INTRODUCTION

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Supplier selection represents one of the most significant functions to be performed by the purchasing decision makers which determines the long-term viability of the firm (Zouggari and Benyoucef, 2011). Several mathematical programming techniques have been proposed for supplier selection in the literature. However, because of the intricacy of the decision-making process involved in supplier selection, all the aforementioned references in supplier selection, except for the Data Envelopment Analysis (DEA) model; rely heavily on some sort of procedure for determining the importance weights associated with the performance criteria. These importance weights are generally subjective and it is often difficult for the decision makers to precisely assign numbers to their preferences. This is especially intimidating for the decision makers when the number of performance criteria is increased. Furthermore, these methods do not consider stochastic data in the supplier selection process (Azadi et al., 2012).

DEA, developed by Charnes et al. (1978), provides a non-parametric methodology for evaluating the efficiency of each of a set of comparable Decision Making Units (DMUs). As Saen (2010) addresses, classical DEA models rely on the assumption that inputs have to be minimized and outputs have to be maximized. However, as Koopmans (1951) discussed earlier, the process of plant may produce bad outputs such as CO₂ emission and effluent.

As Azadi and Saen (2012) addressed, Chance-constrained Programming (CCP) developed by Charnes and Cooper (1963) is an operations research approach for optimization under uncertainty when some or all coefficients in a linear programme are random variables distributed in accordance with some probability law. In CCP, the optimization problem is concerned with identification of the value of the decision variables so that the expected loss in the criterion is minimized subject to the requirement that the probability that any given constraint is violated is not allowed to exceed some a priori specified level (Olesen, 2006). The stochastic input and output variations in DEA have been studied by Sengupta (1998, 2000), Olesen and Petersen (1995), Morita and Seiford (1999), Huang and Li (2001), Cooper et al. (2004) and Olesen (2006). Talluri et al. (2006) utilized the CCP model proposed by Land et al. (1993) for supplier selection.

We use CCP model proposed by Cooper et al. (2004) since it has benefits proposed by Land et al. (1993). It opens possible novel routes for sensitivity analysis. In addition, it can be solved by a deterministic equivalent. However, the model proposed by Talluri et al. (2006) does not consider undesirable factors, while the model proposed in this study takes into account the undesirable factors.

Motivated by those points, the objective of this study is to propose a model for selecting suppliers in the presence of undesirable outputs.

PAST RESEARCHES

Here, various studies on the supplier selection, DEA and undesirable outputs are briefly summarized.

Supplier selection: There are several supplier selection methods available in the literatures such as Analytical Hierarchical Process (AHP) (Chan et al., 2007; Ng, 2008), fuzzy programming model (Sanayei et al., 2010), intelligent model (Das and Shahin, 2003), Multiple Attribute Utility Approach (MAUT) (Min, 1994). Also, there are other methods for supplier selection problem such as fuzzy logic approaches (Bevilacqua and Petroni, 2002; Lee, 2009; Noorul Haq and Kannan, 2006), case-based reasoning (Choy et al., 2005), Multi-objective Programming (MOP) (Arunkumar et al., 2006), mixed integer programming (Hartmut, 2007), chance-constrained and genetic algorithm (He et al., 2008), DEA (Azadi and Saen, 2011; Hosseinzadeh et al., 2011), Analytic Network Process (ANP) (Bayazit, 2006; Gencer and Gurpinar, 2007), integrated approach (Ting and Cho, 2008), total cost of ownership approach (Bhutta and Huq, 2002), hybrid AHP (Sevkli et al., 2008), etc.

Data Envelopment Analysis (DEA): DEA is a non-parametric linear programming method. It has been employed for assessing the relative efficiency of a homogeneous set of DMUs in both profit and non-profit organizations and a number of extensions and applications have been reported (Niknafs and Parsa, 2011; Laha and Kuri, 2011; Koc et al., 2011; Keramidou et al., 2011; Zandieh et al., 2009; Ghorbani et al., 2010; Ergulen and Torun, 2009; Hatami-Marbini et al., 2009; Rayeni et al., 2010; Rayeni and Saljooghi, 2010; Mirhedayatian et al., 2011; Hosseinian et al., 2009; Asharafi and Jaafar, 2011; Jahanshahloo and Afzalinejad, 2007; Taher and Malek, 2009). Amongst the characteristics that make DEA a powerful tool is its ability to deal with multiple outputs and multiple inputs without requiring any assumptions about the functional form relating inputs to outputs; focus on the efficiency frontier and not on the central trend of the production units and free the decision maker from the necessity to use separate indices, such as labor productivity, capital productivity, etc.

