Traditional data envelopment analysis (DEA) models are concerned with measuring the relative efficiency of units with respect to multiple inputs and multiple outputs. These models assume real-valued inputs and outputs. In these models, the role of the measures in relation to the input and/or output is known, and these models also neglect the intervening or linking activities. However, in the real world, there are situations where some inputs and/or outputs can only take integer values, and a measure can also play the role of either an input or an output. In this paper, we propose network DEA models that can handle intermediate products when some inputs and/or outputs have integer values and flexible factors are available.

In Network-Data Envelopment Analysis (NDEA), the values of all data are to be known deterministically, whereas there are many real-world situations in which some uncertain data is available. This paper develops the NDEA models to include uncertain data of interval type. Since the data are uncertain of interval type, it is natural to obtain the ranges of the efficiency measures over the data. The intermediate activities play a central role in measuring the efficiency measures of units with a two-stage structure. The scholars introduced six scenarios to model the intermediate products in NDEA. As the intermediate products have the dual role of outputs of the previous stage and inputs of the next stage, obtaining the efficiency bounds faces some difficulties. This paper provides an explicit formula for computing the lower bound and the upper bound of efficiency measures under different scenarios. The results show that solving only two linear programming problems is needed to derive the bounds of the efficiency measures under some situations, while the computational efforts increase in other cases. Finally, a case study has been used to illustrate the proposed method.

One of the main important issues in Data Envelopment Analysis (DEA) is to recognize the set of anchor points which is the subset of the extreme efficient points of the production possibility set (PPS). An anchor point is an extreme efficient point which is located on the intersection of the strong efficient frontier and the weak efficient frontier. In the other word, each anchor point delineate the strong efficient frontier from the weak efficient frontier. So, if a decision making unit (DMU) is an anchor point, then there is at least one supporting hyperplane whose the gradient vector has some of zero components, and so some input\output factor does not play any role in the performance of the unit under evaluation. The concept of anchor point was used in DEA for the generation of unobserved DMUs in order to extend the DEA efficient frontier and so, this concept plays a critical role in the DEA theory and its applications. Given the importance of the anchor pints in the DEA literature, this study focuses on finding the anchor points and presents a new method to search the anchor points of the PPS under the variable returns to scale (VRS) assumption. For this purpose, we use the definition of the anchor points and present an approach to find the anchor points of the PPS. The proposed method is based on finding the weak and strong defining supporting hyperplanes passing through the unit under evaluation. The main advantage of the proposed method is that it exactly uses the definition of the anchor points to provide the approach and it is very simple to use and the anchor points can be easily identified by solving two simple models. In addition, the proposed approach is such that in addition to determining the anchor points, it also finds two important defining supporting hyperplanes on the PPS, which can be used in many problems in DEA. The potentially of the proposed method is illustrated by some numerical examples, reported in the literature to compare the proposed method with the existing methods.

As an important concept in data envelopment analysis (DEA), elasticity measure has wide theoretical and practical applications in formulating various economic concepts. Anchor points also appear to be particularly interesting and highly useful in DEA, especially for recognizing a decision making unit (DMU) as a benchmark. This paper is an attempt to use left-and right-hand elasticity measures to present a novel definition (characterization) for anchor points. The study results reveal that if there exists an increase in a bundle of input with no rate of change in a bundle of output or if there is a decrease in a bundle of output, but a bundle of input has no rate of change, then such an extreme point is the anchor point.

One of the attractive issues in Data Envelopment Analysis (DEA) literature is to find the anchor points of the production possibility set (PPS). Each extreme efficient unit which is located on the intersection of the strong and weak efficient frontiers of the PPS, is called an anchor point. In the other word, a decision making unit (DMU) is an anchor point, if there is at least one supporting hyperplane at the unit under consideration, in the situation that some components of its gradient vector are equal to zero, and so some input or output factors do not play any role in the performance of that unit. This study presents a new method to identify the anchor points of the PPS under the variable returns to scale (VRS) assumption and in the presence of the uncertain data. The proposed method is based on the robust optimization technique and finding the weak and strong defining supporting hyper-planes passing through the unit under evaluation. The potentially of the proposed method is illustrated by a data set, includes 20 banks in Iran.

