The Pennsylvania State University. The Graduate School ROBUST DESIGN USING LOSS FUNCTION WITH MULTIPLE OBJECTIVES

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1 The Pennsylvania State University The Graduate School The Harold and Inge Marcus Department of Industrial and Manufacturing Engineering ROBUST DESIGN USING LOSS FUNCTION WITH MULTIPLE OBJECTIVES AND PATIENT CLASSIFICATIONS APPLIED TO HEALTH CARE A Thesis in Industrial Engineering and Operations Research by Kristin A. Smith 2010 Kristin A. Smith Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science May 2010

2 ii The thesis of Kristin A. Smith was reviewed and approved* by the following: M. Jeya Chandra Professor of Industrial Engineering Thesis Adviser Russell Barton Professor of Supply Chain and Information Systems Paul Griffin Professor of Industrial Engineering Head of the Department of Industrial Engineering *Signatures are on file in the Graduate School.

3 iii Abstract Health care providers are like any other company in that their concerns include providing quality care to their patients while being conscience of operating costs. If patients feel they have not received the level of care deserved, they could seek out another health care provider. By minimizing the variance in waiting time, the potential of losing a patient can be minimized. This is based on Taguchi s robust design methods. The operating costs are also of interest to the health care provider. The number of servers available to help patients directly relates to the waiting time variance and operating cost. A nonlinear integer programming model is developed to determine the number of servers for each patient class at every station. The methodology uses the common multicriteria decision making method of scaling to solve the multiple objectives of minimizing both the operating cost and the expected loss. The user can also place importance on one objective over the other by specifying the weights applied to each objective in the model. A numerical example is given as a demonstration of the methodology. The model is solved for the optimal number of servers available at five stations throughout a facility. The five stations include check-in, meeting with a nurse, seeing a specialized physician, having tests performed, and finally meeting with another specialized physician to review any results and discuss a treatment plan. Stations do not necessarily require physical movement of the patient. There are also four patient classes considered based on the health condition. Every patient must move through the first two stations where they are classified, and then follow a unique flow for stations three through five based on their classification. By determining the optimal number of servers, both the operating costs and the variance in waiting times can be minimized simultaneously.

4 iv Table of Contents List of Tables... v List of Figures... vi Acknowledgements... vii Chapter 1 Introduction Problem Statement Literature Review Robust design using Loss function Robust design applied to health care Robust design with multiple objectives Robust design with multiple classes/constraints Objective Organization... 7 Chapter 2 Methodology Queuing Theory Applied to Health Care Robust Design Applied to Health Care Incorporating Operating Costs Final Model Solving the Model Chapter 3 Numerical Example Scenario Description Inputs for the Model Relationship between Number of Servers and Waiting Time Variance for Robust Design Solution Step One Solution Step Two Sensitivity Analysis Chapter 4 Conclusions Summary and Conclusions Future Research Bibliography Appendix Percent Increase Calculations... 50

5 v List of Tables Table 3-1: Model Inputs for Mean Inter-arrival and Service Times Table 3-2: Input Values for Unit Resource Costs Table 3-3: Input Values for Waiting Costs per Patient per Unit Time in the Queue Table 3-4: Input Values for Unit Space Cost per Patient in the Queue Table 3-5: Variance Values for Each Patient Classification and Station Combination Table 3-6: Results When Solving for the Expected Loss Objective Function Only Table 3-7: Target Waiting Time Total by Patient Class for the Expected Loss Objective Function Only Table 3-8: Results When Solving for the Operating Cost Objective Function Only Table 3-9: Target Waiting Time Total by Patient Class for the Operating Cost Objective Function Only Table 3-10: Results When Solving Both Objective Functions Weighted Equally Table 3-11: Target Waiting Time Total by Patient Class for Both Objective Functions. 37 Table 3-12: Combinations of Weight Values Used During the Sensitivity Analysis Table 3-13: Number of Servers at Each Station for Each Objective Weight Combination Table 3-14: Target Wait Time at each Station for Each Objective Weight Combination 39 Table 3-15: Target Waiting Time Total by Patient Class for Each Objective Weight Combination Table 3-16: Operating Cost and Expected Loss Values for Each Objective Weight Combination as a Result of the Number of Servers at Each Station Table 3-17: Percent Differences Between the Original Optimal Values from Solving the Objectives Alone and Each Objective Weight Combination Table 3-18: Percent Differences Between the Optimal Values from Solving the Objectives with Equal Weights and Each Objective Weight Combination... 41

6 vi List of Figures Figure 2-1: Health Care Facility Flow Diagram... 9 Figure 2-2: Detailed Health Care Facility Flow Diagram Figure 2-3: Final Model Formulation Figure 2-4: Final Model Formulation to be Solved Figure 3-1: Updated Health Care Facility Flow Diagram... 24

7 vii Acknowledgements I would like to thank Dr. Jeya Chandra for his guidance and advice throughout the completion of this thesis. Also, I would like to thank Dr. Russell Barton for reading and reviewing this work. I would like to acknowledge Thomas Charles, Associate Vice President of Strategy and Business Development at Geisinger Health System, for his input on the uses of the model generated to health care providers. I would also like to acknowledge Dr. Tom Cavalier for his assistance in building the nonlinear integer programming model in Lingo for the numerical example. I would like to thank my family for their love and support. Also, to my friends who have always been there for me with advice and encouragement. You have made my time at Penn State unforgettable.

