ScienceDirect Hierarchical Multiobjective Fuzzy

Available online at www.sciencedirect.com

ScienceDirect Procedia Computer Science 22 (2013) 162 – 171

17th International Conference in Knowledge Based and Intelligent Information and Engineering Systems - KES2013

Hierarchical Multiobjective Fuzzy Random Linear Programming Problems Hitoshi Yanoa , Kota Matsuib b Department

a Graduate School of Humanities and Social Sciences, Nagoya City University,Nagoya, 467-8501, Japan of Computer Science and Mathematical Informatics, Graduate School of Information Science, Nagoya University, Nagoya, 464-8601, Japan

Abstract In this paper, we propose an interactive decision making method for hierarchical multiobjective fuzzy random linear programming problems (HMOFRLP), in which multiple decision makers in a hierarchical organization have their own multiple objective linear functions with fuzzy random variable coefficients. To adress HMOFRLP, it is assumed that each decision maker has fuzzy goals for permissible probability levels in a fractile optimization model. Through a fuzzy decision, two types of membership functions of the original objective functions and the corresponding permissible probability levels are integrated, and a Pareto optimal solution concept is defined. A satisfactory solution is obtained from among a Pareto optimal solution set through the interaction with the decision makers, in which the hierarchical decision structure is reflected through the decision powers. c 2013 2013 The The Authors. © Authors. Published Publishedby byElsevier ElsevierB.V. B.V. Open access under CC BY-NC-ND license. Selection and International. Selection and peer-review peer-reviewunder underresponsibility responsibilityofofKES KES International Keywords: fuzzy random variable; a fractile optimization model; hierarchical multiobjective programming; Pareto optimal solutions; decision power

1. Introduction In the real world decision making situations, it is often required that the goal of the overall system is achieved in the hierarchical structure, where many decision makers who belong to its sections or divisions are in action to seek their own goals independently and are affected each other. The Stackelberg games [1, 20] can be regarded as multilevel programming problems with multiple decision makers. Although many kinds of techniques to obtain a Stackelberg solution have been proposed, almost all of such techniques are unfortunately not efficient in computational aspects. In order to circumvent the computation inefficiency to obtain such a Stackelberg solution and the paradox that the lower level decision power often dominates the upper level decision power, Lai [11], Shih et al.[18] and Lee et al.[13] introduced concepts of memberships of optimalities and degrees of decision powers and proposed fuzzy approaches to multilevel linear programming problems [21] under the assumption that the decision makers are not noncooperative but cooperative each other in the hierarchical decision situations. ∗ Hitoshi

Yano Tel.: +81-52-872-5182; fax: +81-52-872-1531. E-mail address: [email protected].

1877-0509 © 2013 The Authors. Published by Elsevier B.V. Open access under CC BY-NC-ND license. Selection and peer-review under responsibility of KES International doi:10.1016/j.procs.2013.09.092

Hitoshi Yano and Kota Matsui / Procedia Computer Science 22 (2013) 162 – 171

163

On the other hand, in the actual decision making situations, the decision makers often encounter difficulties to deal with vague information or uncertain data. In order to deal with decision problems involving uncertainty, stochastic programming approaches [2],[3],[4],[6] and fuzzy programming approaches [12],[16],[24] have been developed. Recently, mathematical programming problems with fuzzy random variables [10] have been proposed [7],[14],[19], whose concept includes both probabilistic uncertainty and fuzzy one simultaneously. For multiobjective fuzzy random linear programming problems (MOFRLP), Katagiri et al. [8],[9] formulated and proposed interactive methods to obtain a satisfactory solution. As their generalized versions, Yano et al.[22, 23] have proposed fuzzy approaches for MOFRLP. In this paper, we focus on hierarchical multiobjective fuzzy random linear programming problems (HMOFRLP), where multiple decision makers in a hierarchical organization have their own multiple objective linear functions with fuzzy random variable coefficients. In order to deal with HMOFRLP, it is assumed that each decision maker has fuzzy goals for not only permissible probability levels in a fractile optimization model. Using the fuzzy decision, a Pareto optimal solution concept is introduced, in which a proper balance between two kinds of membership functions of the original objective functions and the corresponding permissible objective levels is always attained. A satisfactory solution is obtained from among a Pareto optimal solution set through the interaction with the decision makers, in which the hierarchical decision structure is reflected through the decision power. In section 2, HMOFRLP is formulated as a hierarchical multiobjective stochastic programming problem (HMOSP) by using a concept of a possibility measure [5]. In section 3, through a fractile optimization model for HMOSP, DG -Pareto optimal concept is introduced. After each decision maker specifies the decision powers [11, 13] and the reference membership values [16], the corresponding DG -Pareto optimal solution is obtained by solving the minmax problem. In section 4, an interactive algorithm is proposed to obtain the satisfactory solution from among a DG -Pareto optimal solution set by solving the minmax problem on the basis of the linear programming technique. In section 5, in order to demonstrate the interactive processes under the hypothetical decision maker, HMOFRLP is formulated as a numerical example, and solved by applying the proposed interactive algorithm. Finally, in section 6, we conclude this paper. 2. Hierarchical Multiobjective Fuzzy Random Linear Programming Problems In this section, we focus on a hierarchical multiobjective programming problem involving fuzzy random variable coefficients in objective functions, which is called a hierarchical multiobjective fuzzy random linear programming problem (HMOFRLP). [HMOFRLP] first level decision maker : DM1 c11 x, · · · , c1k1 x) min C1 x = ( x∈X ····················· q-th level decision maker : DMq min Cq x = ( cq1 x, · · · , cqkq x) x∈X

