Hierarchical Multiobjective Fuzzy Random Linear Programming Problems

Procedia Computer Science 22 ( 2013 ) 162 – 171

1877-0509 © 2013 The Authors. Published by Elsevier B.V.Selection and peer-review under responsibility of KES Internationaldoi: 10.1016/j.procs.2013.09.092

ScienceDirectAvailable online at www.sciencedirect.com

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

Hierarchical Multiobjective Fuzzy Random Linear ProgrammingProblems

Hitoshi Yanoa, Kota Matsuib

aGraduate School of Humanities and Social Sciences, Nagoya City University,Nagoya, 467-8501, JapanbDepartment 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 pro-

gramming 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 The Authors. Published by Elsevier B.V.

Selection and peer-review under responsibility of 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 com-

putational 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].

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© 2013 The Authors. Published by Elsevier B.V.Selection and peer-review under responsibility of KES International

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

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 multiob-

jective 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 pro-

posed 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 deci-

sion, 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 de-

cision 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 prob-

lem. 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 formu-

lated 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 vari-

able coefficients in objective functions, which is called a hierarchical multiobjective fuzzy random linear program-

ming problem (HMOFRLP).

[HMOFRLP]first level decision maker : DM1

minx∈X C̃1x = (̃c11x, · · · , c̃1k1x)

· · · · · · · · · · · · · · · · · · · · ·q-th level decision maker : DMq

minx∈X C̃qx = (̃cq1x, · · · , c̃qkq x)

where x = (x1, x2, · · · , xn)T is an n dimensional decision variable column vector, X def= {x ∈ Rn | Ax ≤ b, x ≥ 0},

A is an (m × n) coefficient matrix, b = (b1, · · · , bm)T is an m dimensional column vector. c̃ri = (̃cri1, · · · , c̃rin), i =1, · · · , kr, r = 1, · · · , q, are coefficient vectors of objective function c̃rix, 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 c̃rix, 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, c̃ri j(ω) is a realization of an LR-type fuzzy random variable c̃ri j, which is

an LR fuzzy number [5] whose membership function is defined as follows.

μc̃ri j(ω)(s) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩L(

dri j(ω)−sαri j(ω)

)(s ≤ dri j(ω) ∀ω),

R(

s−dri j(ω)

βri j(ω)

)(s > dri j(ω) ∀ω),


where the function L(t) def= max{0, l(t)} is a real-valued continuous function from [0,∞) to [0, 1], and l(t) is a

strictly decreasing continuous function satisfying l(0) = 1. Also, R(t) def= max{0, r(t)} satisfies the same conditions.

dri j, αri j, βri j are random variables expressed by dri j = d1ri j + trid2

ri j, αri j = α1ri j + triα

2ri j and βri j = β

1ri j + triβ

2ri j. tri

is a random variable whose distribution function is denoted by Tri(·) which is strictly increasing and continuous,

and d1ri j, d

2ri j, α

1ri j, α

2ri j, β

1ri j, β

2ri 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

c̃ri(ω)x becomes an LR fuzzy number characterized by the following membership functions on the basis of the

extension principle [5].

μc̃ri(ω)x(y) =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩L(

dri(ω)x−yαri(ω)x

)y ≤ dri(ω)x

R(

y−dri(ω)xβri(ω)x

)y > dri(ω)x

For the realizations c̃ri(ω)x,i = 1, · · · , kr, r = 1, · · · , q, it is assumed that the decision maker has fuzzy goals

G̃ri,i = 1, · · · , kr, r = 1, · · · , q [16], whose membership functions μG̃ri(y), i = 1, · · · , kr, r = 1, · · · , q are continuous

and strictly decreasing for minimization problems. By using a concept of a possibility measure [5], a degree of

possibility [7] that the objective function value c̃rix satisfies the fuzzy goal G̃ri is expressed as follows.

Πc̃ri x(G̃ri)def= supy min{μc̃ri x(y), μG̃ri

(y)} (1)

Using a possibility measure, HMOFRLP can be transformed into the following hierarchical multiobjective stochas-

tic programming problem (HMOSP).

