Linear optimization on the intersection of two fuzzy relational inequalities defined with Yager family of t-norms

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Linear optimization on the intersection of two

fuzzy relational inequalities defined with Yager

family of t-norms

Amin Ghodousian

and Reza Zarghani

1_{Faculty of Engineering Science, College of Engineering, University of Tehran, P.O.Box}

11365-4563, Tehran, Iran.

2_{School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran,} 11155-4563, Iran.

ABSTRACT ARTICLE INFO

In this paper, optimization of a linear objective func-tion with fuzzy relafunc-tional inequality constraints is in-vestigated where the feasible region is formed as the intersection of two inequality fuzzy systems and Yager family of t-norms is considered as fuzzy composition. Yager family of t-norms is a parametric family of con-tinuous nilpotent t-norms which is also one of the most frequently applied one. This family of t-norms is strictly increasing in its parameter and covers the whole spectrum of t-norms when the parameter is changed from zero to infinity. The resolution of the feasible region of the problem is firstly investigated

Article history:

Received 03, May 2016

Received in revised form 1, April 2017

Accepted 30 April 2017 Available online 01, June 2017

Keyword: Fuzzy relation, fuzzy relational inequality, linear optimization, fuzzy compositions and t-norms.

AMS subject Classification: 05C75.

ABSTRACT Continued

*_{Corresponding author: Amin Ghodousian. Email: [email protected]} †_{[email protected]}

when it is defined with max-Yager composition. Based on some theoretical results, conditions are derived for determining the feasibility. Moreover, in order to simplify the problem, some procedures are presented. It is shown that a lower bound is always attainable for the optimal objective value. Also, it is proved that the optimal solution of the problem is always resulted from the unique maximum solution and a minimal solution of the feasible region. A method is proposed to generate random feasible max-Yager fuzzy relational inequalities and an algorithm is presented to solve the problem.

Finally, an example is described to illustrate these algorithms.

1

Introduction

In this paper, we study the following linear problem in which the constraints are formed as the intersection of two fuzzy systems of relational inequalities defined by Yager family of t-norms:

min Z=cTx Aϕx≤_b1

Dϕx≥_b2

x∈_[0,1]n

(1)

whereI1 ={1,2, .., m1}, I2 ={m1+ 1, m1+ 2, .., m1+m2} andJ ={1,2, .., n}. A= (aij)m1×n

andD = (dij)m2×nare fuzzy matrices such that 0≤aij ≤1 (∀i ∈I1 and∀j ∈J) and 0≤

dij ≤1 (∀i ∈I2 and∀j ∈J). b1 = (b_i1)m1×1 is an m1–dimensional fuzzy vector in [0,1]

m1

(i.e., ₀≤_b1

i≤1,

∀_i ∈ _{I1) , b}2 _{= (b}2

i)m2×1 is an m2–dimensional fuzzy vector in [0,1]

m2 _(i.e.,

0≤b2_i≤1,∀i ∈I2), and c is a vector in Rn. Moreover, “ϕ ” is the max-Yager composition,

that is,ϕ(x, y) =T_Yp(x, y) = maxn1−₍₍₁−_x)p_{+ (1}−_y)p₎1/p_,0o_{in whichp >0.}

By these notations, problem (1) can be also expressed as follows:

min Z=cTx max

j∈_J

{_Tp

Y(aij, xj)} ≤b1_i , i∈I1

max

j∈J

{_Tp

Y(dij, xj)} ≥bi2 , i∈I2

x∈_[0,1]n

(2)

Especially, by settingA=Dandb1=b2, the above problem is converted to max- Yager fuzzy relational equations. As mentioned, the familynT_Ypois strictly increasing inp. It can be easily shown that Yager t-norm T_Yp(x, y) converges to the basic fuzzy intersec-tion min{_{x, y}} _{as p goes to infinity and converges to Drastic product t-norm [8] as p}

approaches zero. Also, it is interesting to note thatT_Y1(x, y) = max{_x+y−_1,0}_{, that is,}

the Yager t-norm is converted to Lukasiewicz t-norm ifp= 1.

