Alternative mixed integer linear programming models for identifying the most efficient decision making unit in data envelopment analysis

A mixed integer linear model for selecting the best decision making unit (DMU) in data envelopment analysis (DEA) has recently been proposed by Foroughi [Foroughi, A. A. (2011a). A new mixed integer linear model for selecting the best decision making units in data envelopment analysis. Computers and Industrial Engineering, 60(4), 550–554], which involves many unnecessary constraints and requires specifying an assurance region (AR) for input weights and output weights, respectively. Its selection of the best DMU is easy to be affected by outliers and may sometimes be incorrect. To avoid these drawbacks, this paper proposes three alternative mixed integer linear programming (MILP) models for identifying the most efficient DMU under different returns to scales, which contain only essential constraints and decision variables and are much simpler and more succinct than Foroughi’s. The proposed alternative MILP models can make full use of input and output information without the need of specifying any assurance regions for input and output weights to avoid zero weights, can make correct selections without being affected by outliers, and are of significant importance to the decision makers whose concerns are not DMU ranking, but the correct selection of the most efficient DMU. The potential applications of the proposed alternative MILP models and their effectiveness are illustrated with four numerical examples.     

一种部分权重信息下两阶段多属性决策方法

研究了只有部分权重信息(区间数),属性值为实数的多属性决策问题。通过建立单目标线性规划模型求出各方案的局部最优权重向量,通过将各方案的局部最优权重向量与全局最佳属性权重的偏差最小化,建立多目标优化模型,并将其转化成一个单目标优化模型计算出全局最优属性权重向量,从而得到了一个两阶段的多属性决策方法。通过实例说明了该方法的可行性和有效性。     

Developing a novel inverse data envelopment analysis (DEA) model for evaluating after-sales units

This paper proposes a novel model of inverse data envelopment analysis (IDEA) based on the slack-based measure (SBM) approach. The developed inverse SBM model can maintain relative efficiency of decision making units (DMUs) with new input and output. This model can also measure the input and output volumes when a decision maker (DM) increases efficiency score. The inverse SBM model is a kind of multi-objective non-linear programming (MONLP) problem, which is not easy to solve. Therefore, we suggest a linear programming model for solving inverse SBM model. In this model efficiency score of DMU under evaluation remains unchanged. Furthermore, we suggest an optimal combination of inputs and outputs in the production possibility set (PPS). A case study is presented to demonstrate the efficacy of our proposed model.     

Improving the discrimination power and weights dispersion in the data envelopment analysis

Data envelopment analysis (DEA) has been a very popular method for measuring and benchmarking relative efficiency of peer decision making units (DMUs) with multiple input and outputs. Beside of its popularity, DEA has some drawbacks such as unrealistic input–output weights and lack of discrimination among efficient DMUs. In this study, two new models based on a multi-criteria data envelopment analysis (MCDEA) are developed to moderate the homogeneity of weights distribution by using goal programming (GP). These goal programming data envelopment analysis models, GPDEA-CCR and GPDEA-BCC, also improve the discrimination power of DEA.     

Fixed-order FIR approximation of linear systems from quantized input and output data

The problem of identifying a fixed-order FIR approximation of linear systems with unknown structure, assuming that both input and output measurements are subjected to quantization, is dealt with in this paper. A fixed-order FIR model providing the best approximation of the input–output relationship is sought by minimizing the worst-case distance between the output of the true system and the modeled output, for all possible values of the input and output data consistent with their quantized measurements. The considered problem is firstly formulated in terms of robust optimization. Then, two different algorithms to compute the optimum of the formulated problem by means of linear programming techniques are presented. The effectiveness of the proposed approach is illustrated by means of a simulation example.     

区间值模糊随机多准则决策方法

针对不完全信息的区间值模糊随机多准则决策问题,提出了两种求解方法。第一种方法利用离差最大化构建区间参数线性规划,通过区间数运算法则和定位规划求得最优准则权重向量、状态集结值区间决策矩阵与期望值区间决策矩阵,根据决策者风险偏好水平得到各方案的期望集结值从而确定排序。第二种方法将区间值模糊数决策矩阵转化为直觉模糊数决策矩阵,利用不完全的准则权重,通过规划模型求解,获取各方案在各自然状态下的加权记分函数值与加权精确函数值的区间,利用不完全的状态概率,得到各方案的记分函数期望值与精确函数期望值的区间,根据决策者风险偏好水平,求得各方案的记分函数与精确函数的期望集结值,进而确定方案的排序结果。算例分析验证了两种方法的有效性和可行性。     

