Data envelopment analysis with common weights: the compromise solution approach

A characteristic of data envelopment analysis (DEA) is to allow individual decision-making units (DMUs) to select the factor weights that are the most advantageous for them in calculating their efficiency scores. This flexibility in selecting the weights, on the other hand, deters the comparison among DMUs on a common base. In order to rank all the DMUs on the same scale, this paper proposes a compromise solution approach for generating common weights under the DEA framework. The efficiency scores calculated from the standard DEA model are regarded as the ideal solution for the DMUs to achieve. A common set of weights which produces the vector of efficiency scores for the DMUs closest to the ideal solution is sought. Based on the generalized measure of distance, a family of efficiency scores called ‘compromise solutions’ can be derived. The compromise solutions have the properties of unique solution and Pareto optimality not enjoyed by the solutions derived from the existing methods of common weights. An example of forest management illustrates that the compromise solution approach is able to generate a common set of weights, which not only differentiates efficient DMUs but also detects abnormal efficiency scores on a common base.     

Ranking decision-making units using common weights in DEA

Data envelopment analysis (DEA) is a linear programming technique that is used to measure the relative efficiency of decision-making units (DMUs). Liu et al. (2008) [13] used common weights analysis (CWA) methodology to generate a CSW using linear programming. They classified the DMUs as CWA-efficient and CWA-inefficient DMUs and ranked the DMUs using CWA-ranking rules. The aim of this study is to show that the criteria used by Liu et al. are not theoretically strong enough to discriminate among the CWA-efficient DMUs with equal efficiency. Moreover, there is no guarantee that their proposed model can select one optimal solution from the alternative components. The optimal solution is considered to be the only unique optimal solution. This study shows that the proposal by Liu et al. is not generally correct. The claims made by the authors against the theorem proposed by Liu et al. are fully supported using two counter examples.     

A DEA approach for fair allocation of common revenue

An issue of considerable importance, how to allocate a common revenue in an equitable manner across a set of competing entities. This paper introduces a new approach to obtaining allocation common revenue on all decision making units (DMUs) in such a way that the relative efficiency is not changed. In this method for determining allocation common revenue dose not need to solving any linear programming. A numerical example is provided to illustrate the results of the analysis.     

Within-group common weights in DEA: An analysis of power plant efficiency

In many real world applications where DEA is applied, DMUs can often be put into groups, such as those which may be under a single management team. This often means that the multipliers used within a group should be common across that group’s members. The case example examined in this regard is one involving a set of power plants, with each containing a set of power units under a common plant management. We develop a goal-programming model for this setting that seeks to derive such a common-multiplier set. The important feature of this multiplier set is that it minimizes the maximum discrepancy among the within-group scores from their ideal levels.     

An epsilon-free approach for finding the most efficient unit in DEA

Data envelopment analysis (DEA), considering the best condition for each decision making unit (DMU), assesses the relative efficiency and partitions DMUs into two sets: efficient and inefficient. Practically, in traditional DEA models more than one efficient DMU are recognized and these models cannot rank efficient DMUs. Some studies have been carried out aiming at ranking efficient DMUs, although in some cases only discrimination of the most efficient unit is desirable. Furthermore, several investigations have been done for finding the most CCR-efficient DMU. The basic idea of the majority of them is to introduce an integrated model which achieves an optimal common set of weights (CSW). These weights help us identify the most efficient unit in an identical condition.     

A note on DEA vs principal component analysis: An improvement to Joe Zhu's approach

This paper considers a previous article published by Zhu in the European Journal of Operational Research which describes a joint use of data envelopment analysis (DEA) and principal component analysis (PCA) in ranking of decision making units (DMUs). In Zhu's empirical study, DEA and PCA yield a consistent ranking. However, this paper finds that in certain instances, DEA and PCA may yield inconsistent rankings. The PCA procedure adopted by Zhu is slightly modified in this article by incorporating other important features of ranking that Zhu has not considered. Numerical results reveal that both approaches show a consistency in ranking with DEA when the data set has a small number of efficient units. But, when a majority of the DMUs in the sample are efficient, only the modified approach produces consistent ranking with DEA.     

均衡及约束凸优化问题公共解的一般迭代算法

梯度投影法在解决约束凸极小化问题中起到了重要的作用. 基于Tian的一般迭代算法, 本文将梯度投影法和平均算子方法相结合, 首次提出隐式和显式的复合迭代算法, 寻求均衡问题和约束凸极小化问题的公共解. 在适当条件下, 获得了强收敛定理.     

