USING STRATIFICATION DATA ENVELOPMENT ANALYSIS FOR THE MULTI- OBJECTIVE FACILITY LOCATION-ALLOCATION PROBLEMS

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1 USING STRATIFICATION DATA ENVELOPMENT ANALYSIS FOR THE MULTI- OBJECTIVE FACILITY LOCATION-ALLOCATION PROBLEMS Jae-Dong Hong, Industrial Engineering, South Carolina State University, Orangeburg, SC 29117, , Ki-young Jeong, Engineering Management, University of Houston Clear Lake, Houston, TX 77058, , ABSTRACT This paper considers the multi-objective facility location-allocation (MOFLA) problems. Facility location-allocation (FLA) decisions provide a basic foundation for designing efficient supply chain network in many practical applications. In this paper, we propose an innovative procedure for finding efficient FLA schemes by utilizing multi-objective programming (MOP) model with MINIMAX objective approach and stratification data envelopment analysis (SDEA) methodology. A case study is presented to illustrate the effectiveness and efficiency of the proposed combining methods. Keywords: Multi-Objective Facility Location-Allocation (MOFLA), Stratification Data Envelopment Analysis (SDEA), Efficient FLA Schemes INTRODUCTION Facility location-allocation (FLA) decisions inherently consist of two kinds of decision plans. One is a strategic decision plan on the facility location, while the other one is an operational decision plan on the facility allocation. Daskin (2013) emphasizes the importance of facility location problems by asserting in his recent book that in short, the success or failure of both private and public sector facilities depends in part on the locations chosen for those facilities. Facility allocation decisions are also as important as the facility location decisions to run supply chain network efficiently. Indeed, many authors have studied FLA problems by modeling them in various forms to answer to a lot of different questions since Cooper (1963) initially set an FLA problem as a mathematical programming model and studied it. Traditional FLA models consider an objective of minimizing the total logistics cost, such as the fixed cost of opening/using the facility plus the transportation or shipping cost. As so many references in Current et al. (1990), Farahani et al. (2010), and Fang and Li (2015) indicate, many researchers have worked on multiobjective/criteria facility location-allocation (MOFLA) problems whose objectives sometimes conflict with each other in nature (Lee et al., 1981). This paper also models the FLA problem as a multi-objective programming (MOP) model and finds efficient FLA decisions utilizing data envelopment analysis (DEA) technology. This paper is motivated by Klimberg and Ratick (2006) and Fang and Li (2015). Their models require the large data required for the inputs and outputs and consequently the huge number of the constraints for their combined location and simultaneous DEA model (SDEA) as the numbers of facilities and their potential sites increase. In addition to those huge data and constraints required by their models, it would be not only difficult to quantify all inputs and

2 outputs for a facility to be located to cover the allocated sites, but also very subjective for a decision maker to decide these magnitudes of such inputs and outputs. In this paper, we propose a practical and realistic approach to finding efficient FLA decisions by generating the inputs and outputs directly through formulating and solving the FLA problem as the MOP model. Combined FLA Model with MOP The following nomenclature is used: BACKGROUND Sets: M: index set of potential facility sites (j =1, 2,, M and m = 1, 2 M). Parameters: c jm : cost of shipping one unit of demand per mile from facility j and demand point m f j : fixed cost for constructing and operating facility j. d jm : distance between facility j and site m C max : maximum number of facilities can be built CAP j max : capacity of facility j h m : demand of site m b j : minimum number of sites that facility j can cover B j : maximum number of sites that facility j can cover Decision Variables: C j : binary variable deciding whether a facility is located at site j y jm : binary variable deciding whether site m is covered by facility j In above nomenclature, facility j denotes the facility located at site j. Also, we assume d jm equal to zero if j = m. Let Ω and Ω - denote the index set of performance metrics for inputs (ω = 1, 2,, κ) and outputs (ω = 1, 2,, γ). Let the nonnegative deviation variables, (δ 1, δ 2, δ κ ) and (δ 1, δ 2, δ γ ), denote the amounts by which each value of performance metrics, X ω or X ω, deviates from the minimum and maximum values, respectively, which are called overachievement and underachievement deviation variables. Then, the deviation variables are expressed as and δ ω = X ω TAR ω δ ω = TAR ω X ω, where TAR ω and TAR ω represent the target value of performance metric, ω and ω, respectively. Now, the minimax objective can be expressed as

