is generated.
5.4.4 Lagrangian Multipliers
Due to the modern software and hardware developments, the linear programming
problems can be efficiently solved by any commercial LP solvers. To take advantage
of the existing tools, we use the dual information of the linear form of the ERSFLCP
problem, denoted by ERSFLCPr, to initialize the Lagrangian multipliers, rather than
starting the algorithms from scratch as traditional approaches. The ERSFLCPr problem
is generated by replacing the constraint sets (5–17) and (5–20) by 0 ≤ xij ≤ 1 and
0 ≤ yjk ≤ 1, respectively. Then, CPLEX is used to solve the ERSFLCPr problem, and the
dual information of the constraint sets 0 ≤ xij ≤ 1 and 0 ≤ yjk ≤ 1 are extracted to set the
initial values of Lagrangian multipliers. Let αr and βr be the dual values associated with
the constraint sets 0 ≤ xij ≤ 1 and 0 ≤ yjk ≤ 1, respectively. The multipliers α0 and β0 of
ERSFLCP-LR are initialized by the equations α0 = −αr and β0 = −βr.
We then apply the method described by Fisher (1981) to update the multipliers. At
each iteration n, the step size tn is obtained by
sn =Bn(θ − θn)∑N
i=1
∑Nj=1(xij −
∑Kk=1 yjk)2 +
∑Nj=1
∑Kk=1(nixij − φjkyjk −Mj(1− yjk))2
,
where ε1 ≤ Bn ≤ 2 − ε2 (ε1, ε2 → 0 ), θ is the best upper bound of the optimal objective
value of ERSFLCP-LR found, and θn is the the optimal objective value of ERSFLCP-LR
at iteration n. Then, the Lagrangian multipliers are reset by
αn+1ij = αn
ij + sn(xij −K∑
k=1
yjk) ∀i, j
βn+1jk = βjkn + sn(nixij − φjkyjk −Mj(1− yjk)) ∀j, k
The iterations are stopped when the number of iteration exceeds a prespecified value,
nmax, or the upper bound is not improved for umax iterations.
106
5.5 Computational Study
In this section, we present results from our computational study, where we investigate
the computational performance of the Lagrangian relaxation approach on the ERSFLCP
problem. The computational experiments are implemented on a workstation with two
Pentium 4 3.2 GHz processor and 6 GB of memory.
5.5.1 Experimental Design
In our study, we conduct experiments on four factors including the maximum number
of facilities opened (p), number of capacity levels (K), the incremental amount of beds
associated with each capacity level (∆), probability of diverting a patient of an opened
facility (γ), and the monetary value for travel time for a patient (t). For each experiment,
we consider three networks consisting of 25, 50 and 100 nodes, respectively, Furthermore,
one of the factor is tested at three levels as listed in Table 5-1, and the other three
parameters are set at level 2.
Table 5-1. Experimental factor settings
Parameters Level 1 Level 2 Level 3Experiment 1 p 5 10 20Experiment 2 K (∆) 5 (8) 8 (5) 10 (4)Experiment 3 γ 0.5% 1% 2%Experiment 4 t ($/ per minute) 25 50 100
For each experimental setting, we generate 30 random instances. Each random
instance is obtained by generating parameters that correspond to patient arrival rates,
service rates, fixed and operating costs and distance between each node pair. Specifically,
each parameter is obtained by using the formula: u(mean value of the parameter), where u
is drawn from the distribution U [0.5, 1.5]. The mean values of the random parameters are
listed as follows:
• Mean arrival rate (ni): 1.5 persons/per hour for each demand node i;
• Mean service rate (µj): 0.5 persons/per hour for the facility opened at location j;
• Mean fixed cost (fj): $5000 /per day for the facility opened at location j;
• Mean operating cost per bed: $1000 /per day for the facility opened at location j;and
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• Mean distance between each node pair (dij): 50 minutes.
Last, the coverage range, d, is set to 50 minutes for all instances.
For each instance, the Lagrangian relaxation algorithm described in Section 5.4 is
applied. The upper limit of the number of LR iterations (nmax) is set at 100,000 and
the upper limit of the number of consecutive iterations fail to improve the best known
feasible solution (umax) is set at 10,000. The parameter B used to modified the step size is
initialized at 2, and divided by 1.5 if the lower bound is not improved for 3N iterations.
