Combined Single-Source and Multi-source Capacitated Facility Location Problems with Data Envelopment Analysis

In this paper, we incorporate an efficiency criterion using data envelopment analysis into the single-source and multi-source capacitated facility location problems. We develop two bi-objective integer programs to find optimal and efficient location patterns, simultaneously. (e proposed models for these capacitated facility location problems have fewer variables and constraints compared to existing models presented in the literature. We use the LP-metric procedure to solve the proposed models on two numerical examples. Results show that new models give better solutions, based on cost and efficiency criteria.


Introduction
Facility location problem (FLP) is concerned with finding optimal locations of facilities and how to allocate them to satisfy customers' demands such that the total fixed opening costs of facilities and the total transportation costs are minimized. It has applications in supply chain network design, telecommunication, public sector services, and distribution network design [1][2][3][4]. In location analysis, patterns with more output and service produced for a given amount of resources are more efficient and less wasteful. Attributes such as availability of infrastructures, resource usage, and receptiveness and perceptions of local population can influence performance of location pattern.
us, considering a multi-objective program which includes different criteria makes sense.
ere are different methods to evaluate efficiency and performance, where, in this paper, we use data envelopment analysis (DEA)-based models. ese models are quantitative and nonparametric mathematical programming-based approaches to evaluate efficiency. ey can be used in combination with FLP models to produce integrated models that can be solved in one step in order to find best locations for facilities and efficient allocation pattern. e DEA technique has extensive applications in different fields such as performance analysis [5], ranking companies or decision-making units [5,6], efficient banking [7], industrial management [6,7], and forecasting profitability in markets [8,9]. Moreover, applications of DEA models with the indeterminacy, impreciseness, vagueness, inconsistent, and incompleteness information are also widely studied [10,11].
In recent works, FLP models are combined with DEA models to achieve best locations and maximum efficiency at the same time. Cook and Green [12] used DEA to select sites for facilities with a single resource restriction on operating budget. ey proposed a mathematical programming model to find locations of sites for a set of retail outlets such that maximize the ratio of benefits to costs. Klimberg and Ratick [13] have utilized concept of DEA in order to formulate and find optimal and efficient facility location/allocation patterns. ey used this concept for both uncapacitated and capacitated facility location problems. ey proposed two bi-objective models for combined DEA and location-allocation models in which the total costs and average DEA efficiency objectives are optimized simultaneously. Karbasian and Dashti [14] used simultaneous DEA for four discrete, deterministic, uncapacitated, and static dispersion facility location problems. ey proposed different multiobjective models to find optimal and efficient facility location patterns and maximize total demands of satisfied customers using the DEA method for p-dispersion problem, p-defense problem, and MaxMinSum and MaxSumSum dispersion and covering location problems in the presence of existing facilities. ey also used a fuzzy goal programming method to solve their multi-objective models. omas et al. [15] used DEA in locating obnoxious facilities and presented two approaches in their study. In the first one, obtained optimum locations of facilities are considered as input of the DEA model. In the second one, a hierarchical process of DEA is used by a single-objective linear programming (LP) model that maximizes the efficiency of p obnoxious facilities to be opened. In their study, the performance is considered in terms of proximity and DEA efficiency scores. Azadeh et al. [16] proposed an integrated hierarchical approach to select the most efficient and best-possible location for solar plants using the DEA approach. e optimum locations of solar power plants are ranked by DEA with respect to some output and input parameters. Also, principal component analysis and taxonomy methods are used to validate results of the model. Mitropoulos et al. [17] used combination of DEA approach to location planning of services and effective allocation of scarce resources such as equipment, funds, or workforce in the health sector. ey proposed a framework to evaluate technical efficiency of existing healthcare centers and location analysis in order to maximize accessibility, utilization, and mean efficiency and select the appropriate number and locations of providers. Location analysis determines which centers will be upgraded and expanded and which ones will be closed. Moheb-Alizadeh et al. [18] studied incorporation of DEA and location-allocation models in a fuzzy environment. ey used the multicriterion form of DEA and simultaneously considered both facility location and demand assignment problems in which the number of facilities to be located was not predetermined. e demand of each product for each customer, the amount of resources that each facility uses, and the output of each facility in each candidate location are assumed as a fuzzy number. Adabi and Omrani [19] studied considering efficiency in design of supply chain and proposed a bi-objective mixed integer linear programming (MILP) where one objective maximized system efficiency of the network and the other one minimized the total setup and transportation costs of the pattern. Mohaghar et al. [20] developed an integration of fuzzy VIKOR and assurance region-DEA for selection and ranking suppliers in a supply chain network. Georgantzinos and Giannikos [21] also considered the incorporation of efficiency in the context of the set covering, set partitioning problem, and set packing problem. Finally, recently Houng and Jeong [22] combined DEA and multi-objective optimization techniques for the efficient facility location-allocation decisions and patterns to help practitioners and decision-makers.
Among FLPs, two widely used and studied problems are single-source capacitated facility location problem (SSCFLP) and multi-source capacitated facility location problem (MSCFLP) [2,3]. In this paper, we combine SSCFLP and MSCFLP models with the CCR model of DEA. e resulting models are integer and mixed integer multi-objective LPs, respectively. In MSCFLP case, our model has less variables and constraints and gives better solutions compared to [13]. In both models, efficiency is defined as weighted sum of the outputs. To solve the proposed multi-objective models, we use the LP-metric method. e remainder of this paper is organized as follows. Section 2 introduces single-source and multi-source capacitated facility location problems. Section 3 presents DEA models which are used in this paper. e proposed combined models of DEA with SSCFLP and MSCFLP are given in Section 4. Section 5 presents the solution procedure for the proposed models. Finally, numerical examples are given in Section 6.