3 Q = Max {α 1 δ 1 δ κ TAR,, α κ 1 TAR, α δ 1 1 κ TAR,, α γ 1 TAR }, (1) γ where α ω and α ω are relative importance weights attached to the overachievement and underachievement deviation variables and the sum of all weights equals one for the purpose of analysis. The formulation for MOFLA model is given as follows: δ γ subject to Minimize Q in (1) (2) α δ 1 1 TAR Q, α κ δ κ 1 TAR Q, α 1 δ 1 κ TAR Q, α δ γ γ Q, (3) 1 TAR γ X 1 δ 1 = TAR 1,, X κ δ κ = TAR κ, X 1 δ 1 = TAR 1, X γ δ γ = TAR γ. (4) Data Envelopment Analysis (DEA) Models Appropriate Constraints for FLA Problem. Among many performance evaluation methods, data envelopment analysis (DEA) has been widely used to evaluate the relative efficiency of a set of peer organizations called decisionmaking units (DMUs). DEA defines relative efficiency as the ratio of the sum of weighted outputs to the sum of weighted inputs. Performance evaluation or measurement often depends upon by the context. Seiford and Zhu (2003) propose the stratification/context-dependent DEA method to measure the attractiveness score and progress of DMUs with respect to a given evaluation context. For this, they stratify DMUs into different efficiency levels. Let J 1 = {DMU j, j = 1, 2,, n} be the whole set of n number of DMUs and iteratively define J l1 = J l E l, until J l1 becomes null. E l consists of all the efficient DMUs on the l th level, that is, E l = {DMU k J l θ (l, k) = 1}, and θ (l, k) is the optimal value to the following CRS model when DMU k is under evaluation. Subject to K θ (l, k) = min λj,θ(l,k) θ(l, k) (5) λ j I ij θ(l, k)i ik 0, i = 1,, p, (6) j F(J l ) K λ j O rj O rk 0, r = 1,, s, (7) j F(J l ) λ j 0, j = 1, n,

4 where j F(J l ) means DMU j εj l, i.e., F( ) represents the correspondence from a DMU set to the corresponding subscript index set. The attractiveness score for each DMU in the l th level (E l ) is computed against DMUs in the (l 1) th and lower levels as the evaluation context (see Zhu [16]). We adopt the DEA-based stratification concept for the FLA problem, find the attractiveness scores for each DMU in the first level E 1 against the DMUs in each lower level, such as E 2, E 3, E L, and compute the average attractiveness score (AAS) to select a few best efficient DMUs. Formulation of FLA Problem CASE STUDY IN SOUTH CAROLINA In 2015, when South Carolina (SC) was stricken by the catastrophic floods, the Federal Emergency Management Agency (FEMA) opened disaster recovery centers (DRCs) in several SC counties to help SC flood survivors. We use the problem of locating DRCs in SC as our case study. The first goal of our case study is to minimize the related logistics costs. Given this problem setting, the total logistics cost (TLC) consists of the fixed cost and the transportation/shipping cost is given by TLC = [ f j C j h m d jm y jm c jm ]. (8) j M j M m M Our second goal is to minimize the maximum coverage distance (MCD) such that each site is covered by one of the DRCs within the endogenously determined distance. Now, MCD is given by MCD = Max{d jm y jm }, j and m. (9) The next goal is to minimize the maximum demand-weighted coverage distance (MDWCD), which is given by MDWCD = Max{h m d jm y jm }, i, j, and m. (10) We assume that if a DRC is disrupted, it can t handle the supplies being delivered to the site of an emergency. Now, our next goal is to maximize the expected number of non-disrupted supplies (ENNDS), which is given by ENNDS = (1 p j ) Z jm h m, (11) j M m M where Z jm = C j y jm and p j denotes the probability that the DRC j is disrupted and we call it a risk probability for site j