[HMOSP]first level decision maker : DM1

maxx∈X (Πc̃11 x(G̃11), · · · ,Πc̃1k1x(G̃1k1

))

· · · · · · · · · · · · · · · · · · · · ·q-th level decision maker: DMq

maxx∈X (Πc̃q1 x(G̃q1), · · · ,Πc̃qkq x(G̃qkq ))

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 p̂ri, i = 1, · · · , kr, r = 1, · · · , q in their subjective manner [8].

[HMOP1( p̂)]first level decision maker : DM1

maxx∈X,h1i∈[0,1],i=1,···,k1

(h11, · · · , h1k1)


maxx∈X,hqi∈[0,1],i=1,···,kq

(hq1, · · · , hqkq )

subject to Pr(ω | Πc̃ri(ω)x(G̃ri) ≥ hri) ≥ p̂ri, i = 1, · · · , kr, r = 1, · · · , q (2)


where p̂r = ( p̂r1, · · · , p̂rkr ), r = 1, · · · , q are vectors of permissible probability levels. Since a distribution function

Tri(·) is continuous and strictly increasing, the constraints (2) can be transformed to the following form.

p̂ri ≤ Pr(ω | Πc̃ri(ω)x(G̃ri) ≥ hri) = Tri

⎛⎜⎜⎜⎜⎜⎜⎜⎝μ−1

G̃ri(hri) − (d1

rix − L−1(hri)α1rix)

d2rix − L−1(hri)α2

rix

⎞⎟⎟⎟⎟⎟⎟⎟⎠⇔ μ−1

G̃ri(hri) ≥ (d1

rix − L−1(hri)α1rix) + T−1

ri (p̂ri)(d2rix − L−1(hri)α

2rix) (3)

Let us define the right-hand side of the inequality (3) as follows.

fri(x, hri, p̂ri)def= (d1


ri ( p̂ri)(d2rix − L−1(hri)α

2rix) (4)

Then, HMOP1( p̂) can be equivalently transformed into the following form.


maxx∈X,h1i∈[0,1],i=1,···,k1

(h11, · · · , h1k1)



(hq1, · · · , hqkq )

subject to μG̃ri( fri(x, hri, p̂ri)) ≥ hri, i = 1, · · · , kr, r = 1, · · · , q (5)

In HMOP2( p̂), let us pay attention to the inequalities (5). fri(x, hri, p̂ri) 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, μG̃ri( fri(x, hri, p̂ri)) = hri,

i = 1, · · · , kr, r = 1, · · · , q at the optimal solution. ¿From such a point of view, HMOP2( p̂) is equivalently expressed

as the following form.


maxx∈X,h1i∈[0,1],i=1,···,k1

(μG̃11( f11(x, h11, p̂11)), · · · , μG̃1k1

( f1k1(x, h1k1

, p̂1k1)))



(μG̃q1( fq1(x, hq1, p̂q1)), · · · , μG̃qkq

( fqkq (x, hqkq , p̂qkq )))

subject to μG̃ri( fri(x, hri, p̂ri)) = hri, i = 1, · · · , kr, r = 1, · · · , q (6)

In order to deal with HMOP3( p̂), the decision makers must specify permissible probability levels p̂ 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 μG̃ri(·). ¿From such a point of view,

we consider the following multiobjective programming problem which can be regarded as a natural extension of

HMOP3( p̂).

[HMOP4]first level decision maker : DM1

maxx∈X,h1i∈[0,1], p̂1i∈(0,1),i=1,···,k1

(μG̃11( f11(x, h11, p̂11)), · · · , μG̃1k1

( f1k1(x, h1k1

, p̂1k1)), p̂11, · · · , p̂1k1

)



maxx∈X,hqi∈[0,1],p̂qi∈(0,1),i=1,···,kq


( fqkq (x, hqkq , p̂qkq )), p̂q1, · · · , p̂qkq )


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

p̂ri as μp̂ri ( p̂ri), i = 1, · · · , kr. Then, HMOP4 can be transformed as the following hierarchical multiobjective

programming problem.