def

where x = (x1 , x2 , · · · , xn )T is an n dimensional decision variable column vector, X = {x ∈ Rn | Ax ≤ b, x ≥ 0}, A is an (m × n) coefficient matrix, b = (b1 , · · · , bm )T is an m dimensional column vector. cri = ( cri1 , · · · , crin ), i = 1, · · · , kr , r = 1, · · · , q, are coefficient vectors of objective function cri x, whose elements are fuzzy random variables [10],[15],[17]), and the symbols "-" and "~" mean randomness and fuzziness respectively. In order to deal with the objective functions cri x, i = 1, · · · , kr , r = 1, · · · , q, Katagiri et al. [8],[9] proposed an LR-type fuzzy random variable which can be regarded as a special version of a fuzzy random variable. Under the occurrence of each elementary event ω, cri j (ω) is a realization of an LR-type fuzzy random variable cri j , which is an LR fuzzy number [5] whose membership function is defined as follows. ⎧ dri j (ω)−s ⎪ ⎪ ⎪ (s ≤ dri j (ω) ∀ω), L ⎪ α (ω) ⎨ ri j μcri j (ω) (s) = ⎪ ⎪ s−dri j (ω) ⎪ ⎪ (s > dri j (ω) ∀ω), ⎩ R β (ω) ri j

164

Hitoshi Yano and Kota Matsui / Procedia Computer Science 22 (2013) 162 – 171 def

where the function L(t) = max{0, l(t)} is a real-valued continuous function from [0, ∞) to [0, 1], and l(t) is a def strictly decreasing continuous function satisfying l(0) = 1. Also, R(t) = max{0, r(t)} satisfies the same conditions. 1 2 dri j , αri j , βri j are random variables expressed by dri j = dri j + tri dri j , αri j = α1ri j + tri α2ri j and βri j = β1ri j + tri β2ri j . tri is a random variable whose distribution function is denoted by T ri (·) which is strictly increasing and continuous, and dri1 j , dri2 j , α1ri j , α2ri j , β1ri j , β2ri j are constants. Similar to Katagiri et al. [8],[9], HMOFRLP can be transformed into a hierarchical multiobjective stochastic programming problem (HMOSP) by using a concept of a possibility measure [5]. As shown in [9], the realizations cri (ω)x becomes an LR fuzzy number characterized by the following membership functions on the basis of the extension principle [5].

⎧ ⎪ ⎪ d ri (ω) x−y ⎪ ⎪ y ≤ dri (ω)x L ⎪ ⎪ ⎨ αri (ω) x

μc (ω) x (y) = ⎪ ⎪ ⎪ ri y− dri (ω) x ⎪ ⎪ y > dri (ω)x ⎪ ⎩ R β (ω) x ri

cri (ω)x,i = 1, · · · , kr , r = 1, · · · , q, it is assumed that the decision maker has fuzzy goals For the realizations ri ,i = 1, · · · , kr , r = 1, · · · , q [16], whose membership functions μ (y), i = 1, · · · , kr , r = 1, · · · , q are continuous G Gri and strictly decreasing for minimization problems. By using a concept of a possibility measure [5], a degree of ri is expressed as follows. possibility [7] that the objective function value cri x satisfies the fuzzy goal G def

ri ) = supy min{μ (y), μ (y)} Πc x (G Gri cri x ri

(1)

Using a possibility measure, HMOFRLP can be transformed into the following hierarchical multiobjective stochastic programming problem (HMOSP). [HMOSP] first level decision maker : DM1 11 ), · · · , Π (G max(Πc x (G c1k1 x 1k1 )) 11 x∈X ·····················

q-th level decision maker: DMq q1 ), · · · , Π (G max(Πc x (G cqkq x qkq )) q1 x∈X 3. A Fractile Optimization Model for HMOSP If the decision makers adopt a fractile optimization model for the objective functions of HMOSP, HMOSP can be converted to the following hierarchical multiobjective programming problem, where the decision makers specify permissible probability levels pˆ ri , i = 1, · · · , kr , r = 1, · · · , q in their subjective manner [8]. [HMOP1( pˆ )] first level decision maker : DM1 (h11 , · · · , h1k1 ) max x∈X,h1i ∈[0,1],i=1,···,k1 ····················· q-th level decision maker: DMq max (h , · · · , hqkq ) x∈X,hqi ∈[0,1],i=1,···,kq q1 subject to Pr(ω | Πc

ri (ω)

x (Gri ) ≥ hri ) ≥ pˆ ri , i = 1, · · · , kr , r = 1, · · · , q

(2)

165


where pˆ r = ( pˆ r1 , · · · , pˆ rkr ), r = 1, · · · , q are vectors of permissible probability levels. Since a distribution function T ri (·) is continuous and strictly increasing, the constraints (2) can be transformed to the following form. ⎞ ⎛ −1 ⎜⎜⎜ μ (hri ) − (d1ri x − L−1 (hri )α1ri x) ⎟⎟⎟ G ri ⎟⎟⎟ ⎜ ri ) ≥ hri ) = T ri ⎜⎜⎜ pˆ ri ≤ Pr(ω | Πc (ω) x (G ⎟⎠ ⎝ ri d2ri x − L−1 (hri )α2ri x 1 −1 1 −1 ˆ ri )(d2ri x − L−1 (hri )α2ri x) ⇔ μG−1 (hri ) ≥ (dri x − L (hri )αri x) + T ri ( p

