Posynomial geometric programming with interval exponents and coefficients

Geometric programming provides a powerful tool for solving nonlinear problems where nonlinear relations can be well presented by an exponential or power function. In the real world, many applications of geometric programming are engineering design problems in which some of the problem parameters are estimates of actual values. This paper develops a solution method when the exponents in the objective function, the cost and the constraint coefficients, and the right-hand sides are imprecise and represented as interval data. Since the parameters of the problem are imprecise, the objective value should be imprecise as well. A pair of two-level mathematical programs is formulated to obtain the upper bound and lower bound of the objective values. Based on the duality theorem and by applying a variable separation technique, the pair of two-level mathematical programs is transformed into a pair of ordinary one-level geometric programs. Solving the pair of geometric programs produces the interval of the objective value. The ability of calculating the bounds of the objective value developed in this paper might help lead to more realistic modeling efforts in engineering optimization areas.     

Qualitative factors in data envelopment analysis: A fuzzy number approach

Qualitative factors are difficult to mathematically manipulate when calculating the efficiency in data envelopment analysis (DEA). The existing methods of representing the qualitative data by ordinal variables and assigning values to obtain efficiency measures only superficially reflect the precedence relationship of the ordinal data. This paper treats the qualitative data as fuzzy numbers, and uses the DEA multipliers associated with the decision making units (DMUs) being evaluated to construct the membership functions. Based on Zadeh’s extension principle, a pair of two-level mathematical programs is formulated to calculate the α-cuts of the fuzzy efficiencies. Fuzzy efficiencies contain more information for making better decisions. A performance evaluation of the chemistry departments of 52 UK universities is used for illustration. Since the membership functions are constructed from the opinion of the DMUs being evaluated, the results are more representative and persuasive.     

Efficiency bounds and efficiency classifications in DEA with imprecise data

This paper is concerned with the use of imprecise data in data envelopment analysis (DEA). Imprecise data means that some data are known only to the extent that the true values lie within prescribed bounds while other data are known only in terms of ordinal relations. Imprecise data envelopment analysis (IDEA) has been developed to measure the relative efficiency of decision-making units (DMUs) whose input and/or output data are imprecise. In this paper, we show two distinct strategies to arrive at an upper and lower bound of efficiency that the evaluated DMU can have within the given imprecise data. The optimistic strategy pursues the best score among various possible scores of efficiency and the conservative strategy seeks the worst score. In doing so, we do not limit our attention to the treatment of special forms of imprecise data only, as done in some of the studies associated with IDEA. We target how to deal with imprecise data in a more general form and, under this circumstance, we make it possible to grasp an upper and lower bound of efficiency. The generalized method we develop in this paper also gives rise to a new scheme of efficiency classifications that is more detailed and informative than the standard efficient and inefficient partition.     

Qualitative and quantitative data envelopment analysis with interval data

In this paper, we investigate DEA with interval input-output data. First we show various extensions of efficiency and that 25 of them are essential. Second we formulate the efficiency test problems as mixed integer programming problems. We prove that 14 among 25 problems can be reduced to linear programming problems and that the other 11 efficiencies can be tested by solving a finite sequence of linear programming problems. Third, in order to obtain efficiency scores, we extend SBM model to interval input-output data. Fourth, to moderate a possible positive overassessment by DEA, we introduce the inverted DEA model with interval input-output data. Using efficiency and inefficiency scores, we propose a classification of DMUs. Finally, we apply the proposed approach to Japanese Bank Data and demonstrate its advantages.     

Measurement of the worst practice of decision-making units in the presence of non-discretionary factors and imprecise data

The objective of the present paper is to propose a novel pair of data envelopment analysis (DEA) models for measurement of relative efficiencies of decision-making units (DMUs) in the presence of non-discretionary factors and imprecise data. Compared to traditional DEA, the proposed interval DEA approach measures the efficiency of each DMU relative to the inefficiency frontier, also called the input frontier, and is called the worst relative efficiency or pessimistic efficiency. On the other hand, in traditional DEA, the efficiency of each DMU is measured relative to the efficiency frontier and is called the best relative efficiency or optimistic efficiency. The pair of proposed interval DEA models takes into account the crisp, ordinal, and interval data, as well as non-discretionary factors, simultaneously for measurement of relative efficiencies of DMUs. Two numeric examples will be provided to illustrate the applicability of the interval DEA models.     

