Chapter 15: Quantitatve Methods in Health Care Management Yasar A. Ozcan 1 Chapter 15. Simulation.

Chapter 15: Quantitatve Chapter 15: Quantitatve Methods in Health Care Methods in Health Care ManagementManagement

Yasar A. OzcanYasar A. Ozcan 11

Chapter 15.Simulation



OutlineOutline

Simulation Process Monte Carlo Simulation Method

– Process– Empirical Distribution– Theoretical Distribution– Random Number Look Up

Performance Measures and Managerial Decisions



When Optimization is not an option. . .When Optimization is not an option. . .

SIMULATE!Simulation can be applied to a wide range of problems in healthcare management and operations.

In its simplest form, healthcare managers can use simulation to explore solutions with a model that duplicates a real process, using a what if approach.



Why Simulate?Why Simulate?Why Simulate?Why Simulate?

To enhance decision making by capturing a situation that is too complicated to model mathematically (e.g., queuing problems)

It is simple to use and understand

Wide range of applications and situations in which it is useful

Software is available that makes simulation easier and faster



Simulation ProcessSimulation ProcessSimulation ProcessSimulation Process1. Define the problem and objectives

2. Develop the simulation model

3. Test the model to be sure it reflects the situation being modeled

4. Develop one or more experiments

5. Run the simulation and evaluate the results

6. Repeat steps 4 and 5 until you are satisfied with the results.



Simulation Basics

We need an instrument to randomly simulate this situation. Let’s call this the “simulator”.

Imagine a simple “simulator” with two outcomes.



So how can it help us?So how can it help us?So how can it help us?So how can it help us?

Waiting Line ServiceSystem

Customers

arrivals

Let’s look at a health care example.

How can we simulate the patient arrivals and service system response?



. . . to simulate patients arrivals in public health clinic!

If the coin is heads, we will assume that one patient arrived in a determined time period (assume 1 hour). If tails, assume no arrivals.

We must also simulate service patterns. Assume heads is two hours of service and tails is 1 hour of service.

Let’s use this simulator. . .



Table 15.1 Simple Simulation Experiment for Public Clinic

Time Coin tossfor arrival

Arrivingpatient

Queue Coin tossfor service

Physician Departingpatient

1) 8:00 - 8:59 H #1 H #1 -

2) 9:00 - 9:59 H #2 #2 T #1 #1

3)10:00 -10:59 H #3 #3 T #2 #2

4)11:00 -11:59 T - - - #3 #3

5)12:00 -12:59 H #4 H #4 -

6) 1:00 - 1:59 H #5 #5 H #4 #4

7) 2:00 - 2:59 T - - - #5 --

8) 3:00 - 3:59 H #6 #6 T #5 #5



Calculation of Performance StatisticsCalculation of Performance StatisticsCalculation of Performance StatisticsCalculation of Performance Statistics

Arrivals Queue (Waiting Line) Service Exit

? ? ? ?

Waiting Line ServiceSystem

Customers

arrivals



Table 15.2 Summary Statistics for Public Clinic Experiment

Patient Queuewait time

Servicetime

Total timein system

#1 0 2 2

#2 1 1 2

#3 1 1 2

#4 0 2 2

#5 1 2 3

#6 1 1 2

Total 4 9 13



Number of Arrivals

Average number waiting

Avg. time in Queue

Service Utilization

Avg. Service Time

Avg. Time in System

Performance Measures



But what if we have multiple arrival patterns?

But what if we have multiple arrival patterns?

Can we use a dice or any other shaped object that could provide random arrival and service times?

We could use Monte Carlo Simulation and a Random Number Table!

Can we use a dice or any other shaped object that could provide random arrival and service times?

We could use Monte Carlo Simulation and a Random Number Table!



