MULTI-CRITERIA MATHEMATICAL PROGRAMMING APPROACHES … · practical instances. The problems of...

Conference Proceedings ICIL 2016

ISBN 978-83-62079-06-3

Page 239

MULTI-CRITERIA MATHEMATICAL PROGRAMMING

APPROACHES FOR ASSIGNMENT OF SERVICES IN HOSPITAL

Bartosz SAWIK1

1AGH University of Science and Technology, Faculty of Management, Department of Applied Computer Science Al. Mickiewicza 30, PL-30-059 Kraków,

Poland

Abstract:

This work presents multi-criteria mathematical programming approaches for optimal allocation

of workers among supporting services in a hospital. The services include logistics,

inventory management, financial management, operations management, medical analysis,

etc. The optimality criteria of the problem are minimization of operational costs

of supporting services subject to some specific constraints. The constraints represent

specific conditions for resource allocation in a hospital. The overall problem is formulated

as a multi-criteria assignment model, where the decision variables represent

the assignment of people to various jobs. Numerical examples are presented and some

computational results modeled on a real data from a hospital in Lesser Poland are

reported. Presented problems have been solved using AMPL programming language with

CPLEX solver, with use of branch and bound method.

Keywords:

multi-criteria mathematical programming, assignment problem, services operations

management, healthcare planning

INTRODUCTION 1.

Hospitals typically lack effective enterprise level strategic and operational planning. Some of them do not have a good personnel organization. In most of the cases they could use some optimization models to try to improve it.

This case shows a real situation where an optimization model is applied to a real hospital in Krakow, Poland. Paper presents discussion about data collected from hospital, about models formulations used and the results obtained as well as the conclusions deducted from them.

Big institutions such as hospital not often use the kind of OR/MS methodologies used in many other service industries to help with capacity planning and management [1]. The assignment of service positions plays an important role in healthcare institutions. Poorly assigned positions in hospital departments or over-employment may result in increased expenses and/or degraded customer service. If too many workers are assigned, capital costs are likely to exceed the desirable value [2]. The supporting services have a strong impact on performance of healthcare institutions such as hospitals. In hospital departments, the supporting services include financial management, logistics, inventory management, analytic laboratories, etc. This paper presents an application of operations research model for optimal supporting service jobs allocation in a public healthcare institution.

International Conference on Industrial Logistics ICIL 2016

September 28 – October 1, 2016, Zakopane, Poland

40

The optimality criterion of the problem is to minimize operations costs of a supporting service subject to some specific constraints. The constraints representing specific conditions for resource allocation in a hospital were modified, compared to previous publications [3–16] according to different optimization models formulation. The overall problem is formulated as a mixed integer program in the literature known as the assignment problem [16,17,18,19,20,21]. The binary decision variables represent the assignment of people to various services.

This paper shows practical usefulness of mathematical programming approaches to optimization of supporting services in healthcare institutions. The results of some computational experiments modeled after a real data from a selected Polish hospital are reported.

INPUT DATA FOR COMPUTATIONS 2.

The data for computations have been gathered from the hospital in Krakow, Poland. This hospital is a modern unit that offers medical care to a huge population. The 97.02% of all the admissions of the Hospital are patients from the Lesser Poland Region and in the last few years the admissions have reach the number of 28000-30000 patients per year. It offers services in clinics and several surgical processes, which are carried out by high-qualified personnel.

Some of the clinics that this hospital has are Neurological Clinic, Multiple Sclerosis Clinic, Dermatological Clinic, Clinic of Plastic Surgery, Otolaryngology Clinic, Radiotherapy Clinic, Urological Clinic, Clinic of Neonatology, etc.

The real data from a selected Polish public healthcare institution from one month period were used for computations. The data include 20 supporting services hospital departments in which there are 88 supporting jobs. Permanent employment is defined as a percent of permanent post between 25% (0.25) to 100% (1.00) according to the size of a job position (part-time or full time) for a selected job in a selected department. Supporting service departments in the hospital consist in total of 214 permanent employments with 221 workers employed before the optimization. Moreover, the maximal amount of money paid monthly for services in each department was used. Specific data consists of the average salaries for selected jobs in the departments defined as costs of assignment of workers to jobs. In addition, the minimum number of permanent employments for each job in each department was given, and the maximal number of positions which can be assigned to a single worker.

Table 1 shows the number of types of supporting services position in departments, the number of permanent employments in departments, the number of employees in each department before optimization and the maximal amount of money monthly paid for services in each department.

