An interactive procedure for linguistic multiple attribute ... - Springer Link

Fuzzy Optim Decis Making (2007) 6:17–27 DOI 10.1007/s10700-006-0022-z

An interactive procedure for linguistic multiple attribute decision making with incomplete weight information Zeshui Xu

© Springer Science+Business Media, LLC 2007

Abstract In this paper, we consider the multiple attribute decision making (MADM) problems, in which the information about attribute weights is partly known and the attribute values are expressed in linguistic labels. We first define the concepts of linguistic positive ideal point, linguistic negative ideal point, and satisfactory degree of alternative. Based on these concepts, we then establish some linear programming models, through which the decision maker interacts with the analyst. Furthermore, we establish a practical interactive procedure for solving the MADM problems considered in this paper. The interactive process can be realized by giving and revising the satisfactory degrees of alternatives till an optimum satisfactory solution is achieved. Finally, a practical example is given to illustrate the developed procedure. Keywords Multiple attribute decision making · Interactive procedure · Satisfactory degree · Linguistic label · Linear programming model

1 Introduction Multiple attribute decision making (MADM) consists of selecting the most desirable alternative from a given alternative set according to a set of attributes. Moreover, a decision maker (DM) usually provides only imprecise estimations of attribute weights, because that: (1) a decision should be made under time

Z. Xu (B) College of Economics and Management, Southeast University, Nanjing, Jiangsu, 210096, China e-mail: [email protected]

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pressure and lack of data, (2) many of the attributes are intangible or nonmonetary because they reflect social and environmental impacts, and (3) a DM has limited attention and information processing capabilities (Park, 2004), a selection is not made in a single step. As a result, some interactive approaches have been developed for solving the MADM problems with incomplete information (Park & Kim, 1997, Kim & Ahn, 1999, Kim, Choi, & Kim, 1999, Xu, 2002, Chen & Lin, 2003, Xu & Chen, 2006). Park & Kim (1997) presented some tools for implementing an interactive procedure in MADM with incomplete information. They described models under both certainty and uncertainty for establishing dominance by using a separable linear programming technique, and presented the characteristic of wear dominance and dominance graph. They also presented an algorithm of generating the dominance graph based on the information of pairwise dominance, which is used for aiding the selection of preferable alternatives. Kim and Ahn (1999) suggested a method utilizing individual decision results to form group consensus. Final group consensus ranking toward more agreement of participants can be built through solving a series of linear programming models, using individual decision results under group members’ possibly different weight constraints. They introduced the aggregated net strength, which is the strength difference between the aggregated strength of an alternative over the others and that of the others over the alternative considered, and then ranked the alternatives by comparing the net strength between alternatives. Kim et al. (1999) presented an interactive procedure, which is described for each DM to make a group consensus by interactively modifying his/her incomplete information to be a concrete or complete one. The procedure has some characteristics including the following: (1) a utility range is calculated based on each group member’s incomplete information, and a preference aggregation method is proposed to get the group’s utility, (2) an interactive procedure is provided to help the group reach a consensus, and (3) the methodology is based on only linear programming models under functionally independent condition, and can handle the tradeoff of decision making time, the quality of group decision making and the burden of group members. Xu (2002) developed an interactive approach based on the alternative achievement scale and alternative comprehensive scale. The approach makes use of the subjective information provided by a DM and the objective information to form single-objective programming models. Chen and Lin 2003 proposed an interactive neural network-based approach, in which the decision neural network (DNN) is used to capture and represent the DM’s preference. They solved an optimization problem by DNN to search for the most desirable solution. Xu and Chen (2006) developed an interactive method for multiple attribute group decision making under fuzzy environment. The method transforms fuzzy decision matrices into their expected decision matrices and constructs the corresponding normalized expected decision matrices by two simple formulas, and then aggregates these normalized expected decision matrices into a complex decision matrix. By solving linear programming models, the method diminishes the given alternative set gradually, and finally finds the most preferred alternative(s). All the above methods can only be suitable for dealing with the MADM

