JOURNAL OF COMPUTERS, VOL. 6, NO. 4, APRIL 2011 711 Analysis of Uncertainty Multi-attribute Group Decision Making Process Based on D-S Evidence Theory Jialin Liu School of Management, Anhui University of Architecture, Hefei, 230022, China Email: [email protected] Bengang Gong School of Management Engineering, Anhui Polytechnic University, Wuhu 241000, China Email: [email protected] Abstract—Group decision-making problem is a common The Dempster-Shafer evidence theory of (the D-S and crucial human activity. By the characteristic of group evidence theory) [8, 9] provides an appropriate decision making process, the model of multi-attribute group framework to model ignorance whilst fuzziness can be decision making (MAGDM) process based on evidence well treated using fuzzy set theory [10-12]. The D-S theory is proposed. The interacting process for MAGDM is theory has been developed by Yang et al and Wang et al. analyzed, and the consensus analysis and the improving for multiple attribute decision analysis under uncertainty approach in group experts’ decision making based on similarity is given. By introducing the concept of experts’ [12-18]. Due to the power of the MADM approach based relative reliability, the Dempster’s rule of combination is on D-S theory in handing and representing uncertainties, improved, and new aggregate method for group experts’ so far, it have been applied to many areas, such as decision making. An example is presented to demonstrate environmental impact assessment [12], pipeline leak the implementation of this improved method. detection[19], bridge condition assessment[20], etc. In addition, a novel reliability prediction technique based on Index Terms—D-S Evidence theory, uncertainty, multi- the evidential reasoning algorithm is developed and attribute group decision making, similarity, relative applied to forecast reliability in turbocharger engine reliability systems [3]. Multi-attributive group decision analysis (MAGDA) problems can be viewed as decision situations where a I. INTRODUCTION group of experts express their preference on multiple Owing to the complexity of construction engineering, a attributes (criteria) to a problem to be solved and try to single expert or decision maker cannot often find a common solution. Similarly, MAGDA problem comprehensively consider all the aspects of one thing, so under various uncertainties can also be modeled using the a complex decision usually has to be made by integrating extensions of the decision making approach based on D-S a group of experts' knowledge and experiences. theory [12]. Different from most conventional MAGDA Therefore, the practice of multiple attribute group methods, the D-S evidence theory approach describes decision making (MAGDM) is to invite internal experts each attribute at an alternative by a distributed assessment or external experts or their combination of related fields using a belief structure [12]. As part of the effort to deal to evaluate each attribute of every alternative with uncertainty MAGDM problems with uncertainties individually. At present, the problem of multiple attribute and subjectivity, the D-S evidence theory has been group decision-making has become a new international devised, developed. Reference [4] proposed incomplete research hotspot [1-3]. Because of the complexity of partial in order to express the uncertain multiple attribute objective things, uncertainty and ambiguity of human decision making information, integrated the uncertainty thinking and other reasons, more multi-attribute group information by using evidential reasoning algorithm. decision is carried out in the uncertain environment. It’s Reference [5, 6] proposed the synthesis of incomplete of theoretical and practical importance to study such a information based on the evidence put forward recursive kind of uncertain multi-attribute decision making algorithm for reasoning to express the uncertain problem [4-7]. information of multi-attribute group decision making. Reference [7] used a number of fuzzy languages into a precise number of methods and proposed multi-attribute Manuscript received April 10, 2010; revised June 10, 2010; accepted group decision making based on evidence theory method October 15, 2010. of decision making to solve the linguistic assessment incomplete information. Reference [3] proposed a GC © 2011 ACADEMY PUBLISHER doi:10.4304/jcp.6.4.711-717 712 JOURNAL OF COMPUTERS, VOL. 6, NO. 4, APRIL 2011 (group consensus) based ER (evidential reasoning) m()∅ = 0 and ∑ mA()= 1 approach on the basis of the ER approach associated with A ⊆Θ belief structures in order to find a GC based solution to a Θ MAGDA problem. where ∅ is an empty set, A is any subset of Θ , and 2 However, the above methods does not take into account is the power set of Θ , which consists of all the subsets of Θ the characteristics of focal element of the similarity which Θ , i.e., 2={∅Θ ,{a1121} ,...,{ aNN} ,{ aa ,} ,...,{ aa ,} ,..., } . given by different