maxx∈X,h1i∈[0,1], p̂1i∈(0,1),i=1,···,k1

(μG̃11( f11(x, h11, p̂11)), · · · , μG̃1k1

( f1k1(x, h1k1

, p̂1k1)), μ p̂11

( p̂11), · · · , μ p̂1k1(p̂1k1

))


maxx∈X,hqi∈[0,1], p̂qi∈(0,1),i=1,···,kq


( fqkq (x, hqkq , p̂qkq )), μ p̂q1( p̂q1), · · · , μ p̂qkq

(p̂qkq ))


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 p̂ri and primax is a sufficiently satisfactory minimum value of p̂ri. Throughout this section, we

make the following assumption.

Assumption 1.μp̂ri ( p̂ri), i = 1, · · · , kr, r = 1, · · · , q are strictly increasing and continuous with respect to p̂ri ∈ Pri, and μp̂ri (primin) =

0, μ p̂ri (primax) = 1.

It should be noted here that, μG̃ri( fri(x, hri, p̂ri)) is strictly decreasing with respect to p̂ri. If the decision makers

adopt the fuzzy decision [16] to integrate μG̃ri( fri(x, hri, p̂ri)) and μp̂ri (p̂ri), HMOP5 can be transformed into the

following form.


maxx∈X, p̂1i∈P1i,h1i∈[0,1],i=1,···,k1

(μDG11

(x, h11, p̂11), · · · , μDG1k1(x, h1k1

, p̂1k1))


maxx∈X, p̂qi∈Pqi,hqi∈[0,1],i=1,···,kq

(μDGq1

(x, hq1, p̂q1), · · · , μDGqkq(x, hqkq , p̂qkq )

)


where

μDGri(x, hri, p̂ri)

def= min{μp̂ri ( p̂ri), μG̃ri

( fri(x, hri, p̂ri))}. (10)


In order to deal with HMOP6, we introduce a DG-Pareto optimal solution concept.

Definition 1.x∗ ∈ X, p̂∗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, p̂ri ∈ Pri, hri ∈ [0, 1], i = 1, · · · , kr, r = 1, · · · , q such that

μDGri(x, hri, p̂ri) ≥ μDGri

(x∗, h∗ri, p̂∗ri),i = 1, · · · , kr, r = 1, · · · , q with strict inequality holding for at least one i, where

μG̃ri( fri(x∗, h∗ri, p̂

∗ri)) = h∗ri, μG̃ri

( fri(x, hri, p̂ri)) = 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 μ̂r = (μ̂r1, · · · , μ̂rkr )

are specified, the corresponding DG-Pareto optimal solution is obtained by solving the following minmax problem.

[MINMAX1(μ̂)]

minx∈X, p̂ri∈Pri,hri∈[0,1],i=1,···,kr ,r=1,···,q,λ∈Λλ

subject to

μ̂ri − μp̂ri (p̂ri) ≤ λ, i = 1, · · · , kr, r = 1, · · · , q (11)

μ̂ri − hri ≤ λ, i = 1, · · · , kr, r = 1, · · · , q (12)

μG̃ri( fri(x, hri, p̂ri)) = hri, i = 1, · · · , kr, r = 1, · · · , q (13)

where Λ = [maxi=1,···,kr ,r=1,···,q μ̂ri − 1,mini=1,···,kr ,r=1,···,q μ̂ri].It should be noted here that, in general, the optimal solution of MINMAX1 (μ̂) 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(μ̂), 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(μ̂) is reformulated as follows.