5.5.2 Experimental Results
Before presenting the experimental results, we use one of the instances in experiments
to show the benefit of utilizing the dual information of ERSFLCPr in LR. The solid lines
in Figure 5-5 depict the upper and lower bounds, respectively, obtained from the LR
iterations with the dual information of ERSFLCPr. The dash lines show the LR results
without applying the dual information of ERSFLCPr. The dual information of ERSFLCPr
provides a good lower bound solution which guides the heuristic to find the best upper
bound solution earlier than the guidance of the lower bound obtained from using the
Lagrangian multipliers generated from scratch. The convergence gaps in Figure 5-5 are
given by
θ − θ
θ× 100, %
where θ and θ are the upper and lower bounds of optimal objective value of ERSFLCP-LR
obtained from the LR algorithm.
Table 5-2 to Table 5-5 report the results of four experiments. In general, LR performs
quite well in solving the ERSFLCP problem in terms of the convergence gap and the
computational effort (i.e., CPU time) required. For the network with 25 nodes, LR takes
less 2 seconds to obtain the solutions with convergence gaps less than 5% on average. For
the largest network tested, i.e., 100 nodes, the average CPU time is increased, although
it is still less than 60 seconds. Moreover, LR does not appear to experience significant
increase in convergence gap due to an increase in problem size.
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0
1000000
2000000
3000000
4000000
5000000
6000000
7000000
8000000
0 5000 10000 15000 20000 25000 30000
Obj
ecti
ve fu
ncti
on v
alue
Iteration
UB (dual)
LB (dual)
UB (scratch)
LB (scratch)
Gap = 15%Gap = 4%Figure 5-5. Convergence speed of the modified LR
The first experiment shows that the smaller the value of p is, the higher the CPU
time required to obtain the solution, since smaller p results in a tighter constraint on
number of facilities opened. In particular, this trend can be observed in the problems with
small- and medium-size networks. Nevertheless, the problem can still be solved within 40
seconds even for large-size networks.
Table 5-2. Experiment 1: effects of maximum number of facilities opened
Network size 25 50 100 AverageGap Time Gap Time Gap Time Gap Time
p (%) (sec.) (%) (sec.) (%) (sec.) (%) (sec.)5 3.0 1.1 4.4 4.9 5.8 41.8 4.4 15.910 3.1 0.9 3.0 4.3 3.8 35.5 3.3 13.620 2.5 0.9 3.3 3.2 2.9 29.4 2.9 11.2
Average 2.9 1.0 3.6 4.1 4.2 35.6 3.5 13.6
In the second experiment, we assume that at most 40 beds are used for an opened
ER, so the incremental amount of beds per capacity level are changed with the setting
of K. A large K value represents that capacity is increased at small scale, i.e., small
batch size, and increases the number of variables of the underlying formulation the
the ERSFLCP problem. As a result, the CPU time is increased as K increases, while
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the growth rate of CPU time is less than the growth rate of K. The impact of K on
convergence gap is not significant, especially, with respect to the problem with large size.
Table 5-3. Experiment 2: effects of capacity setting
Network size 25 50 100 AverageGap Time Gap Time Gap Time Gap Time
K (%) (sec.) (%) (sec.) (%) (sec.) (%) (sec.)5 (8) 0.7 0.8 2.2 4.2 3.6 31.4 2.1 12.18 (5) 3.2 1.1 2.9 4.1 3.9 34.0 3.3 13.110 (4) 3.6 1.1 3.3 4.5 3.9 35.9 3.6 13.8
Average 2.5 1.0 2.8 4.3 3.8 33.8 3.0 13.0
The third experiment investigates the impact of the diversion probability on the
performance of the proposed LR approach. Given the capacity level k, the smaller γ incurs
the smaller value of the maximum arrival rates that a facility is able to serve, i.e., φjk in
constraint set (5–20). That is, the constraint set (5–20) becomes tighter as γ decreases.
As shown in Table 5-4, the CPU time slightly increases as γ decreases from 2.0% to
1.0%. However, when the value of γ is further reduced to 0.5%, the required CPU time
decreases, which is different from the trend that we observe before. The convergence gap
does not appear to be affected by γ significantly; in average, the gaps are 3.6% and 2.9%
when γ’s are 0.5% and 2.0%, respectivley.