Inverse data envelopment analysis model to preserve relative efficiency values: The case of variable returns to scale

This paper studies the inverse Data Envelopment Analysis (inverse DEA) for the case of variable returns to scale (inverse BCC). The developed inverse BCC model can preserve relative efficiency values of all decision making units (DMUs) in a new production possibility set composing of all current DMUs and a perturbed DMU with new input and output values. We consider the inverse BCC model for a resource allocation problem, where increases of some outputs and decreases of the other outputs of the considered DMU can be taken into account simultaneously. The inverse BCC problem is in the form of a multi-objective nonlinear programming model (MONLP), which is not easy to solve. We propose a linear programming model, which gives a Pareto-efficient solution to the inverse BCC problem. However, there exists at least an optimal solution to the proposed model if and only if the new output vector is in the set of current production possibility set. The proposed approach is illustrated via a case study of a motorcycle-part company.     

A new method based on the dispersion of weights in data envelopment analysis

One of the drawbacks of the data envelopment analysis (DEA) is the problem of lack of discrimination among efficient decision making units (DMUs) and hence yielding many numbers of DMUs as efficient. The main purpose of this study is to overcome this inability. In the case in which the minimization of the coefficient of variation (CV) for input–output weights is added to the DEA model, more reasonable and more homogeneous input–output weights are obtained. For this new proposed model based on the CV it is observed that the number of efficient DMUs is reduced, improving the discrimination power. When this new approach is applied to two well-known examples in the literature, and a real-world data of OECD countries, it has been seen that the new model yielded a more balanced dispersion of input–output weights and reduced the number of efficient DMUs. In addition, the applicability of the new model is tested by a simulation study.     

An extended multiple criteria data envelopment analysis model

Several researchers have adapted the data envelopment analysis (DEA) models to deal with two inter-related problems: weak discriminating power and unrealistic weight distribution. The former problem arises as an application of DEA in the situations where decision-makers seek to reach a complete ranking of units, and the latter problem refers to the situations in which basic DEA model simply rates units 100% efficient on account of irrational input and/or output weights and insufficient number of degrees of freedom. Improving discrimination power and yielding more reasonable dispersion of input and output weights simultaneously remain a challenge for DEA and multiple criteria DEA (MCDEA) models. This paper puts emphasis on weight restrictions to boost discriminating power as well as to generate true weight dispersion of MCDEA when a priori information about the weights is not available. To this end, we modify a very recent MCDEA models in the literature by determining an optimum lower bound for input and output weights. The contribution of this paper is sevenfold: first, we show that a larger amount for the lower bound on weights often leads to improving discriminating power and reaching realistic weights in MCDEA models due to imposing more weight restrictions; second, the procedure for sensitivity analysis is designed to define stability for the weights of each evaluation criterion; third, we extend a weighted MCDEA model to three evaluation criteria based on the maximum lower bound for input and output weights; fourth, we develop a super-efficiency model for efficient units under the proposed MCDEA model in this paper; fifth, we extend an epsilon-based minsum BCC-DEA model to proceed our research objectives under variable returns to scale (VRS); sixth, we present a simulation study to statistically analyze weight dispersion and rankings between five different methods in terms of non-parametric tests; and seventh, we demonstrate the applicability of the proposed models with an application to European Union member countries.     

Some properties of Pareto efficiency under the framework of data envelopment analysis

Data envelopment analysis (DEA) has been widely applied to measure the Pareto efficiency of multiple-input and multiple-output decision making units (DMUs). In this paper it is shown that under linear production frontiers DMU efficiency is a weighted arithmetic mean of the efficiencies of the outputs; whereas under loglinear production frontiers DMU efficiency is a weighted geometric mean of the output efficiencies. Furthermore, DMU efficiency can be decomposed with respect to input factors as well, and some results are derived. As a consequence, a modified DEA model is devised, whereby the efficiency of each output (or input) in addition to DMU efficiency is able to be measured in one linear programming solution.     