(3)

ri

Let us define the right-hand side of the inequality (3) as follows. fri (x, hri , pˆ ri )

def

=

(d1ri x − L−1 (hri )α1ri x) + T ri−1 ( pˆ ri )(d2ri x − L−1 (hri )α2ri x)

(4)

Then, HMOP1( pˆ ) can be equivalently transformed into the following form. [HMOP2( pˆ )] first level decision maker : DM1 (h , · · · , h1k1 ) max x∈X,h1i ∈[0,1],i=1,···,k1 11 ····················· q-th level decision maker: DMq max (h , · · · , hqkq ) x∈X,hqi ∈[0,1],i=1,···,kq q1 subject to μGri ( fri (x, hri , pˆ ri )) ≥ hri , i = 1, · · · , kr , r = 1, · · · , q

(5)

In HMOP2( pˆ ), let us pay attention to the inequalities (5). fri (x, hri , pˆ ri ) is continuous and strictly increasing with respect to hri for any x ∈ X. This means that the left-hand-side of (5) is continuous and strictly decreasing with respect to hri for any x ∈ X. Since the right-hand-side of (5) is continuous and strictly increasing with respect to hri , the inequalities (5) must always satisfy the active condition, that is, μGri ( fri (x, hri , pˆ ri )) = hri , i = 1, · · · , kr , r = 1, · · · , q at the optimal solution. ¿From such a point of view, HMOP2( pˆ ) is equivalently expressed as the following form. [HMOP3( pˆ )] first level decision maker : DM1 (μ ( f (x, h11 , pˆ 11 )), · · · , μG1k ( f1k1 (x, h1k1 , pˆ 1k1 ))) max 1 x∈X,h1i ∈[0,1],i=1,···,k1 G11 11

·····················

q-th level decision maker: DMq max (μ ( f (x, hq1 , pˆ q1 )), · · · , μGqkq ( fqkq (x, hqkq , pˆ qkq ))) x∈X,hqi ∈[0,1],i=1,···,kq Gq1 q1 subject to μGri ( fri (x, hri , pˆ ri )) = hri , i = 1, · · · , kr , r = 1, · · · , q

(6)

In order to deal with HMOP3( pˆ ), the decision makers must specify permissible probability levels pˆ in advance. However, in general, the decision makers seem to prefer not only the larger value of a permissible probability level but also the larger value of the corresponding membership functions μGri (·). ¿From such a point of view, we consider the following multiobjective programming problem which can be regarded as a natural extension of HMOP3( pˆ ). [HMOP4] first level decision maker : DM1 max x∈X,h1i ∈[0,1], pˆ 1i ∈(0,1),i=1,···,k1

(μG11 ( f11 (x, h11 , pˆ 11 )), · · · , μG1k ( f1k1 (x, h1k1 , pˆ 1k1 )), pˆ 11 , · · · , pˆ 1k1 ) 1

166


····················· q-th level decision maker: DMq max x∈X,hqi ∈[0,1], pˆ qi ∈(0,1),i=1,···,kq

(μGq1 ( fq1 (x, hq1 , pˆ q1 )), · · · , μGqkq ( fqkq (x, hqkq , pˆ qkq )), pˆ q1 , · · · , pˆ qkq )

subject to μGri ( fri (x, hri , pˆ ri )) = hri , i = 1, · · · , kr , r = 1, · · · , q

(7)

It should be noted in HMOP4 that permissible probability levels are not the fixed values but the decision variables. Considering the imprecise nature of the decision maker’s judgment, we assume that each decision maker (DMr ) has a fuzzy goal for each permissible probability level. Such a fuzzy goal can be quantified by eliciting the corresponding membership function. Let us denote a membership function of a permissible probability level pˆ ri as μ pˆ ri ( pˆ ri ), i = 1, · · · , kr . Then, HMOP4 can be transformed as the following hierarchical multiobjective programming problem. [HMOP5] first level decision maker : DM1 (μ ( f (x, h11 , pˆ 11 )), · · · , μG1k ( f1k1 (x, h1k1 , pˆ 1k1 )), μ pˆ 11 ( pˆ 11 ), · · · , μ pˆ 1k1 ( pˆ 1k1 )) max 1 x∈X,h1i ∈[0,1], pˆ 1i ∈(0,1),i=1,···,k1 G11 11 ····················· q-th level decision maker: DMq (μ ( fq1 (x, hq1 , pˆ q1 )), · · · , μGqkq ( fqkq (x, hqkq , pˆ qkq )), μ pˆ q1 ( pˆ q1 ), · · · , μ pˆ qkq ( pˆ qkq )) max x∈X,hqi ∈[0,1], pˆ qi ∈(0,1),i=1,···,kq Gq1 subject to μGri ( fri (x, hri , pˆ ri )) = hri , i = 1, · · · , kr , r = 1, · · · , q

(8)