Linguistic multiple attribute decision making

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problems with numerical information on attribute values, however, in many real-life situations, such as evaluating the high-technology project investment of venture capital firms (Xu, 2004a), and evaluating the “comfort” or “design” for different kinds of cars, linguistic labels like “good,” “fair,” “poor” are usually be used (Bordogna Fedrizzi, & Passi, 1997; Levrat Voisin, Bombardier, & Bremont, 1997). As a result, a DM is more suitable to provide his/her preferences (attribute values) over alternatives versus various attributes by means of linguistic labels rather than numerical ones. In this paper, we investigate the MADM problems, in which the information about attribute weights is partly known and the attribute values are expressed in linguistic labels. We establish a practical interactive procedure for selecting the most desirable alternative(s). The interactive process can be realized by giving and revising the satisfactory degrees of alternatives till an optimum satisfactory solution is achieved. This paper is organized as follows. Section 2 gives a representation of the problem. Section 3 develops an interactive procedure, which is illustrated with a practical example in Sec. 4. Section 5 concludes this paper.

2 Representation of the problem In real world, human beings are constantly making decisions under linguistic environment (Zadeh & Kacprzyk, 1999; Bustince, Herrera, & Montera, 2006), for example, when evaluating the speed of a car, linguistic labels like “very fast”, “fast”, “slow” are usually be used (Bordogna et al., 1997, Levrat et al., 1997;Herrera, Martinez, & Sanchez, 2005; Herrera-Viedma Martinez, Mata, & Chiclana, 2005; Xu, 2005). Suppose that S = {si | i = −t, ..., t} is a finite and totally ordered discrete label set. Any label, si , represents a possible value for a linguistic label (Yager, 1992, 1995, 1996, Torra, 1996, 2001), and it requires that si < sj iff i < j. The cardinality of S must be small enough so as not impose useless precision on the experts and it must be rich enough in order to allow a discrimination of the performances of each alternative in a limited number of grades (Bordogna et al., 1999). For example, a set of nine labels S could be: S = {s−4 = extremely poor, s−3 = very poor, s−2 = poor, s−1 = slightly poor s0 = fair, 1 = slightly good, s2 = good, s3 = very good, s4 = extremely good} In the process of information aggregating, some results may do not exactly match any linguistic labels in S. To preserve all the given information, Xu (2004a) extended the discrete label set S to a continuous label set S¯ = {sα | α ∈ [−q, q]}, where q(q > t) is a sufficiently large positive integer. If sα ∈ S, then sα is termed an original linguistic label, otherwise, sα is termed a virtual linguistic label. In general, the DM uses the original linguistic labels to evaluate alternatives, and the virtual linguistic labels can only appear in calculation.

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¯ their operational laws are Consider any two linguistic labels sα , sβ ∈ S, defined as follows (Xu, 2004b): (1) sα ⊕ sβ = sα+β ; (2) ρ sα = sρ α , ρ ∈ [0, 1]. Definition 1 Let sα and sβ be two linguistic labels, then we call d(sα , sβ ) = |α − β|

(1)

the distance between sα and sβ . Obviously, the greater the value of d(sα , sβ ), the closer sα to sβ , thus, d(sα , sβ ) can be used as a deviation measure of two linguistic labels. In the following, we represent the MADM problems considered in this paper: Let X = {x1 , x2 , . . . , xn } be a discrete set of alternatives, U = {u1 , u2 , . . . , um } be a discrete set of attributes. Let w = (w1 , w2 , . . . , wm ) ∈ H be the weight m vector of attributes, where wi ≥ 0, i = 1, 2, . . . , m, wi = 1, H is the set of the i=1

known weight information, which can be constructed by the following forms (Park & Kim, 1997, Kim & Ahn, 1999, Xu & Chen, 2006), for i = j: (1) (2) (3) (4) (5)

A weak ranking: {wi ≥ wj }. A strict ranking: {wi − wj ≥ αi }. A ranking with multiples: {wi ≥ αi wj }. An interval form: {αi ≤ wi ≤ αi + εi }. A ranking of differences: {wi − wj ≥ wk − wl }, for j = k = l.

where {αi } and {εi } are non-negative constants. Let A = (aij )m×n be the linguistic decision matrix, where aij ∈ S, which is an attribute value, given by the DM, for the alternative xj ∈ X with respect to the attribute ui ∈ U, and let aj = (a1j , a2j , . . . , amj ) be the vector of attribute values corresponding to the alternatives xj ( j = 1, 2, . . . , n). + + + + Definition 2 Let a+ i = st , for all i , then we call a = (a1 , a2 , ..., am ) the linguistic positive ideal point. − − − − Definition 3 Let a− i = s−t , for all i , then we call a = (a1 , a2 , . . . , am ) the linguistic negative ideal point.

