The Optimistic-Pessimistic Ranking in the Chance Constrained DEA

 Recently, Khodabakhshi and Aryavash have introduced a ranking method [Applied Mathematics Letters, 25 (2012) 2066-2070.], which is based on an optimistic-pessimistic approach of data envelopment analysis (DEA). This method ranks all decision making units according to a combination of their minimum and maximum possible efficiency scores which are determined by solving two linear programming models. This method can only rank decision making units (DMUs) with deterministic input and output indexes, whereas in the real application the data are often imprecise. In this paper, the chance constrained version of this ranking method is presented. KeywordsData envelopment analysis, Stochastic ranking model, Chance constrained programming, Optimistic-pessimistic approach


INTRODUCTION
Data envelopment analysis (DEA) is a decisional technique for performance analysis of decision making units (DMUs).This mathematical instrument was introduced by Charnes et al. (1978) and then developed by other researchers.Nowadays, DEA is widely used in management science, operational research, system engineers, decision analysis and so on.One of the applications of DEA is to rank DMUs according to their efficiencies.The traditional DEA models divide DMUs into two efficient and inefficient groups, whereas in practice, there is often a need to fully rank them.This is one of the shortcomings of the traditional DEA models to rank DMUs.Another drawback of the traditional DEA models is that they cannot discriminate among DMUs when the number of inputs and outputs is too high relative to the number of DMUs.In such situations, the real position of some DMUs cannot be reflected.Furthermore, the traditional DEA models evaluate the DMUs only from the optimistic viewpoint.In the other words, they do not pay attention to the weak points of DMUs.This lack of consideration from the pessimistic attitude misses some important information about the performance of DMUs.
So far, several ranking method have been introduced in the DEA context.A thorough review on ranking methods up 2002 can be found in Adler et al. (2002).Sexton et al. (1986) proposed the ranking method of DMUs based on a cross-efficiency ratio matrix.Cook et al. (1992) developed prioritization models to rank only the efficient units in DEA.Andersen and Petersen (1993) proposed super-efficiency models for ranking only efficient units in the DEA (see also Li et al. (2007)).The benchmarking ranking of efficient DMUs initially developed by Torgersen et al. (1996).Cooper and Tone (1997) ranked the DMUs according to scalar measures of inefficiency in DEA, based on the slack variables.Jahanshahloo et al. (2007) proposed a DEA ranking system based on changing the reference set.Liu and Peng (2008) proposed the common weights analysis (CWA) to determine a set of indices for common weights to rank efficient DMUs of DEA.Khodabakhshi and Aryavash (2012) introduced another ranking method which is based on both pessimistic and optimistic attitudes of DEA.
Although the DEA models have many advantages, they don't allow variations in input and output data.Recently, some models have been presented which incorporate data variation in the DEA models.Jahanshahloo et al. (2009) proposed a method which Ranks DMUs by l 1 -norm with fuzzy data in DEA.Kao and Liu (2003) designed a mathematical programming approach to fuzzy efficiency ranking.Ma and Li (2008) introduced a fuzzy ranking method with range reduction techniques.Also, Zerafat Angiz et al. (2010) proposed another ranking method in the fuzzy environment.
Chance constraints are incorporated in DEA in Olesen and Petersen (1999) leading to a stochastic (chance constrained) programming model.The traditional DEA model envelopes observed input-output vectors for each DMU and the chance constrained DEA model envelopes confidence regions for each DMU.This paper analysis the relationship between the probability levels specified in the chance constraints and the corresponding probability levels defining the confidence regions.The efficiency by stochastic data has been analyzed by Sengupta (2000).In addition Morita and Seiford (1999) Khodabakhshi (2010aKhodabakhshi ( , 2011) ) and Khodabakhshi et al. (2010b) proposed some super efficiency models in the stochastic environment.Khodabakhshi and Asgharian (2009) presented an input relaxation measure of efficiency in stochastic DEA.In addition, Khodabakhshi (2009) estimated most productive scale size with stochastic data in DEA.
Recently, Khodabakhshi andAryavash (2012, 2014a,b) have introduced an optimistic-pessimistic approach in the DEA literature which can be applied for ranking DMUs.In this approach, DMUs are ranked according to a combination of the minimum and maximum possible efficiency scores of DMUs which are computed by solving two linear programming models.The main idea of this method is to estimate the efficiency scores of DMUs under the assumption that the sum of scores of all DMUs equals to one.This method does not have the mentioned limitations of the pervious DEA based ranking methods.In this paper, this ranking method is extended with stochastic inputs and outputs.Then a deterministic equivalent for this model is obtained and converted to a quadratic program.
The rest of the paper has been structured as follows.In Section 2, the optimistic-pessimistic ranking method is briefly demonstrated.In Section 3, this ranking method is extended by according its chance constrained programming formulations.In Section 4, our method is illustrated using an example.In Section 5, we conclude the paper with a summary and a sketch of further research opportunities.

