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Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data



Majid Azadi and Reza Farzipoor Saen
 
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ABSTRACT

Supplier selection has a strategic importance for every company. Nondiscretionary Slacks-based Measure (SBM) model is one of the models in Data Envelopment Analysis (DEA). In many real world applications, data are often stochastic. A successful approach to the address uncertainty in data is to replace deterministic data via random variables, leading to Chance-constrained DEA (CCDEA). In this study, the concept of chance-constrained programming approach is used to develop nondiscretionary SBM model in the presence of stochastic data and also its deterministic equivalent which is a nonlinear program is derived. Furthermore, it is shown that the deterministic equivalent of the stochastic nondiscretionary SBM model can be converted into a quadratic program. Finally, a numerical example demonstrates the application of the proposed model.

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  How to cite this article:

Majid Azadi and Reza Farzipoor Saen, 2012. Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data. Research Journal of Business Management, 6: 103-120.

DOI: 10.3923/rjbm.2012.103.120

URL: https://scialert.net/abstract/?doi=rjbm.2012.103.120
 
Received: April 27, 2012; Accepted: June 30, 2012; Published: December 05, 2012



INTRODUCTION

According to Sonmez (2006), supplier selection is the process of finding the suppliers that are able to present products and/or services to the customer with appropriate quality, at the appropriate cost, quantities and time. As Amid et al. (2011) address, within new strategies for purchasing and manufacturing, suppliers play a key role in achieving corporate competitiveness. Consequently, correct selection of suppliers is a critical element. Main cost of the most industries in manufacturing belongs to cost of raw materials and component parts which in most cases constitutes up to 70% of the total costs. Therefore, purchasing department plays an important role in efficiency and effectiveness of a firm, due to the contribution of supplier performance on expenditure, quality, delivery and service in accomplishing the objectives of a supply chain.

As Chamodrakas et al. (2010) describe, advanced industries should conform to market environment in which accessibility to global competition is an important factor. Consequently, in order to reduce production expenditures, it is important that expenses of companies be logical and reasonable. To this end, reducing the purchasing prices through selection of right supplier can be beneficial. In addition, some advanced production systems like just in time and mass customization manufacturing pay attention to quick provision of raw materials and outsourced components within expected quality and quantity. The fact that many businesses are turning to outside suppliers and manufacturers to obtain universal resources more effectively, emphasizes to importance of requirements of these issues.

Some mathematical programming approaches have been used for supplier selection in the past. Table 1 categorizes the reviewed papers based on applied techniques. Because of the intricacy of the decision making process involved in supplier, all the aforementioned references in Table 1, except for Data Envelopment Analysis (DEA), rely on some form of procedures that assign weights to various performance measures. The primary problem associated with arbitrary weights is that they are subjective and it is often a complex task for the decision maker to precisely assign numbers to preferences. As well, it is a daunting task for the decision maker to assess weighting information as the number of performance criteria is increased. In the meantime, they do not consider stochastic data.

Standard DEA models suppose that Decision Making Units (DMUs) carry out same obligations with same goals, employ similar inputs and create similar outputs. In real world, some factors are out of the control of decision makers and are called non-discretionary factors (Syrjanen, 2004).

Instances from the DEA literature include snowfall or weather in evaluating the efficiency of maintenance units, soil characteristics and topography in different farms, number of competitors in the branches of a restaurant chain and age of facilities in different universities (Saen, 2005).

It is suitable for solving optimization problems with random variables included in constraints and sometimes in the objective function as well (Charnes and Cooper, 1959). As Olson and Swenseth (1987) discuss, CCP was developed as a means of describing constraints in the form of probability levels of attainment. Consideration of chance constraints allows the decision-maker to consider objectives in terms of their attainment probability. If α is a predetermined confidence level desired by the decision-maker, the implication is that a constraint will have a probability of satisfaction of α. The probabilistic nature of this approach lends itself to multi-objective analysis.