experts, and with little consideration to The assigned probability (also called probability mass) the reliability of expert opinion is also an important aspects m (A) measures the belief exactly assigned to A and to measure the importance of experts. This paper was first represents how strongly the evidence supports A. All the given the uncertainty multi-attribute group decision assigned probabilities sum to unity and there is no belief making process model based on D-S evidence theory. This in the empty set ( ∅ ). The assigned probability to Θ , decision making process model are mainly for the “Revised process” and “Synthetic process” in two phases; i.e. m()Θ , is called the degree of ignorance. Each subset then give a analysis and adjustment method based on the A ⊆Θ such that m (A) > 0 is called a focal element of m. consistency of group interaction of focal element of the All the related focal elements are collectively called the similarity; finally, introducing the concept of experts’ body of evidence. relative reliability, making improvements to the Associated with each bpa are a belief measure (Bel) Dempster’s rule of combination, coming up with a new and a plausibility measure (PlS) which are both functions: synthetic method, and showing the synthetic method is m :2Θ → [0,1] , defined by the following equations, reasonable by experiments. respectively: The paper is organized as follows. The uncertain Definition 1. Let Θ be the frame of discernment, each multi-attribute group decision making process model bpa is a belief measure (Bel), which is a based on D-S theory is described in Section 2; the Θ analysis and adjustment of expert group decision function:m : 2→ [0,1] , defined by the following (Revised process) is described in Section 3; The equations: Dempster’s rule of combination of the Θ improvement(Synthetic process) is described in Section 4; Bel() A= ∑ m () B for all ∀∈A 2 the conclusion is presented in Section 5. BA⊆ where A and B are subsets of Θ . Bel (A) represents the II. UNCERTAIN MULTI-ATTRIBUTE GROUP DECISION exact support to A. MAKING PROCESS MODEL BASED ON D-S THEORY Definition 2. Let Θ be the frame of discernment, and The D-S evidence theory [8, 9] provides a technique of each bpa is a plausibility measure (Pls), which is a evaluating a decision alternative’s basic probability function m :2Θ → [0,1] , defined by the following assignment (bpa) even when the decision matrix is equations incomplete [21]. For a MAGDM problem under pls() A=− 1 Bel () A = m () B , for all ∀A∈ 2Θ uncertainty environment, the uncertainty MAGDM ∑ BA∩ ≠Φ process model based on D-S evidence theory is proposed in this paper. Before the introduction, some basic where A and B are subsets of Θ , and A denotes the concepts of the D-S evidence theory and Group decision complement of A. ∀A ⊆Θ , Pls(A) represents the making process model are discussed. possible support to A, i.e. the total amount of belief that could be potentially placed in A. A. Basics of Dempster-Shafer theory The kernel of the D-S evidence theory is the The D-S evidence theory was first developed by Dempster’s rule of combination by which the evidence Dempster in the 1960s and later extended and refined by from different sources is combined. The rule assumes that Shafer in the 1970s [8, 9]. The D-S evidence theory is the information sources are independent and use the related to Bayesian probability theory in the sense that orthogonal sum to combine multiple belief structures they both can update subjective beliefs given new mm= ⊕⊕⊕ m m, where ⊕ represents the evidence [8, 9, 12]. The major difference between the two 12 K theories is that the evidence theory is capable of operator of combination. With two belief structures m1 combining evidence and dealing with ignorance in the evidence combination process. The basic concepts and and m2 , the Dempster’s rule of combination is defined as definitions of the evidence theory relevant to this paper follows: are briefly described as follows. ⎧ 0 C = ∅ Let Θ={aa1 , , N } be a collectively exhaustive and ⎪ []()mmC12⊕=⎨ mAmB12() () mutually exclusive set of decision alternatives, called the ∑ ABC∩= C ≠ ∅ frame of discernment. A basic probability assignment ⎪1()()− mAmB ⎩ ∑ AB∩=∅ 12 (bpa) is a function m: m :2Θ → [0,1], which is called a mass function, satisfying where A and B are both focal elements © 2011 ACADEMY PUBLISHER JOURNAL OF COMPUTERS, VOL. 6, NO. 4, APRIL 2011 713 preferences of individuals reflects the group’s interaction and [mmC12⊕ ]()itself is a bpa. The denominator, process, “Synthetic process” is the group decision making 1()()− mAmB is called the normalization ∑ AB∩=∅ 12 process that individual preference values by amended factor, kmAmB= () () is called the degree of which was synthesis. There is the difference for two main ∑ AB∩=∅ 12 processes as follows: the amendment process of the conflict, which measures the conflict between the pieces preference value is occurred in individual judgment value of evidence.