Incorporating uncertainty in optimal decision making: Integrating mixed integer programming and simulation to solve combinatorial problems

Computers & Industrial Engineering 56 (2009) 106–112

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Incorporating uncertainty in optimal decision making: Integrating mixed integer programming and simulation to solve combinatorial problems Yavuz Acar a,1, Sukran N. Kadipasaoglu b,*, Jamison M. Day b,2 a b

Department of Management, Bogazici University, Istanbul, Turkey C.T. Bauer College of Business, Department of DISC, 334 Melcher Hall, 4800 Calhoun Road, University of Houston, Houston, TX 77204-6021, USA

a r t i c l e

i n f o

Article history: Received 16 June 2007 Received in revised form 10 April 2008 Accepted 10 April 2008 Available online 27 May 2008 Keywords: Simulation Combinatorial optimization Hybrid solution methodology Facility location

a b s t r a c t We introduce a novel methodology that integrates optimization and simulation techniques to obtain estimated global optimal solutions to combinatorial problems with uncertainty such as those of facility location, facility layout, and scheduling. We develop a generalized mixed integer programming (MIP) formulation that allows iterative interaction with a simulation model by taking into account the impact of uncertainty on the objective function value of previous solutions. Our approach is generalized, efficient, incorporates the impact of uncertainty of system parameters on performance and can easily be incorporated into a variety of applications. For illustration, we apply this new solution methodology to the NP-hard multi-period multi-product facility location problem (MPP-FLP). Our results show that, for this problem, our iterative procedure yields up to 9.4% improvement in facility location-related costs over deterministic optimization and that these cost savings increase as the variability in demand and supply uncertainty are increased. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction The negative effect of various sources of uncertainty on operational performance (e.g., cost, profitability, quality, and customer service) is well documented in the literature (Acar, 2007; Hahn, Duplaga, & Hartley, 2000; Lee & Billington, 1995; Lee, Padmanabhan, & Whang, 1997; Qi, Bard, & Yu, 2004; Vidal & Goetschackx, 2000). Yet, inclusion of uncertainties (e.g., demand, lead time, production yield, and raw materials cost) often makes pure mathematical modeling intractable. Therefore, in practice, deterministic mathematical models are widely employed and are followed by multiple sensitivity analyses to evaluate the impact of various types of uncertainty on operational performance. Another widely used business decision tool is simulation where uncertainty in various system parameters can be incorporated; however, it is useful for evaluating the operational performance of only a particular scenario. Simulating all possible scenarios often requires too much computation time for a business problem of realistic size, even with today’s computer power. For example, in a facility location problem (FLP), the total number of possible configurations to be simulated is 2n  1, where n is the number of

* Corresponding author. Tel.: +1 713 743 4707; fax: +1 713 743 4940. E-mail addresses: [email protected] (Y. Acar), [email protected] (S.N. Kadipasaoglu), [email protected] (J.M. Day). 1 Tel.: +90 0532 351 47 58. 2 Tel.: +1 713 743 0402. 0360-8352/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2008.04.003

facilities. Considering the advantages and disadvantages of these two techniques, some researchers have adopted hybrid approaches incorporating both mathematical optimization and simulation techniques (Acar, 2007; Butler, Karwan, & Sweigart, 1992; Dijk & Sluis, 2006; Lee et al., 2006; Lin et al., 2000; Moore, Warmke, & Gorban, 1991; Muriel, Anand, & Yongmei, 2006; Qi & Bard, 2006; Smith et al., 2007; Tang & Liu, 2007). A few researchers have used these two techniques iteratively, by returning values from simulation for re-optimization to solve specific problems, exchanging problem-specific parameters between the two techniques (Byrne & Hossain, 2005; Carlson, Hershey, & Kropp, 1979; De Angelis, Felici, & Impelluso, 2003; Karabakal, Gunal, & Ritchie, 2000; Ko, Ko, & Kim, 2006; Lee, Kim, & Moon, 2002; Leung & Cheung, 2000; Leung, Maheshwari, & Miller, 1993; Nolan & Sovereign, 1972). This paper introduces a novel solution methodology that integrates optimization and simulation and is easily adaptable to various combinatorial problems such as those of facility location, facility layout, and scheduling. Our solution methodology differs from those set forth in previous research in that we present a general solution methodology that can obtain a global optimum for business problems involving various sources of uncertainty. Our solution methodology incorporates a generalized MIP to obtain estimated optimal solutions to the business problem of interest. The paper is organized as follows: In Section 2, we provide a literature review of hybrid methods that iteratively integrate optimization and simulation methodologies. In Section 3, we describe our new hybrid approach. In Section 4, we present the illustration of