5 The maximum effective coverage distance (MECD), denoted byd c, may be shorter than MCD. Our fifth goal is to maximize the covered demands in case of emergency, CDE, which is expressed as where indicator parameter, a jm, are CDE = m M j J h m a jm y jm, (12) a jm = { 1, if d jm D c 0, otherwise. (13) The above five-goal parameters are classified into three inputs, Ω = {TLC, MCD, MDWCD} and two outputs, Ω = {ENNDS, CDE}. Now, let the nonnegative deviation variables, {δ TLC, δ MCD, δ MDWCD, δ ENNDS, δ CDE } denote the amounts by which each value of TLC, MCD, MDWCD, ENNDS, and CDM deviates from each target value. Then, we formulate a multiobjective linear programming (MOLP) model with the minimax objective as stated before. Solving MOLP and Applying DEA We use major disaster declaration records in South Carolina from FEMA database. Forty-six counties are clustered based on proximity and populations into twenty counties. Then, we choose one city from each clustered county based on a centroid approach and assume that all population within the clustered county exists in that city. The distance between these cities is considered to be the distance between counties. Based on the historical record and the assumption, the risk probability for each site (a county or a clustered county) is calculated in Table 1. For the case study, we hypothetically pre-determine the following parameters. The maximum numbers of DRCs to be built are 5. The minimum and maximum number of demand points that each DRC can handle at least 2 (b j = 2) and at most 7 (B j = 7) sites. The capacity of each DRC, CAP j max, is set to 1,500 K in terms of the quantity of supplies. The fixed cost for a DRC is listed in Table 1 and the shipping cost, c jm, per 1 K unit of demand per mile is $1.00. We also set the maximum effective coverage distance in case of emergency, D c, in (30), equal to 35 miles. No City County Table 1. Data for Locations of DRCs Population (K) Risk Probability Fixed Cost, f j (M) 1 Aiken Aiken/Barnwell Anderson Anderson/Oconee/Pickens Beaufort Beaufort/Jasper Bennettsville Marlboro/Darlington/Chesterfield Charleston Charleston

6 6 Columbia Richland/Fairfield/Kershaw Conway Horry Florence Florence/Dillon/Marion Georgetown Georgetown/Williamsburg Greenville Greenville/Laurens Greenwood Greenwood/Abbeville Hampton Hampton/Allendale Lexington Lexington/Newberry/Saluda McCormick McCormick/Edgefield Moncks Corner Berkeley Orangeburg Orangeburg/Bamberg/Calhoun Rock Hill York/Chester/Lancaster Spartanburg Spartanburg/Cherokee/Union Sumter Sumter/Clarendon/Lee Walterboro Colleton/Dorchester Before solving the MOLP model, it is necessary to determine the target values for each goal parameter. Each of these target values could be obtained by setting the corresponding weight equal to 1 and solving the MOLP model. Now, we solve and summarize the target values of five performance metrics, TLC min, MCD min, MDWCD min, ENNDS max, and CDE max, in Table 2. Using the values in Table 2, the MOLP model is solved for various values of the weight, α g, where each weight changes between 0 and 1 with an increment of 0.1. There are 1,001 configurations arising out of the combinations of the setting of α under the condition α 1 α 2 α 3 α 4 α 5 =1. After we solve the model, we reduce 1,001 configurations into 615 consolidated configurations, based upon the values of the five performance measures. Each of 615 configurations is considered as a DMU. Table 2. The target values of five performance metrics TLC min MCD min MDWCD min ENNDS max CDE max $376, miles K 4027K 3158K Now, using the SDEA method, we stratify 615 DMUs into 22 levels. We identify the average attractiveness score (AAS) and rank DMUS in the first level (Level 1). In Table 3, we present the top DMU in the first level (Level 1) as well as the DMU in the last level (Level 22), along with the combination of weights, value of each performance metric, CRS efficient score (ES), and the average attractiveness score (AAS) and its ranking (R). From Table 3, we see that DMU 250 ranked number one with the highest AAS would be the best option, whereas DMU 10 in Level 22 with the lowest ES of the least option. Note that the four DMUs in Level 22 are considered as the least efficient. In Table 4, we present the optimal location-allocation of DRCs for the DMUs with the top two DMUs in Level 1 and in Level 22. A notable observation in Table 4 is that all three DMUs in Level 1 selects {Charleston, Conway, Greenville, Lexington, Rock Hill} as the optimal locations for DRCs, but allocations are a little different. In the meantime, the optimal location-allocation