[MINMAX2(μ̂,w)]

minx∈X, p̂ri∈Pri,λ∈Λ,hri∈[0,1],i=1,···,kr ,r=1,···,q,λ

subject to

μ̂ri − μ p̂ri (p̂ri) ≤ λ/wr, i = 1, · · · , kr, r = 1, · · · , q (15)

μ̂ri − hri ≤ λ/wr, i = 1, · · · , kr, r = 1, · · · , q (16)

μG̃ri( fri(x, hri, p̂ri)) = hri, i = 1, · · · , kr, r = 1, · · · , q (17)

where Λ = [maxi=1,···,kr ,r=1,···,q wi(μ̂ri − 1),mini=1,···,kr ,r=1,···,q wiμ̂ri]. In the constraints (16) and (17), it holds that

hri = μG̃ri( fri(x, hri, p̂ri)) ≥ μ̂ri − λ/wr,

⇔ μ−1

G̃ri(hri) = (d1


ri (p̂ri) · (d2rix − L−1(hri)α

2rix) ≤ μ−1

G̃ri(μ̂ri − λ/wr). (18)

In the right hand side of (18), because of L−1(hri) ≤ L−1(μ̂ri − λ/wr) and α1rix + T−1

ri (p̂ri)α2rix > 0, it holds that

(d1rix − L−1(hri)α

1rix) + T−1

ri ( p̂ri) · (d2rix − L−1(hri)α

2rix)

= (d1rix + T−1

ri ( p̂ri)d2rix) − L−1(hri)

(α1

rix + T−1ri ( p̂ri)α

2rix)

≥ (d1rix + T−1

ri ( p̂ri)d2rix) − L−1(μ̂ri − λ/wr)

(α1

rix + T−1ri ( p̂ri)α

2rix). (19)


Using (18) and (19), it holds that

μ−1

G̃ri(μ̂ri − λ/wr) ≥ (d1

rix + T−1ri ( p̂ri)d2

rix) − L−1(μ̂ri − λ/wr)(α1

rix + T−1ri (p̂ri)α

2rix)

= (d1rix − L−1(μ̂ri − λ/wr)α

1rix) + T−1

ri (p̂ri) · (d2rix − L−1(μ̂ri − λ/wr)α

2rix). (20)

Moreover, because of p̂ri ≥ μ−1p̂ri

(μ̂ri − λ/wr), (20) can be transformed into the following form.

Tri

⎛⎜⎜⎜⎜⎜⎜⎜⎝μ−1

G̃ri(μ̂ri − λ/wr) − (d1

rix − L−1(μ̂ri − λ/wr)α1rix)

d2rix − L−1(μ̂ri − λ/wr)α2

rix

⎞⎟⎟⎟⎟⎟⎟⎟⎠ ≥ p̂ri ≥ μ−1p̂ri

(μ̂ri − λ/wr),

⇔ μ−1

G̃ri(μ̂ri − λ) ≥ (d1

rix − L−1(μ̂ri − λ)α1rix) + T−1

ri (μ−1p̂ri

(μ̂ri − λ))(d2rix − L−1(μ̂ri − λ)α2

rix) (21)

Therefore, MINMAX2(μ̂,w) can be reduced to the following minmax problem.

[MINMAX3(μ̂,w)]

minx∈X,λ∈Λ λ

subject to

μ−1

G̃ri(μ̂ri − λ/wr) ≥ (d1

rix − L−1(μ̂ri − λ/wr)α1rix) + T−1

ri (μ−1p̂ri

(μ̂ri − λ/wr)) · (d2rix − L−1(μ̂ri − λ/wr)α

2rix),

i = 1, · · · , kr, r = 1, · · · , q (22)

It should be noted here that MINMAX3(μ̂,w) is equivalent to MINMAX3(μ̂,w). The relationships between

the optimal solution (x∗, λ∗) of MINMAX3(μ̂,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(μ̂,w), then x∗ ∈ X, p̂∗ri = μ

−1p̂ri

(μ̂ri − λ∗/wr) ∈Pri, h∗ri = μ̂ri − λ∗/wr ∈ [0, 1], i = 1, · · · , kr, r = 1, · · · , q is a DG-Pareto optimal solution.

(2) If x∗ ∈ X, p̂∗ri ∈ Pri, h∗ri ∈ [0, 1], i = 1, · · · , kr, r = 1, · · · , q is a DG-Pareto optimal solution, then x∗ ∈ X,λ∗ = wr(μ̂ri − μp̂ri (p̂∗ri)), i = 1, · · · , kr, r = 1, · · · , q is an optimal solution of MINMAX3(μ̂,w) for some reference

membership values μ̂ = (μ̂r1, · · · , μ̂rkr ).