Y. Acar et al. / Computers & Industrial Engineering 56 (2009) 106–112

our solution methodology on the multi-period multi-product facility location problem (MPP-FLP). Finally, in Section 5, we present our conclusions. 2. Literature review Several researchers have developed iterative solution approaches for various types of problems that integrate optimization and simulation methodologies. To date, these solution methods have focused on solving very specific problems when incorporating uncertainty factors. The earliest solution approach involving iterative use of simulation and optimization was developed by Nolan and Sovereign (1972). Their recursive approach involves an allocation of resources by a linear programming (LP) model at an aggregate level and a revision of productivity estimates by simulation of the schedules generated by optimization. They applied their solution approach to the strategic mobility system problem of the US military transportation system. Leung et al. (1993) developed an iterative approach for flexible manufacturing systems planning. The inputs of their integer programming optimization model include system utilization, makespan, and vehicle utilization. These parameters are updated by simulating the output of the optimization before resolving again in an iterative fashion. Their procedure terminates when the simulation outcomes comply with the results from the optimization. Leung and Cheung (2000) developed a hybrid iterative solution methodology for the DHL Hong Kong distribution network. Their simulation model is used to evaluate the daily operational performance of the network configuration suggested by their MIP optimization. If service coverage or service reliability is unacceptable, or if the utilization and cost estimates differ significantly from those used in the optimization model, the input parameters are updated and the optimization model is solved again. Karabakal et al. (2000) developed a hybrid solution approach that iterates between a simulation and MIP model with an objective to find the optimum configuration for the Volkswagen of America’s vehicle distribution system. Two major input parameters to the MIP are demand and truck load factors, both of which depend on the location policy. Therefore, simulation is used to generate demand and truck load factor estimates as a result of implementing a particular location policy obtained by the MIP model. They used these estimates to update the input parameters (demand and truck load factor) of the MIP in an iterative manner until both the MIP and simulation agreed on a particular location policy. Lee et al. (2002) developed a hybrid solution approach that combines analytic LP and simulation models to solve multi-product and multi-period production–distribution problems. Their LP model minimizes the overall cost of production, distribution, inventory holding, and shortage costs and determines production and shipment quantities between production facilities and retailers. Their simulation model is used to adjust the production time and distribution lead-time in the LP model. The iteration stops when the difference between the preceding and current simulation runs in the production and distribution lead-times is deemed small enough. De Angelis et al. (2003) developed a solution methodology that interactively uses simulation and optimization to determine the estimated optimal configuration of servers in a health care facility. In their iterative solution approach, simulation is used to generate a ‘‘training set” from which a relationship between the input parameters (number of servers at each service location) and resulting service performance (average time spent in system) is estimated and then used as an objective function in the optimization. They adopted a radial basis function, a particular type of neural network, to estimate this relationship. The configu-