Characterizing an equitable allocation of shared costs: A DEA approach

In many applications to which DEA could be applied, there is often a fixed or common cost which is imposed on all decision making units. This would be the case, for example, for branches of a bank which can be accessed via the numerous automatic teller machines scattered throughout the country. A problem arises as to how this cost can be assigned in an equitable way to the various DMUs. In this paper we propose a DEA approach to obtain this cost allocation which is based on two principles: invariance and pareto-minimality. It is shown that the proposed method is a natural extension of the simple one-dimensional problem to the general multiple-input multiple-output case.     

Supporting rural telecommunications: A compromise solutions approach

Rural telecommunication networks can foster the development of needed infrastructures for rural residents in an economically viable way. Thanks to emerging broadband fiber optics technology, rural telecommunications can be established via hub cities that function as service centers for neighboring smaller rural areas. Determining hub locations typically requires trade-offs among conflicting criteria. Policy makers typically set their goals as target values. The main aim of this paper is to present a zero—one compromise programming model that reflects such policy makers' target-setting behavior under a multiple criteria environment. A case in which the model is applied to the location of hub telecommunication centers in an American state further illustrates the applicability of the method.     

A viscosity approximation method for finding a common solution of fixed points and equilibrium problems in Hilbert spaces

In this paper, we introduce an iterative method for finding a common element of the set of fixed points of a nonexpansive mapping and the set of common fixed points of a countable family of nonexpansive mappings in Hilbert spaces. Using the result we consider a strong convergence theorem in variational inequalities and equilibrium problems. The result present in this paper extend and improve the corresponding result of Qin et al. (Nonlinear Anal 69:3897–3909, 2008), Plubtieng and Punpaeng (J Math Anal Appl 336:455–469, 2007) and many others.     

A novel heuristic approach to determine compromise management for end-of-life electronic products

The European Union directive of Waste Electrical and Electronic Equipment for recycling end-of-life (EOL) products has had a significant impact on global enterprises. Recent studies have shed light on optimization of the EOL process. In addition to identifying the most economical EOL process, identifying the EOL process with the smallest environmental load is equally important, and this is thus a bi-criteria optimization problem. This study attempts to optimize EOL processes for electronic products based on a three-stage heuristic approach, which simultaneously minimizes cost and environmental impact. The proposed heuristic approach then assesses the most common disassembly and recycling processes by using the characteristics of electronic product recycling. Next, the best process for this bi-criteria optimization problem is identified by using the compromise programming method. The empirical analysis is based on data for notebook, and the potential impact on best EOL processes when notebook adopt new product designs is also discussed.     

A non-radial DEA approach to measuring environmental performance

Data envelopment analysis (DEA) has gained great popularity in environmental performance measurement because it can provide a synthetic standardized environmental performance index when pollutants are suitably incorporated into the traditional DEA framework. Past studies about the application of DEA to environmental performance measurement often follow the concept of radial efficiency measures. In this paper, we present a non-radial DEA approach to measuring environmental performance, which consists of a non-radial DEA-based model for multilateral environmental performance comparisons and a non-radial Malmquist environmental performance index for modeling the change of environmental performance over time. A case study of OECD countries using the proposed non-radial DEA approach is also presented. It is found that the environmental performance of OECD countries as a whole has been improved from 1995 to 1997.     

A von Neumann alternating method for finding common solutions to variational inequalities

By modifying von Neumann’s alternating projections algorithm, we obtain an alternating method for solving the recently introduced Common Solutions to Variational Inequalities Problem (CSVIP). For simplicity, we mainly confine our attention to the two-set CSVIP, which entails finding common solutions to two unrelated variational inequalities in Hilbert space.     

A subjective and objective integrated approach to determine attribute weights

This paper proposes an integrated approach to determine attribute weights in the multiple attribute decision making (MADM) problems. The approach makes use of the subjective information provided by a decision maker and the objective information to form a two-objective programming model. Thus the resultant attribute weights and rankings of alternatives reflect both the subjective considerations of a decision maker (DM) and the objective information. An example is used to illustrate the applicability of the proposed approach.     