In order to elicit the membership functions appropriately, we suggest the following procedures. First of all, each decision maker (DMr ) sets the intervals Pri = [primin , primax ], i = 1, · · · , kr , where primin is an unacceptable maximum value of pˆ ri and primax is a sufficiently satisfactory minimum value of pˆ ri . Throughout this section, we make the following assumption. Assumption 1. μ pˆ ri ( pˆ ri ), i = 1, · · · , kr , r = 1, · · · , q are strictly increasing and continuous with respect to pˆ ri ∈ Pri , and μ pˆ ri (primin ) = 0, μ pˆ ri (primax ) = 1. It should be noted here that, μGri ( fri (x, hri , pˆ ri )) is strictly decreasing with respect to pˆ ri . If the decision makers adopt the fuzzy decision [16] to integrate μGri ( fri (x, hri , pˆ ri )) and μ pˆ ri ( pˆ ri ), HMOP5 can be transformed into the following form. [HMOP6] first level decision maker : DM1 μDG11 (x, h11 , pˆ 11 ), · · · , μDG1k (x, h1k1 , pˆ 1k1 ) max 1 x∈X, pˆ 1i ∈P1i ,h1i ∈[0,1],i=1,···,k1 ····················· q-th level decision maker: DMq μDGq1 (x, hq1 , pˆ q1 ), · · · , μDGqkq (x, hqkq , pˆ qkq ) max x∈X, pˆ qi ∈Pqi ,hqi ∈[0,1],i=1,···,kq

where

subject to μGri ( fri (x, hri , pˆ ri )) = hri , i = 1, · · · , kr , r = 1, · · · , q def

μDGri (x, hri , pˆ ri ) = min{μ pˆ ri ( pˆ ri ), μGri ( fri (x, hri , pˆ ri ))}.

(9)

(10)

167


In order to deal with HMOP6, we introduce a DG -Pareto optimal solution concept. Definition 1. x∗ ∈ X, pˆ ∗ri ∈ Pri , h∗ri ∈ [0, 1], i = 1, · · · , kr , r = 1, · · · , q is said to be a DG -Pareto optimal solution to HMOP6, if and only if there does not exist another x ∈ X, pˆ ri ∈ Pri , hri ∈ [0, 1], i = 1, · · · , kr , r = 1, · · · , q such that μDGri (x, hri , pˆ ri ) ≥ μDGri (x∗ , h∗ri , pˆ ∗ri ),i = 1, · · · , kr , r = 1, · · · , q with strict inequality holding for at least one i, where μGri ( fri (x∗ , h∗ri , pˆ ∗ri )) = h∗ri , μGri ( fri (x, hri , pˆ ri )) = hri , i = 1, · · · , kr , r = 1, · · · , q. For generating a candidate of a satisfactory solution which is also DG -Pareto optimal, each decision maker is asked to specify the reference membership values [16]. Once the reference membership values μˆ r = (μˆ r1 , · · · , μˆ rkr ) are specified, the corresponding DG -Pareto optimal solution is obtained by solving the following minmax problem. [MINMAX1(μ)] ˆ min λ x∈X, pˆ ri ∈Pri ,hri ∈[0,1],i=1,···,kr ,r=1,···,q,λ∈Λ subject to μˆ ri − μ pˆ ri ( pˆ ri ) ≤ λ, μˆ ri − hri ≤ λ, μGri ( fri (x, hri , pˆ ri )) = hri ,

i = 1, · · · , kr , r = 1, · · · , q i = 1, · · · , kr , r = 1, · · · , q i = 1, · · · , kr , r = 1, · · · , q

(11) (12) (13)

where Λ = [maxi=1,···,kr ,r=1,···,q μˆ ri − 1, mini=1,···,kr ,r=1,···,q μˆ ri ]. It should be noted here that, in general, the optimal solution of MINMAX1 (μ) ˆ does not reflect the hierarchical structure between q decision makers where the upper level decision maker can take priority over the lower level decision makers. In order to cope with such a hierarchical preference structure between q decision makers in MINMAX1(μ), ˆ we introduce the concept of the decision powers w = (w1 , · · · , wq ) for the membership functions (10), where the r-th level decision maker (DMr ) can specify the decision power wr+1 in his/her subjective manner and the last decision maker (DMq ) has no decision power. In order to reflect the hierarchical preference structure between multiple decision makers, the decision powers w = (w1 , · · · , wq )T have to satisfy the following inequality condition. w1 = 1 ≥ w2 ≥ · · · · · · ≥ wq−1 ≥ wq > 0

(14)

Then, the corresponding modified MINMAX1(μ) ˆ is reformulated as follows. [MINMAX2(μ, ˆ w)] min λ x∈X, pˆ ri ∈Pri ,λ∈Λ,hri ∈[0,1],i=1,···,kr ,r=1,···,q, subject to μˆ ri − μ pˆ ri ( pˆ ri ) ≤ λ/wr , μˆ ri − hri ≤ λ/wr , μGri ( fri (x, hri , pˆ ri )) = hri ,

i = 1, · · · , kr , r = 1, · · · , q i = 1, · · · , kr , r = 1, · · · , q

(15) (16)

i = 1, · · · , kr , r = 1, · · · , q

(17)

where Λ = [maxi=1,···,kr ,r=1,···,q wi (μˆ ri − 1), mini=1,···,kr ,r=1,···,q wi μˆ ri ]. In the constraints (16) and (17), it holds that hri = μGri ( fri (x, hri , pˆ ri )) ≥ μˆ ri − λ/wr ,