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ration obtained by solving the optimization model is simulated. If the difference in the solution value obtained by simulation and optimization models is small enough, the procedure terminates. Otherwise, the configuration is added to the training set and a new estimation of the objective function is calculated to be used in the next optimization model. Byrne and Hossain (2005) developed an extended linear programming model for the hybrid modeling approach first proposed by Byrne and Bakir (1999). Their hybrid solution approach iteratively applies simulation and LP to solve a multi-period multiproduct production planning problem. They obtained job workload and resource utilization through simulation and used LP to obtain the estimated optimal production plan that minimizes total costs. Byrne and Bakir (1999) demonstrated that their solution approach outperforms an LP approach alone. Ko et al. (2006) developed a hybrid optimization-simulation approach to design a distribution network for third party logistics (3PL) providers. They used a genetic algorithm to solve the optimization model that determines the distribution network structure. Subsequently, the simulation model is applied to capture the uncertainty in client demand, order-picking time, and travel time for the capacity plans of the warehouses. The simulation is used to estimate the average service time at each warehouse. Then, the service time is used to define appropriate throughput capacity constraints to be incorporated into subsequent optimization runs. If the simulation outputs satisfy the required performances, the procedure is terminated. Each of the iterative solution procedures described above is specific to a particular problem and exchange problem-specific parameters between simulation and optimization models. We present and examine a general solution methodology that obtains an estimated global optimum for combinatorial optimization problems with components of uncertainty.

3. Hybrid solution methodology We propose a novel solution methodology that can be employed to solve various combinatorial problems with components of uncertainty. A general MIP formulation is developed to obtain the estimated optimal solution to the deterministic problem. Then, a simulation of the deterministic solution determines the impact of uncertainty on the objective. The difference between the deterministic and simulated objective function is incorporated back into the MIP formulation. As a new solution is searched and alternatives are evaluated by the MIP model, solutions that were previously obtained and simulated are evaluated by taking into account the impact of uncertainty on the objective. The process iterates until a previously simulated solution with uncertainty impact is found to be optimal in the current MIP formulation. The steps of the solution procedure are summarized in Fig. 1. For simplicity, we assume that the objective function is a cost minimization. 3.1. Step 1: Run MIP optimization The following generalized MIP formulation incorporates the necessary additional components for integrating the resulting uncertainty impact into the optimization procedure. The formulation below assumes a cost minimization, but a similar formulation is easily created for profit maximization or any other objective. We assume that for any potential solution, the deterministic objective will be less than the objective obtained under uncertainty. Although it is uncommon that uncertainty will, in fact, lower cost, this result is possible. Nonetheless, a deterministic objective less than that found under uncertainty can be ensured by using prob-

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upper bound in accordance with the smallest known objective value for a previous candidate solution. Consider the simplistic example depicted in Fig. 2. The Y-axis represents the total cost (Z) and the X-axis represents the variable of interest (a set of Xi’s). Points A through L represent potential scenarios. In Step 1, the MIP model obtains scenario A as optimal for the function depicted in Fig. 2A. When the MIP model in Step 1 obtains a previously simulated P candidate solution (i.e., when nZn > 0), our procedure terminates with this scenario as the estimated global optimal solution under uncertainty. Alternatively, if the MIP model obtains a scenario that has not been simulated before, the procedure proceeds to Step 2.

Run MIP optimization. (Step 1) Has the solution already been simulated?

No

Yes

Run simulation. (Step 2)

Calculate difference between deterministic cost and average simulation cost.

Terminate with optimal solution.

Update formulation with solution uncertainty impact. (Step 3)

3.2. Step 2

Fig. 1. General procedure for optimization under uncertainty.

lem-specific cost parameters that provide a reasonable lower bound. MIP optimization parameters cij, aij, and bi are problem specific parameters binary parameter (1, if binary variable Xi in candidate Tni solution n is 1; 0, otherwise) impact of uncertainty found by simulation for candidate Nn solution n Qmin minimum total expected cost with uncertainty obtained so far from simulations M large number

Decision variable Xi Zn

binary decision variable binary variable for incorporating the cost of uncertainty for candidate solution n (1, if the scenario currently considered by the MIP procedure was suggested in a previous iteration; 0: otherwise)