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(μ̂,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(μ̂,w) are active at the optimal solution (x∗, λ∗), i.e.,

μ−1

G̃ri(μ̂ri − λ∗/wr) − (d1

rix∗ − L−1(μ̂ri − λ∗/wr)α

1rix∗)

= T−1ri (μ−1

p̂ri(μ̂ri − λ∗/wr)) · (d2


2rix∗), i = 1, · · · , kr, r = 1, · · · , q. (23)

If some constraint of (22) is inactive, i.e.,

μ−1

G̃ri(μ̂ri − λ∗/wr) − (d1


1rix∗)

> T−1ri (μ−1



2rix∗),

⇔ μ−1

G̃ri(μ̂ri − λ∗/wr) > fri(x∗, μ̂ri − λ∗/wr, μ

−1p̂ri

(μ̂ri − λ∗/wr)), (24)

we can convert the inactive constraint (24) into the active one by applying the bisection method for the reference

membership value μ̂ri ∈ [λ∗/wr, λ∗/wr + 1].

For the optimal solution (x∗, λ∗) of MINMAX3(μ̂,w), where the active conditions (23) are satisfied, we solve

the DG-Pareto optimality test problem defined as follows.

[DG-Pareto optimality test problem]


maxx∈X,εri≥0,i=1,···,kr ,r=1,···,q

w =q∑

r=1

kr∑i=1

εri (25)

subject to

T−1ri (μ−1


rix − L−1(μ̂ri − λ∗/wr)α2rix) + (d1

rix − L−1(μ̂ri − λ∗/wr)α1rix) + εri

= T−1ri (μ−1



2rix∗) + (d1


1rix∗),

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 x̌, ε̌ri, i = 1, · · · , kr, r = 1, · · · , q of DG-Pareto optimality test problem, if w = 0 (equiv-

alently, ε̌ri = 0, i = 1, · · · , kr, r = 1, · · · , q), x∗ ∈ X, μ−1p̂ri

(μ̂ri − λ∗/wr) ∈ Pri, μ̂ri − λ∗/wr ∈ [0, 1], i = 1, · · · , kr, r =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 μG̃ri

(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 p̂ri and primax is a sufficiently satisfactory minimum value of p̂ri. According to

Assumption 1, each decision maker sets his/her membership function μp̂ri ( p̂ri).

Step 3: Set the initial reference membership values as μ̂ri = 1, i = 1, · · · , kr, r = 1, · · · , q.

Step 4: Set the initial decision powers wr = 1, r = 1, · · · , q.

Step 5: Solve MINMAX3(μ̂,w) by combined use of the bisection method with respect to λ ∈ Λ and the first-

phase 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, p̂

∗ri), i = 1, · · · , kr, where p̂∗ri = μ

−1p̂ri

(μ̂ri − λ∗/wr), h∗ri = μ̂ri − λ∗/wr, i = 1, · · · , kr, r =1, · · · , q, then stop. Otherwise, let the s-th level decision maker (DMs) be the uppermost of the decision makers

who are not satisfied with the current values of his/her membership functions μDGsi(x∗, h∗si, p̂

∗si), i = 1, · · · , ks.

Considering the current values, DMs updates his/her decision power ws+1 and/or his/her reference membership

values μ̂si = 1, i = 1, · · · , ks according to the following two rules, and return to Step 5.

Rule 1.:In order to guarantee the inequality conditions (14), ws+1 ≤ ws must be satisfied. After updating ws+1, if there

exists some index t > s + 1 such that ws+1 < wt, then the corresponding decision power wt is set as wt ← ws+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., μ̂ri = μDGri(x∗, h∗ri, p̂∗ri), i = 1, · · · , kr, r = 1, · · · , q, r � s. After that, DMs updates

his/her reference membership values μ̂si, i = 1, · · · , ks. Here, it should be stressed for DMs 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 hierar-

chical two-objective linear programming problem with fuzzy random variable coefficients under two hypothetical

decision makers.