In 1978, basic DEA model was proposed by Charnes, Cooper and Rhodes (CCR). The CCR model is presented in Model 1. Table 1 presents the nomenclatures used in this study. By solving Model 1 n times (each time evaluating a different DMU), relative efficiency scores for all the DMUs are obtained. These measures divide the DMUs into two categories: those with score of 1 (efficient) and those with scores less than 1 (inefficient):

(1)

DEA models may be generally classified into radial and non-radial models. The radial models include the CCR ratio form (the radial model under constant RTS technology, where RTS stands for returns to scale) and the BCC model (the radial model under variable RTS). This type of efficiency measure needs a separate treatment between output-orientation and input-orientation.

Table 1:	The nomenclatures

The non-radial models include an additive model, multiplication model, Range-adjusted Measure (RAM) (Cooper et al., 1999) and slack-based measure. This group of efficiency measure does not need any special treatment on the output/input orientation. Both are aggregated into a single efficiency measure (Sueyoshi and Sekitani, 2007).

Undesirable outputs: In accordance with the global environmental conservation awareness, undesirable outputs of productions and social activities, e.g., air pollutants and hazardous wastes, are being increasingly recognized as dangerous and undesirable. Thus, development of technologies with less undesirable outputs is an important subject of concern in every area of production. DEA usually assumes that producing more outputs relative to less input resources is a criterion of efficiency. In the presence of undesirable outputs, however, technologies with more good (desirable) outputs and less bad (undesirable) outputs relative to less input resources should be recognized as efficient (Cooper et al., 2007).

Authors believe that this study has a significant contribution to an important and very much under-researched topic. The contributions of proposed model are as follows:

•	The proposed model considers undesirable outputs
•	The proposed model considers stochastic data
•	The proposed model considers both undesirable outputs and stochastic data, simultaneously

PROPOSED MODEL

As Cooper et al. (2007) address, Russell Measure (RM) reflects nonzero slacks in inputs and outputs when they are present. In this way we avoid limitations of the radial measures which cover only some of the input or output inefficiencies and hence measure only weak efficiency. Following, Fare and Lovell (1978) and Cooper et al. (2007); the Russell Measure (RM) for the jth DMU (j = 1, 2, …, n) can be formulated as follows:

(2)

where, the variables (θ_i and φ_r) indicate the level of efficiency related to the ith input and the rth output, respectively. The variables (λ_j for j = 1, …, n) are used for a structural connection among DMUs in the input-output space.

In examining Model 2, Cooper et al. (1999, 2007) discuss that it is difficult to compute and to interpret the RM. Then, they proposed to use Enhanced Russell Graph Measure (ERGM) to overcome such difficulties. The ERGM is formulated as follows:

(3)

The difference between Model 2 and 3 can be found in only their objective functions. It is clear that Model 3 is a nonlinear programming problem and hence, it is still difficult to solve the problem. To enhance the computational capability of Model (3), Cooper et al. (2007) have proposed a transformation from Model 3 to a linear programming equivalence via the well-known treatment of fractional programming (Charnes and Cooper, 1962). To briefly review their treatment applied to Model 3, a new variable:

is included into Model 3. Here, the variable satisfies both 0≤β≤1 and:

Then, all the variables in Model 3 can be transformed as follows: u_i = βθ_i (i = 1,…, m), v_r = βφ_r (r = 1,…, s) and t_j = βλ_j (j = 1,…, n). Using these transformed variables, Model 3 can be reformulated as follows:

(4)

As a result of the transformation, Model 4 is reformulated as a linear programming problem. Therefore, it can be easily solved by any linear programming software. However, the ERGM proposed by Cooper et al. (1999) and Cooper et al. (2007) can provide an approximate of efficiency score for RM. Such an effort cannot perfectly solve the computation issue of the RM measurement. Sueyoshi and Sekitani (2007) proposed to use Second-order Cone Programming (SOCP) that can directly solve the RM without depending upon the ERGM approximation. Moreover, the SOCP approach makes it possible to formulate a dual model of the RM. After the SOCP is applied to reformulate the RM, then the primal and dual models can be established within the computational framework of the interior point method (not Simplex method). As a result of the dual development, the type of RTS is determined which is an economic implication under the RM. The model proposed by Sueyoshi and Sekitani (2007) is as follows:

(5)

