Fuzzy data envelopment analysis and its application to location problems

Abstract

In this paper, fuzzy DEA (data envelopment analysis) models are proposed for evaluating the efficiencies of objects with fuzzy input and output data. The obtained efficiencies are also fuzzy numbers that reflect the inherent ambiguity in evaluation problems under uncertainty. An aggregation model for integrating fuzzy attribute values is provided in order to rank objects objectively. Using the proposed method, a case study involving a restaurant location problem is analyzed in detail. Rent of establishment, traffic amount, level of security, consumer consumption level and competition level are identified as the primary factors in determining an ideal location for a Japanese-style rotisserie restaurant. Based on field investigation, the uncertain information on primary factors is represented by fuzzy numbers. Using the fuzzy aggregation model, the best location of restaurant is determined. The case study shows that fuzzy DEA models can be quite useful for solving business problems under uncertainty.

Introduction

Data envelopment analysis (DEA) is a non-parametric method in which multiple inputs and outputs could be used to measure an entity’s performance. DEA is a mathematical programming technique aiming at the measurement of decision-making units’ (DMUs) relative efficiencies [4], [7], [10], [23], [28], [35]. Most of DEA researchers assume that input and output data are crisp and without any variation. In fact, inputs and outputs of DMUs are ever-changing. For example, in evaluating operation efficiencies of airlines, the inputs include seat-kilometers available, cargo-kilometers available, fuel and labor while the output involves passenger-kilometers [5]. It is common knowledge that these inputs and output can easily change because of factors such as the weather, seasons of nature, and operating state. As DEA is a ‘boundary’ method sensitive to outliers, it is very difficult to evaluate the efficiency of a DMU with varying inputs and outputs by conventional DEA models. Some researchers have proposed models such as stochastic frontier models [1], [2], [13], [25] to deal with the variation of data in efficiency evaluation problems.

Uncertainty is an attribute of information, and commonly exists in many decision-making problems. Uncertainty is linked to information through the concept of granular structure characterized by indistinguishability, equivalence, similarity, proximity or functionality. The concept of granularity underlies the concept of linguistic variable [33]. For example, an expert can make a general conclusion that the output capacity of airline A is about 200 passenger-kilometers and mileage is high based on his considerable experience. These linguistic variables are used to characterize the general situation of inputs and outputs and reflect the ambiguity of the experts’ judgment. Few researchers have discussed DEA models under uncertainty [6], [8], [12], [14], [15], [16], [17], [18], [19], [20], [21], [24], [27], [29], [30], [34].

Aggregation operator plays an important role in information integration and decision analysis. It integrates information from higher dimensions to one dimension to facilitate an overall judgment in the decision-making procedure. Several kinds of aggregation operators have been discussed in the literature [3], [9], [11], [22], [26], [31], [32]. In essence, in these methods, aggregation is represented as a generalized weighted sum where the weights of attributes are predetermined by decision-makers to represent their preferences or thresholds. Clearly, different weights can lead to different aggregation results. In fact, it is very difficult to choose suitable weights because of the inherent uncertainty and subjectivity in determining them. Sometimes there is no specific authority (decision-maker) who can determine the weights of attributes.

Let us consider an example of a motorcycle manufacturing company. The company has designed several new products and is interested in selecting the most popular products for mass-production. In the process, information about attributes such as price, style, comfort, mileage, etc. are collected through a questionnaire administered to possible customers. In this case, it is highly unlikely that this company can predetermine the weights of attributes because the decision to buy is probably not been reached by many customers. However, it is certain that the company can give some suggestions on attributes such as “the price is the most important attribute for a good selling”. Customers also cannot determine the weights of attributes because producing the type of motorcycle is completely decided by the company rather than the individual preference of some customer. However, customers can express their comments on the attributes of motorcycles. Hence, in some situations, there is no clear authority to determine the weights of attributes. This type of evaluation system is called agent–clients evaluation (ACE) system.

In ACE systems, the agent (company) can collect some information on the evaluated objects from clients (customers) and decide which action should be taken to meet clients’ preference. The ACE systems greatly differ from conventional multi-criteria decision-making problems in the sense that there is an agent rather than an authority that has right to specify weights of attributes. In such a system, the weights need to reflect the inherent intention of the clients. A self-organizing fuzzy aggregation model is proposed in [15] for ACE.

In this research, a fuzzy DEA model is proposed. This model is an extension of CCR model for evaluating the fuzzy efficiency of DMU with fuzzy input and output data. The crisp efficiency in CCR model is generalized to a fuzzy number that reflect the inherent uncertainty in real evaluation problems. Based on the proposed fuzzy DEA model, a fuzzy aggregation model for integrating fuzzy attribute values of objects is provided to rank objects objectively. Using the proposed method, a case study involving a restaurant location problem is presented. Rent of establishment, traffic amount, level of security, consumer consumption level and competition level are identified as the primary factors in determining an ideal location for a Japanese-style rotisserie restaurant. Based on field investigation, the uncertain information on primary factors is represented by fuzzy numbers. Using the proposed fuzzy aggregation model, the best location of restaurant is determined.

This paper is organized as follows: In Section 2, fuzzy DEA models are proposed for obtaining the fuzzy efficiencies of DMUs with symmetrical L–L fuzzy input and output data. In Section 3, the methods for evaluating the objects with fuzzy attribute values are provided. In Section 4, a case study involving a restaurant location problem is presented in detail. Section 5 makes some concluding remarks for this research.