A new framework for the solution of DEA models

We provide an alternative framework for solving data envelopment analysis (DEA) models which, in comparison with the standard linear programming (LP) based approach that solves one LP for each decision making unit (DMU), delivers much more information. By projecting out all the variables which are common to all LP runs, we obtain a formula into which we can substitute the inputs and outputs of each DMU in turn in order to obtain its efficiency number and all possible primal and dual optimal solutions. The method of projection, which we use, is Fourier–Motzkin (F–M) elimination. This provides us with the finite number of extreme rays of the elimination cone. These rays give the dual multipliers which can be interpreted as weights which will apply to the inputs and outputs for particular DMUs. As the approach provides all the extreme rays of the cone, multiple sets of weights, when they exist, are explicitly provided. Several applications are presented. It is shown that the output from the F–M method improves on existing methods of (i) establishing the returns to scale status of each DMU, (ii) calculating cross-efficiencies and (iii) dealing with weight flexibility. The method also demonstrates that the same weightings will apply to all DMUs having the same comparators. In addition it is possible to construct the skeleton of the efficient frontier of efficient DMUs. Finally, our experiments clearly indicate that the extra computational burden is not excessive for most practical problems.     

A revised and generalized model with improved discrimination for finding most efficient DMUs in DEA

In this paper we first propose a generalized model which in its special case revises a recently proposed model for finding most BCC-efficient DMU. Then, by explaining about the drawbacks of existing approaches, an algorithm will be developed to consider all possible alternative optimal solutions and determine the set of most efficient DMUs. Another model also will be proposed to provide more discrimination which can be used to select a single most efficient DMU, when it is desirable. The proposed approaches of this paper are applicable for all assumptions about returns-to-scale, and can be utilized to select a DMU or provide a full ranking of DMUs. The contents of the paper are illustrated through numerical examples.     

A positive approach to estimating the weights for quadratic multiple objective programming

This paper integrates positive and normative approaches to modelling. The normative approach uses assumptions associated with multiple objective programming. The positive approach uses past observations to estimate the weights associated with each objective criteria. The technique encompasses both linear and non-linear objectives such as profit, cost and risk as well as quadratic calibration terms. The proposed methodology minimizes the sum of squared errors about the ideal multiple objective function, that is one that would reproduce observed results, rather than to minimize errors between fitted and observed activity levels. The technique removes the need to rely upon the use of abstract restraints normally applied to mathematical programming methods and provides a more objective means of testing the appropriateness of a model than previously. The technique has many applications in the field of mathematical modelling such as forecasting and analysing changes in decision-making and behaviour.     

Proposing a method for fixed cost allocation using DEA based on the efficiency invariance and common set of weights principles

One of the applications of data envelopment analysis is fixed costs allocation among homogenous decision making units. In this paper, we first prove that Beasley’s method (Eur J Oper Res 147(1):198–216, 2003), whose infeasibility has been claimed by Amirteimoori and Kordrostami (Appl Math Comput 171(1):136–151, 2005), always has a feasible solution and the efficiency invariance principle does not necessarily satisfy in Amirteimoori and Kordrostami’s method (Appl Math Comput 171(1):136–151, 2005). Hence, we present two equitable methods for fixed cost allocation based on the efficiency invariance and common set of weights principles such that, if possible, they help meet these two principles. In the first method, the costs are allocated to DMU in such a way that the efficiency score of DMUs does not change, and simultaneously this allocation has the minimum distance from the allocation that has been obtained with a common set of weights. However, in the second method, the costs are allocated in such a way that input and output of all units have a common set of weights and it has the minimum distance from the allocation that satisfies the efficiency invariance principle. Moreover, both methods, consider the satisfaction of each unit of the allocated cost. Finally, the proposed method is illustrated by two real world examples.     

Revising an iterative algorithm for finding a common solution of a system of variational inequalities for two inverse strongly accretive operators

In this paper, we give a short and simple proof of the recent result of Katchang and Kumam (Positivity 15:281–295, 2011). In our proof, we do not assume the uniform convexity of a space.     

A DEA game model approach to supply chain efficiency

Data envelopment analysis (DEA) is a useful method to evaluate the relative efficiency of peer decision making units (DMUs). Based upon the definitions of supply chain efficiency, we investigate the efficiency game between two supply chain members. It is shown that there exist numerous Nash equilibriums efficiency plans for the supplier and the manufacturer with respect to their efficiency functions. A bargaining model is then proposed to analyze the supplier and manufacturer's decision process and to determine the best efficiency plan strategy. DEA efficiency for supply chain operations is studied for the central control and the decentralized control cases. The current study is illustrated with a numerical example.