Figure 1 – Ludwik Rydygier Hospital in Krakow [23]


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Table 1 – Input data used in the model

Number

of types

of supporting

services

position in

department

Number

of permanent

jobs in

department

Number

of employees

in department

– before

optimization

Maximal

amount

of money

monthly paid

for services in

department

1.Central Heating Department 5 15.5 16 29250

2.Power Department 3 15 15 31050

3.Medical Bottled Gases Department 2 6 6 11400

4.Ventilation & Air-condition Department 4 8 8 16650

5.Heating & Hydraulic Department 4 11 11 21200

6.Distribution Department 3 6 6 13600

7.Medical Equipment Department 4 6.75 8 17500

8.Technical Department 5 11 11 20950

9.Economy Department 5 21 21 31360

10.Hospital Pharmacy 11 19.5 20 43400

11.Sterilization Department 5 27 27 41500

12.Stuff Monitoring Department 5 13 13 27150

13.Information Department 4 6.5 7 16100

14.Business Executive Department 5 8 8 15450

15.Technical Executive Department 4 3.5 4 7150

16.Law Regulation Department 3 7 7 16100

17.Attorneys-at-law Department 2 2.5 4 7950

18.Hospital Management Cost Section 5 9 9 15550

19.Salary Section 5 6.75 9 15800

20.Accounting Section 4 11 11 26950

TOTAL 88 214 221 426060

The hospital provides a lot of services. First of all, it has a lot of different departments, such as department of anesthesiology and intensive care, oncology, obstetrics or radiotherapy, among others. Then, for having a diagnostic they have a laboratory, with a pathology sub-department. They offer several fields of investigation: General analysis, Clinical chemistry, Hematology and coagulation, Immunochemistry, Microbiology, Toxicology, Gynecological cytology, Blood group serology. They also have some imaging machines, which are basically radiology, mammography, ultrasound and computer tomography machines.

PROBLEM FORMULATION 3.

Mathematical programming approach deals with optimization problems of maximizing or minimizing a function of many variables subject to inequality and equality constraints and integrality restrictions on some or all of the variables. In particular, 0-1 variables represent binary choice. Therefore, the model presented in this paper is defined as a mixed integer programming problem.



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Suppose there are m people and p jobs, where pm ≠ . Each job must be done by at least one

person; also, each person can do at least, one job. The problem objective is to assign the people to the jobs so as to minimize the total cost of completing all of the jobs.

The optimality criterion of the defined problem is to minimize operations costs of a supporting service subject to some specific constraints. The constraints represent specific conditions for resource allocation in a hospital. The overall problem is formulated as a modified assignment problem. The decision variables represent the assignment of people among various services. Compared to previously published papers [3–16] decision variables and constraints were modified according to different optimization models formulation.

OPTIMIZATION MODELS 4.

The problem of optimal assignment is formulated as a triple objective integer program, which allows commercially available software (e.g. AMPL/CPLEX [22]) to be applied for solving practical instances.

The problems of assignment are handled in two different ways and three different approaches. Firstly, each problem is solved as a bi-objective mixed integer program with use of weighted-sum approach, lexicographic approach and reference point approach. Secondly, problems are formulated as a triple-objective with the same approaches. For solving all models CPLEX solver is used.

The bi-objective models are used for optimization of allocation of personnel of a hospital. The objective is to minimize operational costs of supporting services and maximizing (or minimizing) the total number of employees. Maximization or minimization of the total number of employees depends on decision maker preferences, like service level, and cost.

The triple objective models by mixed integer programming consist of criteria as follows: operational costs of supporting services, total number of employees, total number of permanent employments. Constraints are defined to secure that the total cost of assignment of all the employees of a selected department must be less than or equal the maximal monthly budget for salaries in that department. Constraints also ensures that the total number of permanent employments in selected department must be less than or equal the maximum number of permanent employments in that department. Constraints guarantees that the minimal number of permanent employments is less than or equal the given maximum number of permanent employments. Constraints assume that the assignment of workers to selected permanent jobs type is not more than maximal requirements. Constraints also show the relation of variables, which assures that at least one staff member must be working in one job of one department of the hospital. Constraints also assures that the number of permanent employment of a selected job is less than or equal to the maximal number of permanent employments of that job of selected department.

COMPUTATIONAL RESULTS 5.

In this section numerical examples and some computational results are presented to illustrate possible applications of the proposed formulations of integer programming of optimal assignment of service positions. Selected problem instances with the examples are modeled on a real data from a Polish hospital.