8 1 Chapter 1 Introduction 1.1 Problem Statement Successful businesses strive to provide excellent services to their customers while simultaneously keeping their operating costs at a minimum. A company risks losing a customer s business if the services they provide do not meet the customer s standards. At the same time, the company must operate efficiently in order to maximize their revenues. Health care facilities face similar problems. If a patient is required to wait an extreme length of time before receiving care, then the health care provider risks losing that patient to another local provider. In a manufacturing environment, a part is made up of components that fit together to make the final product. This assembly can be stated as a mathematical expression of the components. By setting each component s process at a particular setting, the number of final products that are outside the specification limits can be minimized. Taguchi s robust design using loss function method allows for the expected loss of a part to be minimized by determining the target values for the quality characteristic of each component while taking into consideration that there is some known variability in the processes used to make the components. This method is most often applied to manufacturing and production environments, but can also be applied to service systems where customer satisfaction is important. Considering a patient s stay at a health care provider s office is made up of multiple steps including exams, tests, and procedures, the total length of stay includes

9 2 these steps as well as any waiting time between them. The length of stay can be stated as a sum of each step during the visit as well as the waiting times between each step just as the components of a final product can be described as a summation of their measurements. By determining the target value for each segment of time, the expected loss of a customer can be minimized for the hospital. At the same time, the cost of providing care to patients should be minimized. By solving these two objectives simultaneously, the optimal number of resources can be determined so as to minimize the expected cost of losing a patient to another health care facility. Patients arriving to a provider are initially examined to determine the health condition to be treated. This allows for a patient s illness to determine the physicians he or she needs to meet with and the tests that must be done. This is to be expected since every patient cannot be treated in the same manner. Depending on the patient s condition, an appropriate amount of time can be predetermined in which the patient should receive care. This means there is more than one final product in this scenario, so target values for each patient classification will be determined. 1.2 Literature Review Robust design using Loss function When a product fails, it must be replaced or fixed. The costs associated with this failure include tracking down the product, transporting it, and compensating for the failure. No matter what the losses of poor quality are believed to be, they are generally six times greater (Taguchi & Clausing, 1990). Losses are incurred when a product manufacturer deviates from a target value; the more deviation, the higher the loss. If a

10 3 characteristic exactly meets the target value established, the loss equates to zero. This loss can be captured in a simple quadratic loss function since loss increases by the squared deviation from the target value. This means the focus of quality control should be placed on meeting a target value as opposed to simply operating within specification limits (Ryan, 2000). Developed by Genichi Taguchi, this robust design method using loss functions allows the loss associated with a deviation from the target value to be determined while relating cost to variability (Taner & Antony, 2006). By controlling the means of the components that make up the quality characteristic, the mean and variance of the quality characteristic can be controlled (Chandra, 2001) Robust design applied to health care Taguchi s loss function technique has been suggested as a quality improvement technique in previous works. Godfrey (1992) suggested this common technique used in manufacturing environments be applied to health care problems. Just as the loss function technique has helped improve manufactured components, health care providers can also benefit from the implementation. Godfrey parallels the two types of environments to show that the popular method can cross over to the health care setting. Taguchi s methods have been applied to health care scenarios in the past in order to study the factors affecting a patient s length of stay (LOS) in an emergency department (ED). Rinderer (1996) applied Taguchi s design of experiments methodology to determine the most significant effects on LOS in an attempt to reduce the performance measure. Of the eleven factors explored in this study, the three most significant factors

11 4 on the response of LOS were found to be having an extra physician in the ED, having a dedicated laboratory staff, and implementing an auto-hold policy where a patient could be held while trying to contact his or her private physician. The number of physicians or resources available to server patients directly affects the amount of time a patient spends in a health care facility. This leads to the use of queuing theory as a tool to determine these times and queue lengths. Taguchi s robust design methods have been used in many commercial and industrial applications (Taner & Antony, 2006). Taguchi sees the loss a defective product or service has on society with regards to monetary loss, customer dissatisfaction, time lost, and possible hazards to the surrounding environment. With this lose-to-society mentality, the importance of a quality characteristic being on target is stressed for both the patients and the healthcare providers. By implementing Taguchi s loss function, Taner and Antony propose that quality management and the measurement of the outcomes in health care can be improved with regards to total costs. They also focus on the manpower available in the environment, leading to the use of queuing theory. Chandra (2001) describes three possible types of characteristics for describing the relationships between quality components. Nominal-the-better (N-type) is appropriate when a quality characteristic has a target value mean. Smaller-the-better (S-type) is used when the quality characteristic target value is zero. Similarly, larger-the-better (L-type) is applied when the quality characteristic target value is infinity. Since having patients wait in a health care facility is not ideal, this would qualify the quality characteristic as a smaller-the-better type. This means the sum of variances in the waiting times should be minimized.