Table 1: A summary of methods for suppliers selection
Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data

The selection of α can be a managerial decision. Chance constraints for stochastic functions based upon sampling information would often be normally distributed. Sampling information has long been used in business as a means of determining the expected value of functional coefficients in linearly constrained systems.

A main contribution on the stochastic DEA might be found in the work of Sengupta (1982, 1987, 1990, 1997, 1998, 2000) who has widely studied the research theme, using the CCP proposed by Charnes and Cooper (1963). An important feature of his studies is that stochastic variables are incorporated into DEA and afterward the stochastic DEA is reformulated into a deterministic equivalent. Land et al. (1993) utilized CCP (Charnes and Cooper, 1961; Cooper et al., 1996) to develop efficient frontiers which envelop a given of observation most of time. Olesen and Petersen (1995) proposed a Chance-constrained DEA (CCDEA) model that uses piecewise linear envelopment of confidence regions for observed stochastic multiple inputs and multiple outputs. Cooper et al. (1996) incorporated Simon (1957) “satisficing concepts” into DEA model with chance constrained. Also, stochastic DEA approaches can be found in but not limited to Cooper et al. (1998), Huang and Li (1996) and Li (1998). Morita and Seiford (1999), proposed a measure for reliability of efficient DMUs as the amount of stochastic variations that remain the efficient DMU being efficient. A minimum efficiency score at a specified probability level is also used as a robustness measure. Moreover, they discussed some stochastic measures such as an expected efficiency score, a probability being efficient, an α percentile of efficiency score. Sueyoshi (2000) proposed a “DEA future analysis” that incorporates future information on outputs into its analytical framework. A stochastic DEA model is used as an initial starting formulation and then it is reformulated by both CCP and the estimation technique of PERT/CPM. Besides Huang and Li (2001) generalize two conventional DEA model by incorporating two conventional DEA model by incorporating random disturbances into input and output data. Cooper et al. (2002) proposed CCP models that are directed to determine where efficient and inefficient behavior will occur with associated probabilities. Their method replaces ordinary DEA formulations with stochastic counterparts in the form of a series of CCP models. Emphasis is on technical efficiency and inefficiencies which do not require costs or prices but which are nevertheless basic in that the achievement of technical efficiency is necessary for the attainment of “allocative”, “cost” and other types of efficiencies.

Talluri et al. (2006) utilized the CCP model proposed by Land et al. (1993) for supplier selection, since it is a well-established methodology and provides an innovative and simple method to incorporate variability in input and output measures into the decision making process. The developed model in this paper uses CCP model proposed by Cooper et al. (2004). Since it has the advantages of model proposed by Land et al. (1993), it opens possible new routes for “sensitivity analysis”. Additionally, it can be solved by a deterministic equivalent. Also, model utilized by Talluri et al. (2006) does not consider nondiscretionary factors while model utilized in this paper takes into account the nondiscretionary factors.

In summary, the approach presented in this study has some distinctive contributions that are as below:

A stochastic nondiscretionary SBM model is developed and also its deterministic equivalent which is a nonlinear program is derived
It is shown that the deterministic equivalent of the stochastic nondiscretionary SBM model can be converted into a quadratic program
Sensitivity analysis of the stochastic nondiscretionary SBM model is discussed with respect to changes in parameters
For the first time, the proposed model is used for the problem of supplier selection
The proposed model deals with stochastic data in a direct manner

The objective of this paper was to propose a new stochastic SBM nondiscretionary model for supplier selection.

PROPOSED MODEL

DEA is a decision technique that has been widely used for performance analysis in public and private sectors. DEA developed by Charnes et al. (1978), is a nonparametric estimation method, in the sense that no choice of a parametric functional form is needed in the estimation of the frontier. DEA can be practical to any organization/industry where a rationally homogeneous set of DMUs use the identical set of inputs to produce an identifiable range of outputs. Traditional DEA models can merely measure radial efficiency (weak efficiency). To measure strong efficiency in DEA, Tone (2001) proposed SBM. This model deals directly with the input excesses and output shortfalls. SBM uses the Additive model and provides a scalar measure ranging from 0 to 1 that encompasses all of the inefficiencies that the model can identify (Cooper et al., 2007). SBM does not deal with stochastic data and assume that all input and output data are exactly known.

