Mathematical Problems in Engineering

Volume 2018, Article ID 2108726, 14 pages

https://doi.org/10.1155/2018/2108726

## A Multiple Criteria Decision Analysis Method for Alternative Assessment Results Obeying a Particular Distribution and Application

^{1}Jinling Institute of Technology, Nanjing, Jiangsu 211169, China^{2}College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 211106, China

Correspondence should be addressed to Wang He-Hua; nc.ude.tij@hhw

Received 6 October 2017; Revised 8 March 2018; Accepted 26 March 2018; Published 3 May 2018

Academic Editor: Josefa Mula

Copyright © 2018 Wang He-Hua 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

A multiple criteria decision analysis (MCDA) problem is studied in this paper, for which the evaluation results obey a particular distribution. First, when solving a multiple criteria decision analysis (MCDA) problem, a grey target decision analysis framework is proposed to determine uncertain parameters and criteria weights. A measurement for comprehensive off-target distance is defined, which includes the undetermined parameters. Second, to satisfy the requirements of a specific distribution (such as a normal distribution) in the assessment results, an optimization model that incorporates the off-target distance constraints is proposed by considering the skewness and kurtosis test method. Third, a particle swarm optimization (PSO) algorithm is extended to solve the proposed model by seeking the appropriate parameters and weights. Fourth, a numerical example is applied to demonstrate the feasibility and application of the proposed method. In the end, the proposed model is extended to other distribution requirements.

#### 1. Introduction

Over the past few decades, multiple criteria decision analysis (MCDA) tool has been adopted to enable a decision-maker (DM) to make a decision from a finite set of alternatives with respect to multiple criteria [1]. MCDA is defined by a set of alternatives (denoted by ), from which a DM can select the optimal alternative according to the identified set of criteria (denoted by ). The evaluation value of the alternative with respect to the criterion given by the DMs is represented as . The key tasks of the MCDA methods are to study the weight of each criterion, integrate the multiple criteria, and aggregate the information from different DMs. On that basis, many methods are developed to extend the possible applications for MCDA.

In the process of the MCDA, decision-makers prefer that the alternatives obey a certain distribution when faced with more decisions, such as the normal distribution. On one hand, if the alternative is evaluated to obey the normal distribution, the corresponding evaluation results are reasonable. On the other hand, when the attribute weight and the decision rules are not very clear, a normal distribution is often also shown for the alternatives. Based on the framework of MCDA, the comprehensive value can be obtained by aggregating the values and weights of criteria. In most practical cases, the MCDA problems often involve multiple dimensions, in which the consideration of criteria interactions is essential. In this paper, we consider a new situation when the assessment result obeys a normal distribution with criteria independence. Under such conditions, the comprehensive value still obeys the normal distribution, which is related to the weights of criteria. Then the multidimensional problem can be reduced to a weighting problem. Thus, we mainly study the issue of how to derive the weights of the criteria and make a decision under a normal distribution. As many decision-making problems pertain to the MCDA category, we propose a method for deriving the weights of the criteria as a decision-making parameter. Considering the simplicity of the grey target decision among the MCDA, this paper proposes some parameters for the normally designed problems based on the grey target decision analysis framework. Our main contribution is to propose an analysis framework for the MCDA problem with a normal distribution being included based on the grey target decision method and then design an algorithm for the model. First, we propose an improved grey target decision method to optimize the target-setting and criterion weights. Second, we establish an optimization model to test the normal distribution of criteria based on the skewness and kurtosis. Third, we develop a particle swarm optimization (PSO) algorithm to solve the model. According to the results of the model, we can verify the rationality of the target center, obtain the criteria weights, and select the optimal alternative. Furthermore, the analysis method is applied in a case and extended to two other cases.

The structure of this paper is presented as follows. Section 2 reviews the related literature. In Section 3, the grey target decision method is described. In Section 4, we describe the random variable settings of the target, the measurement of the combined off-target distance, and the constraints of normal distribution assessment. Then, we propose a model to determine the optimal target and criteria weights. In addition, an intelligent optimization algorithm is developed to solve this model. In Section 5, we apply the proposed methodology by using real data, and some extensions are suggested in Section 6. Finally, conclusions in the paper are given in Section 7.