12 5 A patient s condition is a common basis for patient classification. Since the types of physicians and equipment depend on the condition, a patient s classification dictates the flow he or she follows in a health care facility. This initial classification can be determined in a triage stage (Saunders, 1987). Once a classification is assigned, the patient will meet with a dedicated set of nurses and physicians Robust design with multiple objectives Applying Taguchi s robust design method to problems with multiple objectives has been done in previous studies. It is recommended that any multi-criteria problem be transformed into a single objective problem by M. Chen (1998). This allows the Taguchi method to be applied to the problem as intended, instead of possibly deleting some of the objectives of interest. When any number of objectives need to be deleted from a model, some engineering judgment is necessary to determine which one to get rid of. If only one objective is considered when solving a robust design problem, the results can lead to the worst solution based on another objective. Chen, Chiang, and Lin (2000) state that when a method is limited to a single objective in a multiple objective problem, it is possible to find a solution that is not a global optimal. The techniques generated allow multiple objective problems to be solved while eliminating the uncertainty generated by engineering judgment. Since there is generally more than one objective when designing a process, multiple objectives should be considered when evaluating the quality of the process (Chen, Allen, Tsui, and Mistree, 1996). Song (1994) points out that the optimal solution in a multi-objective problem is usually a set, unlike a single objective problem where the problem has a single solution.

13 6 Sundaresan, Ishii, and Houser (1993) suggest utilizing a single weighted objective to solve a multiple objective problem. Ramakrishnan and Rao (1991) introduce the use of a nonlinear program to solve a robust design problem using Taguchi s loss function (expected loss) as the objective Robust design with multiple classes/constraints Wu and Yeh (2004) discuss the use of multiple quality characteristics when applying robust design to a problem. Traditionally, robust design is used with one quality characteristic, but they argue that in application, most parts have more than one quality characteristic. Elsayed and Chen (1993) propose a multiple characteristic model using the loss function. With their model, they can determine the optimal process settings when more than one quality characteristic is present. The model considered adjustment factors that were both independent and dependent. The loss function was used to estimate the performance measure on quality (PerMQ). Including multiple quality characteristics allows the designer to place special emphasis on some characteristics if desired (Chen, 1997) 1.3 Objective This thesis will apply Taguchi s robust design using loss function method to a health care environment. Multiple objectives will be solved so as to find the optimal number of resources (nurses, physicians, or pieces of equipment) necessary to provide quality health care to patients. The first objective in the model will include minimizing the operating costs incurred by the health care provider including resource costs, waiting

14 7 costs, and queue space costs. The second objective will minimize the total variance in the waiting time which can be related to the expected loss within the system. These multiple objectives will be solved with weights and scales applied, which is a common multi-criteria optimization technique. Multiple classifications of patients will be included as each patient is treated depending on their condition and require varying resources. This means multiple constraints will be used in the robust design model so the all patient types will be taken into consideration. The constraints in the model will ensure each patient class does not wait more than a given amount while visiting the health care facility. The robust design using loss function method generated in this thesis will present health care providers with target values for the wait times of each customer classification as a result of the number of servers (resources) available to help patients in order to meet the multiple objectives discussed. 1.4 Organization Chapter one of this thesis provides an introduction to the problem. This chapter also contains a review of work done by various researchers using Taguchi s robust design using loss function method, applications of the method to health care, the robust design method applied to multiple objectives, and the method with multiple classifications as well as the objective. Chapter two outlines the methodology used to solve the problem described, including necessary inputs and assumptions of the model. A numerical example will be presented in chapter three to show how the model can help health care providers make important decisions. Chapter four summarizes the thesis, provides conclusions, and includes several areas for future work.

15 8 Chapter 2 Methodology Balancing patient satisfaction with total operating expenses at health care centers is an important goal for health care providers. Patient satisfaction can be related to the length of time a patient is asked to wait before receiving care. Shortening the waiting time before a procedure is usually done by increasing the number of physicians or other resources available at that time. This in turn increases the operating costs. By minimizing both the average total waiting time of a patient while at the center and the costs of the resources, the optimal operating parameters can be determined. Health care providers see different types of patients with various ailments. The health problem a patient is experiencing dictates the procedures needed and the resources used. To differentiate patients in a health care center, a patient classification system will be introduced. Since all patients must be treated regardless of the services required, this will allow all the patient types to be taken into consideration when optimizing the system. Average length of stay (LOS) is one of the most common performance measures analyzed by health care providers. Patient satisfaction is important in order to keep them coming back as well as efficiency in providing the services. This can be done by minimizing the amount of time patients are required to wait between procedures and meetings with physicians. Health care providers are interested in minimizing the expected loss of a patient to another practice. This measure can be related to the mean and the variance of the patients wait time. By minimizing the variance in the multiple