1 −1 −1 1 ⇔ μG−1 ˆ ri − λ/wr ). ˆ ri ) · (d2ri x − L−1 (hri )α2ri x) ≤ μG−1 (μ (hri ) = (dri x − L (hri )αri x) + T ri ( p ri

(18)

ri

In the right hand side of (18), because of L−1 (hri ) ≤ L−1 (μˆ ri − λ/wr ) and α1ri x + T ri−1 ( pˆ ri )α2ri x > 0, it holds that (d1ri x − L−1 (hri )α1ri x) + T ri−1 ( pˆ ri ) · (d2ri x − L−1 (hri )α2ri x) = (d1ri x + T ri−1 ( pˆ ri )d2ri x) − L−1 (hri ) α1ri x + T ri−1 ( pˆ ri )α2ri x ≥ (d1ri x + T ri−1 ( pˆ ri )d2ri x) − L−1 (μˆ ri − λ/wr ) α1ri x + T ri−1 ( pˆ ri )α2ri x .

(19)

168


Using (18) and (19), it holds that ˆ ri − λ/wr ) μG−1 (μ ri

≥ (d1ri x + T ri−1 ( pˆ ri )d2ri x) − L−1 (μˆ ri − λ/wr ) α1ri x + T ri−1 ( pˆ ri )α2ri x

=

(d1ri x − L−1 (μˆ ri − λ/wr )α1ri x) + T ri−1 ( pˆ ri ) · (d2ri x − L−1 (μˆ ri − λ/wr )α2ri x).

(20)

μ−1 ˆ ri pˆ ri (μ

Moreover, because of pˆ ri ≥ − λ/wr ), (20) can be transformed into the following form. ⎞ ⎛ −1 ⎜⎜⎜ μ (μˆ ri − λ/wr ) − (d1ri x − L−1 (μˆ ri − λ/wr )α1ri x) ⎟⎟⎟ Gri ⎟⎟⎟ ≥ pˆ ≥ μ−1 (μˆ − λ/w ), ⎜ T ri ⎜⎜⎜⎝ r ri ⎟⎠ pˆ ri ri 2 2 −1 dri x − L (μˆ ri − λ/wr )αri x

ˆ ri − λ))(d2ri x − L−1 (μˆ ri − λ)α2ri x) ˆ ri − λ) ≥ (d1ri x − L−1 (μˆ ri − λ)α1ri x) + T ri−1 (μ−1 ⇔ μG−1 pˆ ri (μ (μ

(21)

ri

Therefore, MINMAX2(μ, ˆ w) can be reduced to the following minmax problem. [MINMAX3(μ, ˆ w)] min λ x∈X,λ∈Λ subject to ˆ ri − λ/wr ) μG−1 (μ ri

≥ (d1ri x − L−1 (μˆ ri − λ/wr )α1ri x) + T ri−1 (μ−1 ˆ ri − λ/wr )) · (d2ri x − L−1 (μˆ ri − λ/wr )α2ri x), pˆ ri (μ i = 1, · · · , kr , r = 1, · · · , q

(22)

It should be noted here that MINMAX3(μ, ˆ w) is equivalent to MINMAX3(μ, ˆ w). The relationships between the optimal solution (x∗ , λ∗ ) of MINMAX3(μ, ˆ w) and DG -Pareto optimal solutions can be characterized by the following theorem. Theorem 1. (1) If x∗ ∈ X, λ∗ ∈ Λ is a unique optimal solution of MINMAX3(μ, ˆ w), then x∗ ∈ X, pˆ ∗ri = μ−1 ˆ ri − λ∗ /wr ) ∈ pˆ ri (μ ∗ ∗ Pri , hri = μˆ ri − λ /wr ∈ [0, 1], i = 1, · · · , kr , r = 1, · · · , q is a DG -Pareto optimal solution. (2) If x∗ ∈ X, pˆ ∗ri ∈ Pri , h∗ri ∈ [0, 1], i = 1, · · · , kr , r = 1, · · · , q is a DG -Pareto optimal solution, then x∗ ∈ X, λ∗ = wr (μˆ ri − μ pˆ ri ( pˆ ∗ri )), i = 1, · · · , kr , r = 1, · · · , q is an optimal solution of MINMAX3(μ, ˆ w) for some reference membership values μˆ = (μˆ r1 , · · · , μˆ rkr ). 4. An Interactive Algorithm In this section, we propose an interactive algorithm to obtain a satisfactory solution from among a DG -Pareto optimal solution set. ¿From Theorem 1, it is not guaranteed that the optimal solution (x∗ , λ∗ ) of MINMAX3(μ, ˆ w) is DG -Pareto optimal, if it is not unique. In order to guarantee the DG -Pareto optimality, we first assume that all of the constraints (22) of MINMAX3(μ, ˆ w) are active at the optimal solution (x∗ , λ∗ ), i.e., μG−1 ˆ ri − λ∗ /wr ) − (d1ri x∗ − L−1 (μˆ ri − λ∗ /wr )α1ri x∗ ) (μ ri

=

ˆ ri T ri−1 (μ−1 pˆ ri (μ

− λ∗ /wr )) · (d2ri x∗ − L−1 (μˆ ri − λ∗ /wr )α2ri x∗ ), i = 1, · · · , kr , r = 1, · · · , q.