knowledge can be treated as FRE problems [36]. In addition to the preceding appli-cations, FRE theory has been applied in many fields, including fuzzy control, discrete dynamic systems, prediction of fuzzy systems, fuzzy decision making, fuzzy pattern recognition, fuzzy clustering, image compression and reconstruction, fuzzy informa-tion retrieval, and so on. Generally, when inference rules and their consequences are known, the problem of determining antecedents is reduced to solving an FRE [34]. We refer the reader to [26] in which the authors provided a good overview of FRE and classified basic FREs by investigating the relationship among operators used in the definition of fuzzy relational equations.

The solvability determination and the finding of solutions set are the primary (and the most fundamental) subject concerning with FRE problems. Di Nola et al. proved that the solution set of FRE (if it is nonempty) defined by continuous max-t-norm composi-tion is often a non-convex set that is completely determined by one maximum solucomposi-tion and a finite number of minimal solutions [5]. This non-convexity property is one of two bottlenecks making major contribution to the increase of complexity in problems that are related to FRE, especially in the optimization problems subjected to a system of fuzzy relations. The other bottleneck is concerned with detecting the minimal so-lutions for FREs. Chen and Wang [2] presented an algorithm for obtaining the logical representation of all minimal solutions and deduced that a polynomial-time algorithm to find all minimal solutions of FRE (with max-min composition) may not exist. Also, Markovskii showed that solving max-product FRE is closely related to the covering problem which is an NP-hard problem [33]. In fact, the same result holds true for a more general t-norms instead of the minimum and product operators [2,3,29,30]. Over the last decades, the solvability of FRE defined with different max-t composi-tions have been investigated by many researchers [35,37,38,41,43,44,46,49,52]. More-over, some researchers introduced and improved theoretical aspects and applications of fuzzy relational inequalities (FRI)[13,15,16,22,27,51]. Li and Yang [27] studied a FRI with addition-min composition and presented an algorithm to search for minimal solutions. They applied FRI to meet a data transmission mechanism in a BitTorrent-like Peer-to-Peer file sharing systems. Ghodousian and Khorram [13] focused on the algebraic structure of two fuzzy relational inequalities Aϕx ≤ _b1_{and Dϕx} ≥_b2_{, and}

studied a mixed fuzzy system formed by the two preceding FRIs, where ϕis an oper-ator with (closed) convex solutions. Generally, if ϕis an operator with closed convex solutions, the solutions set ofDϕx≥_b2 _{is determined by a finite number of maximal}

solutions as well as the same number of minimal ones. In particular, ifϕis a contin-uous non-decreasing function (specially, a contincontin-uous t-norm), all maximal solutions overlap each other [13]. Guo et al. [15] investigated a kind of FRI problems and the relationship between minimal solutions and FRI paths. They also introduced some rules for reducing the problems and presented an algorithm for solving optimization problems with FRI constraints.

sub-jected to FRE constraints with max-min operation into an integer programming prob-lem and solved it by branch and bound method using jump-tracking technique. In [24] an application of optimizing the linear objective with max-min composition was employed for the streaming media provider seeking a minimum cost while fulfilling the requirements assumed by a three-tier framework. Wu et al. [45] improved the method used by Fang and Li, by decreasing the search domain and presented a simpli-fication process by three rules resulted from a necessary condition. Chang and Shieh [1] presented new theoretical results concerning the linear optimization problem con-strained by fuzzy max–min relation equations. They improved an upper bound on the optimal objective value, some rules for simplifying the problem and proposed a rule for reducing the solution tree. The topic of the linear optimization problem was also investigated with max-product operation [11,18,32]. Loetamonphong and Fang defined two sub-problems by separating negative and non-negative coefficients in the objective function and then obtained the optimal solution by combining those of the two problems [32]. The maximum solution of FRE is the optimum of the sub-problem having negative coefficients. Another sub-problem was converted into a bi-nary programming problem and solved by branch and bound method. Also, in [18] and [11] some necessary conditions of the feasibility and simplification techniques were presented for solving FRE with max-product composition. Moreover, some gen-eralizations of the linear optimization with respect to FRE have been studied with the replacement of max-min and max-product compositions with different fuzzy compo-sitions such as max-average composition [21,47], max-star composition [14,23] and max-t-norm composition [19,28,42]. For example, Li and Fang [28] solved the linear optimization problem subjected to a system of sup-t equations by reducing it to a 0-1 integer optimization problem. In [19] a method was presented for solving linear op-timization problems with the max-Archimedean t-norm fuzzy relation equation con-straint. In [42], the authors solved the same problem whit continuous Archimedean t-norm and used the covering problem rather than the branch-and-bound methods for obtaining some optimal variables.