基于公共权重的区间DEA效率评价及其排序方法研究

针对传统区间数据包络分析方法,在确定每一个决策单元区间效率的上界和下界时,存在的评价尺度不一致且计算复杂等问题,本文提出了一种同时最大化所有决策单元的效率上界和下界的公共权重区间DEA模型,并给出了一种考虑决策者偏好信息的可能度排序方法,用以解决区间效率的全排序问题。最后,以中国大陆11个沿海省份工业生产效率测算为例说明了所提方法的有效性和实用性。     

An extended slacks-based measure model for data envelopment analysis with negative data

Data envelopment analysis (DEA) is a non-parametric approach based on linear programming that has been widely applied for evaluating the relative efficiency of a set of homogeneous decision-making units (DMUs) with multiple inputs and outputs. The original DEA models use positive input and output variables that are measured on a ratio scale, but these models do not apply to the variables in which negative data can appear. However, with the widespread use of interval scale data and undesirable data, the emphasis has been directed towards the simultaneous consideration of the positive and negative data in DEA models. In this paper, using the slacks-based measure, we propose an extended model to evaluate the efficiency of DMUs, even if some variables are measured on an interval scale and some on a ratio scale. Moreover, the extended model allows for the presence of all interval-scale variables, which are capable of taking both negative and positive values.     

Fuzzy efficiency ranking in fuzzy two-stage data envelopment analysis

Data envelopment analysis (DEA) is a useful tool for efficiency measurement of firms and organizations. Many production systems in the real world are composed of two processes connected in series. Measuring the system efficiency without taking the operation of each process into consideration will obtain misleading results. Two-stage DEA models show the performance of individual processes, thus is more informative than the conventional one-stage models for making decisions. When input and output data are fuzzy numbers, the derived efficiencies become fuzzy as well. This paper proposes a method to rank the fuzzy efficiencies when the exact membership functions of the overall efficiencies derived from fuzzy two-stage model are unknown. By incorporating the fuzzy two-stage model with the fuzzy number ranking method, a pair of nonlinear program is formulated to rank the fuzzy overall efficiency scores of DMUs. Solving the pair of nonlinear programs determines the efficiency rankings. An example of the ranking of the 24 non-life assurance companies in Taiwan is illustrated to explain how the proposed method is applied.     

Evaluating the performances of decision-making units based on interval efficiencies

Efficiency could be not only the ratio of weighted sum of outputs to that of inputs but also that of weighted sum of inputs to that of outputs. When the previous efficiency measures the best relative efficiency within the range of no more than one, the decision-making units (DMUs) who get the optimum value of one perform best among all the DMUs. If the previous efficiency is measured within the range of no less than one, the DMUs who get the optimum value of one perform worst among all the DMUs. When the later efficiency is measured within the range of no more than one, the DMUs who get the optimum value of one perform worst among all the DMUs. If the later efficiency is measured within the range of no less than one, the DMUs who get the optimum value of one perform best among all the DMUs. This paper mainly studies an interval DEA model with later efficiency, in which efficiency is measured within the range of an interval, whose upper bound is set to one and the lower bound is determined by introducing a virtual ideal DMU, whose performance is definitely superior to any DMUs. The efficiencies, obtained from interval DEA model, turn out to be all intervals and are referred to as interval efficiencies, which combine the best and the worst relative efficiency in a reasonable manner to give an overall assessment of performances for all DMUs. Assessor's preference information on input and output weights is also incorporated into interval DEA model reasonably and conveniently. Through an example, some differences are found from the ranking results obtained from interval DEA model and bounded DEA model using the Hurwicz criterion approach to rank the interval efficiencies.     

An inverse optimization model for imprecise data envelopment analysis

Inverse data envelopment analysis (InDEA) is a well-known approach for short-term forecasting of a given decision-making unit (DMU). The conventional InDEA models use the production possibility set (PPS) that is composed of an evaluated DMU with current inputs and outputs. In this paper, we replace the fluctuated DMU with a modified DMU involving renewal inputs and outputs in the PPS since the DMU with current data cannot be allowed to establish the new PPS. Besides, the classical DEA models such as InDEA are assumed to consider perfect knowledge of the input and output values but in numerous situations, this assumption may not be realistic. The observed values of the data in these situations can sometimes be defined as interval numbers instead of crisp numbers. Here, we extend the InDEA model to interval data for evaluating the relative efficiency of DMUs. The proposed models determine the lower and upper bounds of the inputs of a given DMU separately when its interval outputs are changed in the performance analysis process. We aim to remain the current interval efficiency of a considered DMU and the interval efficiencies of the remaining DMUs fixed or even improve compared with the current interval efficiencies.     

Interval data without sign restrictions in DEA

Conventional DEA models assume deterministic, precise and non-negative data for input and output observations. However, real applications may be characterized by observations that are given in form of intervals and include negative numbers. For instance, the consumption of electricity in decentralized energy resources may be either negative or positive, depending on the heat consumption. Likewise, the heat losses in distribution networks may be within a certain range, depending on e.g. external temperature and real-time outtake. Complementing earlier work separately addressing the two problems; interval data and negative data; we propose a comprehensive evaluation process for measuring the relative efficiencies of a set of DMUs in DEA. In our general formulation, the intervals may contain upper or lower bounds with different signs. The proposed method determines upper and lower bounds for the technical efficiency through the limits of the intervals after decomposition. Based on the interval scores, DMUs are then classified into three classes, namely, the strictly efficient, weakly efficient and inefficient. An intuitive ranking approach is presented for the respective classes. The approach is demonstrated through an application to the evaluation of bank branches.     