(23)

If some constraint of (22) is inactive, i.e., μG−1 ˆ ri − λ∗ /wr ) − (d1ri x∗ − L−1 (μˆ ri − λ∗ /wr )α1ri x∗ ) (μ ri

ˆ ri − λ∗ /wr )) · (d2ri x∗ − L−1 (μˆ ri − λ∗ /wr )α2ri x∗ ), > T ri−1 (μ−1 pˆ ri (μ

⇔ μG−1 ˆ ri − λ∗ /wr )), ˆ ri − λ∗ /wr ) > fri (x∗ , μˆ ri − λ∗ /wr , μ−1 pˆ ri (μ (μ

(24)

ri

we can convert the inactive constraint (24) into the active one by applying the bisection method for the reference membership value μˆ ri ∈ [λ∗ /wr , λ∗ /wr + 1]. For the optimal solution (x∗ , λ∗ ) of MINMAX3(μ, ˆ w), where the active conditions (23) are satisfied, we solve the DG -Pareto optimality test problem defined as follows. [DG -Pareto optimality test problem]

169


max

x∈X,ri ≥0,i=1,···,kr ,r=1,···,q

w=

q kr

ri

(25)

r=1 i=1

subject to T ri−1 (μ−1 ˆ ri − λ∗ /wr )) · (d2ri x − L−1 (μˆ ri − λ∗ /wr )α2ri x) + (d1ri x − L−1 (μˆ ri − λ∗ /wr )α1ri x) + ri pˆ ri (μ = T ri−1 (μ−1 ˆ ri − λ∗ /wr )) · (d2ri x∗ − L−1 (μˆ ri − λ∗ /wr )α2ri x∗ ) + (d1ri x∗ − L−1 (μˆ ri − λ∗ /wr )α1ri x∗ ), pˆ ri (μ i = 1, · · · , kr , r = 1, · · · , q

(26)

For the optimal solution of DG -Pareto optimality test problem, the following theorem holds. Theorem 2. For the optimal solution xˇ , ˇri , i = 1, · · · , kr , r = 1, · · · , q of DG -Pareto optimality test problem, if w = 0 (equivalently, ˇri = 0, i = 1, · · · , kr , r = 1, · · · , q), x∗ ∈ X, μ−1 ˆ ri − λ∗ /wr ) ∈ Pri , μˆ ri − λ∗ /wr ∈ [0, 1], i = 1, · · · , kr , r = pˆ ri (μ 1, · · · , q is a DG -Pareto optimal solution. Now, following the above discussions, we can present the interactive algorithm in order to derive a satisfactory solution from among a DG -Pareto optimal solution set. [An interactive algorithm] Step 1: The decision makers (DMr , r = 1, · · · , q) set the membership functions μGri (y), i = 1, · · · , kr for the fuzzy goals of the objective functions in HMOFRLP. Step 2: Each decision maker (DMr ) sets the intervals Pri = [primin , primax ], i = 1, · · · , kr , where primin is an unacceptable maximum value of pˆ ri and primax is a sufficiently satisfactory minimum value of pˆ ri . According to Assumption 1, each decision maker sets his/her membership function μ pˆ ri ( pˆ ri ). Step 3: Set the initial reference membership values as μˆ ri = 1, i = 1, · · · , kr , r = 1, · · · , q. Step 4: Set the initial decision powers wr = 1, r = 1, · · · , q. Step 5: Solve MINMAX3(μ, ˆ w) by combined use of the bisection method with respect to λ ∈ Λ and the firstphase of the two-phase simplex method of linear programming, and obtain the optimal solution (x∗ , λ∗ ). For the optimal solution (x∗ , λ∗ ), The corresponding DG -Pareto optimality test problem is formulated and solved. Step 6: If each of the decision makers (DMr , r = 1, · · · , q) is satisfied with the current values of the DG -Pareto optimal solution μDGri (x∗ , h∗ri , pˆ ∗ri ), i = 1, · · · , kr , where pˆ ∗ri = μ−1 ˆ ri − λ∗ /wr ), h∗ri = μˆ ri − λ∗ /wr , i = 1, · · · , kr , r = pˆ ri (μ 1, · · · , q, then stop. Otherwise, let the s-th level decision maker (DM s ) be the uppermost of the decision makers who are not satisfied with the current values of his/her membership functions μDG si (x∗ , h∗si , pˆ ∗si ), i = 1, · · · , k s . Considering the current values, DM s updates his/her decision power w s+1 and/or his/her reference membership values μˆ si = 1, i = 1, · · · , k s according to the following two rules, and return to Step 5. Rule 1.: In order to guarantee the inequality conditions (14), w s+1 ≤ w s must be satisfied. After updating w s+1 , if there exists some index t > s + 1 such that w s+1 < wt , then the corresponding decision power wt is set as wt ← w s+1 . Rule 2.: At first, the reference membership values of DMr , r = 1, · · · , q, r s must be set as the current values of the membership functions, i.e., μˆ ri = μDGri (x∗ , h∗ri , pˆ ∗ri ), i = 1, · · · , kr , r = 1, · · · , q, r s. After that, DM s updates his/her reference membership values μˆ si , i = 1, · · · , k s . Here, it should be stressed for DM s that any improvement of one membership function can be achieved only at the expense of at least one of the other membership functions. 5. A Numerical Example In order to demonstrate the proposed method and the interactive processes, we consider the following hierarchical two-objective linear programming problem with fuzzy random variable coefficients under two hypothetical decision makers. [HMOFRLP] first level decision maker : DM1 min x∈X