#### 2. Literature Review

A number of well-known MCDA methods have been studied in recent years [1], such as the multiattribute utility theory (MAUT), the analytic hierarchy process (AHP), linear programming technique for multidimensional analysis of preference (LINMAP), and the elimination and choice translating reality (ELECTRE) method. The MAUT is an effective tool for decision analysis and quantitative analysis of decision-making problems, which is carried out by utility analysis [1, 2]. The AHP, introduced by Saaty [3], is an effective tool for dealing with complex decision-making, helping decision-makers set priorities and select the best alternative. The LINMAP aims to obtain the best alternative, which is closest to the ideal alternative through pairwise comparisons of alternatives [4]. The ELECTRE method is modeled by using binary outranking relations between two alternatives, which considers four situations [5]. These methods are widely used in MCDA problems as the basic analysis tools. Next, we review the literatures according to the steps of MCDA. First of all, the multiple preferences are obtained in many studies because preference structuring and modeling are greatly important. The ordinal information was studied and preference programming was suggested in [6]. The pairwise judgment in the AHP is widely applied as a preference style [3]. To solve the uncertain decision-making problems, linguistic preference [7], interval numbers [8], and triangular fuzzy matrix [9] were developed in many studies. From these literatures, we can see that various preference styles are used to express the DM’s judgments. The combination use of different uncertain styles will be a research trend for MCDA problems. Second, many aggregation methods of MCDA are developed to aggregate the information effectively, such as the utility-based individual preference [2], the nonlinear information aggregation method [10], the prioritized weighted aggregation [11], the ordered weighted averaging (OWA) operator and the families of OWA [12], the Choquet-Mahalanobis operator [13], and the methods of determining OWA weights [14]. The optimization model and aggregation operator are the main methods. In addition, the applications of MCDA are much increasingly common in practice. For example, the supplier selection was studied with the fuzzy technique for order of preference by similarity to an ideal solution (TOPSIS) in [15]; the global e-government was evaluated in [16]; the energy policy support was researched in [17].

Uncertain situations often arise when DMs make a decision. Thus, many methods were developed to solve the uncertainty. Podinovski examined the ranking order of decision alternatives under uncertainty with unknown utility functions and rank-ordered probabilities [18]. Jiménez et al. introduced a dominance intensity measuring method to derive a ranking order of alternatives on the basis of the multiattribute utility theory (MAUT) and fuzzy sets theory [19]. The interval AHP was applied to solve the complex case [20]. Zhao et al. aggregated the hesitant triangular fuzzy information method based on Einstein operations and applied it to multiple criteria decision-making [21]. Lee and Chen [22] proposed a new fuzzy decision-making method based on the proposed likelihood-based comparison relations of the hesitant fuzzy linguistic term sets. Due to the complexity and uncertainty in decision-making problems and the inherent subjective characteristics of decision-makers’ judgments, the descriptions of criteria and alternatives involve uncertainty [23]. In the problems with criteria interactions, classification methods are studied [13] for some particular distributions. The problem can be deduced as a particular MCDA case with uncertain parameters, which should be analyzed from the perspective of the modeling or optimization of some related parameters. In most methodologies, it is assumed that each of the criteria is independent of the others. In this paper, we also focus on the discussion of the MCDA method with the assumption of independent criteria. But, in recent years, many methods focus on the criteria interactions. Different from the methods of independent criteria, the method of criteria interaction also involves the appropriate selection of aggregation function and distance. Fuzzy measures can be used to express redundancy, complementariness, and interactions among criteria [24, 25]. Choquet integral [26] and Sugeno integral [27] provide two main aggregation functions to deal with criteria interactivity. Distance functions based on fuzzy integral are also studied [25]. The Mahalanobis distance and the Euclidean distance are used to measure the distance between objects [28]. Mahalanobis distance [13] uses the inverse of the variance-covariance matrix to measure the covariance distance of data, which is useful in problems considering the dependence between criteria. The Euclidean distance is only applied to deal with independent variables. But the Euclidean distance is simple to compute and interpret. Under the assumption of independent criteria, the research of criteria interactions is an interesting work to be studied in the future. In this paper, we focus on the discussion of the MCDA method for which the assessment results of alternatives obey a particular distribution under the assumption of independent criteria. The relevant research involves the determination of criteria weights, the selection of decision model, and the decision parameter settings, as well as the information reasoning and assembling. For example, criterion weight was determined based on the feedback model in multiple criteria group decision analysis problems with requirements for group consensus in an evidential reasoning context [29]. Ahn [30] presented a simple method for finding the extreme points of various types of incomplete criteria weights. Chun [31] solved the multicriteria decision problem with sequentially presented decision alternatives based on the assumption that the decision-maker has a major criterion that must be “optimized” and minor criteria that must be “satisfied.” Yang and Wang proposed a linguistic decision aiding technique based on incomplete preference information [32]. However, few studies focused on the problems in which alternatives obey a specific distribution.