Recently, many interesting generalizations of the linear programming subject to a sys-tem of fuzzy relations have been introduced and developed based on composite opera-tions used in FRE, fuzzy relaopera-tions used in the definition of the constraints, some devel-opments on the objective function of the problems and other ideas [6,10,16, 25,31,48]. For example, Wu et al. [48] represented an efficient method to optimize a linear frac-tional programming problem under FRE with max-Archimedean t-norm composition. Dempe and Ruziyeva [4] generalized the fuzzy linear optimization problem by consid-ering fuzzy coefficients. Dubey et al. studied linear programming problems involving interval uncertainty modeled using intuitionistic fuzzy set [6]. The linear optimiza-tion of bipolar FRE was studied by some researchers where FRE defined with max-min composition [10] and max-Lukasiewicz composition [25,31]. In [31], the authors pre-sented an algorithm without translating the original problem into a 0-1 integer linear problem.

literature as well [12,13,15,16,22,50,51]. Yang [50] applied the pseudo-minimal in-dex algorithm for solving the minimization of linear objective function subject to FRI with addition-min composition. Xiao et al. [51] introduced the latticized linear pro-gramming problem subject to max-product fuzzy relation inequalities with applica-tion in the optimizaapplica-tion management model of wireless communicaapplica-tion emission base stations. Ghodousian and Khorram [12] introduced a system of fuzzy relational in-equalities with fuzzy constraints (FRI-FC) in which the constraints were defined with max-min composition. They used this fuzzy system to convincingly optimize the edu-cational quality of a school (with minimum cost) to be selected by parents.

The remainder of the paper is organized as follows. In section 2, some preliminary notions and definitions and three necessary conditions for the feasibility of problem (1) are presented. In section 3, the feasible region of problem (1) is determined as a union of the finite number of closed convex intervals. Two simplification operations are introduced to accelerate the resolution of the problem. Moreover, a necessary and sufficient condition based on the simplification operations is presented to realize the feasibility of the problem. Problem (1) is resolved by optimization of the linear objec-tive function considered in section 4. In addition, the existence of an optimal solution is proved if problem (1) is not empty. The preceding results are summarized as an algorithm and, finally in section 5 an example is described to illustrate. Additionally, in section 5, a method is proposed to generate feasible test problems for problem (1).

2. Basic properties of max-Yager FRI

This section describes the basic definitions and structural properties concerning prob-lem (1) that are used throughout the paper. For the sake of simplicity, letS_Tp

Y(A, b

1_{) and}

S_Tp Y(D, b

2_{) denote the feasible solutions sets of inequalitiesAϕx≤b}1 _andDϕx≥b2_,

re-spectively, that is,S_Tp Y(A, b

1_{) =}n

x∈_[0,1]n _{: Aϕx}≤_b1o_andS

T_Yp(D, b2) =

n

x∈_[0,1]n _{: Dϕx}≥_b2o_.

Also, letS_Tp

Y(A, D, b

1_{, b}2_{) denote the feasible solutions set of problem (1). Based on the}

foregoing notations, it is clear thatS_Tp

Y(A, D, b

1_{, b}2_{) =S}

T_Yp(A, b1)

T S_Tp

Y(D, b

2_).

Definition 1.For eachi∈_{I1and eachj} ∈_{J, we defineS}

T_Yp(aij, b_i1) =

n

x∈_{[0,1] : T}p

Y(aij, x)≤b_i1

o .