In this paper, the suspicious units including anchor, terminal, and exterior units are investigated as important subsets of vertex units. Based on the concept of separating hyperplanes, an alternative definition of vertex units in data envelopment analysis is presented. Moreover, an advanced mathematical model for obtaining the separating hyperplane that splits up a vertex unit from the other units is proposed. Utilizing the core concept of separating hyperplanes, the special geometry of terminal units enables us to introduce a new definition for terminal units. Thus, some theorems have been proved which provide necessary and sufficient conditions for obtaining terminal units. We made use of the concept of supporting hyperplanes to provide a basic definition for exterior units and present a careful model for discovering exterior units. Also, based on the concept of supporting hyperplanes, different definitions of anchor units have been represented. Finally, the relationship between the sets of exterior, terminal and anchor units have been demonstrated in a theorem.

Received 24 February 2016, Revised 28 April 2016, Accepted 8 May 2016 Abstract The problem of utilizing undesirable (bad) outputs in DEA models often need replacing the assumption of free disposability of outputs by weak disposability of outputs. The Kuosmanen technology is the only correct representation of the fully convex technology exhibiting weak disposability of bad and good outputs. Also, there are some specific features of non-radial data envelopment analysis (DEA) models for obtaining all projections of a decision making unit (DMU) on the boundary of production possibility set (PPS) or efficient frontier. Production technologies in DEA are modeled by polyhedral sets that envelop the observed DMUs. Because the efficient frontiers of DEA technologies are generally non-smooth and are characterized by different faces, thus, all projections of a DMU on efficient frontier can not belong to different faces that do not have common points. The rationale behind abovementioned statement is as follows: if all projections of a DMU belong to different faces then the interior points of PPS will become efficient that contradicts the principles of optimality conditions in linear programming models. Therefore all projections would belong to a unique face that is called minimum face. In this paper we propose a procedure to find minimum face and so all projections of a DMU on efficient frontiers in non-radial DEA models with undesirable outputs. This leads us to an interesting algorithm to obtain minimum face.

In this paper, an extension of the generalized data envelopment analysis (GDEA) model has been introduced. In this generalization different cases of production possibility set (PPS) of the extended generalized data envelopment analysis (EGDEA) model are investigated. The proposed EGDEA model is used to evaluate decision’s making units (DMUs) either with full convexity assumption imposed on inputs and outputs (CCR and BCC models) or without considering convexity assumption for each input and output (FDH model). We have proposed a model in the EGDEA, which can treat each input and output individually with respect to the convexity assumption.

In this paper characteristics of defining hyperplanes of constant returns to scale technology in DEA have been investigated. A defining hyperplane namely H is a type of hyperplane that with the elimination of H, the production possibility set (PPS) will be enlarged (In this paper a defining hyperplane exactly is the full dimensional efficient facet (FDEF) and may be found in Olesen and Peterson (1996, 2003)). The point of view of some of the characteristics is conceptual and the interpretation of defining hyperplanes of constant returns to scale technology can be achieved by these conceptual characteristics. However, some of the characteristics are practical and one can easily utilize them in practice. Some parts of topology and convex analysis have been considered to show the truth of characteristics.

In this study we consider the problem of analyzing the performance of a decision making unit in to two directions: at first stage, we consider resource utilization versus output performance and at second stage, we investigate inter-unit comparison or group evaluation when the data are interval. A famous method that deals with the second stage of the forgoing proposed method is data envelopment analysis (DEA). Although comparing DMUs only with each other in DEA have some shortcomings that we are interested to overcome these shortcomings in the case when the data are interval. Some parts of game theory and optimization are considered to propose a solution.

The conventional game theory is based on known payoffs. In the real situations, usually the payoffs are not known and have to be approximated. In this paper, a method for the two-person zero-sum and non-zero-sum games that the payoffs are represented by fuzzy data, has been investigated. The procedure is based on Linear Complementarity Problem (LCP) which unifies bimatrix games.

The relative efficiency of a DMU is the result of comparing the inputs and outputs of the DMU and those of other DMUs in the PPS (production possibility set). If the inputs and outputs are fuzzy, the DMUs cannot be easily evaluated and ranked using the obtained efficiency scores. In this paper, presenting a new idea for ranking of DMUs with fuzzy data. And finally, we introduce a numerical example.

In this paper we investigate problems of consensus-making among individuals or organizations with multiple criteria for evaluating performance when the players are supposed to be egoistic and the score for each criterion for a player is supposed to be a triangular fuzzy number. Egoistic means that each player sticks to his superiority regarding the criteria. The concept that is developed in optimization leads the problem to a dilemma called ‘egoist’s dilemma’. Some parts of game theory are considered to propose a solution.