Min Z ¼

X i

ci X i þ

X

Nn Z n

ð1Þ

n

subject to

X i X i X

aij X i 6 bi

for all j

ð2Þ

ð2T ni X i  T ni  X i Þ 6 Z n  1 for all n

ð3Þ

ð2T ni X i  T ni  X i Þ P MðZ n  1Þ for all n

ð4Þ

i

Z P Q min

ð5Þ

The deterministic MIP model obtains an optimal deterministic scenario and yields a total cost. Initially, there are no previous candidate solutions, so the MIP objective summation over n and constraints (3) and (4) do not impact the first optimization. Similarly, it is important to choose the initial value of Qmin to be large enough that the deterministic optimal is feasible. When there are previously obtained candidate solutions with simulated uncertainty impacts (Nn) the objective incorporates this impact into the objective for that particular solution. Constraints (3) and (4) serve to ensure the uncertainty impact of a candidate solution is incorporated only when that candidate solution is being considered. Constraint (5) is incorporated into the model to set an

When a new candidate solution is obtained by the MIP in Step 1, it is simulated while incorporating the various uncertainties of interest for the problem. This simulation may use several replications to obtain an expected value for the objective value under uncertainty, Zsim. The difference between the deterministic objective, Z, and the expected objective under uncertainty from simulation, Zsim, is considered the uncertainty impact, Nn = Zsim  Z. These values are depicted graphically in Fig. 2B by the points A and A0 as well as the distance between them. The procedure proceeds to Step 3. 3.3. Step 3 After simulating the candidate solution in Step 2, the MIP formulation is updated by updating Qmin as well as including new constant parameters, Nn, Zn, and the set of variables, Tni, for the candidate solution and its uncertainty impact. For the new candidate solution value n, each is set to the current value of Xi, Nn is incorporated, and Qmin is updated if the new Zsim improves upon the previous best known solution under uncertainty. The procedure then proceeds back to Step 1. Continuing with the example in Fig. 2B, the use of Qmin effectively eliminates points in the shaded area from the search space since they represent costs higher than that of the best known solution under uncertainty. In Fig. 2C, the second iteration of the MIP model obtains a new optimal deterministic candidate solution, represented by point B. Since scenario B has not been simulated in previous iterations, the procedure proceeds to Step 2. In Step 2, scenario B is simulated and the uncertainty impact for scenario B is obtained (the distance between B and B0 in Fig. 2C). Since B0 is greater than the best known solution under uncertainty (A0 ), Qmin is not updated. The procedure proceeds to Step 3 where the cost of uncertainty for scenario B is incorporated into the objective function of the MIP model. After returning to Step 1, the MIP model obtains the new optimal candidate solution represented by point C. Since candidate solution C has not been simulated in previous iterations, the uncertainty impact for scenario C (the distance between C and C0 in Fig. 2D) is obtained through simulation. The parameters and search space are updated in the MIP model in Step 3. Next, the procedure proceeds to the Step 1 of the forth iteration; the MIP model is resolved and point C, a previously simulated solution, is found to be optimal. Since scenario C was simulated in earlier iterations, the procedure terminates with C as the estimated global optimal solution under uncertainty.

4. Illustration of the generalized solution methodology: MPPFLP In order to test the usefulness of our procedure, we employed the procedure on the multi-product, multi-period facility location problem (MPP-FLP). We assume that open and close decisions,

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A

900

B

900

800

800

700

700

600 K

600

L

500

500 J

I

400 300

400 H

G

300

F E

200

A`

200

D

D C

100

A

A

0

B

0 1

C

C

100

B

5

1

9 13 17 21 25 29 33 37 41 45 49 53

D

900

800

700

700

600

600

500

500

400

400

B`

9 13 17 21 25 29 33 37 41 45 49 53

900

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5

300

200

C`

200

D

D

C

100

100

C

B 0

0 1

5

1

9 13 17 21 25 29 33 37 41 45 49 53

5

9 13 17 21 25 29 33 37 41 45 49 53

Fig. 2. A simple illustrative example of the hybrid MIP-simulation solution method. (In graphs A–D, the X-axes represent an independent variable and the Y-axes represent a dependent variable.)