16 9 waiting times of a patient and the expected waiting times, the practice can minimize the expected loss of a patient. Another performance metric of interest to health care providers is the total cost of providing the necessary care to their patients. By minimizing the patients wait times, the number of resources and space necessary to accommodate all the patients would increase. This will ultimately raise the costs to operate the practice. The operating costs should also be minimized simultaneously to ensure the provider can benefit from the center. The system to be analyzed will cover from the time a patient arrives to the health care center, receives care, and is then released. Once a patient arrives and checks in, he or she will proceed to a number of stations to meet with nurses, physicians, and complete any necessary tests. Each of these steps will be considered its own station even if it is not necessary for the patient to physically move to a new location. At the final station, the patient will be released once a diagnosis and treatment plan have been determined. Figure 2-1 depicts a general system flow for all patients. Figure 2-1: Health Care Facility Flow Diagram 2.1 Queuing Theory Applied to Health Care Queuing theory is a common method used to study parameters of systems including periods of service and waiting. In order to relate the number of resources available for serving patients to the amount of time spent waiting to be treated, this

17 10 method will be implemented. Using this method, some assumptions about the system must be made, including: 1. Each station is independent of the others. 2. The inter-arrival rate to a station follows an exponential distribution. 3. The service times follow a general distribution. 4. First two stations serve all patients in the system and have general resources. 5. Patients belong to one of multiple classes depending on the type of health problem. Each class follows its own flow and has its own dedicated set of resources after the first two stations. 6. The servers (nurses, physicians, etc) available at any station are identical with respect to the service provided at the station. 7. Each station follows a first-in-first-out service for each patient class. 8. There is infinite waiting space at each station for each patient class. 9. Average arrival rate at any station is less than the average service rate at that station. Following these assumptions, each station in the health care center is treated as an M/G/s queuing system. Notation used for this model includes: 1. Patient class notation: i=a, B,, N 2. Number of stations in the facility: j=1, 2,, M 3. Waiting time in the queue for patient class i at station j: w qij, j = 3,, M 4. Waiting time in the queue for any patient class at station j: w q j, j = 1,2

18 11 5. Average length of queue for patient class i at station j: L qij, j = 3,, M 6. Average length of queue for any patient class at station j: L q j, j = 1,2 7. Number of servers for patient class i at station j: s ij, j = 3,, M 8. Number of servers for any patient class at station j: s j, j = 1,2 9. Mean inter-arrival time for patient class i at station j: 1 λ, j = 3,, M ij 10. Mean inter-arrival time for any patient class at station j: 1 λ j, j = 1,2 11. Mean service time for each server for patient class i at station j: 1 μ ij, j = 3,, M 12. Mean service time for each server for any patient class at station j: 1 μ j, j = 1,2 13. Standard deviation of service time for patient class i at station j: σ ij, j = 3,, M 14. Standard deviation of service time for any patient class at station j: σ j, j = 1,2 The waiting time for a station depends upon the number of servers available at the station. Once the patients check in and see a nurse, they move to different stations depending on the condition. Therefore, there are general resources for the first two stations, and then dedicated ones for the remaining stations. This can be seen in Figure 2-2. The dot ( ) in the subscript of the variables for stations 1 and 2 means for all the patient classes. This is used when there are not dedicated servers for the station.

19 12 Figure 2-2: Detailed Health Care Facility Flow Diagram From Drayer (2007), the average waiting time in the queue and average length of the queue can be quantified by equations 2.1 and 2.2, respectively.

20 σ ij μ ij 2 s ij λ ijμ ij λ ij μ ij s ij 1! (s ij μ ij λ ij ) 2 w q ij = λ ij n λ ij s ij + 1 μ ij 2λ ij s ij 1 μ ij n=0 + n! μ ij s ij! s ij μ ij s ij μ ij λ ij i = A, B,, N and j = 1, 2,, M (eq. 2-1) and σ ij μ ij 2 s ij λ ijμ ij λ ij μ ij s ij 1! (s ij μ ij λ ij ) 2 L q ij = λ ij n λ ij s ij s ij 1 μ ij n=0 + n! μ ij s ij! s ij μ ij s ij μ ij λ ij i = A, B,, N and j = 1, 2,, M (eq. 2.2) The purpose of this model is to determine the optimal number of servers for each patient class at each station, which minimizes the operating cost and the mean and variance of the total waiting time for each patient. 2.2 Robust Design Applied to Health Care Generally applied to assemblies in manufacturing settings, the robust design method determines the optimal mean value of a quality characteristic component by