The formulation for the SBM nondiscretionary model given by Saen (2005) is as below:

Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data
(1)

We can replace model (1) with:

Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data
(2)

Now, the new nondiscretionary SBM model is developed which permits the possible existence of stochastic variability in the data. 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, the output oriented nondiscretionary SBM 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:

Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data
(3)

Definition 1: (Stochastic efficiency) DMUo is DEA stochastic efficient if and only if the following two conditions are both satisfied:

γ* = 1
S¯* = S+* = 0

Now assume ςr 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:

Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data
(4)

There must then exist a positive number sr+>0 such that:

Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data
(5)

This positive value of sr+ permits a still further raise in Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data for any set of sample observations devoid of worsening any other input or output. It is easy to see that ςr = 0 if and only if sr+ = 0.

In an analogous manner, presume ξi>0 represents 'external slack' for the ith input chance-constraint. We select its value to satisfy:

Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data
(6)

There must then exist a positive number s¯i>0 such that:

Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data
(7)

Such a positive value of s¯i permits a decrease in Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data for any sample without worsening any other input or output to the indicated probabilities. It is easy to show that ξi = 0 if and only if s¯i = 0.

Consequently for the constraint 3 of Model (3) we have:

Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data
(8)

and

Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data
(9)

Using relations (4-9), can replace Model (3) with following model:

Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data
(10)

We can replace the first constraint of Model (10) with:

Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data
(11)

This reorients the inequality in the braces and replaces (1-α) with α. It next follows that:

Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data
(12)

Where:

Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data
(13)

We can also write Eq. 13 as:

Φ(a) = α

Where:

Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data

This comes from:

Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data
(14)

Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data
(15)

with f the density function for the standard normal variable. From Eq. 15, we have:

Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data
(16)

Therefore:

Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data
(17)

We can replace the constraint 2 of Model (10) with following relation:

Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data
(18)

It follows that:

Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data
(19)

i.e.:

Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data
(20)

Where:

Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data

This comes from:

Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data
(21)

Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data
(22)

Now -Φ-1 (α) = Φ-1 (1-α), by virtue of the symmetry related with the normal distribution, so from relation (20) we have:

Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data
(23)

Thus:

Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data
(24)

Therefore, the deterministic equivalent for model (10) is as below:

Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data
(25)

To derive equations for σIi(λ) note that:

Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data

Therefore:

Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data

Similarly, for the constraints 2 and 3 of model (25) we have:

Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data

It is obvious, from the forms of σor (λ), σIi (t, λ) and σor (t, λ) that model (25) is a non-linear program. We demonstrate that this non-linear program can be transformed to a quadratic program. Assume that, wor, wIi are nonnegative variables. Replacing wor, wIi, respectively, by σor (λ), σIi (λ) and adding the following quadratic equality constraints:

Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data

Hence, model (25) is transformed to a quadratic programming problem:

Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data

Where:

Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data

NUMERICAL EXAMPLE

The idea for this example is taken from Saen (2009c) and Maital and Vaninsky (2001). The example contains specifications on twenty suppliers (DMUs). These DMUs consume two inputs to produce two outputs. The data set are in Table 2. Distance and cost were used as inputs for the DEA model. The outputs utilized in the study are supplier variety and R and D expenditures. Moreover, assume that cost, supplier variety and distance are nondiscretionary variables, i.e., these factors are exogenously fixed and cannot be changed by suppliers (at least in the short term).

In summary, the suppositions are as below:

Distance is not controllable
Cost is 50% under control
Supplier variety is not controllable
R and D expenditure is controllable

Table 2: Related attributes for 20 suppliers
Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data
DMU: Decision-making units, R and D: Research and development

Table 3 reports the results of efficiency assessments for the 20 suppliers obtained by model 25 which are calculated with α = 0.05. The efficient suppliers are 4, 5, 6, 9, 10, 11 and 19. These suppliers are efficient because the following two conditions are both satisfied:

γ* = 1
S¯* = S+* = 0

This example shows the applicability of the proposed model using chance-constrained DEA with non-discretionary factors and stochastic data in SBM model context. As is seen, in Table 3 suppliers were selected in uncertain environment with α = 0.05. Supplier selection in such uncertain environment reduces the material purchasing cost and enhances company competitiveness which is why many experts suppose that the supplier selection is the most significant activity of a purchasing department.