once made, are fixed within the tactical horizon (12 periods – months). Our consideration of multiple periods allows capturing operational costs when demand is seasonal by incorporating aggregate planning strategies such as inventory carrying, back ordering, and overtime production. The MIP formulation below obtains the optimal solution to the deterministic MPP-FLP. We used the ILOG CPLEX optimization engine to solve the MIP formulation in each iteration of our procedure. Sets i j k t n

plant demand location product period (1, . . . , N) iteration

Parameters Djkt Fi

demand for product k in location j in period t fixed cost of plant i

Ci Oi Kik Lik Rijk Mik H W V

regular capacity of plant i overtime capacity of plant i regular production cost per hour for product k in plant i overtime production cost per hour for product k in plant i shipment cost of product k between plant i and demand location j production time required to produce a unit of product k in plant i inventory carrying cost per period (set to 1% of product value) unit cost of unmet order (set to a large number) unit cost of backordering by one period (set to 1.5 times production cost)

The following parameters are obtained in the second step of our procedure. Tni Nn Qmin

binary parameter (1, if facility i in iteration n is open; 0, otherwise) cost of uncertainty for the scenario in iteration n minimum total cost obtained so far from the simulations in previous iterations

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Variables Xi Pikt Gikt Sijkt Bijkt Iikt Ujkt Zn

binary variable (1, if facility i is permanently open; 0, otherwise) production quantity of product k in plant i in period t over production quantity of product k in plant i in period t shipment quantity of product k from plant i to demand location j in period t backorder quantity of product k produced in plant i for demand location j in period t Ending inventory of product k in plant i in period t Unmet demand of product k in plant i in period t binary variable to force the cost of uncertainty (1, if the scenario currently considered by the MIP model was suggested in a previous iteration; 0, otherwise)

Objective function : XXX X FiXi þ M ik K ik Pikt Min Z ¼ i

þ

i

XXX

t

k

M ik Lik Gikt

t

i

k XX XX þ ðSijkt þ Bijkt ÞRijk i

þ

j

k XX X j

þ

ð6Þ

t

M ik K ik HIikt

t

i

þ

k

XXX

k

WU jkt

t

XXXX i

j

k

VBijkt þ

t

Subject to : X X Sijkt þ Bijkt þ U jkt Djkt ¼ i

i

X

Pikt þ Gikt þ Iikðt1Þ ¼ S þ j ijkt X M ik Pikt 6 C i X i for all i; t k X k X

M ik Gikt 6 Oi X i

X

Nn Z n

n

for all j; k; t X j

þ Bijkðt1Þ þ Iikt

for all i; t

ð7Þ for all i; k; t ð8Þ ð9Þ

4.1. Experimental design We designed an experiment with three levels of demand uncertainty, setting the standard deviation of monthly demand to 10%, 20%, and 30% of mean demand. Similarly, we used three levels of supply uncertainty, setting the standard deviation of monthly capacity to 10%, 20%, and 30% of mean capacity. This resulted in nine factor level combinations. The experimental factors are summarized in Table 1 below. We randomly generated facility location problems in order to compare the solution quality of our approach to that of the deterministic MIP alone. Twenty random problems were generated for each of the nine factor level combinations resulting in 180 random problems in total. The details of random problem generation are given in Appendix A.

ð10Þ

4.2. Results

ð2T ni X i  X i  T ni Þ 6 Z n  1 for all n

ð11Þ

ð2T ni X i  X i  T ni Þ P MðZ n  1Þ for all n

ð12Þ

Each of the 180 problems was solved to estimated global optimality under uncertainty in accordance with our solution procedure. The comparable cost of the deterministic MIP solution is obtained by simply simulating the scenario suggested by the MIP solution under demand and supply uncertainty. Therefore, 1800 simulation runs (10 replications of 180 problems) were made to obtain these results. Our solution comparison is made in terms of average percent cost improvement (APCI) where: APCI = (Cost of MIP solution  Cost of Iterative Procedure)  100/Cost of MIP solution. In both the MIP and simulation cases, cost is calculated by summing up all resulting cost components as shown in the MPPFLP objective function. We expect our solution procedure to outperform the deterministic MIP model particularly when there is a high level of demand and/or supply uncertainty.