Capacitated Facility Location Problems
In what follows, we give MSCFLP and SSCFLP models. e MSCFLP can be formulated as the following MILP: x hl ≥ 0, ∀l, ∀h, where h � 1, . . . , H is the index of facility locations, l � 1, . . . , L is the index of demand points, c hl is the cost of shipping one unit of demand from facility h to demand point l, a l is the number of units of demand at l, b h is the capacity of facility h, f h is the fixed cost of opening facility h, and x hl is the fraction of demand at point l satisfied from facility h: Objective function (1) calculates the total fixed opening and transportation costs. e transportation costs are calculated as the product of the per unit transportation costs and the amount shipped from facility h to demand l. Constraints (2) ensure that every demand point is satisfied, and constraints (3) ensure that only open facilities can supply demand points considering their capacity. Finally, constraints (4) and (5) are binary and nonnegative constraints on variables, respectively.
In SSCFLP, customers are forced to be served only from a single facility which is applicable for real-world situations where multiple deliveries may increase the cost of maintaining and updating the inventory [23]. It can be formulated as the following ILP: where Objective function (7) calculates the total fixed opening and transportation costs. e transportation costs are calculated as the product of the per unit transportation costs and the amount shipped from facility h to demand l. Constraints (8) ensure that every demand point is satisfied, and constraints (9) ensure that only open facilities can supply demand points considering their capacity. Finally, constraints (10) are binary constraints on variables.

Data Envelopment Analysis
DEA is a nonparametric method based on mathematical programming for measuring relative efficiency of multiple homogeneous decision-making units (DMUs) with the same inputs and outputs. DEA is an effective tool for performance benchmarking when multiple performance measures exist and a priori information on the tradeoffs among these measures is completely available [6]. Farrell proposed a nonparametric approach for evaluating the efficiency of DMUs applying two inputs and one outputs, and later, Charnes et al. developed this approach for several inputs and outputs [24,25]. Evaluation of DMUs is done by the ratio of the weighted sum of outputs to the weighted sum of inputs which is relative efficiency of them. In this manner, DEA finds the weights such that the efficiency of each DMU is maximized rather than the other DMUs. e multiplier form of the CCR (CCRm) DEA model evaluating the efficiency of o-th DMU is as follows: where n is number of DMUs, m is number of inputs, s is number of outputs, I ij is the i-th input value for j-th DMU, O rj is the r-th output value for j-th DMU, u r is the weight values of the r-th output, and v i is the weight values of the ith input. e objective function, which is the efficiency score of under evaluation DMU, cannot exceed unity, and DMUs are classified into two types based on their scores. e DMUs with an efficiency score 1 are called efficient, and others are called inefficient [5]. Charnes et al. transformed the CCRm to an LP as follows [25]: where ε is non-Archimedean infinitesimal value to prevent numerous zeros in input and output weights. e CCRm is always feasible [5], for example, if ε � 0, then because where u rj is the weight values of the r-th output in the j-th DMU and v ij is the weight values of the i-th input in the j-th DMU. Similar to model (15)- (18), models (19)- (22) are also feasible, for example, v � 1/I 1,1 1/I 1,2 · · · 1/I 1,n 0

Combined MSCFLP-DEA and SSCFLP-DEA Models
In this section, DEA methodology has been used as a tool for incorporating concept of efficiency into MSCFLP and SSCFLP models as another objective to provide optimum location pattern with respect to the total transportation and fixed costs and also the performance of facilities at different potential sites. e SSCFLP is an important variant of CFLPs in practical situations that a combined model for which is proposed in this section. In order to incorporate DEA in CFLP models, the DEA efficiency scores of all DMUs should be calculated in one LP. We develop combined CFLP-DEA models using models (19) x hl ≤ M hl z hl , ∀h, ∀l, z hl ≤ x hl , ∀h, ∀l, O rhl u rhl ≤ z hl , ∀h, ∀l, ∀r, u rhl ≥ εz hl , ∀r, ∀h, ∀l, v ihl ≥ εz hl , ∀i, ∀h, ∀l, x hl ≥ 0, ∀l, ∀h,  (23) and constraints (25) and (26) are related to the MSCFLP model in (1)- (5). Moreover, we need extra binary variables z hl to show allocation status between facilities and demand points which is defined as follows: Constraints (27) and (28) O rhl u rhl ≤ x hl , ∀h, ∀l, ∀r, u rhl ≥ εx hl , ∀r, ∀h, ∀l, v ihl ≥ εx hl , ∀i, ∀h, ∀l, First objective function (37) and constraints (39) and (40) are related to the SSCFLP similar to the model in