Similarly, for eachi ∈_{I2 and eachj} ∈_J,S

T_Yp(dij, b2i) =

n

x∈_{[0,1] : T}p

Y(dij, x)≥b_i2

o . Fur-thermore, the notations

J_i1 =nj ∈_{J : S}

T_Yp(aij, b1i),∅

o

,∀_i ∈_{I1, and J}2

i =

n

j ∈_{J : S}

T_Yp(dij, bi2),∅

o

,∀_i ∈_{I2, are used}

in the text.

Remark 1.From the least-upper-bound property of R, it is clear that inf

x∈_[0,1]

n S_Tp

Y(aij, b

1

i)

o

and sup

x∈_[0,1]

n S_Tp

Y(aij, b

1

i)

o

exist, ifS_Tp Y(aij, b

1

i),∅. Moreover, sinceT p

Y is a t-norm, its

mono-tonicity property implies thatS_Tp Y(aij, b

1

i) is actually a connected subset of [0,1].

inf

x∈_[0,1]

n S_Tp

Y(aij, b

1

i)

o

= min

x∈_[0,1]

n S_Tp

Y(aij, b

1

i)

o

and sup

x∈[0,1]

n S_Tp

Y(aij, b

1

i)

o

= max

x∈_[0,1]

n S_Tp

Y(aij, b

1

i)

o

.

There-fore,

S_Tp Y(aij, b

1

i) =

" min

x∈_[0,1]

n S_Tp

Y(aij, b

1

i)

o , max

x∈_[0,1]

n S_Tp

Y(aij, b

1

i)

o #

, i.e., S_Tp Y(aij, b

1

i) is a closed

sub-interval of [0,1]. By the similar argument, ifS_Tp Y(dij, b

2

i),∅, then we haveST_Yp(dij, bi2) =

" min

x∈_[0,1]

n S_Tp

Y(dij, b

2

i)

o , max

x∈_[0,1]

n S_Tp

Y(dij, b

2

i)

o #

⊆_[0,1].

From Definition 1 and Remark 1, the following two corollaries are resulted.

Corollary 1.For eachi ∈_{I1and eachj}∈_J,

S

T_Yp(aij,b

1

i),∅. Also,ST p Y(aij, b

1

i) =

"

0, max

x∈_[0,1]

n S_Tp

Y(aij, b

1

i)

o #

.

Proof. Since T_Yp(aij,0) = 0, we have T p

Y (aij,0) ≤ bi1, ∀i ∈ I1 and ∀j ∈ J. Therefore,

0 ∈ _S

T_Yp(aij, bi1) and then min_x∈[0,1]

n S_Tp

Y(aij, b

1

i)

o

= 0, ∀_i ∈ _{I1 and} ∀_j ∈ _{J. Now, by noting}

Remark 1 we also have, S_Tp Y(aij, b

1

i) =

"

0, max

x∈_[0,1]

n S_Tp

Y(aij, b

1

i)

o #

, ∀_i ∈ _{I1 and} ∀_j ∈ _{J. This}

completes the proof.

Corollary 2.If_S T_Yp(dij,b

2

i),∅for somei

∈_I2andj∈_{J, thenS}

T_Yp(dij, bi2) =

" min

x∈[0,1]

n S_Tp

Y(dij, b

2 i) o ,1 # .

Proof. Noting Remark 1, it is sufficient to show that 1 ∈ _S

T_Yp(dij, b

2

i). Suppose that

S_Tp Y(dij, b

2

i),∅. Therefore, there exists somex ∈ [0,1] such thatT p

Y (dij, x)≥b2i. Now,

the monotonicity property of T_Yp implies T_Yp(dij,1) ≥ T p

Y(dij, x) ≥ b2_i that means 1 ∈

S_Tp Y(dij, b

2

i).

Remark 2.Corollary 1 together with Definition 1 impliesJ_i1=J,∀_i∈_I1.