3 An interactive procedure for linguistic MADM with incomplete weight information Based on the linguistic decision matrix A = (aij )m×n , the overall value of the alternative xj can be expressed as zj (w) = w1 a1j ⊕ w2 a2j ⊕ · · · ⊕ wm amj ,

j = 1, 2, . . . , n

(2)


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Obviously, the greater the value zj (w), the better the alternative xj is. Especially, we get the overall values corresponding to the linguistic positive ideal point and the linguistic negative ideal point as follows, respectively: z+ (w) = w1 st ⊕ w2 st ⊕ · · · ⊕ wm st = (w1 + w2 + · · · + wm ) st = st , − z (w) = w1 s−t ⊕ w2 s−t ⊕ · · · ⊕ wm s−t = (w1 + w2 + · · · + wm ) s−t = s−t .

(3)

(4)

By (1), we let d(zj (w), z− (w)) be the distance between the overall value zj (w) of the alternative xj and the overall value z− (w) corresponding to the linguistic negative ideal point a− , then the greater the value d(zj (w), z− (w)), the better the alternative ej is. Definition 4 Let d(z+ (w), z− (w)) be the distance between the overall value z+ (w) corresponding to the linguistic positive ideal point a+ and the overall value z− (w) corresponding to the linguistic negative ideal point a− , then we call µ(zj (w)) =

d(zj (w), z− (w)) d(z+ (w), z− (w))

(5)

the satisfactory degree of the alternative xj . From Definition 4, we know that the satisfactory degree µ(zj (w)) of the alternative xj is the ratio of the distance between the overall value zj (w) of the alternative xj and the overall value z− (w) of the linguistic negative ideal point a− to the distance between the overall value z+ (w) of the linguistic positive ideal point a+ , and the overall value z− (w) of the linguistic negative ideal point a− . Since d(z+ (w), z− (w)) = d(st , s−t ) = | t − (−t)| = 2t

(6)

then (5) can be rewritten as: µ(zj (w)) =

1 d(zj (w), z− (w)). 2t

(7)

Obviously, the greater the distance between the overall value zj (w) of the alternative xj and the overall value z− (w) of the linguistic negative ideal point a− , the higher the satisfactory degree µ(zj (w)) of the alternative xj is, that is, the satisfactory degree µ(zj (w)) of the alternativexj is a strictly monotone increasing function with respect to d(zj (w), z− (w)). Therefore, the higher the satisfactory

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degree µ(zj (w)), the better the alternative ej is. As a result, we establish the following optimization model: (M-1) Maximize µ(w) = (µ(z1 (w)), µ(z2 (w)), . . . , µ(zm (w))). Subject to: w ∈ H m wi = 1. wi ≥ 0, i = 1, 2, . . . , m, i=1

We utilize the max–min operator proposed by Zimmermann and Zysno (1980) to integrate the satisfactory degrees of all alternatives, i.e., we get the following optimization model: (M-2)

Maximize λ Subject to: µ(zj (w)) ≥ λ, w∈H wi ≥ 0,

i = 1, 2, . . . , m,

j = 1, 2, . . . , n, m

wi = 1,

i=1

where λ = min µ(zj (w)). j

By solving the model (M-2), we get the original optimal solution w(0) = (0) (0) T (0) (w1 , w(0) 2 , . . . , wm ) , and then calculate the satisfactory degrees µ(zj (w )) (j = 1, 2, . . . , n) of the alternatives xj (j = 1, 2, . . . , n). In the course of decision (0) making, the DM provides the lower bounds λj (j = 1, 2, . . . , n) of the satisfactory degrees of the alternatives xj (j = 1, 2, . . . , n) according to µ(zj (w(0) ))(j = 1, 2, . . . , n). Then, we establish the following optimization model: n (M-3) Maximize J = λj j=1 (0)

Subject to: µ(zj (w)) ≥ λj ≥ λj , j = 1, 2, ..., n w ∈ H, m wi ≥ 0, i = 1, 2, ..., m, wi = 1. i=1