Table 5-4. Experiment 3: effects of diversion probability
Network size 25 50 100 AverageGap Time Gap Time Gap Time Gap Time
γ (%) (sec.) (%) (sec.) (%) (sec.) (%) (sec.)0.5% 3.7 0.9 3.3 4.1 3.8 34.9 3.6 13.31.0% 3.1 1.0 3.0 4.3 3.9 56.6 3.3 20.62.0% 2.4 0.9 2.7 3.6 3.5 48.3 2.9 17.6
Average 3.1 0.9 3.0 4.0 3.7 46.6 3.3 17.2
The last experiment explores the effects of patients’ value of time t on the performance
of the proposed LR approach. Table 5-5 shows that the higher the patients’ time value is,
the smaller the convergence gap is. We also observe that the CPU time is not significantly
affected by the change of t.
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Table 5-5. Experiment 4: effects of time value
Network size 25 50 100 AverageGap Time Gap Time Gap Time Gap Time
t (%) (sec.) (%) (sec.) (%) (sec.) (%) (sec.)25 4.9 1.1 4.9 5.8 5.6 45.8 5.1 17.650 3.0 1.1 2.9 4.7 3.8 49.0 3.2 18.2100 1.8 1.1 1.7 5.4 2.7 50.8 2.1 19.1
Average 3.2 1.1 3.2 5.3 4.0 48.5 3.5 18.3
To illustrate the performance of the heuristic developed for obtaining the upper
bound (feasible) solution at each iteration, we use CPLEX to solve the ERSFLCP
problem. The parameters, p, K, γ and t are set to their respective values for level 2 in
Table 5-1. For each test instance, CPLEX is stopped if the relative stopping tolerance of
0.01% is satisfied, or CPU time of 3,600 seconds is used. Table 5-6 summarizes the results,
where the relative error of each instance is given by
Obj of CPLEX - Upper bound of LR
Obj of CPLEX× 100%.
Table 5-6 shows that the developed heuristic is very effective. For small- and median-size
problems, the upper bounds obtained from the heuristic are very close to the optimal
objective values; the average and maximum relative errors are less than 0.3% and 1.5%,
respectively. For the problem with largest size, CPLEX is not able to find the optimal
solution within 3,600 seconds. Therefore, we use the best results that CPLEX can find
within 3,600 seconds to compare with the results obtained by heuristic. The comparison
further ensures the effectiveness of the heuristic developed here. The heuristic not only
spends much less time (36 seconds in average) than CPLEX, but also find solutions with
better quality in 50% of the instances.
5.6 Concluding Remarks and Future Research Directions
In this chapter, we developed a model for the emergency room services facility
location and capacity planning (ERSFLCP) problem, in which each facility is modeled
as an M/M/c/c queueing system to consider the impact of the uncertainties associated
with patient arrivals and lengths of stay. The model was designed to simultaneously
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Table 5-6. Performance of the heuristic
Nodes 25 50 100CPLEX Heuristic Err. CPLEX Heuristic Err. CPLEX Heuristic Err.
Instance (sec.) (sec.) (%) (sec.) (sec.) (%) (sec.) (sec.) (%)1 0.4 0.0 0.8 31.8 3.3 0.1 >3600 26.6 0.42 0.7 0.0 1.4 84.3 4.5 0.1 >3600 59.7 -1.33 0.3 0.0 1.1 44.9 3.5 0.0 >3600 26.0 0.44 0.5 0.0 1.1 109.9 4.7 0.2 >3600 39.7 0.25 0.8 0.0 0.8 82.4 5.1 0.5 >3600 31.9 0.26 0.7 0.0 0.8 47.9 7.5 0.9 >3600 33.3 0.17 0.2 0.0 0.9 115.0 3.5 0.1 >3600 29.4 -0.58 0.2 0.0 0.8 18.7 4.1 0.0 >3600 30.5 0.19 0.9 0.0 1.0 27.7 3.5 0.2 >3600 51.4 -0.410 0.2 0.0 0.9 32.6 4.6 0.2 >3600 27.8 0.711 0.8 0.1 1.1 130.9 5.1 0.5 >3600 65.6 0.112 0.3 0.0 0.9 65.2 5.3 0.2 >3600 30.1 -0.313 0.6 0.0 0.9 65.5 3.0 0.1 >3600 22.8 -1.814 0.3 0.0 0.9 39.9 4.8 0.6 >3600 39.5 -0.415 0.6 0.0 0.9 25.6 3.4 0.3 >3600 25.9 -0.316 0.5 0.0 0.9 456.4 3.7 0.3 >3600 55.3 -0.817 0.6 0.0 1.0 47.9 4.3 0.6 >3600 34.8 0.418 0.8 0.0 1.6 34.7 3.2 0.1 >3600 41.5 0.219 0.7 0.0 0.8 47.3 3.5 0.3 >3600 19.5 0.220 0.8 0.0 0.8 119.8 3.8 1.2 >3600 24.4 -0.221 0.4 0.0 0.8 53.5 3.8 0.1 >3600 25.7 0.222 0.4 0.0 0.8 46.7 3.4 0.0 >3600 25.4 -1.223 0.2 0.0 0.8 17.4 7.5 0.5 >3600 33.7 0.324 0.3 0.0 0.8 62.8 3.3 0.1 >3600 49.6 -1.125 0.5 0.0 0.9 502.1 3.5 0.2 >3600 28.1 0.526 0.8 0.0 0.9 53.6 3.6 0.3 >3600 23.6 -1.427 0.6 0.0 1.0 27.4 3.7 0.4 >3600 19.9 -0.828 0.6 0.0 0.9 146.6 4.8 0.3 >3600 37.1 0.229 0.2 0.0 0.9 39.9 5.8 1.2 >3600 60.2 -1.230 0.1 0.0 0.9 176.9 4.1 0.0 >3600 46.2 -0.2