c1i x =

10 j=1

c1i j x j , i = 1, 2

170


second level decision maker : DM2 min x∈X

c2i x =

10 j=1

c2i j x j , i = 1, 2 10

cri j (ω) ak j x j ≤ bk , 1 ≤ k ≤ 7}, and it is assumed that a realization of an LR-type fuzzy random variable cri j is an LR fuzzy number whose membership function is defined as ⎛ 1 ⎛ ⎞ ⎞ ⎜⎜⎜ (dri j + tri (ω)dri2 j ) − s ⎟⎟⎟ ⎜⎜⎜ s − (dri1 j + tri (ω)dri2 j ) ⎟⎟⎟ ⎟⎟⎠ for s > dri j (ω), μcri j (ω) (s) = L ⎜⎜⎝ ⎟⎟⎠ for s ≤ dri j (ω), and μcri j (ω) (s) = R ⎜⎜⎝ α1ri j + tri (ω)α2ri j β1ri j + tri (ω)β2ri j where L(t) = R(t) = max{0, 1 − t}, and the parameters dri1 j , dri2 j , α1ri j , α2ri j , β1ri j , β2ri j are given in Table 1. Similarly, the parameters of the left-hand side of the constraints are shown in Table 1 and the parameters of the right-hand-side are b1 = 140, b2 = −220, b3 = −190, b4 = 75, b5 = −160, b6 = 130, b7 = 90. For simplicity, we assume that tri , r = 1, 2, i = 1, 2 are standard Gaussian random variables defined as tri ∼ N(0, 1). In HMOFRLP, let us assume that the hypothetical decision makers set their membership functions for the objective fri (x, hri , pˆ ri ) − frimax functions as μGri ( fri (x, hri , pˆ ri )) = , i = 1, 2, r = 1, 2, where [ f11min , f11max ] = [2400, 2600], frimin − frimax [ f12min , f12max ] = [1400, 1500], [ f21min , f21max ] = [−1000, −800], [ f22min , f22max ] = [−700, −500] (Step 1). We also assume that the hypothetical decision makers set their membership functions for the permissible probability levels primin − pˆ ri as μ pˆ ri ( pˆ ri ) = , i = 1, 2, r = 1, 2, where [p11min , p11max ] = [0.3, 0.9], [p12min , p12max ] = [0.2, 0.85], primin − primax [p21min , p21max ] = [0.3, 0.87], [p22min , p22max ] = [0.4, 0.91] (Step 2). Set the initial reference membership values as (μˆ 11 , μˆ 12 ) = (μˆ 21 , μˆ 22 ) = (1, 1) (Step 3) and decision powers as (w1 , w2 ) = (1, 1) (Step 4). At Step 5, solve MINMAX3(μ, ˆ w) to obtain the corresponding DG -Pareto optimal solution as μGri ( fri (x, hri , pˆ ri )) = 0.804, r = 1, 2, i = 1, 2. For the current value of the DG -Pareto optimal solution, DM1 updates his/her decision power as w2 = 0.7 in order to improve his/her own membership functions μDG1i (·), i = 1, 2 at the expense of DM2 ’s membership functions μDG2i (·), i = 1, 2 (Step 6). Then, the corresponding DG -Pareto optimal solution is obtained as μG1i ( f1i (x, h1i , pˆ 1i )) = 0.838, i = 1, 2, μG2i ( f2i (x, h2i , pˆ 2i )) = 0.769, i = 1, 2. Then, DM1 is satisfied with the current value of the membership functions, but DM2 is not satisfied with the current values. Therefore, in order to improve μDG22 (·) at the expense of μDG21 (·), DM2 updates his/her reference membership values as (μˆ 21 , μˆ 22 ) = (0.730, 0.790) according to Rule 2 (Step 6). Then, the corresponding DG -Pareto optimal solution is obtained as μG1i ( f1i (x, h1i , pˆ 1i )) = 0.852, i = 1, 2 μG21 ( f21 (x, h21 , pˆ 21 )) = 0.750, μG22 ( f22 (x, h22 , pˆ 22 )) = 0.810. Since both DM1 and DM2 are satisfied with current values of the above membership functions, stop the interactive processes (Step 6). where X = {(x1 , · · · , x10 ) ≥ 0 |

j=1

6. Conclusion In this paper, we have proposed an interactive decision making method for HMOFRLP to obtain a satisfactory solution from among a DG -Pareto optimal solution set. In the proposed method, each decision maker is required to specify the membership functions for the fuzzy goals of not only the original objective functions but also the permissible probability levels. After each decision maker updates not only the reference membership values but also the decision powers according to two kinds of updating rules, the corresponding DG -Pareto optimal solution is obtained by solving the corresponding minmax problem MINMAX3(μ, ˆ w) on the basis of linear programming technique. At any DG -Pareto optimal solution, a proper valance between permissible probability levels and the corresponding objective functions is attained. Moreover, a hierarchical decision structure between multiple decision makers will be reflected at a satisfactory solution through the decision powers. References [1] Anandalingam G. A mathematical programming model of decentralized multi-Level systems, Journal of Operational Research Society, 1988; 39; 1021-1033. [2] Birge,JR, Louveaux,F. Introduction to stochastic programming, Springer; 1997. [3] Charnes A, Cooper WW. Chance constrained programming, Management Science 1959; 6; 73-79.

