Interval-valued fuzzy TOPSIS method with leniency reduction and an experimental analysis

Abstract

The purpose of this paper is to present a new TOPSIS (the technique for order preference by similarity to ideal solution) for estimating the importance of criteria and reducing the leniency bias in multiple-criteria decision analysis based on interval-valued fuzzy sets. Several types of net predispositions are defined to represent an aggregated effect of interval-valued evaluations. The relative closeness of each alternative to the ideal solution is then determined by net predispositions. Because positive or negative leniency may exist when most criteria are assigned unduly high or low ratings, respectively, some deviation variables are introduced to mitigate the effects of overestimated and underestimated ratings on criterion importance. Considering the two objectives of maximal closeness coefficient and minimal deviation values, an integrated programming model is proposed to compute optimal weights for the criteria and corresponding closeness coefficients for alternative rankings. A flexible algorithm using interval-valued fuzzy TOPSIS methods is established by considering both objective and subjective information to compute optimal multiple-criteria decisions. The feasibility and effectiveness of the proposed methods are illustrated by a numerical example. Finally, an experimental analysis of interval-valued fuzzy rankings given different conditions for the criterion weights is conducted with discussions on average Spearman correlation coefficients and contradiction rates.

Introduction

TOPSIS (the technique for order preference by similarity to ideal solution), developed by Hwang and Yoon [23], is a well-known MCDA (multiple-criteria decision analysis) method. The basic concept of the TOPSIS method is that the chosen alternative should have the shortest distance from the positive ideal solution and the farthest distance from the negative ideal solution. There are necessary steps in utilizing TOPSIS involving numerical measures of the relative importance of criteria and the performance of each alternative on these criteria. However, exact data may be difficult to be precisely determined since human judgments are often vague under many conditions. Thus, an extension of TOPSIS to the fuzzy environment is a natural generalization of TOPSIS methods [8], [24].

The fuzzy TOPSIS theory and applications have been comprehensively developed. On the basis of triangular fuzzy numbers, Chen [8] extended the TOPSIS method for group decision-making in a fuzzy environment. Tiryaki and Ahlatcioglu [45] presented a fuzzy ranking and weighting algorithm to investigate the fuzzy stock selection problem. They used triangular fuzzy numbers to define the fuzzy positive and negative ideal solutions. Jahanshahloo et al. [24] developed an algorithmic method to extend TOPSIS for decision-making problems with fuzzy data. Combining the concepts of grey relation and pair-wise comparison, Kuo et al. [28] proposed a new fuzzy decision-making method for multi-criteria analysis. They determined a referential sequence (positive ideal solution) using a series of L–R fuzzy numbers. Xu and Chen [49] extended TOPSIS and proposed an interactive method for fuzzy multiple-attribute group decision-making. Yang and Hung [51] explored the use of TOPSIS and fuzzy TOPSIS in solving a plant layout design problem. Li [30] developed a compromise ratio methodology to handle fuzzy multi-attribute group decision-making problems and conducted a comparative analysis with a fuzzy extension of TOPSIS. Li [31] introduced a multi-attribute ranking index based on a fuzzy closeness measure in a compromise programming method. Chen and Tsao [13] extended the TOPSIS based on interval-valued fuzzy sets (IVFSs) and conducted an experimental analysis on distance measures. Lin et al. [39] proposed a dynamic MCDA model that uses the TOPSIS technique to deal with uncertain information and to aggregate multi-period evaluations. Liu [40] proposed a method to resolve the multi-attribute decision-making problem using a TOPSIS method based on criterion weights where the criterion values are all vague interval values. Ashtiani et al. [3] presented an interval-valued fuzzy TOPSIS method for solving MCDA problems. Li [29] constructed several optimization models, and further, Li and Wang [37] and Li et al. [38] developed an interval fractional programming model for MCDA analysis. In the context of interval-valued intuitionistic fuzzy sets, Li [32] constructed a pair of nonlinear fractional programming models to calculate the relative closeness coefficient intervals of alternatives to the ideal solutions. In a similar manner, Li [33] developed TOPSIS-based nonlinear-programming methodology. Chen and Lee [11] presented an interval type-2 fuzzy TOPSIS method to handle fuzzy multi-attribute group decision-making problems based on interval type-2 fuzzy sets. Based on the concept of the closeness coefficient which is similar to that in the TOPSIS, Li [34] developed auxiliary nonlinear programming models for solving multi-attribute decision-making problems in which ratings of alternatives on attributes are expressed using interval-valued intuitionistic fuzzy sets and preference information on attributes is incomplete. Considering anchored judgment with displaced ideals and solution precision with minimal hesitation, Chen [9] proposed several auxiliary optimization models to obtain the optimal weights of the attributes and to acquire the corresponding TOPSIS index for alternative rankings.