Type-2 fuzzy multi-objective DEA model: An application to sustainable supplier evaluation

In the last 10 years, sustainable supply chain management (SSCM) has become one of the important topics in business and academe. Sustainable supplier performance evaluation and selection play a significant role in establishing an effective SSCM. One of the techniques that can be used for evaluating sustainable supplier performance is data envelopment analysis (DEA). The conventional DEA methods require accurate measurement of both input and output variables present in the problem. In practice, the observed values of the input and output data present in real-world problems are often imprecise. To cope with this situation, fuzzy DEA models were constructed for expressing relative fuzzy efficiencies of decision-making units (DMUs). However, fuzzy DEA is still limited to fuzzy input/output data while some inputs and outputs might be affected by various factors of uncertainty and information granularity, meaning that they could be better modeled in terms of fuzzy sets of type-2. In this paper, we develop a multi-objective DEA model in a setting of type-2 fuzzy modeling to evaluate and select the most appropriate sustainable suppliers. In the proposed model, both efficiency and effectiveness are considered to describe the integrated productivity of suppliers. In sequel, chance constrained programming, critical value-based reduction methods and equivalent transformations are considered to solve the problem. A detailed case study is employed to show the advantages of the proposed model in terms of measuring effectiveness, efficiency and productivity in an uncertain environment expressed at different confidence levels. At the same time, the results demonstrate that the model is capable of helping decision makers to balance economic, social, and environmental factors when selecting sustainable suppliers.     

基于前景和量子理论的模糊多属性决策方法

针对属性权重不完全确定且属性偏好值为区间直觉模糊数的多属性决策问题,提出一种基于前景理论和量子进化算法的模糊多属性决策方法。该方法根据前景理论及模糊数距离公式,定义区间直觉模糊数的前景价值函数,同时将决策者对方案的风险偏好纳入决策行为中,以此来构建方案综合前景值最大化的非线性规划模型。通过引入量子进化算法,求解模型得出最优权重向量。最终根据方案前景值确定出方案的排序。该方法适用于模糊决策环境,能满足决策者不提供确定属性权重的要求,并充分考虑决策者风险心理因素对决策行为的影响,具有广泛的应用价值。数值算例说明了该方法的有效性和可行性。     

Attribute weights determination models for consensus maximization in multiple attribute group decision-making

Multiple attribute group decision making (MAGDM) is an important research field of decision science. A critical aspect of MAGDM is to determine the weights of attributes. In this paper, we study the MAGDM problem in which the attributes are given in real numbers or interval numbers, and the information about attribute weights is completely unknown or partially known. We first get the group opinion by fusing all individual opinion with each decision-makers' importance and introduce the deviation variable of each individual opinion and the group opinion. Then, we develop a quadratic programming model by means of minimizing the sum of all the deviation values, and a simple and straightforward formula for determining attribute weights can be derived from solving the developed models. We also establish a generalized model for solving MAGDM problems with partial weight information on attributes. In addition, we establish some similar models for MAGDM with interval attribute values. At last, we apply our models to a practical problem of a military unit purchasing new artillery weapons.     

The effect of concave and convex weight adjustments onself-organizing maps

Two nonlinear models of weight adjustments of self-organizing maps are derived to obtain desirable densities of output units, one that approaches the probability distribution p(xi) of the inputs and one that approaches a uniform distribution. If a convex model is used to adjust weights, the density of output units can be made to approach p(xi) instead of the p(xi)(2/3) which results from the linear weight adjustment of Kohonen's self-organizing maps. If a concave model of weight adjustments is used, the density approaches a uniform distribution and the winner frequency distribution of output units is proportional to p(xi). The former can provide more efficient data representations for vector quantization, while the latter can provide more meaningful measures for cluster analysis. Numerical demonstrations validate the mathematical derivations. The convergence of the concave model is faster than the linear and convex models while the convergence of the convex model is comparable to that of the linear model.     

Ranking efficient decision-making units in data envelopment analysis using fuzzy concept

This paper introduces a new mathematical method for improving the discrimination power of data envelopment analysis and to completely rank the efficient decision-making units (DMUs). Fuzzy concept is utilised. For this purpose, first all DMUs are evaluated with the CCR model. Thereafter, the resulted weights for each output are considered as fuzzy sets and are then converted to fuzzy numbers. The introduced model is a multi-objective linear model, endpoints of which are the highest and lowest of the weighted values. An added advantage of the model is its ability to handle the infeasibility situation sometimes faced by previously introduced models.     