MONTE CARLO METHODMONTE CARLO METHODMONTE CARLO METHODMONTE CARLO METHOD

A probabilistic A probabilistic simulation techniquesimulation technique

Used only when a Used only when a process has a random process has a random componentcomponent

Must develop a Must develop a probability distribution probability distribution that reflects the that reflects the random component of random component of the system being the system being studiedstudied

A probabilistic A probabilistic simulation techniquesimulation technique

Used only when a Used only when a process has a random process has a random componentcomponent

Must develop a Must develop a probability distribution probability distribution that reflects the that reflects the random component of random component of the system being the system being studiedstudied



Step 1: Selection of an appropriate probability distribution

Step 2: Determining the correspondence between distribution and random numbers

Step 3: Obtaining (generating) random numbers and run simulation

Step 4: Summarizing the results and drawing conclusions

MONTE CARLO METHOD



If managers have no clue pointing to the type of probability distribution to use, they may use an empirical distribution, which can be built using the arrivals log at the clinic.

For example, out of 100 observations, the following frequencies, shown in table below, were obtained for arrivals in a busy public health clinic.

Empirical Distribution

Table 15.3 Patient Arrival Frequencies

Number

of arrivals

Frequency 0 180 1 400 2 150 3 130 4 90

5 & more 50 Sum 1000



Table 15.4 Probability Distribution for Patient Arrivals

Number of arrivals

Frequency

Probability

Cumulative probability

Corresponding random numbers

0 180 .180 .150 1 to 180 1 400 .400 .580 151 to 580 2 150 .150 .730 581 to 730 3 130 .130 .860 731 to 860 4 90 .090 .950 861 to 950

5 & more 50 .050 1.00 951 to 000



The second popular method for constructing arrivals is to use known theoretical statistical distributions that would describe patient arrival patterns.

From queuing theory, we learned that Poisson distribution characterizes such arrival patterns. However, in order to use theoretical distributions, one must have an idea about the distributional properties for the Poisson distribution, namely its mean.

In the absence of such information, the expected mean of the Poisson distribution can also be estimated from the empirical distribution by summing the products of each number of arrivals times its corresponding probability (multiplication of number of arrivals by probabilities).

In the public health clinic example, we get

Theoretical Distribution

λ = (0*.18)+(1*.40)+(2*.15)+(3*.13)+(4*.09)+(5*.05) = 1.7



Table 15.5 Cumulative Poisson Probabilities for λ=1.7

Arrivalsx

Cumulativeprobability

Correspondingrandom numbers

0 .183 1 to183

1 .493 184 to 493

2 .757 494 to 757

3 .907 758 to 907

4 .970 908 to 970

5 & more 1.00 970 to 000



Table 15.6 Cumulative Poisson Probabilities for

Arrivals: λ=1.7 Patients arrived



0 .183 1-183 1 .493 184-493 2 .757 494-757 3 .907 758-907

4 & more 1.000 908 to 000

Service: μ =2.0 Patients served



0 .135 1 to135 1 .406 136 to 406 2 .677 407 to 677 3 .857 678 to 857

4& more 1.000 858 to 000



Finding Random NumbersFinding Random NumbersFinding Random NumbersFinding Random Numbers Numbers must be Numbers must be

both uniformly both uniformly distributed and must distributed and must not follow any patternnot follow any pattern

Always avoid starting Always avoid starting at the same spot on a at the same spot on a random number tablerandom number table

Numbers must be Numbers must be both uniformly both uniformly distributed and must distributed and must not follow any patternnot follow any pattern

Always avoid starting Always avoid starting at the same spot on a at the same spot on a random number tablerandom number table

2419

4572



Figure 15.1 Random Numbers*

* Random numbers are generated using Excel



Table 15.7 Monte Carlo Simulation Experiment for Public Health Clinic

Time

Random numbers

& (arrivals)

Arriving patients

Queue

Random numbers &

(service)

Physician

Departing Patients

1) 8:00 - 8:59 616 (2) #1,#2 - 764 (2) #1,#2 #1,#2 2) 9:00 - 9:59 862 (3) #3,#4,#5 #4,#5 180 (1) #3 #3 3)10:00 -10:59 56 (0) - - 903 (4+) #4,#5 #4,#5 4)11:00 -11:59 583 (2) #6,#7 - 780 (3) #6,#7 #6,#7 5)12:00 -12:59 908 (4) #8,#9,#10,#11 #9,#10,#11 164 (1) #8 #8 6) 1:00 - 1:59 848 (3) #12,#13,#14 #11,#12,#13,#14 546 (2) #9,#10 #9,#10 7) 2:00 - 2:59 38 (0) - #12,#13,#14 351 (1) #11 #11 8) 3:00 - 3:59 536 (2) #15,#16 900 (4+) #12,#13,#14,#15,#16 #12,#13,#14,#15,#16