Min. 0.1 0.0 0.8 17.4 3.0 0.0 >3600 19.5 -1.8Avg. 0.5 0.0 0.9 91.8 4.3 0.3 >3600 35.5 -0.3Max. 0.9 0.1 1.6 502.1 7.5 1.2 >3600 65.6 0.7
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determine the number of facilities opened and their respective locations as well as the
capacity levels of the facilities so that the probability of diverting patients is not larger
than a particular threshold. A Lagrangian relaxation approach was proposed to obtain
solutions of the ERSFLCP problem. The relaxation scheme proposed yield a separable
Lagrangian problem that is easy to solve. The upper bound at each LR iteration was
obtained by a heuristic developed in this work. In addition, to speed up the convergence
process, we suggested to use the dual information of the linear programming relaxation
of the ERSFLCP problem to generate the initial set of Lagrangian multipliers. The
computational study demonstrated that the ERSFLCP problem can be efficiently solved
by the Lagrangian relaxation algorithm, and the developed heuristic provides upper bound
solutions with good quality.
An immediate extension of our model is to include the closest-assignment constraints
which ensure that each demand point is allocated to the closest open facility. Some other
practically relevant variations are concerned with the alternative objective functions, such
as profit maximization, and travel and service times minimization. Another closely related
problem is that given some ER service facilities are opened already and B new beds are
considered to be added at existing facilities or newly opened facilities. We note that this
can be defined by adding the constraints on capacity available, distinguishing the set of
existing and new ER locations, and replacing the fixed cost to capacity expansion cost of
existing facilities.
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CHAPTER 6CONCLUSIONS AND FUTURE RESEARCH DIRECTIONS
In this work, we have presented the integrated use of optimization and queueing
theory to determine the optimal capacity plan for health care systems.
Chapter 2 detailed the aggregate hospital bed capacity planning (AHBCP) problem
and a network flow approach to specify the optimal bed capacity planning decisions
throughout a finite planning horizon for hospitals. In this chapter, a hospital was modeled
as a G/G/c queueing system with a single bed type and a single patient class. We
demonstrated that for realistic-sized capacity planning problems, our network formulation
is not computationally intensive, and allows us to obtain optimal bed capacity plans
quickly.
Chapter 3 introduced the health care team capacity planning (HCTCP) problem, in
which the underlying queueing system was more complex than the one used for AHBCP.
In particular, we considered a queueing system where there are two classes of patients
and two types of care teams, where service rates are patient-class dependent and one
type of care team can substitute for the other. We developed queueing models for both
preemptive and non-preemptive cases, and developed approximation procedures to
estimate the average time that each patient class spends in the system. The results from
approximation method were then incorporated into the optimization model to determine
the minimal cost capacity plan of health care teams throughout a finite planning horizon.
