Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 6145196, 14 pages

http://dx.doi.org/10.1155/2016/6145196

## A Modified TOPSIS Method Based on Numbers and Its Applications in Human Resources Selection

^{1}School of Computer and Information Science, Southwest University, Chongqing 400715, China^{2}Big Data Decision Institute, Jinan University, Tianhe, Guangzhou 510632, China^{3}Institute of Integrated Automation, School of Electronic and Information Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China^{4}School of Engineering, Vanderbilt University, Nashville, TN 37235, USA

Received 29 February 2016; Accepted 28 April 2016

Academic Editor: Rita Gamberini

Copyright © 2016 Liguo Fei et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Multicriteria decision-making (MCDM) is an important branch of operations research which composes multiple-criteria to make decision. TOPSIS is an effective method in handling MCDM problem, while there still exist some shortcomings about it. Upon facing the MCDM problem, various types of uncertainty are inevitable such as incompleteness, fuzziness, and imprecision result from the powerlessness of human beings subjective judgment. However, the TOPSIS method cannot adequately deal with these types of uncertainties. In this paper, a -TOPSIS method is proposed for MCDM problem based on a new effective and feasible representation of uncertain information, called numbers. The -TOPSIS method is an extension of the classical TOPSIS method. Within the proposed method, numbers theory denotes the decision matrix given by experts considering the interrelation of multicriteria. An application about human resources selection, which essentially is a multicriteria decision-making problem, is conducted to demonstrate the effectiveness of the proposed -TOPSIS method.

#### 1. Introduction

Multicriteria decision-making (MCDM) or multiple-criteria decision analysis is an important branch of operations research that definitely uses multiple-criteria in decision-making environments [1, 2]. In daily life and professional learning, there exist generally multiple conflicting criteria which need to be considered in making decisions and optimization [3, 4]. Price and spend are typically one of the main criteria with regard to a large amount of practical problems. However, the factor of quality is generally another criterion which is in conflict with the price. For example, the cost, safety, fuel economy, and comfort should be considered as the main criteria upon purchasing a car. It is the most benefit for us to select the safest and most comfortable one which has the bedrock price simultaneously. The best situation is obtaining the highest returns while reducing the risks to the most extent with regard to portfolio management. In addition, the stocks that have the potential of bringing high returns typically also carry high risks of losing money. In service industry, there is a couple of conflicts between customer satisfaction and the cost to provide service. Upon making decision, it will be compelling if multiple-criteria are considered even though they came from and are based on subjective judgment of human. What is more, it is significant to reasonably describe the problem and precisely evaluate the results based on multiple-criteria when the stakes are high. With regard to the problem of whether to build a chemical plant or not and where the best site for it is, there exist multiple-criteria that need to be considered; also, there are multiple parties that will be affected deeply by the consequences.

Constructing complex problems properly as well as multiple-criteria taken into account explicitly results in more reasonable and better decisions. Significant achievements in this field have been made since the beginning of the modern multicriteria decision-making (MCDM) discipline in the early 1960s. A variety of approaches and methods have been proposed for MCDM. In [5], a novel MCDM method named FlowSort-GDSS is proposed to sort the failure modes into priority classes by involving multiple decision-makers, which has the robust advantages in sorting failures. In the field of multiple objective mathematical programming, Evans and Yu [6, 7] proposed the vector maximization method aimed at approximating the nondominated set which is originally developed for multiple objective linear programming problems. Torrance [8] used elaborate interview techniques to deal with the problem in MCDM, which exist for eliciting linear additive utility functions and multiplicative nonlinear utility functions. And there are many other methods, such as best worst method [9], characteristic objects method [10], fuzzy sets method [11–13], rough sets [14], and analytic hierarchy process [15–17]. In [18], the authors aim to systematically review the applications and methodologies of the MCDM techniques and approaches, which is a good guidance for us to fully understand the MCDM. Technique for order preference by similarity to ideal solution (TOPSIS), which is proposed in [19–23], is a ranking method in conception and application. The standard TOPSIS methodology aims to select the alternatives which have the shortest distance from the positive ideal solution and the longest distance from the negative ideal solution at the same time. The positive ideal solution maximizes the benefit attributes and minimizes the cost attributes, whereas the negative ideal solution maximizes the cost attributes and minimizes the benefit attributes. The TOPSIS methodology is applied widely in MCDM field [24–27], especially in the fuzzy extension of linguistic variables [28–31].

It is obvious that the mentioned approaches play a role under some specific circumstances, but, in the practical applications, they show more uncertainties due to the subjective judgment of experts’ assessment. In order to effectively handle various uncertainties involved in the MCDM problem, a new representation of uncertain information, called numbers [32], is presented in this paper. It is an extension of Dempster-Shafer evidence theory. It gives the framework of nonexclusive hypotheses, applied to many decision-making problems under uncertain environment [33–38]. Comparing with existing methods, numbers theory can efficiently denote uncertain information and more coincide with the actual conditions.