i

X

We assume the business environment requires the shipment and production schedules to be determined at each location at the beginning of each month. The simulation model executes the current month’s production and shipment schedule. If, due to the stochastic nature of demand and supply, a less than planned quantity is shipped in one period, the remaining quantity cannot be scheduled for shipment until the beginning of next month. At the end of each period, the simulator records inventory levels, backorders, actual quantities shipped, and calculates inventory cost, production cost, and shipment cost. We used 10 replications as suggested by the precision test set forth by Law (2007) and our relative error never exceeded 10% (at a 95% confidence level). Unique random number seeds are used in each simulation run and our simulation model was run in Automod 10.1. Our MIP formulation parameters, Nn, Tni, and Qmin are updated at the end of each iteration. The solution, obtained in each Step 1 and simulated in each Step 2, is described by a certain set of open and close facilities and allocation of resources to customer demand. In each Step 3, the MIP is updated and the procedure iterates back to Step 1 until a previously simulated solution is obtained as the optimal solution from the MIP.

i

Z 6 Q min

ð13Þ

Objective function (6) is the sum of fixed facility, regular and overtime production, shipment, backorder, inventory holding, unmet demand costs, and the cost of uncertainty as previously explained. Constraint (7) is the demand constraint. Constraint (8) maintains the production flow from one period to next. Constraints (9) and (10) are supply related constraints making sure total production including over production does not exceed capacity. Constraints (11) through (13) establish a link between simulation and deterministic models. We incorporated both demand and supply uncertainties into the simulation model. Demand uncertainty is expected to cause an increase in total cost, mainly due to the increase in inventory carrying and/or backordering costs. When actual demand turns out to be greater than the forecast, backorders occur. On the other hand, over-forecasting causes the company to produce more than required, resulting in unnecessary inventory. Similarly, supply uncertainty causes an increase in backordering cost when the actual capacity turns out to be less than the expected capacity.

Table 1 Experimental factors Experimental factors

Factor levels

Demand uncertainty Standard deviation of demand

10%, 20%, 30% of mean demand

Supply (plant capacity) uncertainty Standard deviation of available capacity

10%, 20%, 30% of mean capacity

Y. Acar et al. / Computers & Industrial Engineering 56 (2009) 106–112 Table 2 Average percent cost improvement obtained by the iterative procedure over the deterministic approach Supply uncertainty

Demand uncertainty 10%

20%

30%

10% 20% 30%

1.15 2.94 4.27

3.66 6.05 7.70

5.76 7.54 9.38

Table 3 ANOVA results on percent cost improvement Source

DF

Type I SS

Mean square

F Value

Pr > F

Demand Supply Demand  supply

2 2 4

0.069941 0.038882 0.000546

0.034971 0.019441 0.000137

11.29 6.28 0.04

<.0001 0.0023 0.9963

The results show that use of our solution procedure in the selection of facilities results in better decisions than the deterministic approach. The average percent cost improvement is presented in Table 2 below. Results show that our solution procedure provides cost savings up to an average 9.38% over those that are obtained by simply using the MIP model. As the uncertainty (variability) in demand and supply increase, the cost savings from our procedure also increase. The higher the uncertainty, the more beneficial it is to incorporate its impact into decision making. Table 3 shows the results of the ANOVA procedure comparing our approach with the deterministic approach at various levels of demand and supply uncertainty. The cost improvement achieved with our iterative procedure is statistically significant at the 1% level for the main effects of both demand and supply uncertainty. The interaction effect, on the other hand, is not statistically significant. In other words the rate of improvement at each level of demand uncertainty remains the same across different levels of supply uncertainty and vice versa. 5. Conclusion We developed a generalized solution procedure that obtains estimated global optimal solutions to combinatorial problems with uncertainty and illustrated its use by applying it to the MPP-FLP. Our experimental results illustrate how solutions that optimize while also incorporating sources of uncertainty outperform the deterministic optimal and the amount of improvement increases as uncertainty increases. These results confirm the importance of incorporating uncertainty in decision making; especially in strategic decision areas such as facility location, where a 9% improvement in costs can have a significant impact on long term profitability. It should also be noted that the extent of cost improvement depends on the relative significance of the different costs that are involved in the decision being made. The higher the costs caused by uncertainties, the more important it becomes to incorporate those sources of uncertainty when solving the problem. This insight is important for many businesses, such as those in the fashion and high-tech industries, where the cost of mismatch between demand and supply is very high. Several additional sources of uncertainty such as transportation lead-times and exchange rate fluctuations can easily be incorporated into our solution procedure and we expect our approach to yield even better improvements as more sources of uncertainty are taken into consideration. The primary findings reported in this paper are promising, however, there are some open research issues that remain to be examined. This paper only examines the use of our estimated global optimal procedure on facility location problems with aggregate