At this juncture, the new model is developed. Assume we have n DMUs each consuming m inputs and producing p outputs. The outputs corresponding to indices 1, 2,..., k are desirable and the outputs corresponding to indices k+1, k+2,..., p are undesirable outputs. We like to produce desirable outputs as much as possible and not to produce undesirable outputs. Let XεR₊^mxn and YεR₊^Pxn be the matrices, consisting of non-negative elements, containing the observed input and output measures for the DMUs. Korhonen and Luptacik (2004) decomposed matrix Y into two parts:

where a kxn matrix Y^g is standing for desirable outputs (good) and a (p-k)xn matrix Y^b is standing for undesirable outputs (bad). We further assume that there are no duplicated units in the data set. We denote by x_j (the jth column of X) the vector of inputs consumed by DMU_j and by x_ij the quantity of input i consumed by DMU_j. A similar notation is used for outputs. Occasionally, we decompose the vector y_j into two parts:

where the vectors y^g_j and y^b_j refer to the desirable and undesirable output. When it is not necessary to emphasize the different roles of inputs and (desirable/undesirable) outputs, we denote

and:

Furthermore, we denote 1 = [1,…, 1]^T and refer by e_i to the ith unit vector in Rⁿ We consider set T = {u|u = Uλ, λεΛ}, where Λ = {λ|λε and Aλ≤b},e_iεΛ, i = 1,…, n. Furthermore, consider matrix Aε and vector Bε which are used to specify the feasible values of λ variables.

Model (5) is combined with undesirable output:

(6)

Model 6 is structured under variable RTS technology, depending upon β. Now, the novel model of stochastic ERGM-undesirable output is developed which permits the possible existence of stochastic variability in the data. The proposed model can deal with both undesirable outputs and stochastic data in ERGM context, simultaneously. There is not any model that discusses supplier selection in the presence of both undesirable outputs and stochastic data in ERGM context. The proposed model is the first and unique model.

As we know, the typical DEA models do not permit stochastic variations in input and output; hence, DEA efficiency measurement may be sensitive to such variations. For instance, a DMU which is measured as efficient relative to other DMUs, might turn inefficient if such random variations are considered. In what follows, stochastic version of the output-oriented undesirable model is presented which allows for the possibility of stochastic alterations in input and output data. We suppose that all inputs and outputs are random variables with a multivariate normal distribution and known parameters.

Assume that ξ_i represents ‘external slack’ for the ith input. We select its value to satisfy:

(7)

There must then exist a positive number such that:

(8)

Such a positive value of permits a reduce in for any sample devoid of worsening any other input or output to the indicated probabilities. It is easy to demonstrate that ξ_i = 0 if and only if =0.

In a similar manner, ζ_r is the external slack for the rth output. Via external slack, we refer to slack outside the braces. We can select the value of this external slack which is not stochastic, so it satisfies:

(9)

There must then exist a positive number s^g_r>0 such that:

(10)

This positive value of s^g_r permits a still further raise in for any set of sample observations devoid of worsening any other input or output. It is easy to see that ζ_r = 0 if and only if s^g_r = 0.

Also, for constraint 4 of Model 6 we have:

(11)

Consequently:

(12)

Using Relations 7-12, can replace Model 6 with following model:

(13)

For the 3rd constraint in Model 13, we have:

Note that the conversion process is discussed for constraint 3 in Model 13 and the same process could be repeated for constraints 2 and 4.

For the sake of simplicity, we indicate:

by σ_r^go (t). Hence:

In other words:

where, Z is a normal standard variable and we have:

Or:

The deterministic equivalent for Model 13 is as follows:

(14)

To derive equations for σ_i^I (t) note that:

Therefore:

Similarly, for the constraints 3 and 4 of Model 14, we have:

It is obvious, from the forms of σ^I_i (t), σ_r^go (t) and σ_r^bo (t), that Model 14 is a nonlinear programme.

NUMERICAL EXAMPLE

The idea for this example is taken from Azadi and Saen (2012). The example contains specifications on twenty suppliers (DMUs). These DMUs consume two inputs to produce two outputs. The data are available in Table 2.

Table 2:	Related attributes for 20 suppliers

Table 3:	The efficiency scores for the 20 suppliers with α = 0.05

The performance measures utilized were number of personnel, average time for serving customers, profit margin and number of dissatisfied customers. Number of personnel and average time for serving customers were used in some way as inputs for the DEA model. The desirable output utilized in the study is profit margin. The undesirable output is number of dissatisfied customers. Note that the inputs and outputs selected in this study are not exhaustive by any means but are some general measures that can be utilized to evaluate suppliers.

The computational results from using Model 14 with α = 0.05 are shown in Table 3. The efficient suppliers are Khozestan, Mazandaran, Gharb and Omran. These suppliers are efficient.