Sensitivity analysis is the study of how the variation (uncertainty) in the output of a mathematical model can be apportioned to different sources of variation in the input of a model. Table 4 shows the sensitivity of results in terms of different α values. In fact, sensitivity analysis performed in Table 4 shows that how the uncertainty in the output of a model can be apportioned to different source of uncertainty in the model input.

With respect to definition 1, Table 5 implies that the DMUs 3, 6, 9, 11, 12, 13, 17 and 18 are efficient. The ranking results of Tables 3 and 5 depict there are some differences among the ranks.

Table 3: The efficiency scores for the 20 suppliers with α = 0.05
Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data
DMU: Decision-making units, R and D: Research and development

Table 4: Efficiency scores (γ*) for different α
Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data
DMU: Decision-making units, R and D: Research and development

Therefore, stochastic data leads to different results. This shows that if there are stochastic data, then we must apply stochastic models. Note that the results rely on the specified probability level α.

Table 5: The efficiency scores for 20 suppliers using Model (6)
Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data

The stochastic model applied in this numerical example permits the data errors and provides probabilistic results. In general, if the data are under uncertainty and probabilistic situations and a rough estimate is required, the stochastic models might be favored.

CONCLUSION

Supplier plays an important role in company successes. Though, as the marketplace becomes more global, supplier is now seen as a significant area where industries can cut expenditures and improve their patron service quality. In order to raise their competitive advantages, many firms consider Supply Chain Management (SCM) outsourcing as very significant. A successful supplier choice plays a significant role in building the long-term relationships between the outsourcing firm and a supplier.

In this study, nondiscretionary SBM model was discussed. In addition to developing stochastic version of the nondiscretionary SBM model, we attained the deterministic equivalent of the stochastic version which can be changed to a quadratic problem. As a numerical example, the proposed approach was also applied to data of twenty suppliers. Sensitivity analysis of the proposed model was illustrated.

The problem considered in this study is at initial stage of investigation and further researches can be done based on the results of this study. Some of them are as follow:

Similar research can be repeated for supplier selection in the presence of both deterministic data and fuzzy data
Similar research can be repeated for supplier selection in the presence of both stochastic data and slightly non-homogeneous DMUs
This study used the proposed model for supplier selection. It seems that more fields (e.g., market selection, technology selection, personnel selection, etc.) can be applied

NOMENCLATURE

j = I, … , n collection of suppliers (DMUs)
r = I, … , s the set of outputs
i = I, … , m the set of inputs
A∈Rmxn = A matrix with m rows and n columns:

X = [x1, …, xn]∈Rmxn

Y = [y1, …, yn]∈Rmxn

DMUo = The DMU under investigation
yrj = The rth output of jth DMU
xij = The ith input of jth DMU
yro = The rth output of the DMUo
xio = The ith input of the DMUo
~ = Used to identify the inputs and outputs as random variables with a known joint probability distribution
γ = The best possible relative efficiency achieved by DMUo
Φ-1 = Inverse of standard normal distribution function
Rm = Excesses in inputs
S+ = Shortage in outputs
Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data = rth output shortfalls
Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data = ith input excesses
Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data = Standard deviation of rth output
Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data = Standard deviation of ith input
α = Risk that is between zero and 1
Varyro = rth output variance of the DMUo
Varx1O = ith input variance of the DMUo
ξ, ς and ζ = The external slacks
Image for - Developing a Nondiscretionary Slacks-based Measure Model for Supplier Selection in the Presence of Stochastic Data = Standard normal random variable
βi, γγ = Represent parameters (to be prescribed)
t = A variable which helps a nonlinear model to be converted to a linear model
= j] vector of DMU loadings, determining best practice for the DMUo
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