In the computational experiments the historical data is considered. Computational time takes only a fraction of a second to find optimal solution if any exists. The computational experiments have


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been performed using AMPL programming language [22] and the CPLEX v.11 solver on a laptop with Intel® Core 2 Duo T9300 processor running at 2.5GHz and with 4GB RAM.

Analysis of computational results shows the total budget, the number of employees and the number of permanent employments for each case. Values of β1, β2, β3 are responsible for setting priority of each objective. For instance β1 is connected with the total budget objective, β2

with the number of employees and β3 with the number of permanent employments.

Figure 2 – Total budgets, number of employees and number of permanent jobs

In the Figure 2. it is shown that are computed values of budget, number of employees and permanent jobs. In addittion, when the value of β2 is 0.7 we have the lowest values for budget, number of employees and permanent jobs. We can also see, that when β2 is 1 all the values are 0 due to the fact that we do not have any constraint for the number of employees or the number of permanent employments. Moreover, when β1 = 1000 or β3 = 1000 the results are more or less the same with the maximum number of employees and permanent jobs and the maximal total budget too.



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Figure 3 – Objectives’ relation in the triple objective model

By looking at this graph (Figure 3) we are able to see that the relation between the number of positions and total budget with the number of employees is a linear relation. Logically if we increase the number of workers, the number of permanent positions and the total budget increase as well.

CONCLUSIONS 6.

Operations research techniques, tools and theories have long been applied to a wide range of issues and problems in healthcare.

The allocation of personnel in a hospital is an important factor. It is possible to use an optimization models to find optimal solutions for these types of problems. One possible method is the branch and bound method with the weighted-sum, lexicographic or reference point approach. It is possible to solve the problem with different objective functions and the results must be studied and analyzed deeply.

This paper proves the practical usefulness of mathematical programming approach to optimization of supporting service in a hospital. The results of computational experiments modeled after a real data from a hospital in Lesser Poland indicate that the number of hired workers can be reduced in almost all departments of the hospital.

The proposed modified multi-objective assignment problem and weighted-sum, lexicographic or reference point approach can be easily implemented for management of supporting services in another institution, not only healthcare. Obtained results consist of the monthly expenses for salaries, the number of workers and the amount of permanent employments needed for jobs in all considered supporting service departments.

Computational time takes only a fraction of a second to find the optimal solution because of a relatively small size of the input data. Presented optimization model is NP-hard, but computable. Implementation of reference point method ensures to obtain results with non-dominated set of solutions. The global optimums for considered three objective functions are presented.


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Considered approaches used models with the weighted-sum functions. Computational results shown how varying the values of the weights of the objective function can make a lot of changes in the results obtained. In the bi-objective model it is more easily to see the variations of values due to the changes of weights but in the triple-objective is not so easy to conclude some solutions.

Health care is a really important issue in the society, what is why it is too important to have health care institutions well developed and organized. There are a lot of problems in hospitals such as delays in the Emergency Rooms, low bed occupancy levels, wrong allocation of treatments or disagreement of the personnel with the schedule, as well as nurse rostering problems.

All over the years these problems have been treated with solutions carried out by manually processes. Nowadays, to make this easier we can take advantage of computational methods and approaches, which use optimization models to improve health care institutions’ work.

Therefore, to realize the allocation of personnel in a hospital it is a good idea to use an optimization model. Applied method in this research was the branch and bound algorithm with the weighted sum approach which helps to fix the assignment of services in one Polish hospital. It is possible to solve the problem with different objective functions and the results must be studied and analyzed deeply.

ACKNOWLEDGEMENT 7.

The author is grateful to the anonymous reviewers for their comments. This work has been partially supported by NCN grant # DEC-2013/11/B/ST8/04458 and by AGH University of Science and Technology.

REFERENCES 8.

[1] Green, L.V.: Capacity planning and management in hospitals in Operations Research and Health

Care, (Eds.: Brandeau, M. L. and Sainfort, F, Pierskalla W. P.), Kluwer Academic Publishers, Boston, USA, 2004.

[2] Brandeau, M.L., Sainfort, F, Pierskalla, W.P.: Operations Research and Health Care, Kluwer Academic Publishers, Boston, USA, 2004.