21 14 minimizing the variance associated with the mean setting. It is assumed in this method that the mean setting of a process affects the variance of the process. Therefore, a relationship between the resulting mean value of a variable and the variance of the variable must be defined. Without this relationship, the robust design method cannot be applied to the specified problem. In a health care environment, the patient s LOS is one of the most important performance measures. Since a patient s total time in a health care facility consists of all service and waiting times, all times are added together to find the total LOS. Service times are determined by the procedure needed to be performed, so it is assumed these times cannot be changed. However, the waiting times between each station can be changed so as to minimize the LOS. Therefore, the total LOS can be minimized by minimizing each period of waiting before receiving service. The quality characteristic of interest is the total waiting time in queue for each patient class. The quality characteristic s relationship between characteristics of components, which are the individual waiting times at each station, can be explained by equation 2.3. There is one relationship for each patient class in the system. w qi = e w qi1, w qi2, w qi3,, w qim = w qi1 + w qi2 + w qi3 + + w qim i = A, B,, N (eq. 2.3) Additional inputs for the robust design model include: 1. The relationship between the number of servers and the variance in the waiting time for patient class i at station j: g ij s ij, j = 3,, M

22 15 2. The relationship between the number of servers and the variance in the waiting time for any patient class at station j: g j s j, j = 1,2 3. The total waiting time in the system for patient class i: w qi 4. Standard deviation of waiting time for patient class i at station j: σ w ij, j = 3,, M 5. Standard deviation of waiting time for any patient class at station j: σ w j, j = 1,2 6. Maximum total waiting time in the facility for patient class i: w qi0 As stated earlier, the expected loss of a patient can be minimized by minimizing the sum of the variances of the quality characteristic and the sum of the mean waiting times. Minimizing the sum of the mean waiting times is taken care of in a constraint which has a maximum target value. In the health care environment, this would be the sum of the variances of the waiting times in the system. The first objective function of the model becomes the minimization of equation 2.4. The equation is simplified as every patient goes to the first two stations where there are general resources. N i=a,b, 2 σ w i = σ w A + σ w B + + σ w N 2 = g j s j j=1 M + g Aj s Aj j=3 M + + g Nj s Nj j=3 i = A, B,, N and j = 1, 2,, M (eq 2.4)

23 16 It is assumed that in real-life problems, either the function g ij s ij is available analytically or can be obtained using simulation. Also, it is assumed the variance in the waiting times will decrease as the number of servers increases for each class at each station. Since the quality characteristic relationship is linear, the constraints of the model are simply the sum of the waiting times must not exceed a set target value. This model is considered a smaller-the-better (S-type) problem. Since waiting times do not help the patient in any way, they should be minimized and as close to zero as possible. Since having zero waiting time in the system would drive up the cost of operating the practice, upper limits on the total waiting times, w qi0, are set for each patient class. A sample constraint can be seen in equation 2.5; there is one constraint for each patient class. The first two values for wait time will be the same in every constraint since every patient class goes through the first two stations. w q 1 + w q 2 + w qi3 + + w qim w qi0 i = A, B,, N (eq. 2.5) 2.3 Incorporating Operating Costs Health care providers are businesses that are looking to make money like any other company. While providing great care to their patients, they are also interested in minimizing the costs associated with providing their services. Providers must worry about costs associated with the resources located at the facility as well as the costs to allocate space for patients to wait between stations.

24 17 Some additional inputs associated with costs include the following: 1. Unit resource cost for patient class i at station j: c rij, j = 3,, M 2. Unit resource cost for any patient class at station j: c r j, j = 1,2 3. Waiting cost per patient per unit time for patient class i at station j: c qij, j = 3,, M 4. Waiting cost per patient per unit time for any patient class at station j: c q j, j = 1,2 5. Unit space cost per patient for patient class i at stations 3,, M: v qi 6. Unit space cost per patent for any patient class at stations 1 and 2: v q 7. Fixed cost for queue space for patient class i: f i The resource located at a station is dependent on the patient class and the station. This could include a physician, nurse, or a piece of equipment. To determine the cost of the resources in the facility, the number of each resource type is multiplied by the cost of the resource. The cost per unit time per patient is multiplied by the waiting time of each patient class at each station to calculate the waiting cost of a patient. This corresponds to the costs incurred for having a patient wait to be served at a station. In order to keep patients happy and coming back to the practice, entertainment should be provided (magazines, books, etc) to the patients to keep them busy. The cost of these items is included here. The third cost to be considered is the cost associated with the amount of space necessary for the patients to wait between stations. There is a fixed cost for constructing the area as well as a variable cost for each unit of space for each patient. The average

25 18 length of the queue is multiplied by the unit cost and then added to the fixed cost to determine the space costs. Taking all three types of costs into account, the cost function to be minimized can be seen in equation 2.6. It is simplified since the first two stations of the flow include general resources. N M N M N M i=a,b, j=1 c rij s ij + w q ij c q ij + f i + v qi L qij i=a,b, j=1 i=a,b, j=1 N M = c r 1 s 1 + c r 2 s 2 + c rij s ij i=a,b, j=3 + w q 1 c q 1 + w q 2 c q 2 N M 2 N M + w q ij c q ij i=a,b, j=3 + Nf i + v q L q j + v qi L qij j=1 i=a,b, j=3 (eq. 2-6) Since the fixed cost of constructing a waiting area is a constant, it can be dropped from the equation. The objective then becomes equation 2.7. c r 1 s 1 + c r 2 s 2 + N M i=a,b, j=3 c rij s ij N M + w q 1 c q 1 + w q 2 c q 2 + w q ij c q ij i=a,b, j=3 2 N M + v q L q j + v qi L qij j=1 i=a,b, j=3 (eq. 2-7)