Solving the model (M-3), if there exists no optimal solution, then the DM needs to reconsider the lower bounds λ(0) j (j = 1, 2, . . . , n) of the satisfactory degrees of the alternatives xj (j = 1, 2, . . . , n) till the optimal solution is obtained. Theorem 1 The optimal solution of the model (M-3) is the Pareto solution of the model (M-1). Proof Suppose that w∗ is the optimal solution of the model (M-3), and w∗ is not the Pareto solution of the model (M-1), then there exists w such that µ(zj (w )) ≥ µ(zj (w∗ )), for all xj ∈ X, and there exists xj0 ∈ X, such that (0) µ(zj0 (w )) ≥ µ(zj0 (w∗ )). Then µ(zj (w )) ≥ λj ≥ λj , for all xj ∈ X, and there (0)

exists λj0 , such that µ(zj0 (w )) ≥ λj0 > λj0 ≥ λj . Thus, n j=1,j=j0

λj + λj0 >

n j=1

λj ,

(8)


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which contradicts to the supposition condition. This completes proof of Theorem 1. Based on the above models and Theorem 1, we give a practical interactive procedure for the MADM problems considered in this paper. The procedure is described as follows: Step 1 Use the model (M-2) to derive the original optimal solution w(0) = (0) (0) (0) (w1 , w2 , . . . , wm )T , and then calculate the satisfactory degrees µ(zj (w(0) )) (j = 1, 2, . . . , n) of the alternatives xj (j = 1, 2, . . . , n). The DM gives the lower bounds λ(0) j (j = 1, 2, . . . , n) of the satisfactory degrees of the alternatives xj (j = 1, 2, . . . , n) according to the satisfactory degrees µ(zj (w(0) ))(j = 1, 2, . . . , n). Let k = 1. (k)

(k)

Step 2 Utilize the model (M-3) to derive the weight vector w(k) = (w1 , w2 , . . . , (k)

wm )T and calculate the satisfactory degrees µ(zj (w(k) ))(j = 1, 2, . . . , n) of the alternatives xj (j = 1, 2, . . . , n).

Step 3 If the DM is satisfied with the result obtained by Step 2, then calculate the overall values zj (w)(j = 1, 2, . . . , n) of the alternatives xj (j = 1, 2, . . . , n) by using (2), and rank all alternatives according to the values of zj (w)(j = 1, 2, . . . , n), and then go to step 4; if there exists no solution for the model (M-3) or the result does not satisfy the DM, then the DM should increase the satisfactory degrees of some alternatives, and decrease the satisfactory degrees of some other alternatives. Let k = k + 1, and return to Step 2. Step 4 End. 4 Illustrative example In this section, a MADM problem of evaluating university faculty for tenure and promotion (adapted from Bryson and Mobolurin, 1995) is used to illustrate the developed procedure. A practical use of the developed procedure involves the evaluation of university faculty for tenure and promotion. The criteria (attributes) used at some universities are u1 : teaching, u2 : research, and u3 : service. Five faculty candidates (alternatives) xj (j = 1, 2, 3, 4, 5) are to be evaluated using the linguistic label set S = {s−4 = extremely poor, s−3 = very poor, s−2 = poor, s−1 = slightly poor, s0 = fair, s1 = slightly good, s2 = good, s3 = very good, s4 = extremely good} by the DM under these three attributes, as listed in Tables 1. Suppose that the known weight information is as follows: H = {0.25 ≤ w1 ≤ 0.40, 0.15 ≤ w2 ≤ 0.30, w3 > w2 }.

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Table 1 Linguistic decision matrix A

ui

x1

x2

x3

x4

x5

u1 u2 u3

s2 s3 s0

s3 s4 s−2

s0 s3 s2

s4 s−1 s2

s2 s3 s1

To get the most desirable alternative(s), the following steps are involved: Step 1 Utilize the model (M-2) to establish the following optimization model: Maximize λ, 1 Subject to : (2w1 + 3w2 + 4) ≥ λ, 8 1 (3w1 + 4w2 − 2w3 + 4) ≥ λ, 8 1 (3w2 + 2w3 + 4) ≥ λ, 8 1 (4w1 − w2 + 2w3 + 4) ≥ λ, 8 1 (2w1 + 3w2 + w3 + 4) ≥ λ, 8 0.25 ≤ w1 ≤ 0.40, 0.15 ≤ w2 ≤ 0.30, w3 > w2 , wj ≥ 0, j = 1, 2, 3, w1 + w2 + w3 = 1. By solving this model, we get the original optimal solution w(0) = (0.3751, 0.3000, 0.3429)T and obtain the satisfactory degrees µ(zj (w(0) ))(j = 1, 2, 3, 4, 5) of the alternatives xj (j = 1, 2, 3, 4, 5): µ(z1 (w(0) )) = 0.7063, µ(z2 (w(0) )) = 0.7049,

