Shiang-Tai Liu

Fuzzy efficiency measures in data envelopment analysis

Abstract The existing data envelopment analysis (DEA) models for measuring the relative efficiencies of a set of decision making units (DMUs) using various inputs to produce various outputs are limited to crisp data. To deal with imprecise data, the notion of fuzziness has been introduced. This paper develops a procedure to measure the efficiencies of DMUs with fuzzy observations. The basic idea is to transform a fuzzy DEA model to a family of conventional crisp DEA models by applying the α-cut approach. A pair of parametric programs is formulated to describe that family of crisp DEA models, via which the membership functions of the efficiency measures are derived. Since the efficiency measures are expressed by membership functions rather than by crisp values, more information is provided for management. By extending to fuzzy environment, the DEA approach is made more powerful for applications.

Solving fuzzy transportation problems based on extension principle

Abstract Transportation models play an important role in logistics and supply chain management for reducing cost and improving service. This paper develops a procedure to derive the fuzzy objective value of the fuzzy transportation problem, in that the cost coefficients and the supply and demand quantities are fuzzy numbers. The idea is based on the extension principle. A pair of mathematical programs is formulated to calculate the lower and upper bounds of the fuzzy total transportation cost at possibility level α . From different values of α , the membership function of the objective value is constructed. Two different types of the fuzzy transportation problem are discussed: one with inequality constraints and the other with equality constraints. It is found that the membership function of the objective value of the equality problem is contained in that of the inequality problem. Since the objective value is expressed by a membership function rather than by a crisp value, more information is provided for making decisions.

A fuzzy DEA/AR approach to the selection of flexible manufacturing systems

Flexible Manufacturing System (FMS) offers opportunities for manufacturers to improve their technology, competitiveness, and profitability through a highly efficient and focused approach to manufacturing effectiveness. Data envelopment analysis (DEA) has been utilized as a multiple criteria tool for evaluation of FMSs. The concept of the assurance region (AR) is restricting the ratio of any two weights to some range to avoid the evaluated alternatives from ignoring or relying too much on any criterion in evaluation. In this paper, we develop a fuzzy DEA/AR method that is able to evaluate the performance of FMS alternatives when the input and output data are represented as crisp and fuzzy data. Based on Zadehs extension principle, a pair of two-level mathematical programs is formulated to calculate the lower and upper bounds of the fuzzy efficiency score of the alternatives. We transform this pair of two-level mathematical programs into a pair of conventional one-level DEA/AR method to evaluate the FMS performance. An example illustrates the application of the proposed methodology.

Fractional programming approach to fuzzy weighted average

This paper proposes a fractional programming approach to construct the membership function for fuzzy weighted average. Based on the α-cut representation of fuzzy sets and the extension principle, a pair of fractional programs is formulated to find the α-cut of fuzzy weighted average. Owing to the special structure of the fractional programs, in most cases, the optimal solution can be found analytically. Consequently, the exact form of the membership function can be derived by taking the inverse function of the α-cut. For other cases, a discrete but exact solution to fuzzy weighted average is provided via an efficient solution method. Examples are given for illustration.

Data envelopment analysis with missing data: an application to University libraries in Taiwan

In measuring the relative efficiencies of a set of decision making units (DMUs) via data envelopment analysis (DEA), detailed inputs and outputs are usually involved. However, there are cases where some DMUs are unable to provide all the necessary data. This paper adopts the concept of a membership function used in fuzzy set theory for representing imprecise data. The smallest possible, most possible, and largest possible values of the missing data are derived from the observed data to construct a triangular membership function. With the membership function, a fuzzy DEA model can be utilized to calculate the efficiency scores. Since the efficiency scores are fuzzy numbers, they are more informative than crisp efficiency scores calculated by assuming crisp values for the missing data. As an illustration, the efficiency scores of the 24 University libraries in Taiwan, with three missing values, are calculated to show the extent that the actual amount of resources and services provided by each University is away from the technically efficient amount of resources and services. This methodology can also be applied to calculate the relative efficiencies of the DMUs with imprecise linguistic data.