26 Final Model This model will be solving for the optimal goal waiting time for each patient class at each station as well as the number of servers each patient class has available to them at each station. The two objectives described in previous sections will be pulling the resulting waiting times and number of servers in different directions. As the waiting times are decreased, the number of servers will need to increase, resulting in a higher total operating cost. By reducing the number of servers, the operating cost will decrease but the variance of the waiting times will increase. This will lead to a higher expected loss of a patient. The two objective functions must be balanced to find the optimal results. The final model is presented in Figure 2-3.

27 20 Find s 1, s 2, s A3,,, s AM, s B3,,, s BM,, s N3,,, s NM so as to Objective function I: min expected loss = M N i=a,b, 2 σ w i = σ w A + σ w A + + σ w N M Objective function II: = g Aj s Aj j=1 N M min cost = c rij s ij i=a,b, j=1 + + g Nj s Nj j=1 N M N M + w q ij c q ij + v qi L qij i=a,b, j=1 i=a,b, j=1 N M = c r 1 s 1 + c r 2 s 2 + c rij s ij i=a,b, j=3 + w q 1 c q 1 + w q 2 c q 2 N M 2 N M + w q ij c q ij i=a,b, j=3 + v q L q j + v qi L qij j=1 i=a,b, j=3 subject to w qa 1 + w qa 2 + w qa w qam w qa 0 w qb 1 + w qb 2 + w qb w qbm w qb 0 w qn 1 + w qn 2 + w qn w qnm w qn 0 Figure 2-3: Final Model Formulation

28 Solving the Model This model will be solved using a scaling method commonly used when dealing with multiple objectives. Each objective in this model will be multiplied by weights of importance which will sum up to one. Since both objectives are not of the same magnitude, the objective resulting in a higher value would be favored over the lower objective value without applying the scaling method. This would not allow the objectives to be treated based on the desired weights. Hence, instead of simply applying weights to each objective, they will be scaled. To determine the scale applied to each objective, the objectives must be solved one at a time to find the optimal value for each objective. Let the resulting optimal values be U I for objective function I and U II for objective function II. Each objective function will be divided by the respective U k value and added together. This will ensure the deviations from the optimal are minimized. The two objective functions can still be minimized when adding the two modified objectives together. If one of the optimal values of an objective function found in step one is zero, the optimal value used in solving the objectives together will be set equal to a very small positive number. Weights, Y I for objective I and Y II for objective II, can still be applied to each objective function to specify the importance of each objective to the overall objective function. These weights are determined by the user which reflect the relative importance of these objectives to the user. Each objective function will be multiplied by the appropriate weight, with the sum of the weights adding up to one. This can be seen in Figure 2-4. This model will be solved using the LINGO optimization package.

29 22 Find s 1, s 2, s A3,,, s AM, s B3,,, s BM,, s N3,,, s NM so as to min Y I M j=1 g Aj s Aj + + j=1 g Nj s Nj M U I + Y II N M N M i=a,b, j=1 c rij s ij + i=a,b, j=1 w q ij c q ij + i=a,b, v qi j=1 L qij N M U II subject to w qa 1 + w qa 2 + w qa w qam w qa 0 w qb 1 + w qb 2 + w qb w qbm w qb 0 w qn 1 + w qn 2 + w qn w qnm w qn 0 where Y I + Y II = 1.0 U I = optimum value when objective I is solved alone U II = optimum value when objective II is solved alone Figure 2-4: Final Model Formulation to be Solved

30 23 Chapter 3 Numerical Example 3.1 Scenario Description For the implementation of the presented model, an example problem is created similar to a real life health care provider, in which there are four possible patient classes and five total stations. The first two stations for checking in and seeing a nurse will be the same for every patient class and have general resources. After meeting with a nurse, patients meet with specialized physicians and have certain tests completed depending on the health condition they are at the health care center for. This is considered to be done in three stations, for a total of five stations each patient class must visit before they leave the facility with a diagnosis and treatment plan. Figure 3-1 shows an updated flow of the system.

31 24 Figure 3-1: Updated Health Care Facility Flow Diagram It is assumed that the patient meets with a specialized physician depending on the class at station 3. Next, at station 4, patients will have testing completed using dedicated equipment. Finally, patients meet with another physician to discuss the results, diagnosis, and treatment plan at station 5 before leaving the facility.