Table 15.8 Summary Statistics for Public Clinic Monte Carlo Simulation Experiment

Patient

Queue wait time

Service time

Total time in system

#1 0 0.5 0.5 #2 0 0.5 0.5 #3 0 1.0 1.0 #4 1 0.5 1.5 #5 1 0.5 1.5 #6 0 0.5 0.5 #7 0 0.5 0.5 #8 0 1.0 1.0 #9 1 0.5 1.5

#10 1 0.5 1.5 #11 2 1.0 3 #12 2 0.2 2.2 #13 2 0.2 2.2 #14 2 0.2 2.2 #15 0 0.2 0.2 #16 0 0.2 0.2

Total 12 8 20



Using information from Tables 15.7 and 15.8, we can delineate the performance measures for this simulation experiment as:

Number of arrivals: There are total of 16 arrivals.Average number waiting: Of those 16 arriving patients; in 12 instances patients were counted as waiting during the 8 periods, so the average number waiting is 12/16=.75 patients.Average time in queue: The average wait time for all patients is the total open hours, 12 hours ÷ 16 patients = .75 hours or 45 minutes.Service utilization: For, in this case, utilization of physician services, the physician was busy for all 8 periods, so the service utilization is 100%, 8 hours out of the available 8: 8 ÷ 8 = 100%.Average service time: The average service time is 30 minutes, calculated by dividing the total service time into number of patients: 8 ÷ 16 =0.5 hours or 30 minutes.Average time in system: From Table 15.8, the total time for all patients in the system is 20 hours. The average time in the system is 1.25 hours or 1 hour 15 minutes, calculated by dividing 20 hours by the number of patients: 20 ÷ 16 = 1.25.

Performance Measures



Figure 15.2 Excel-Based Simulated Arrivals



Figure 15.3 Excel Program for Simulated Arrivals



r1 < rt r1 >= rt

r2 < rt

r2 >= rt

Marketing andreferral systemsto increase businessvolume

Appointment Scheduling

Increase Capacity

Busy time during regular hoursr1 = ---------------------------------

Total busy time, including during over time

Total regular hours open

Total regular hours open

r2 = ------------------------------------------

rt = Target utilization rate (e.g., 90%)

Figure 15.4 Performance-Measure-Based Managerial Decision Making

Status Quo



Advantages of SimulationAdvantages of SimulationAdvantages of SimulationAdvantages of Simulation

Used for problems difficult to solve Used for problems difficult to solve mathematicallymathematically

Can experiment with system Can experiment with system behavior without experimenting behavior without experimenting with the actual systemwith the actual system

Compresses timeCompresses time Valuable tool for training decision Valuable tool for training decision

makersmakers

Used for problems difficult to solve Used for problems difficult to solve mathematicallymathematically

Can experiment with system Can experiment with system behavior without experimenting behavior without experimenting with the actual systemwith the actual system

Compresses timeCompresses time Valuable tool for training decision Valuable tool for training decision

makersmakers



LimitationsLimitations

Does not produce an optimumDoes not produce an optimum Can require considerable effort to Can require considerable effort to

develop a suitable modeldevelop a suitable model Monte Carlo is only applicable when Monte Carlo is only applicable when

situational elements can be described situational elements can be described by random variablesby random variables

Does not produce an optimumDoes not produce an optimum Can require considerable effort to Can require considerable effort to

develop a suitable modeldevelop a suitable model Monte Carlo is only applicable when Monte Carlo is only applicable when

situational elements can be described situational elements can be described by random variablesby random variables



The End

Chapter 15: Quantitatve Methods in Health Care Management Yasar A. Ozcan 1 Chapter 15. Simulation.

Documents

Transcript of Chapter 15: Quantitatve Methods in Health Care Management Yasar A. Ozcan 1 Chapter 15. Simulation.