THE OPTIMISTIC-PESSIMISTIC APPROACH
Assume that there are n decision making units DMU under the assumption that the sum of scores of all DMUs equals to unity, this means that: Based on this assumption, they have derived the following linear programming model: Min and Max y u This model must be run two times.First, is minimized to determine its minimum value ( ) and then is maximized to determine its maximum value ( ).In fact, these scores are obtained using the pessimistic and optimistic viewpoints, respectively.Now, we turn to the dual (envelopment) model associated with model (2).Considering the objective function of this model, two dual models must be obtained, one for minimizing problem and the other for maximizing problem.The dual models of (2) can be written as follows: and, The minimum ( ) q and maximum ( ) q possible efficiency scores of 0 DMU are respectively estimated using the models (3,4).The models (3,4) are respectively called optimistic and pessimistic models.In fact, the ( ) q and ( ) q are obtained using the pessimistic and optimistic viewpoints, respectively.Hence, we have the following interval for the efficiency score of j DMU : The intervals (5) can be rewritten as following convex combinations: To aggregate L j q and U j q into a single score, a value of interval [0, 1] must be assigned to parameter j l .To determine the efficiency of DMUs in an equitable way, the values of all j l must be equally selected.So, we must have Hence, the system of n equations in the 2n variables equations ( 6) is transformed to the following system of n equations with 1 n + variables: On the other hand, based on assumption (1), we have . Therefore, the values of all j q can be determined by solving the following system of n + 1 linear equations with n + 1 variables: We rank DMUs according the answer of this system which are depicted by M j q ( 1, , ) j n =  .

THE CHANCE CONSTRAINED RANKING
In what follows, we introduce stochastic version of the models (3) and ( 4 In this model, P means probability, α is a predetermined value between 0 and 1 which specifies the significance level.Now, we exploit the normality assumption to introduce a deterministic equivalent to this model.For each {1,..., } r s Î we have The inequality (10) can be rewritten as follows: In inequality (12), j is the cumulative distribution function (cdf) of a standard Normal random variable and 1 jis its inverse.
Similarly, for each {1,..., } i m Î and for each {1,..., } j n Î , we have: ( , ) The inequality ( 13) can be rewritten as follows: Therefore, the deterministic equivalent of model ( 9) can be represented by: , Also, it can be shown that the corresponding stochastic version of the model ( 4) is as follows: In a similar way, it can be shown that the deterministic equivalent of model ( 17) can be represented by: 1 Using the aforementioned property of normal distribution, and replacing non-negative variables I ij w and O rj w , respectively, by ( , ) s l q and ( ) s l we obtain the following deterministic models which are quadratic programs: , , 0, and , , 0.

A numerical example
In this section, our method is illustrated using a numerical example.In this example, there are four DMUs with one input and two outputs.We assume that all indexes are stochastic variables with normal distribution.The mean values of these variables are shown in the columns (2-4) of Table 1.Also, we assume that the input and outputs of all DMUs have the same variance To compute the results of models (19,20) we have chosen 0.55 a = for which 1 0.12.j -= Furthermore, we assume that the input and outputs for different DMUs are independent.This independence assumption then implies that Running the models (19,20) on these data, the intervals of scores of DMUs have been estimated which can be seen in the fifth column of Table 1.Then, the integrated scores of DMUs have been determined by solving the system (8) and depicted in the sixth column of the table.Finally, DMUs have been ranked according to their aggregated scores as can be seen in the last column of the table.
The interval score of the first DMU is [0.001, 0.546] , this means that the minimum and maximum possible efficiency scores of this DMU at the significance level 0.55 a = are respectively 1 0.001 L q = and 1 0.546 U q = .According to these values, this DMU has the worst performance from the pessimistic viewpoint and the best performance from the optimistic viewpoint.So, only one of these attitudes cannot portray the real position of this DMU, so we should use both attitudes.To this end, these values are aggregated to the score 1 0.357 M q = by using the system (8), and so, gain the second position of ranking.The score 1 M q obviously depicts the relative efficiency of 1 DMU using both of its week and strong points.Hence, it is more logical and reliable than the efficiency scores which are based on only one of these attitudes.

Conclusion
The proposed method of this study has some advantages in comparison with the other ranking methods.In this approach, the weights of inputs and outputs are endogenous.Also, it is based on the both optimistic and pessimistic attitudes of DEA, so its results can be more reliable and equitable than the ranking methods which are based on only one of these attitudes.This model has high discrimination power, and can be easily used when the number of inputs and outputs is too high relative to the number of DMUs.Furthermore, one can get a full rank of all DMU using this approach.Finally, in the future researches, the presented method can be developed to rank the DMUs with ordinal data.
inputs (i = 1,…, m) into s outputs (r = 1,…, s) and 0 DMU is an under evaluation DMU.Also, suppose that all inputs and outputs are non-negative deterministic numbers.Khodabakhshi and Aryavash (2012, 2014a, b) introduced a new DEA models which determines the efficiency scores of 0 have discussed DEA efficiency reliability and probability.Also, Cooper et al. (1996) and Cooper et al. (1998) introduced stochastic formulation of the original models which incorporate possible uncertainty in the inputs and/or outputs.Jess et al. (2001) introduced a semi-infinite programming model in DEA to study an interesting chemical engineering problem.Khodabakhshi (2007) presented a super-efficiency model based on improved outputs in DEA.Also, ) (

Table 1 :
The data and results of example