32 Inputs for the Model The inter-arrival times for each patient class to each station follow an exponential distribution, so the mean value must be determined. A uniform distribution is used for the service times of each patient class at each station, so a mean value must first be decided upon and then minimum and maximum values around that mean. The deviations from the mean of the uniform distributions used will be 10% of the mean value. The service times at station 4 are slightly higher values since this station includes testing of some sort. The service times at stations 3 and 5 are similar since they both involve meeting with a physician. These two stations, while similar, will still have dedicated physicians. Later, different values will be used to determine the sensitivity of the model. Table 3-1 presents the mean input values used for this created example problem, all in minutes.

33 26 Table 3-1: Model Inputs for Mean Inter-arrival and Service Times Station 1 Station 2 Station 3 Station 4 Station 5 All Classes All Classes Class A Class B Class C Class D Class A Class B Class C Class D Class A Class B Class C Class D Measure Value Arrival 7 Service 5 Arrival 8 Service 6 Arrival 10 Service 9 Arrival 10 Service 8 Arrival 10 Service 9 Arrival 10 Service 7 Arrival 25 Service 20 Arrival 30 Service 25 Arrival 22 Service 20 Arrival 18 Service 15 Arrival 9 Service 8 Arrival 18 Service 15 Arrival 18 Service 10 Arrival 15 Service 12 Input values for costing are also required for this example problem. With regards to the resources, it is assumed that the check-in workers are the least expensive to provide since these people will require less training to perform their job. The nurses at station 2 will cost the health care provider more money to supply since they have a higher education. Likewise, the physicians located at stations 3 and 5 will cost the provider

34 27 even more since they have a higher education level than the nurses. It is assumed that the equipment located at station 4 is expensive to manage and operate, so the resources at this station have the highest costs to the provider. The input values for this created example problem can be seen in Table 3-2. Table 3-3 includes the input values for costs associated with having a patient in a queue, while the unit space cost per patient waiting in a queue can be found in Table 3-4. Table 3-2: Input Values for Unit Resource Costs Cost Value c r 1 10 c r 2 25 c ra3 50 c rb3 45 c rc3 55 c rd3 50 c ra4 100 c rb4 120 c rc4 135 c rd4 130 c ra5 55 c rb5 60 c rc5 65 c rd5 60

35 28 Table 3-3: Input Values for Waiting Costs per Patient per Unit Time in the Queue Cost Value c q 1 12 c q 2 10 c qa3 8 c qb3 6 c qc3 10 c qd3 9 c qa4 7 c qb4 6 c qc4 8 c qd4 8 c qa5 6 c qb5 4 c qc5 5 c qd5 6 Table 3-4: Input Values for Unit Space Cost per Patient in the Queue Cost Value v q. 5 v qa 5 v qb 4 v qc 6 v qd 5

36 Relationship between Number of Servers and Waiting Time Variance for Robust Design To be able to apply the robust design technique to this scenario, the relationship between the number of servers at a particular station and the variance of the waiting time for that station must be known. This can come from analytical data or determined through simulation. For this example, Rockwell s Arena simulation package (version 12.0) was used to determine the variance of the waiting time before a station for each patient class for different number of servers. One simulation was created for each unique station and patient class combination. With the first two stations being general and the last three being specific to the patient class, there were fourteen total simulations created. A scenario was run with the number of servers ranging from one to five, as the number of necessary servers is not expected to exceed five at any one station. Within a simulation model, the mean inter-arrival time for that station and patient class will be incorporated where the entity (i.e.: patient) is created (i.e.: arrives to the station). Once an entity is created, it must seize the necessary resource at that station to receive service, following the given distribution for service time, before it can leave that station. Each simulation will be run for 100 replications. This ensures a large sample size necessary for the variance calculation. The same distributions will be used each time a replication is run, however different random numbers will be generated following the given distribution. Each replication is run for twelve hours with one hour of warm up time where statistics are not collected. This ensures the system is in steady state. This is

37 30 one limitation of this work, as a health care facility may never experience steady state. Future work could be performed using a Non-Homogeneous Poisson Process (NHPP). The average waiting time before getting served for each replication was used to determine the variance of the waiting time for each station and patient class. The resulting variance was calculated for each of the scenarios with the number of servers equal from one to five. These values were used as the g ij s ij function in the first objective function relating the number of servers at a station to the variance of the waiting time. Since the number of servers is a discrete number between one and five, the ideal number of servers dictates the waiting time variance value used at that station. Table 3-5 presents the variance values for each of the patient classification and station combinations. For example, if the number of servers at the second station (a general station) is determined to be 2, then the variance value used in the expected loss equation will equal

38 31 Table 3-5: Variance Values for Each Patient Classification and Station Combination Servers σ 2 w 1 σ 2 w 2 σ 2 wa3 σ 2 wb3 σ 2 wc3 σ 2 wd3 σ 2 wa4 σ 2 wb4 σ 2 wc4 σ 2 wd4 σ 2 wa5 σ 2 wb5 σ 2 wc5 σ 2 wd E E E E E E E E E E E E-07