A mathematical programming approach to fuzzy efficiency ranking

Abstract Data envelopment analysis is a widely applied approach for measuring the relative efficiencies of a set of decision-making units (DMUs) which use multiple inputs to produce multiple outputs. When some observations are fuzzy, the efficiencies become fuzzy as well. This paper devises a method to rank the fuzzy efficiency scores without knowing the exact form of the membership functions. The idea is to apply the maximizing set–minimizing set method, which is normally applied when membership functions are known. Via a skillful modeling technique, the requirement of the membership functions is avoided. The efficiency rankings are consequently determined by solving a pair of nonlinear programs for each DMU. To illustrate how the proposed method is applied, the ranking of the 24 university libraries in Taiwan with fuzzy observations is exemplified.

Stochastic data envelopment analysis in measuring the efficiency of Taiwan commercial banks

Conventional data envelopment analysis (DEA) for measuring the efficiency of a set of decision making units (DMUs) requires the input/output data to be constant. In reality, however, many observations are stochastic in nature; consequently, the resulting efficiencies are stochastic as well. This paper discusses how to obtain the efficiency distribution of each DMU via a simulation technique. The case of Taiwan commercial banks shows that, firstly, the number of replications in simulation analysis has little effect on the estimation of efficiency means, yet 1000 replications are recommended to produce reliable efficiency means and 2000 replications for a good estimation of the efficiency distributions. Secondly, the conventional way of using average data to represent stochastic variables results in efficiency scores which are different from the mean efficiencies of the presumably true efficiency distributions estimated from simulation. Thirdly, the interval-data approach produces true efficiency intervals yet the intervals are too wide to provide valuable information. In conclusion, when multiple observations are available for each DMU, the stochastic-data approach produces more reliable and informative results than the average-data and interval-data approaches do.

Fuzzy measures for correlation coefficient of fuzzy numbers

Correlation coefficient of random variables has wide applications in statistical analysis. This paper extends the applications to fuzzy environment, with a methodology for calculating the correlation coefficient of fuzzy numbers developed. Different from previous studies, the correlation coefficient calculated in this paper is a fuzzy number, rather than a crisp value. The idea is based on Zadehs extension principle. A pair of nonlinear programs is formulated to find the α-cut of the fuzzy correlation coefficient. From different values of α, the membership function of the fuzzy correlation coefficient is constructed. To illustrate how to interpret the fuzzy correlation coefficient in real world applications, the correlation between the technology level and management achievement from a sample of 15 machinery firms in Taiwan is exemplified. All indications show that the correlation between technology and management in Taiwan machinery firms is rather low.

A numerical solution method to interval quadratic programming

Quadratic programming has been widely applied to solving real world problems. The conventional quadratic programming model requires the parameters to be known constants. In the real world, however, the parameters are seldom known exactly and have to be estimated. This paper discusses the interval quadratic programming problems where the cost coefficients, constraint coefficients, and right-hand sides, are represented by interval data. Since the parameters are interval-valued, the objective value is interval-valued as well. A pair of two-level mathematical programs is formulated to calculate the upper bound and lower bound of the objective values of the interval quadratic program. Based on the duality theorem and by applying the variable transformation technique, the pair of two-level mathematical programs is transformed into conventional one-level quadratic program. Solving the pair of quadratic programs produces the interval of the objective values of the problem. An example illustrates the whole idea and sheds some light on interval quadratic programming.

Fuzzy efficiency measures in fuzzy DEA/AR with application to university libraries

Data envelopment analysis (DEA) allows individual decision-making unit (DMU) to select the weights that are most favorable to them in calculating the ratio of the aggregated output to the aggregated input. The concept of the assurance region (AR) is restricting the ratio of any two weights to some range to avoid the evaluated DMUs from ignoring or relying too much on any criterion in evaluation. In this paper we develop a fuzzy DEA/AR method that is able to calculate the fuzzy efficiency score when the input and output data are represented as convex fuzzy numbers. Based on Zadehs extension principle, a pair of two-level mathematical programs is formulated to calculate the lower and upper bounds of the fuzzy efficiency score. We transform this pair of two-level mathematical programs into a pair of conventional DEA/AR method to derive the bounds of the efficiency. The dual models of the fuzzy DEA/AR for efficiency improvement are also considered. To illustrate how the proposed method is applied, the measurement of the efficiency of the university libraries in Taiwan with fuzzy observations is exemplified.