Solution Procedure
Since MSCFLP-DEA and SSCFLP-DEA models are bi-objective with inconsistent objective functions, we use the LPmetric method [26] which is one of the famous and useful methods for solving multi-objective problems with conflicting objectives.
is method considers each objective function separately and reformulate a single objective to minimize the sum of the normalized differences between the objectives and their optimal values. e two objective functions are denoted by Z 1 and Z 2 . Based on the LP-metric method, MSCFLP-DEA and SSCFLP-DEA should be solved for each one of Z 1 and Z 2 separately. Objective function of the LP-metrics can be formulated as follows: where 0 ≤ ω ≤ 1 is the weight which indicates relative importance of the two objective functions and Z * 1 and Z * 2 are optimum values of Z 1 and Z 2 , respectively. Using LP-metric objective function Z 3 , we have single-objective models which can be solved by efficient solvers such as CPLEX and Gurobi [27,28].

Numerical Examples
In this section, we demonstrate the MSCFLP-DEA and SSCFLP-DEA models on two examples. In both examples, each MSCFLP-DEA and SSCFLP-DEA models are optimally solved three times for objective functions Z 1 , Z 2 , and Z 3 . Objective function Z 1 minimizes the total fixed and transportation costs. e function Z 2 maximizes the sum of efficiencies of DMUs. e third objective function Z 3 is the LP-metric objective function which uses best values of Z 1 and Z 2 to make a tradeoff between costs and efficiency scores of the facility-demand links. We solve the models using IBM ILOG CPLEX 12.5 on a CORE2Duo CPU of 2 GHZ and 2GB of RAM.

Example 2.
is example is taken from [18] and includes five facilities (F 1 − F 5 ), eight demand points (D 1 − D 8 ), three inputs (I 1 − I 3 ), and two outputs (O 1 − O 2 ). Results are given in Tables 3 and 4 for the MSCFLP-DEA and the SSCFLP-DEA, respectively. Figure 1 illustrates the tradeoff between the total costs and the sum of efficiency scores of DMUs for w � 1, 0.9, 0.8, 0.7, 0.6, 0.5, and 0, respectively. is curve is an efficient frontier, and the decision-maker can select a suitable value for w. As it is shown, sum of efficiencies of facility-demand links increases by increasing the total costs. It is worthy that the slope of increasing efficiency is higher at the first and then it decreases. In example 1, the average of efficiency scores %84 increases with only %12 increase in the costs. So, it means that we can find efficient patterns with a little more costs which is valuable. Figure 2 and 3 show the effect of weight of LP-metric objective function on the total costs and efficiency scores, respectively. Figure 4 shows the relative variation of efficiency scores by relative variation of total costs. As we see in Figure 4, %1.44 increase in the costs of locations pattern leads to %81.93 increase in the efficiency of pattern. e maximum efficiency is achieved by about %8.87 increase in the costs.
Results of solving the SSCFLP-DEA model in example 1 are shown in Table 2. Figure 5 represents the tradeoff between the total costs and the sum of efficiency scores for w � 1, 0.8, 0.6, 0.5, 0.4, 0.2, and 0, respectively. As it is shown, sum of efficiencies of facility-demand links increases by increasing the total cost. e slope of increasing efficiency decreases by costs. e average of efficiency scores % 71.31 increases by changing some assignments and without any additional costs. Figure 6 and 7 show the effect of weight of LP-metric objective function on the total costs and efficiency scores, respectively. Total costs decrease by increasing the weight, and sum of efficiencies increases by decreasing it. Figure 8 shows that relative variation of efficiency scores by relative variation of total cost. % 11.38 increase in the costs of Similar results hold for example 2. Sum of efficiencies of the links increases by increasing the total costs, and an efficient pattern can be achieved by a little additional cost.

Conclusions
In this paper, we have incorporated the concept of the efficiency into two kinds of capacitated facility location problems, SSCFLP and MSCFLP, using DEA models. We have proposed two bi-objective integer programs for SSCFLP-DEA and MSCFLP-DEA models using the CCR         model of DEA for performance evaluation and measuring efficiency. e developed model for MSCFLP-DEA has less variables and constraints compared with the one proposed by Klimberg and Ratick in [13]. In both SSCFLP-DEA and MSCFLP-DEA models, the number of facilities to be located is not predetermined and the models should specify it. Our models can determine location pattern including location of facilities and assignment of customers to them and also consider efficiency scores simultaneously. We used an LPmetric procedure to solve the bi-objective models, and computational results on two numerical examples show that these models are more reliable and efficient. Results of numerical examples show that the high level of efficiency can be achieved by only little increase in the costs. Due to the uncertainty of data in real-world applications, studying this problem under uncertainty can be considered as a future research direction.