A comprehensive fuzzy DEA model for emerging market assessment and selection decisions

The changing economic conditions have challenged many financial institutions to search for more efficient and effective ways to assess emerging markets. Data envelopment analysis (DEA) is a widely used mathematical programming technique that compares the inputs and outputs of a set of homogenous decision making units (DMUs) by evaluating their relative efficiency. In the conventional DEA model, all the data are known precisely or given as crisp values. However, the observed values of the input and output data in real-world problems are sometimes imprecise or vague. In addition, performance measurement in the conventional DEA method is based on the assumption that inputs should be minimized and outputs should be maximized. However, there are circumstances in real-world problems where some input variables should be maximized and/or some output variables should be minimized. Moreover, real-world problems often involve high-dimensional data with missing values. In this paper we present a comprehensive fuzzy DEA framework for solving performance evaluation problems with coexisting desirable input and undesirable output data in the presence of simultaneous input–output projection. The proposed framework is designed to handle high-dimensional data and missing values. A dimension-reduction method is used to improve the discrimination power of the DEA model and a preference ratio (PR) method is used to rank the interval efficiency scores in the resulting fuzzy environment. A real-life pilot study is presented to demonstrate the applicability of the proposed model and exhibit the efficacy of the procedures and algorithms in assessing emerging markets for international banking.     

An integer‐valued data envelopment analysis model with bounded outputs

Integer‐valued data envelopment analysis (DEA) with alternative returns to scale technology has been introduced and developed recently by Kuosmanen and Kazemi Matin. The proportionality assumption of their introduced “natural augmentability” axiom in constant and nondecreasing returns to scale technologies makes it possible to achieve feasible decision‐making units (DMUs) of arbitrary large size. In many real world applications it is not possible to achieve such production plans since some of the input and output variables are bounded above. In this paper, we extend the axiomatic foundation of integer‐valued DEA models for including bounded output variables. Some model variants are achieved by introducing a new axiom of “boundedness” over the selected output variables. A mixed integer linear programming (MILP) formulation is also introduced for computing efficiency scores in the associated production set.     

Optimal input/output reduction in production processes

While conventional Data Envelopment Analysis (DEA) models set targets for each operational unit, this paper considers the problem of input/output reduction in a centralized decision making environment. The purpose of this paper is to develop an approach to input/output reduction problem that typically occurs in organizations with a centralized decision-making environment. This paper shows that DEA can make an important contribution to this problem and discusses how DEA-based model can be used to determine an optimal input/output reduction plan. An application in banking sector with limitation in IT investment shows the usefulness of the proposed method.     

Establishing dominance and potential optimality in multi-criteria analysis with imprecise weight and value

The model presented in this paper does not require exact estimations of decision parameters such as attribute weights and values that may often be considerable cognitive burden of human decision makers. Information on the decision parameters is only assumed to be in the form of arbitrary linear inequalities which form constraints in the model. We consider two criteria, dominance and potential optimality, to check whether or not each alternative is outperform for a fixed feasible region denoted by the constraints. In particular, we develop a method to identify potential optimality of alternatives when all (or subsets) of the attribute values as well as weights are imprecisely know. This formulation becomes a nonlinear programming problem hard to be solved generally so that we provide in this paper how this problem is transformed into a linear programming equivalent.Scope and purposeMost managerial decisions involve choosing an optimal alternative from a number of available alternatives. Researchers have proposed a lot of methods to assist decision makers in choice making with a set of, usually conflicting, criteria or attributes. Many of these approaches require exact (or precise) information about either or both attribute values and/or trade-off weights. In some practice, however, it is not easy for decision makers to provide such exact data because, for example, intangible attributes to reflect social and environmental impacts may be included. To cope with such problem, a mathematical programming model-based approach to multi-criteria decision analysis is presented in this paper when both attribute weights and marginal values are imprecisely identified. A weighted additive rule is used to evaluate the performance of alternatives. We then show how to obtain non-dominated and potentially optimal alternatives in order to support choice making.     

基于证据理论-DEA 交叉效率的混合型多属性决策方法

针对含有投入产出指标的混合型多属性决策问题,提出一种基于证据理论和数据包络分析(DEA)交叉效率的决策方法.首先运用DEA对决策系统中投入产出指标进行处理,得到DEA交叉效率矩阵,并运用证据理论集结其交叉效率得分;然后将效率得分作为决策系统指标值,与系统中其他指标进行模糊等级转换,通过证据理论对指标值融合,进而得到决策单元的期望效用,据此对决策单元进行排序;最后通过实例与其他文献方法进行对比分析,以表明所提出方法的可行性和有效性.