Therefore, in this paper, to well address these issues in TOPSIS method, an extended version is presented based on numbers named -TOPSIS, which considers the interrelation of multicriteria and handles the fuzzy and uncertain criteria effectively. The -TOPSIS method can represent uncertain information more effectively than other group decision support systems based on classical TOPSIS method, which cannot adequately handle these types of uncertainties. An application has been conducted using the -TOPSIS method in human resources selection, and the result can be more reasonable because of its consideration about the interrelation of multiple-criteria.

The remainder of this paper is constituted as follows. Section 2 introduces the Dempster-Shafer theory and its basic rules and some necessary related concepts about numbers theory and its distance function and TOPSIS. The proposed -TOPSIS method is presented in Section 3. Section 4 conducts an application in human resources selection based on -TOPSIS. Conclusion is given in Section 5.

#### 2. Preliminaries

##### 2.1. Dempster-Shafer Evidence Theory

Dempster-Shafer evidence theory [39, 40], which is first developed by Dempster and later extended by Shafer, is used to manage various types of uncertain information [41–44], belonging to the category of artificial intelligence. As a theory widely applied under the uncertain environment, it needs weaker conditions and has a wider range of use than the Bayesian probability theory. When the ignorance is confirmed, Dempster-Shafer theory could convert into Bayesian theory, so it is often regarded as an extension of the Bayesian theory. Dempster-Shafer theory has the advantage to directly express the “uncertainty” by assigning the probability to the subsets of the union set composed of multiple elements, rather than to each of the single elements. Besides, it has the ability to combine pairs of bodies of evidence or belief functions to generate a new evidence or belief function [45, 46].

The decision-making or optimization in real system is very complex with incomplete information [47–49]. With the superiority in dealing with uncertain information and the practicability in engineering, a number of applications of evidence theory have been published in the literature indicating its widespread for fault diagnosis [50, 51], pattern recognition [52–54], supplier selection [55, 56], and risk assessment [57, 58]. Also, it exerts a great effect on combining with other theories and methods such as fuzzy numbers [59], decision-making [60], and AHP [61–63]. Moreover, based on the Dempster-Shafer theory, the generalized evidence theory has been proposed by Deng to develop the classical evidence theory [64] to handle conflicting evidence combination [65]. It should be noted that the combination of dependent evidence is still an open issue [66, 67]. For a more detailed explanation of evidence theory, some basic concepts are introduced as follows.

*Definition 1 (frame of discernment). *A frame of discernment is a set of alternatives perceived as distinct answers to a question. Suppose is the frame of discernment of researching problem, a finite nonempty set of elements that are mutually exclusive and exhaustive, indicated byand denote as the power set composed of elements of , and each element of is regarded as a proposition. Based on the two conceptions, mass function is defined as below.

*Definition 2 (mass function). *For a frame of discernment , a mass function is a mapping from to , formally defined bysatisfyingwhere is an empty set and represents the propositions.

In Dempster-Shafer theory, is also named as basic probability assignment (BPA), and is named as assigned probability, presenting how strong the evidence supports . is regarded as a focal element when , and the union of all focal elements are called the core of the mass function.

Considering two pieces of evidence from different and independent information sources, denoted by two BPAs and , Dempster’s rule of combination is used to derive a new BPA from two BPAs.

*Definition 3 (Dempster’s rule of combination). *Dempster’s rule of combination, also known as orthogonal sum, is expressed by , defined as follows:withwhere is a normalization constant, called conflict coefficient of two BPAs. Note that the Dempster-Shafer evidence theory is only applicable to such two BPAs which satisfy the condition .

##### 2.2. Number Theory

number theory, proposed by Deng [32], is a generalization of Dempster-Shafer evidence theory. A wide range of applications have been published based on it, especially in the uncertain environment and MCDM [33]. In the classical Dempster-Shafer theory, there are several strong hypotheses on the frame of discernment and basic probability assignment. However, some shortcomings still exist which limit the representation of some types of information as well as the restriction of the application in practice. number theory, considered as an extension and developed method, makes the following progress.

First, Dempster-Shafer evidence theory deals with the problem about the strong hypotheses, which means that elements in the frame of discernment are required to be mutually exclusive. In general, the frame of discernment is determined by experts, always involving human being’s subjective judgments and uncertainty. Hence, the hypothesis is hard to meet. For example, there are five anchor points “zero dependence [ZD],” “low dependence [LD],” “moderate dependence [MD],” “high dependence [HD],” and “complete dependence [CD]” corresponding to dependence levels available to analysts to make judgments. It is inevitable that there exist some overlaps of human being’s subjective judgments. number theory is more suitable to the actual situation based on the framework of nonexclusive hypotheses. The difference between Dempster-Shafer theory and number theory about this is shown in Figure 1.