39 Solution Step One Once the model was coded into the Lingo optimization software package, each objective had to be solved separately to determine the scaling factors. When solving for the minimum expected cost of losing a patient to another provider, the resulting loss value was This means the value of U I will be equal to when solving both objective functions together. The number of servers at each area as well as the target waiting times for each station can be seen in Table 3-6. Table 3-6: Results When Solving for the Expected Loss Objective Function Only Optimal Number Target Waiting of Servers Time (min) Station 1 All Classes Station 2 All Classes Station 3 Station 4 Station 5 Class A Class B Class C Class D Class A Class B Class C Class D Class A Class B Class C Class D The operating cost resulting from this number of servers at each station equals $4, The total amount of waiting time for each patient class can be seen in Table

40 Each patient class total waiting time includes the times at stations 1 and 2 since every patient must go through these two stations. Table 3-7: Target Waiting Time Total by Patient Class for the Expected Loss Objective Function Only Patient Class Target Waiting Time Total (min) Class A Class B Class C Class D Next, the objective function minimizing the total operating cost was solved alone. This resulted in a minimum operating cost of $3, Therefore, the value of U II will be set equal to 3, when both objective functions are solved together. The cost is incurred when the numbers of servers at each station are assigned as seen in Table 3-8. This table also presents the target waiting times in each queue when taking into consideration only the operating cost objective function.

41 34 Table 3-8: Results When Solving for the Operating Cost Objective Function Only Optimal Number of Servers Target Waiting Time (min) Station 1 All Classes Station 2 All Classes Class A Station 3 Class B Class C Class D Class A Station 4 Class B Class C Class D Class A Station 5 Class B Class C Class D The expected loss value (total variance in the waiting times) was when solving for this objective function only. The total amount of waiting time by patient class can be seen in Table 3-9. Table 3-9: Target Waiting Time Total by Patient Class for the Operating Cost Objective Function Only Patient Class Target Waiting Time Total (min) Class A Class B Class C Class D Each of the total average waiting time values in Table 3-9 does not exceed the limit of 30 minutes. This makes it so that, on average, no customer waits in the facility for more than 30 minutes to be treated. This does not include treatment times, only waiting times.

42 35 The resultant values for the number of servers at each station differ when solving for each objective function separately. This is because the two objective functions are pulling the number of servers in different directions. The expected loss objective function is minimizing the sum of the variances associated with the number of servers at each station. The variance value at each station is smallest when the number of servers is equal to five. One the other hand, the operating cost objective function is optimal when the number of servers at each station is less than five. This is because each server has a unit cost associated with it, so limiting the number of servers will minimize the operating cost. The resource cost alone would result in the number of servers to be equal to one at every station; however, there are also costs placed on the amount of time a patient waits in a queue as well as the space needed for patients to wait in a queue. This causes the number of resources to increase, so the waiting time and queue lengths are decreased. There are also limits on the total amount of time a patient class can spend waiting while at the health care facility, which will cause the number of servers to increase. Therefore, the number of servers as a result of only minimizing the operating cost is a balance between resource costs and queuing costs. 3.4 Solution Step Two Once the optimal values for each objective function where found individually, both objective functions could be solved together. This was done by scaling the two objective functions and adding them together. The weights used when solving the model the first time were such that each objective function was treated equally; so Y I = 0.5 and

43 36 Y II = 0.5. Table 3-10 presents the resulting values for number of servers that should be available and the target waiting times for each patient class at each station. Table 3-10: Results When Solving Both Objective Functions Weighted Equally Optimal Number of Servers Target Waiting Time (min) Station 1 All Classes Station 2 All Classes Class A Station 3 Class B Class C Class D Class A Station 4 Class B Class C Class D Class A Station 5 Class B Class C Class D The values shown in Table 3-10 result in an operating cost of $4, and a total variance value of Both of these values are greater than those when the objectives are solved individually. However, these values are less than those calculated when the other objective function was optimized. When the total variance in waiting time was minimized, the operating cost was found to be $4, This is % greater than the operating cost found when both objective functions were considered. Likewise, the total variance in waiting time was found to be when optimizing the operating cost. This is % greater than the total variance value when both objectives were included. Calculations for percent increases can be found in the Appendix.

44 37 The total waiting times found during this step are much closer to those found when minimizing the expected loss objective function only. This is because the values for the optimal number of servers at each station are much closer to the number of servers found when optimizing the total variance objective function only. The target waiting time totals can be found in Table Table 3-11: Target Waiting Time Total by Patient Class for Both Objective Functions Patient Class Target Waiting Time Total (min) Class A Class B Class C Class D Sensitivity Analysis A number of weight combinations were also used to determine the sensitivity of the model. By applying weights that are not equal to one another, the user is able to place higher importance on one of the objective functions over the other. The weights used will still sum up to one. Four total combinations of weight values were used to conduct the sensitivity analysis, with the importance of one objective over the other varying in each case. Table 3-12 lists out the 4 combinations of weight values, where Y I is the weight applied to objective function I and Y II is the weight applied to objective function II.