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production planning strategies. Since our approach is generalized and can be applied to solve different types of combinatorial problems, it would be interesting to consider different problems with various components of uncertainty and differing cost structures in future research. Appendix A. Appendix Random problem generation Random problems included 10 demand points, 10 potential facility locations, and 10 products. We used uniform and normal distribution to obtain a random problem. The parameters for random problems are: demand, capacity, production rate, production, shipment, and fixed facility costs. The choice of parameter values is explained below. - The mean demand for products is generated randomly between 0 and 100 U/month based on a uniform distribution. Demand is assumed to have monthly seasonality, which is derived by a uniform distribution and ranged between 0 and 20 U/month. Monthly demand is assumed to be normally distributed. The standard deviation of monthly demand is set at three levels: 10%, 20%, and 30% of the mean. - Available capacity is assumed to be normally distributed. The mean capacity of the plants ranged between 18,000 and 32,000 machine h/month based on a uniform distribution. Overtime capacity is set to 10% of the mean capacity. The standard deviation of capacity is set at three levels: 10%, 20%, and 30% of the mean. - The machine hour requirement for each product ranged between 7 and 13 h. - The fixed cost of plants ranged between 7 and 13 million dollars. - Production cost is calculated as machine hour requirement multiplied by machine hour cost, which is generated randomly between $20 and $40 per hour based on a uniform distribution. - Shipment cost between plants is generated randomly between $10 and $30 based on a uniform distribution. We set the annual inventory carrying cost to 12% of the production cost and backordering cost to the production cost. References Acar, Y. (2007). Supply chain modeling and forecasting method selection. Ph.D. thesis. Houston, TX: University of Houston. Butler, T. W., Karwan, K. R., & Sweigart, J. R. (1992). Multi-level strategic evaluation of hospital plans and decisions. The Journal of the Operational Research Society, 43(7), 665–675. Byrne, M. D., & Bakir, M. A. (1999). Production planning using a hybrid simulation– analytical approach. International Journal of Production Economics, 59(1-3), 305–311. Byrne, M. D., & Hossain, M. M. (2005). Production planning: An improved hybrid approach. International Journal of Production Economics(93/94), 225–229. Carlson, R. C., Hershey, J. C., & Kropp, D. H. (1979). Use of optimization and simulation models to analyse outpatients’ health care settings. Decision Sciences, 10(3), 412–433. De Angelis, V., Felici, G., & Impelluso, P. (2003). Integrating simulation and optimisation in health care centre management. European Journal of Operational Research, 150(1), 101–115. Dijk, N. M., & Sluis, V. D. E. (2006). Check-in computation and optimization by simulation and IP in combination. European Journal of Operational Research, 171(3), 1152–1168. Hahn, C. K., Duplaga, E. A., & Hartley, J. L. (2000). Supply-chain synchronization: Lessons from Hyundai motor company. Interfaces, 30(4), 32–45. Karabakal, N., Gunal, A., & Ritchie, W. (2000). Supply-chain analysis at Volkswagen of America. Interfaces, 30(4), 46–55. Ko, H. J., Ko, C. S., & Kim, T. (2006). A hybrid optimization/simulation approach for a distribution network design of 3PLs. Computers & Industrial Engineering, 50(4), 440–449. Law, A. M. (2007). Simulation modeling and analysis. New York: McGraw Hill. Lee, E. K., Maheshwary, S., Mason, J., & Glisson, W. (2006). Large-scale dispensing for emergency response to bioterrorism and infectious-disease outbreak. Interfaces, 36(6), 591–607.

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