Our computational study showed that our approximation approach provides sufficiently
accurate results that can be used in practice to make long-term health care team service
capacity planning decisions.
After the aggregate bed capacity was specified in Chapter 2, we developed the
hospital bed allocation (HBA) model to obtain the balanced bed allocation among
different medical care units, which were modeled as M/M/c/c queueing systems.
To efficiently solve the problem, we proposed three solution approaches including
114
genetic algorithm(GA), greedy randomized adaptive search procedure (GRASP) and
a hybridization of GA and GRASP (HA). Our computational study showed that the
proposed algorithms can solve the problem within a short time while providing solutions
with high quality for large-sized, realistic test instances.
Chapter 5 developed a emergency room services facility location and capacity planning
(ERSFLCP) model, where each emergency room service facility is viewed as a M/M/c/c
queueing system. The model was designed to simultaneously determine the number of
facilities opened and their respective locations as well as the capacity levels of the facilities
(capture in terms of number of beds) so that the probability that the facility is full and
incoming patients have to be diverted (i.e., diversion probability) is not larger than a
particular threshold. A Lagrangian relaxation approach was proposed to obtain facility
locations and capacity plan. The experimental results illustrated that the Lagrangian
relaxation algorithm is very efficient in solving the problem, and the developed heuristic
provides solutions with good quality.
In our work to date, we primarily focused on strategic level, long-term planning
decisions where we made some simplifying assumptions as to the capabilities of the
resources and the arrival rates and lengths of stay for the patients. Specifically, we
assumed that the available beds in a service (in Chapters 3, 4, and 5) or the hospital (in
Chapter 2) are identical. Similarly, we assumed that the arrival rates and length of stay
for the patients are identical (in Chapters 2 and 5). But we also considered the case where
patients can be grouped into two classes (each of which corresponds to an acuity level) or
into multiple classes (each of which corresponds to a particular speciality) to model arrival
rates and lengths of stay. As we mainly focused on strategic level decision making, these
assumptions are justifiable. However, for more detailed planning there is a need to take
other realistic considerations into account.
There are typically multiple types of beds (e.g., adult intensive care beds, pediatric
intensive care beds, burn intensive care beds, medical/surgical beds), i.e., multiple
115
units, in a service or a hospital. Typically, different units are used to accommodate the
patients during different phases of the treatment. Therefore, for more detailed capacity
analysis, there is a need to distinguish between these units and model how the patients
flow through these units. Another important concern in this context is the consideration
of interaction between different units as well as services. That is, if there is no enough
capacity in a speciality service, the patient can be accommodated in another speciality
service. This, in turn, increases the effective service rate of the speciality and the traffic of
another speciality. Similarly, if there is not enough capacity in the downstream unit within
a service, then the patient can continue treatment in the upstream unit, i.e., blocking
Therefore, for more detailed capacity planning, there is a need to consider multiple types
of resources, multiple types of patients, and multiple modes of interaction between units
and services.
In our work, we have primarily focused on the objective of minimizing total costs (in
Chapters 2, 3, and 5) and balancing work load across units (in Chapter 4). Nowadays,
hospitals are focusing on revenue management practices to improve their financial
situation. This modern revenue management culture requires hospital administrators
to focus on maximizing profit rather than on minimizing operating costs. Therefore,
future work should examine the revenue aspects of hospital operations and focus on profit
maximization type objectives. However, this shift in emphasis from cost minimization to
profit maximization should not ignore the quality aspects associated with the delivery of
health care services. In our research, we focused on timeliness (e.g., average service time,
average waiting time) and access (e.g., diversion probability) to quantify service quality.
Future research in the area should concentrate on developing other metrics to model other
aspects of service quality, such as patient safety and service effectiveness.
116
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BIOGRAPHICAL SKETCH
Chin-I Lin was born in Taipei, Taiwan. She received her B.S. and M.S in civil
engineering from the National Central University in Taiwan in 1994 and 1996, respectively.
From 1997 to 2002, she worked for China Airlines, and her major tasks included demand
forecast, market analysis, route analysis, and fleet planning. She pursued her master
and doctoral degrees in the Department of Industrial and Systems Engineering at
the University of Florida since 2002. Chin-I’s main research interest is Operations
Research, and topics of special interest are health care management and airline flight/crew
scheduling. Thus far, her research has focused on capacity management in health care
delivery.
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