The grey system theory is also widely used in MCDA, which usually deals with the uncertain problems. Since Deng first introduced the grey system, its theory and application have been developed rapidly in recent years [33]. The grey target decision method is an important application in the grey system theory in decision-making problems. The method refers to the targeted value setting in the satisfactory effect region, which is currently used in project evaluation, supplier selection, production efficiency evaluation, and other fields [34]. Chen et al. [35] presented an approach for monitoring equipment conditions based on the grey target decision. The measurement for target distance and the weight optimization model were also studied for adaptation to the three-parameter interval grey number [36]. Considering the multiple stages linguistic label, the grey target decision method is proposed to measure the target distance [37]. When the value distribution of information is asymmetrical, a multiscale extended grey target decision method is proposed to deal with the problems with interval grey numbers [38].

Information is often imprecise in most complex decision-making problems, which cannot possibly be predicted with certainty on the alternative performances. From the above review, the existing research of MCDA focuses on fuzzy and uncertain information processing, the weight setting method and optimization model, and the preference data aggregation methods. However, most of MCDA researches rarely discuss the multiple criteria decision analysis problem of alternative information obeying a specific distribution. Therefore, it is necessary to study such decision-making problems when the decision-making object is subject to a distribution.

#### 3. Grey Target Decision Analysis Theory

The grey systems theory was developed by Deng to study the problems involving “small samples and poor information” [33]. Its research objects can be classified into the black ones, the white ones, and the grey ones, according to certain cognitive hierarchy. In this theory, a system is usually defined as a black box if its internal structures and features are completely unknown, whereas white box indicates that the internal features of the system have been fully explored. If the internal features of a system are partly known and partly unknown, it is known as a grey system. The differences between fuzzy and grey methods are summarized in a previous study [34]. Unlike fuzzy mathematics, the grey systems theory focuses on research objects that have a clear extension and unclear intension, whereas fuzzy mathematics has its strength in the study of problems with “cognitive uncertainties.” All objects of fuzzy mathematics have the characteristic of “having a clear interior extent without a clear exterior extent.”

In addition to controlling the intrinsic social, economic, agricultural, and ecological characteristics, the main research tasks of grey system theory are forecasting and decision-making. The nonuniqueness principle is an important fundamental principle of grey theory. This implies that the solution to any problem with incomplete and nondeterministic information is not unique. Strategically, the principle of nonuniqueness is realized through the concept of grey targets. The idea and analysis of the grey target decision are presented in Figure 1. When the optimal or satisfactory goal is set to be the grey target, the area between the grey target and the alternative is essentially an optimal or satisfactory area, which is not a strict optimal result. In practical applications, the grey target decision method is flexible. In many cases, it is impossible to achieve absolute optimality, so it is often desirable to have a satisfactory outcome. In the actual application process, the grey target can gradually shrink and reduce to a point.