Cost efficiency in data envelopment analysis under the law of one price

To impose the law of one price (LoOP) restrictions, which state that all firms face the same input prices, Kuosmanen, Cherchye, and Sipiläinen (2006) developed the top-down and bottom-up approaches to maximizing the industry-level cost efficiency. However, the optimal input shadow prices generated by the above approaches need not be unique, which influences the distribution of the efficiency indices at the individual firm level. To solve this problem, in this paper, we developed a pair of two-level mathematical programming models to calculate the upper and lower bounds of cost efficiency for each firm in the case of non-unique LoOP prices while keeping the industry cost efficiency optimal. Furthermore, a base-enumerating algorithm is proposed to solve the lower bound models of the cost efficiency measure, which are bi-level linear programs and NP-hard problems. Lastly, a numerical example is used to demonstrate the proposed approach.     

Interval cross efficiency for fully ranking decision making units using DEA/AHP approach

Data envelopment analysis (DEA) is a popular technique for measuring the relative efficiency of a set of decision making units (DMUs). Fully ranking DMUs is a traditional and important topic in DEA. In various types of ranking methods, cross efficiency method receives much attention from researchers because it evaluates DMUs by using self and peer evaluation. However, cross efficiency score is usual nonuniqueness. This paper combines the DEA and analytic hierarchy process (AHP) to fully rank the DMUs that considers all possible cross efficiencies of a DMU with respect to all the other DMUs. We firstly measure the interval cross efficiency of each DMU. Based on the interval cross efficiency, relative efficiency pairwise comparison between each pair of DMUs is used to construct interval multiplicative preference relations (IMPRs). To obtain the consistency ranking order, a method to derive consistent IMPRs is developed. After that, the full ranking order of DMUs from completely consistent IMPRs is derived. It is worth noting that our DEA/AHP approach not only avoids overestimation of DMUs’ efficiency by only self-evaluation, but also eliminates the subjectivity of pairwise comparison between DMUs in AHP. Finally, a real example is offered to illustrate the feasibility and practicality of the proposed procedure.     

Fuzzy profit measures for a fuzzy economic order quantity model

Changing economic conditions make the selling price and demand quantity more and more uncertain in the market. The conventional inventory models determine the selling price and order quantity for a retailer’s maximal profit with exactly known parameters. This paper develops a solution method to derive the fuzzy profit of the inventory model when the demand quantity and unit cost are fuzzy numbers. Since the parameters contained in the inventory model are fuzzy, the profit value calculated from the model should be fuzzy as well. Based on the extension principle, the fuzzy inventory problem is transformed into a pair of two-level mathematical programs to derive the upper bound and lower bound of the fuzzy profit at possibility level α. According to the duality theorem of geometric programming, the pair of two-level mathematical programs is transformed into a pair of conventional geometric programs to solve. By enumerating different α values, the upper bound and lower bound of the fuzzy profit are collected to approximate the membership function. Since the profit of the inventory problem is expressed by the membership function rather than by a crisp value, more information is provided for making decisions.     

Measuring the performances of decision-making units using interval efficiencies

Efficiency is a relative measure because it can be measured within different ranges. The traditional data envelopment analysis (DEA) measures the efficiencies of decision-making units (DMUs) within the range of less than or equal to one. The corresponding efficiencies are referred to as the best relative efficiencies, which measure the best performances of DMUs and determine an efficiency frontier. If the efficiencies are measured within the range of greater than or equal to one, then the worst relative efficiencies can be used to measure the worst performances of DMUs and determine an inefficiency frontier. In this paper, the efficiencies of DMUs are measured within the range of an interval, whose upper bound is set to one and the lower bound is determined through introducing a virtual anti-ideal DMU, whose performance is definitely inferior to any DMUs. The efficiencies turn out to be all intervals and are thus referred to as interval efficiencies, which combine the best and the worst relative efficiencies in a reasonable manner to give an overall measurement and assessment of the performances of DMUs. The new DEA model with the upper and lower bounds on efficiencies is referred to as bounded DEA model, which can incorporate decision maker (DM) or assessor's preference information on input and output weights. A Hurwicz criterion approach is introduced and utilized to compare and rank the interval efficiencies of DMUs and a numerical example is examined using the proposed bounded DEA model to show its potential application and validity.     