171

Hitoshi Yano and Kota Matsui / Procedia Computer Science 22 (2013) 162 – 171 Table 1. Parameters of the objective functions and the constraints 1 d11 2 d11 1 d12 2 d12 1 d21 2 d21 1 d22 2 d22 α111 α211 α112 α212 α121 α221 α122 α222 a1 a2 a3 a4 a5 a6 a7

[4] [5] [6] [7]

19 3 12 1 -12 1 -12 1 0.312 0.098 0.756 0.042 0.605 0.083 0.272 0.085 12 -2 3 -11 -4 5 -3

48 2 46 2 -38 2 -36 2 0.759 0.042 0.704 0.058 0.107 0.074 0.313 0.056 -2 5 -16 6 7 -3 -4

21 2 23 4 -23 4 -27 4 0.225 0.074 0.295 0.06 0.564 0.058 0.442 0.065 4 3 -4 -5 -6 14 -6

10 1 38 2 -33 2 -30 2 0.990 0.052 0.508 0.095 0.885 0.06 0.583 0.096 -7 16 -8 9 -5 -3 9

18 4 33 2 -33 2 -33 2 0.248 0.045 0.216 0.081 0.652 0.032 0.341 0.066 13 6 -8 -1 13 -9 6

35 3 48 1 -45 1 -45 1 0.951 0.024 0.859 0.068 0.957 0.096 0.641 0.043 -1 -12 2 8 6 -7 18

46 1 12 2 -12 2 -11 2 0.643 0.06 0.692 0.096 0.464 0.042 0.892 0.026 -6 12 -12 -4 -2 4 11

11 2 8 1 -9 1 -12 1 0.984 0.036 0.741 0.091 0.733 0.076 0.432 0.091 6 4 -12 6 -5 -4 -9

24 4 19 2 -19 2 -19 2 0.340 0.082 0.641 0.011 0.230 0.081 0.611 0.052 11 -7 4 -9 14 -5 -4

33 2 20 1 -20 1 -8 1 0.465 0.035 0.569 0.09 0.416 0.083 0.147 0.035 -8 -10 -3 6 -6 9 7

Danzig GB. Linear programming under uncertainty, Management Science, 1955; 1; 197-206. Dubois D, Prade H. Fuzzy sets and systems, Academic Press; 1980. Kall P, Mayer J. Stochastic linear programming models, theory, and computation, Springer; 2005. Katagiri H, Ishii H, Itoh T. Fuzzy random linear programming problem, In: Proceedings of Second European Workshop on Fuzzy Decision Analysis and Neural Networks for Management, Planning and Optimization; 1997, p.107-115. [8] Katagiri H, Sakawa M, Kato K, Ohsaki S. An interactive fuzzy satisficing method for fuzzy random multiobjective linear programming problems through the fractile optimization model using possibility and necessity measures, In: Proceedings of the Ninth Asia Pacific Management Conference; 2003, p.795-802. [9] Katagiri H, Sakawa M, Kato K, Nishizaki I. Interactive multiobjective fuzzy random linear programming: Maximization of possibility and probability, European Journal of Operational Research 2008; 188; 530-539. [10] Kwakernaak H. Fuzzy random variable-1, Information Sciences 1978; 15; 1-29. [11] Lai YJ. Hierarchical optimization : a satisfactory solution, Fuzzy Sets and Systems 1996; 77; 321-335. [12] Lai VJ, Hwang CL. Fuzzy mathematical programming, Springer; 1992. [13] Lee ES, Shih H. Fuzzy and multi-level decision making Springer; 2001. [14] Luhandjula MK, Gupta MM. On fuzzy stochastic optimization, Fuzzy Sets and Systems, 81, pp.47-55, (1996). [15] Puri ML, Ralescu DA. Fuzzy random variables, Journal of Mathematical Analysis and Applications 1986; 14; 409-422. [16] Sakawa M. Fuzzy sets and interactive multiobjective optimization Plenum Press; 1993. [17] Sakawa M, Nishizaki I, Katagiri H. Fuzzy stochastic multiobjective programming Springer; 2011. [18] Shih H, Lai YJ, Lee ES. Fuzzy approach for multi-level programming problems, Computers and Operations Research 1996; 23; 73-91. [19] Wang GY, Qiao Z. Linear programming with fuzzy random variable coefficients, Fuzzy Sets and Systems 1993; 57; 295-311. [20] Wen UP, Hsu ST. Linear bi-level programming problems - a review, Journal of Operational Research Society 1991; 42; 125-133. [21] Yano H. A fuzzy approach for hierarchical multiobjective linear programming problems, Lecture Notes in Engineering and Computer Science: Proceedings of The International MultiConference of Engineers and Computer Scientists 2010, 2010, p.2098-2103. [22] Yano H, Matsui K. Fuzzy approaches for multiobjective fuzzy random linear programming problems through a probability maximization model, Lecture Notes in Engineering and Computer Science: Proceedings of The International MultiConference of Engineers and Computer Scientists 2011, 2011, p.1349-1354. [23] Yano H, Sakawa M. Interactive multiobjective fuzzy random linear programming through fractile criteria, Advances in Fuzzy Systems, Hindawi Publishing Corporation, 2012, Article ID 521080. [24] Zimmermann HJ. Fuzzy Sets, Decision-Making and Expert Systems, Kluwer Academic Publishers; 1987.