µ(z3 (w(0) )) = 0.6982

µ(z4 (w(0) )) = 0.7358, µ(z5 (w(0) )) = 0.7491 (0)

The DM gives the lower bounds λj (j = 1, 2, 3, 4, 5) of the satisfactory degrees of the alternatives xj (j = 1, 2, 3, 4, 5) according to the satisfactory degrees µ(zj (w(0) )) (j = 1, 2, 3, 4, 5): (0)

λ1 = 0.7000,

(0)

λ2 = 0.7100,

(0)

λ3 = 0.6900,

(0)

λ4 = 0.7300,

(0)

λ4 = 0.7300


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Step 2 By the model (M-3), we establish the following optimization model: Maximize J =

5

λj ,

j=1

Subject to :

1 (2w1 + 3w2 + 4) ≥ λ1 ≥ 0.7000, 8 1 (3w1 + 4w2 − 2w3 + 4) ≥ λ2 ≥ 0.7100, 8 1 (3w2 + 2w3 + 4) ≥ λ3 ≥ 0.6900, 8 1 (4w1 − w2 + 2w3 + 4) ≥ λ4 ≥ 0.7300, 8 1 (2w1 + 3w2 + w3 + 4) ≥ λ5 ≥ 0.7300, 8 0.25 ≤ w1 ≤ 0.40, 0.15 ≤ w2 ≤ 0.30, w3 > w2 wj ≥ 0 , j = 1, 2, 3, w1 + w2 + w3 = 1.

Solving this model, we get the attribute weight vector w(1) = (0.39, 0.30, 0.31)T , and calculate the satisfactory degrees µ(zj (w(1) )(j = 1, 2, 3, 4, 5) of the alternatives xj (j = 1, 2, 3, 4, 5): µ(z1 (w(1) ) = 0.7100,

µ(z2 (w(1) )) = 0.7188, µ(z3 (w(1) )) = 0.6900 µ(z4 (w(1) )) = 0.7350, µ(z5 (w(1) )) = 0.7487.

Step 3 The DM is satisfied with this result. Therefore, we can calculate the overall values zj (w(1) )(j = 1, 2, 3, 4, 5) of the alternatives xj (j = 1, 2, 3, 4, 5) by using (2): z1 (w(1) ) = s1.68 ,

z2 (w(1) ) = s1.75 ,

z3 (w(1) ) = s1.52 ,

z4 (w(1) ) = s1.88 ,

z5 (w(1) ) = s1.99 and rank all alternatives according to the values of zj (w(1) )(j = 12, 3, 4, 5): x5 x4 x2 x1 x3 . Hence, the most desirable candidate is x5 . 5 Concluding remarks In MADM, the DM sometimes may provide incomplete information about attribute weights, and the attribute values are also usually expressed in linguistic labels because of time pressure, lack of knowledge, and his/her limited

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attention and information processing capabilities. Thus, it is necessary to pay attention to this issue. In this paper, we have defined some new concepts, such as the linguistic positive ideal point, linguistic negative ideal point, and satisfactory degrees of alternatives under linguistic environment. Based on these concepts, we have established an interactive procedure for solving the problems. The interactive process can be realized by giving and revising the satisfactory degree of alternative till an optimum satisfactory solution is achieved. The procedure has been applied to the evaluation of university faculty for tenure and promotion. The theoretical analysis and the computational results have showed that the interactive procedure developed in this paper is an encouraging and robust method for solving the MADM problems with linguistic information. Much work still remains to be done in the future to make the interactive procedure more effective, especially developing a decision support system (DSS) based on the framework proposed in this paper. Acknowledgements The work was supported by the National Natural Science Foundation of China under Grant 70571087.

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