[3] Sawik, B.: Weighted-sum approach to health care optimization, Applications of Management Science, 17(2015), 163–180, Emerald Group Publishing Limited, ISSN: 0276-8976/doi:10.1108/S0276-897620140000017012

[4] Sawik, B.: Lexicographic approach to health care optimization, Applications of Management Science, 17(2015), 181–201, Emerald Group Publishing Limited, ISSN: 0276-8976/doi:10.1108/S0276-897620140000017013

[5] Sawik, B.: Health care optimization problem formulations with weighted-sum, lexicographic and

reference point approaches, MOTSP 2015, Brela, Makarska, Croatia 2015. [6] Sawik, B.: Application of multi-criteria mathematical programming models for assignment

of services in a hospital, Applications of Management Science, 16(2013), 39–53, Emerald Group Publishing Limited, ISSN: 0276-8976/doi:10.1108/S0276-897620130000016006.

[7] Sawik, B.: A multi-objective mathematical programming model with conditional value-at-risk for

assignment of services in a health care institution, Management of Technology Step to Sustainable Production (MOTSP) 2013, May 29–13, 2013, Novi Vinodolski, Croatia.

[8] Sawik, B.: A single and triple-objective mathematical programming models for assignment

of services in a health care institution, Int. J. Logistics Systems and Management, 15(2–3)(2013), 249–259.



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[9] Sawik, B.: Multi-Criteria Mathematical Programming for Assignment of Services in a Healthcare

Institution, Conference Proceedings – The International Conference on Industrial Logistics (ICIL) 2012, June 14-16, 2012, University of Zagreb, Zadar, Croatia, 161–167.

[10] Sawik, B.: Assignment of Supporting Services in a Healthcare Institution using Multi-

Criteria Mathematical Programming, The International Conference on Industrial Logistics (ICIL) 2012, June 14–16, 2012, Zadar, Croatia.

[11] Sawik, B.: A bi-objective portfolio model for supporting services in a hospital, INFORMS 2012 Annual Conference – Informatics rising, October 14–17, 2012, Institute for Operations Research and the Management Science, Phoenix, USA, 2012.

[12] Sawik B.: A Reference Point Method to Triple-Objective Assignment of Supporting Services in

a Healthcare Institution, Decision Making in Manufacturing and Services, 4(1–2)(2010), 37–46. [13] Sawik B.: Reference Point Approach to Assignment of Supporting Services in a Healthcare

Institution, chapter in: Świerniak A., Krystek J. (Eds.) Automatics of Discrete Processes, Theory

and Applications, Vol. 2, Monographic of the Silesian University of Technology, PKJS, Gliwice, Poland 2010, 187–196.

[14] Sawik B., Mikulik J.: An Optimal Assignment of Workers to Supporting Services in a Hospital, Polish Journal of Medical Physics and Engineering, 14(4)(2008), 197–206.

[15] Sawik B., Mikulik J.: Model for Supporting Service Optimization in a Public Health Care

Institution, EFOMP council and officer's meeting, 14th general assembly of the Polish Society of Medical Physics, European Conference Medical Physics and Engineering 110 years after the discovery of Polonium and Radium, 17–21 September 2008, Kraków, AGH, Polish Academy of Science, Poland.

[16] Sawik B.: Bi-Objective Integer Programming Approach to Optimal Assignment of Supporting

Services in a Health-Care Institution, The 21st Conference of the European Chapter on Combinatorial Optimization (ECCO XXI) May 29–31, 2008, Dubrovnik, Croatia.

[17] Bowman Jr, V.J.: On the relationship of the Tchebycheff norm and the efficient frontier of multi-

criteria objectives, In: Thiriez H., Zionts S. (Eds.) Multiple Criteria Decision Making, Lecture Notes in Economics and Mathematical Systems, Vol. 130. Springer-Verlag, Berlin, Germany 1976, 76–86.

[18] Burrkard R, Dell’Amico M, Martello S.: Assignment Problems, SIAM, Philadelphia, USA, 2008.

[19] Ehrgott M.: Multicriteria Optimization, Second edition, Springer, Berlin, Germany, 2000. [20] Nemhauser G.L., Wolsey L.A.: Integer and Combinatorial Optimization, John Wiley & Sons,

Toronto, Canada, 1999.

[21] Steuer R.E.: Multiple Criteria Optimization: Theory, Computation and Application, John Wiley & Sons, New York, USA, 1986.

[22] Fourer R., Gay D.M., Kernighan B.W.: A Modeling Language for Mathematical Programming, Management Science, 36(1990), 519–554.

[23] www.szpitalrydygier.pl Last access Jun 2016

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