[HMOFRLP]first level decision maker : DM1

minx∈X c̃1ix =10∑j=1

c̃1i j x j, i = 1, 2


second level decision maker : DM2

minx∈X c̃2ix =10∑j=1

c̃2i j x j, i = 1, 2

where X = {(x1, · · · , x10) ≥ 0 | ∑10j=1 ak jx j ≤ bk, 1 ≤ k ≤ 7}, and it is assumed that a realization c̃ri j(ω)

of an LR-type fuzzy random variable c̃ri j is an LR fuzzy number whose membership function is defined as

μc̃ri j(ω)(s) = L

⎛⎜⎜⎜⎜⎜⎝ (d1ri j + tri(ω)d2

ri j) − s

α1ri j + tri(ω)α2

ri j

⎞⎟⎟⎟⎟⎟⎠ for s ≤ dri j(ω), and μc̃ri j(ω)(s) = R

⎛⎜⎜⎜⎜⎜⎝ s − (d1ri j + tri(ω)d2

ri j)

β1ri j + tri(ω)β2

ri j

⎞⎟⎟⎟⎟⎟⎠ for s > dri j(ω),

where L(t) = R(t) = max{0, 1 − t}, and the parameters d1ri j, d

2ri j, α

1ri j, α

2ri j, β

1ri j, β

2ri j are given in Table 1. Sim-

ilarly, 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 sim-

plicity, 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

functions as μG̃ri( fri(x, hri, p̂ri)) =

fri(x, hri, p̂ri) − frimax

frimin − frimax

, i = 1, 2, r = 1, 2, where [ f11min, f11max] = [2400, 2600],

[ 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

as μp̂ri (p̂ri) =primin − p̂ri

primin − primax

, i = 1, 2, r = 1, 2, where [p11min, p11max] = [0.3, 0.9], [p12min, p12max] = [0.2, 0.85],

[p21min, p21max] = [0.3, 0.87], [p22min, p22max] = [0.4, 0.91] (Step 2). Set the initial reference membership values

as (μ̂11, μ̂12) = (μ̂21, μ̂22) = (1, 1) (Step 3) and decision powers as (w1,w2) = (1, 1) (Step 4). At Step 5, solve

MINMAX3(μ̂,w) to obtain the corresponding DG-Pareto optimal solution as μG̃ri( fri(x, hri, p̂ri)) = 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 ob-

tained as μG̃1i( f1i(x, h1i, p̂1i)) = 0.838, i = 1, 2, μG̃2i

( f2i(x, h2i, p̂2i)) = 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. There-

fore, in order to improve μDG22(·) at the expense of μDG21

(·), DM2 updates his/her reference membership values

as (μ̂21, μ̂22) = (0.730, 0.790) according to Rule 2 (Step 6). Then, the corresponding DG-Pareto optimal solution

is obtained as μG̃1i( f1i(x, h1i, p̂1i)) = 0.852, i = 1, 2 μG̃21

( f21(x, h21, p̂21)) = 0.750, μG̃22( f22(x, h22, p̂22)) = 0.810.

Since both DM1 and DM2 are satisfied with current values of the above membership functions, stop the interactive

processes (Step 6).

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(μ̂,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 deci-

sion makers will be reflected at a satisfactory solution through the decision powers.

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Table 1. Parameters of the objective functions and the constraints

d111 19 48 21 10 18 35 46 11 24 33

d211 3 2 2 1 4 3 1 2 4 2

d112

12 46 23 38 33 48 12 8 19 20

d212 1 2 4 2 2 1 2 1 2 1

d121 -12 -38 -23 -33 -33 -45 -12 -9 -19 -20

d221 1 2 4 2 2 1 2 1 2 1

d122 -12 -36 -27 -30 -33 -45 -11 -12 -19 -8

d222 1 2 4 2 2 1 2 1 2 1

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α221

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Hierarchical Multiobjective Fuzzy Random Linear Programming Problems