The difference between TOPSIS and fuzzy TOPSIS chiefly lies in rating approaches. The merit of fuzzy TOPSIS is to estimate the performance of given alternatives with respect to defined criteria by using fuzzy data to express value of alternatives. Because the decision information given by decision-makers is often imprecise or uncertain due to a lack of data, time pressure, or the decision-makers’ limited attention and information processing capabilities [50], available information is not sufficient for the exact definition of a degree of membership for certain elements. That is, the exact value for the membership degree of an element to a given set cannot be easily identified [35]. There may be some hesitation degree between membership and non-membership. In view that there are many real life situations where due to insufficiency in information availability [35], IVFSs [44], [52] with ill-known membership grades are appropriate to deal with such problems. IVFSs are defined by an interval-valued membership function, and an element's degree of membership in a set is characterized as a closed subinterval of [0, 1]. IVFSs approximate the real but unknown membership grades [35] and involve more uncertainties than ordinary fuzzy sets. Because it may be difficult for decision-makers to exactly quantify their opinions of subjective judgments as a number within the interval [0, 1], it is better to represent the degree of membership by an interval rather than a single number. Therefore, IVFSs can be used to capture imprecise or uncertain decision information, and they are suitable for expressing the degrees of membership in MCDA problems. In this study, IVFS was used to develop MCDA methods due to its ability to easily reflect the ambiguous nature of subjective judgments.

Substantial research has focused on the extended TOPSIS methods underlying interval-valued fuzzy information in recent years. What seems to be lacking, however, is the investigation of leniency bias in the context of IVFSs. A direct method of survey research can be used to collect IVFS data. The methods used for constructing degrees of membership are almost invariably based on data representing the judgment of the evaluator. However, a critical issue may emerge from the investigation process of subjective judgments concerning appraisal behaviors: the error of positive or negative leniency [20]. Positive leniency is defined as the rater's tendency to assign scores that are higher than the true scores, while negative leniency refers to the rater's tendency to give scores that are lower than the true scores. Leniency bias is an error commonly found in rating behavior [22], [25], [43]; numerous laboratory and field studies have demonstrated a significant leniency bias on subjective ratings. For instance, leniency has been found to be a common rater bias on student grades and average ratings [17], [19], job performance ratings [22], whistle blowing [2], [4], interaction with defendants [1], self-efficacy [5], attitudes and self-monitoring [25], and assessment of communication skills [21].

Because some decision-makers tend to give positive connotations to the importance of most criteria, a major difficulty of most appraisal systems of subjective criterion weights seems to be the pervasiveness of positive leniency. Indeed, many surveys and interview-based studies have indicated leniency in rating to be the norm rather than the exception in most practical cases [6], [25], [41]. Consider linguistic ratings as an example. Li [36] introduced 5-scale linguistic values and their corresponding intuitionistic fuzzy sets. Because the IVFS is mathematically equivalent to the intuitionistic fuzzy set [35], we can use the same way to transform linguistic variables to IVFSs. If a decision-maker provide importance ratings of five criteria on linguistic terms of “very high”, “high”, “very high”, “high”, and “very high”, respectively. Based on the transformation of linguistic variables by Li [36], the subjective importance of each criterion can be expressed as the following IVFS values: [0.95, 0.95], [0.70, 0.75], [0.95, 0.95], [0.70, 0.75], and [0.95, 0.95]. The high ratings of criterion importance will result in overestimation of the priority weights of criteria. More specifically, the sum of the five lower membership degrees is as large as 4.25, which means that the criterion weights can hardly be normalized to sum to one. Clearly the example indicates that positive leniency in criterion importance leads to sub-additivity of degrees of membership. In addition, the degrees of importance for all criteria have no discrimination power in the decision-making process. Thus, researchers should recognize the high likelihood of leniency bias when conducting an investigation concerning criterion importance. To control for the spurious influence of positive and negative leniency response biases, we introduce several deviation variables to mitigate the effects of overestimated and underestimated ratings, respectively, on criterion importance.

Using an interval-valued fuzzy framework, the purpose of this study is to establish a TOPSIS method for handling leniency biases and for estimating criterion weights in MCDA. In addition, a series of net predispositions for interval-valued evaluations is proposed from various perspectives to identify the mixed results of the outcome expectations. Based on the net predispositions, the relative degree of closeness of each alternative with respect to the positive ideal solution is defined. Because the information available on the relative importance of the multiple criteria for decision-making is often incomplete, this study proposes several optimization models with closeness coefficients for ill-known membership grades. To cope with the problem of leniency response bias in rating behavior, an integrated programming model is developed, utilizing both deviation variables and weighted closeness coefficients. Furthermore, objective information in the decision matrix and subjective information of the criterion importance are combined to construct algorithms using the TOPSIS method for acquiring optimal decisions. Next, a numerical example is shown to illustrate the proposed method. Finally, a large set of random MCDA problems are generated, and computational studies are undertaken to compare preference orders determined by interval-valued fuzzy TOPSIS with different conditions for the criterion weights.