7 scheme of DRCs for the two DMUs in Level 22 is quite different from the schemes for the three best DMUs. The locations of DRCs and their allocation schemes for DMU 250 and DMU 10 are depicted in Figure 1. In Figure 1 where FLA scheme for DMU 250 is depicted, we note that, instead of DRCs Lexington or Charleston, DRC Greenville covers Hampton, which is 190 miles away from Greenville. From Table 3, we see that MCD for DMU 250 is 190 miles, but MWDCD is the target (minimum) value of The reason is that since Hampton has a small demand of 33K from Table 1, a relatively high value of MCD does not affect MWDCD. The two DRC locations, Lexington and Charleston have higher risk probabilities, and 0.25, respectively, than Greenville with In addition, Hampton is and 78 miles away from Lexington and Charleston, which exceed the maximum effective coverage distance,d c, of 35 miles. Thus, without sacrificing MWDCD and CDE, DRC Greenville for DMU 250 covers Hampton to enhance ENNDS at the sacrifice of MCD and TLC. Note that finding the location of DRC is a strategic decision plan. CONCLUSION In this study, we consider the multi-objective facility location-allocation (MOFLA) problems and present a combining procedure to find efficient location-allocation schemes. We accomplish by first using a multiple objective linear programming (MOLP), which would yield more balanced options, to generate all inputs and outputs for each configuration arising out of the combinations of the weight factor α. Each configuration is considered as a decision making unit (DMU). For the generated DMUs, we apply data envelopment analysis (DEA) to identify efficient locationallocation schemes, and then we use the stratification/context-dependent DEA to select the best ones. Through a case study using actual major disaster declaration records in South Carolina, we demonstrate the applicability of our procedure and observe that our combined approach with MOLP model and DEA for the FLA problem performs well. ACKNOWLEDGEMENTS This material is based upon work that is supported by the National Institute of Food and Agriculture, U.S. Department of Agriculture, under project number SCX REFERENCES References are available upon request from Jae Hong

8 Table 3. DMUs in Level 1 and Level 22 Level Level DM U # α=(α 1, α 2, α 3, α 4, α 5 ) TLC( $) MCD (miles) MDWCD (K) miles) ENNDS (K) CDE (K) Input Input Input Output Output ES AAS (R) (0.1, 0.0, 0.0, 0.1, 0.8) $433, * (1) (0.0, 0.0, 0.0, 1.0, 0.0) $592, N/A AAS (ES) (1.000) (1.000) N/A (0.829) Table 4. DRC Location and Allocation for Some Efficient and Inefficient DMUs DMU DRC Site 1 Site 2 Site 3 Site 4 Site 5 Site Charleston Beaufort Georgetown Moncks Corner Orangeburg Walterboro Conway Florence Greenville Anderson Greenwood Hampton McCormick Spartanburg Lexington Aiken Sumter Columbia Rock Hill Bennettsville Charleston Beaufort Hampton Moncks Corner Walterboro Conway Bennettsville Florence Georgetown Greenville Anderson Greenwood Spartanburg Lexington Aiken Columbia Orangeburg Sumter Rock Hill McCormick Florence Bennettsville Conway Georgetown Moncks Corner Sumter Greenwood Anderson Lexington Orangeburg Aiken Columbia Spartanburg Greenville McCormick Rock Hill Walterboro Beaufort Charleston Hampton Anderson Conway Hampton Lexington Orangeburg Rock Hill Beaufort Bennettsville Columbia Charleston Georgetown Moncks Corner Walterboro N/A (0.727) Greenwood Florence Sumter Spartanburg Greenville Aiken McCormick 38

9 Figure 1. Most Efficient and Inefficient Facility Location-Allocation Schemes (DMU 250 ) (DMU 10 ) 39