Effects of fuzzy data on decision making in a competitive supply chain

The essence of data plays a critical role in decision making in supply chain management (SCM). When data embedded in supply chains (SCs) are fuzzy, the associated equilibrium and performance measures also become fuzzy. This paper investigates the effects of fuzzy data on decision making in a two-echelon SC with a supplier and duopolistic retailers playing a Stackelberg strategic game in both intra- and inter-echelons. In contrast to existing approaches, this paper devises an analysis method to provide a likely interval of the fuzzy maximal profit with a known possibility level (degree of certainty) rather than a singleton (crisp value). The idea is based on the extension principle to reformulate the two-level optimization problem as a pair of parametric quadratic programs in order to calculate the lower and upper bounds of the leader’s fuzzy maximal profit at each possibility level of the obtained information. The analytic results indicate that the higher the degree of uncertainty, the smaller (larger) the lower (upper) bound of the maximum profit of each SC member. Moreover, the main results obtained from eight scenarios show that when the degree of demand diversity between the two retailers is significantly high, the Stackelberg leader is most likely to obtain lower profit and the marginal contribution of the primary demand to the total profit of the duopolistic retailers will exceed that of the powerful supplier’s maximum profit.     

Dominance relations and ranking of units by using interval number ordering with cross-efficiency intervals

This paper proposes an approach to the cross-efficiency evaluation that considers all the optimal data envelopment analysis (DEA) weights of all the decision-making units (DMUs), thus avoiding the need to make a choice among them according to some alternative secondary goal. To be specific, we develop a couple of models that allow for all the possible weights of all the DMUs simultaneously and yield individual lower and upper bounds for the cross-efficiency scores of the different units. As a result, we have a cross-efficiency interval for the evaluation of each unit. Existing order relations for interval numbers are used to identify dominance relations among DMUs and derive a ranking of units based on the cross-efficiency intervals provided. The approach proposed may also be useful for assessing the stability of the cross-efficiency scores with respect to DEA weights that can be used for their calculation.     

The mean-absolute deviation portfolio selection problem with interval-valued returns

In real-world investments, one may care more about the future earnings than the current earnings of the assets. This paper discusses the uncertain portfolio selection problem where the asset returns are represented by interval data. Since the parameters are interval valued, the gain of returns is interval valued as well. According to the concept of the mean-absolute deviation function, we construct a pair of two-level mathematical programming models to calculate the lower and upper bounds of the investment return of the portfolio selection problem. Using the duality theorem and applying the variable transformation technique, the pair of two-level mathematical programs is transformed into a conventional one-level mathematical program. Solving the pair of mathematical programs produces the interval of the portfolio return of the problem. The calculated results conform to an essential idea in finance and economics that the greater the amount of risk that an investor is willing to take on the greater the potential return.     

Sensitivity and stability analysis on the first and second levels of efficiency score relative to data error

In data envelopment analysis (DEA), efficient decision making units (DMUs) are of primary importance as they define the efficient frontier. The current paper develops a new sensitivity analysis approach for a category DMUs and finds the stability radius for all efficient DMUs. By means of combining some classic DEA models and with the condition that the efficiency scores of efficient DMUs remain unchanged, we are able to determine what perturbations of the data can be tolerated before efficient DMUs become inefficient. Our approach generalizes the conventional sensitivity analysis approach in which the inputs of efficient DMUs increase and their outputs decrease, while the inputs of inefficient DMUs decrease and their outputs increase. We find the maximum quantity of perturbations of data so that all first level efficient DMUs remain at the same level.     

A conic relaxation model for searching for the global optimum of network data envelopment analysis

Network data envelopment analysis (DEA) models the internal structures of decision-making units (DMUs). Unlike the standard DEA model, multiplier-based network DEA models are often highly non-linear and cannot be converted into linear programs. As such, obtaining a non-linear network DEA's global optimal solution is a challenge because it corresponds to a nonconvex optimization problem. In this paper, we introduce a conic relaxation model that searches for the global optimum to the general multiplier-based network DEA model. We reformulate the general network DEA models and relax the new models into second order cone programming (SOCP) problems. In comparison with linear relaxation models, which is potentially applicable to general network DEA structures, the conic relaxation model guarantees applicability in general network DEA, since McCormick envelopes involved are ensured to be finite. Furthermore, the conic relaxation model avoids unnecessary linear relaxations of some nonlinear constraints. It generates, in a more convenient manner, feasible approximations and tighter upper bounds on the global optimal overall efficiency. Compared with a line-parameter search method that has been applied to solve non-linear network DEA models, the conic relaxation model keeps track of the distances between the optimal overall efficiency and its approximations. As a result, it is able to determine whether a qualified approximation has been achieved or not, with the help of a branch and bound algorithm. Hence, our proposed approach can substantially reduce the computations involved.