Definition 2. For eachi ∈_{I1and eachj} ∈_{J, we define}

Uij=

      

1 aij< b1i

1− p

q (1−_b1

i)p−(1−aij)p aij≥bi1

Also, for eachi ∈_{I2 and eachj}∈_{J, we set}

Lij=

          

+∞ _{dij< b}2

i

0 b2_i = 0, dij≥b2i

1− qp (1−_b2

i)p−(1−dij)p bi2,0, dij ≥bi2

Remark 3. From Definition 2, if aij = b1i, then Uij = 1. Also, we have Lij = 1, if

Lemma 1 below shows thatUijandLijstated in Definition 2, determine the maximum

and minimum solutions of setsS_Tp Y(aij, b

1

i) (i∈I1) andST_Yp(dij, b2i)(i ∈I2), respectively.

Lemma 1. (a) Uij = max x∈_[0,1]

n S_Tp

Y(aij, b

1

i)

o

, ∀_i ∈_{I1 and} ∀_j ∈_{J. (b) If}

S_Tp Y(dij,b

2

i),∅ for some i ∈_I2andj ∈_{J, thenLij= min}

x∈_[0,1]

n S_Tp

Y(dij, b

2

i)

o .

Proof. (a) Let i ∈ _I1,j ∈ _{J and x} ∈ _S

T_Yp(aij, b1_i). Firstly, suppose that aij ≤ b1_i. In this

case,Uij=1 from Definition 2 and Remark 3. Sincex∈S_Tp Y(aij,b

1

i), thenx∈[0,1]and therefore x ≤_{Uij. Hence, it is su}ffi_{cient to show thatUij} ∈ _S

T_Yp(aij, b1_i). But, the identity law of

T_Yp impliesT_Yp(aij, Uij) =T p

Y(aij,1) =aij ≤b1i. Therefore, Uij ∈ST_Yp(aij, bi1) and x≤Uij

(∀_x∈_S

T_Yp(aij,b

1

i)) that mean Uij= max x∈[0,1]

S

T_Yp(aij,b

1

i)

. Otherwise, suppose that a

ij > bi1. In this

case,Uij= 1− p

q (1−_b1

i)p−(1−aij)p. SinceT p

Y (aij, Uij) =bi1andT p

Y has the monotonicity

property, we have_U ij∈S_Tp

Y(aij,b

1

i) andT

p

Y(aij, x)≥b1_i for eachx > Uij. Therefore,Uij must be the maximum of the set_S

T_Yp(aij,b

1

i). (b) Leti ∈_I2,j ∈_{J andx}∈_S

T_Yp(dij, bi2). SinceS

T_Yp(dij,b

2

i),∅, then we must havedij

≥_b2

i

(be-cause, ifdij< b2i, thenT p

Y(dij, x)≤T p

Y(dij,1) =dij < b2i,∀x∈[0,1]). Ifbi2= 0,thenLij = 0

from Definition 2 and Remark 3. Therefore, T_Yp(dij, Lij) = T p

Y(dij,0) = 0 = b2iand

ob-viously Lij = 0 ≤ x, ∀_x∈_S

T_Yp(dij,b

2

i). Consequently, Lij = min_x∈_[0,1]

n S_Tp

Y(dij, b

2

i)

o

. Otherwise,

suppose that b2_i , 0. In this case, we have Lij = 1− p

q (1−_b2

i)p−(1−dij)p. Again,

sinceT_Yp(dij, Lij) =b_i2 andT p

Y has the monotonicity property, we haveLij∈S_Tp Y(dij,b

2

i) and T_Yp(dij, x)≤b2i for eachx < Lij. Therefore,Lij must be the minimum of the setS_Tp

Y(dij,b

2

i). This completes the proof.

Lemma 1 together with the corollaries 1 and 2 results in the following consequence.

Corollary 3.(a) For eachi ∈_{I1 andj}∈_J,S

T_Yp(aij, b_i1) = [0, Uij]. (b) IfS_Tp Y

(dij,bi2),∅for some i ∈_I2andj ∈_{J, thenS}

T_Yp(dij, bi2) = [Lij,1]. Definition 3. For eachi ∈_{I1, letS}

T_Yp(ai, b1i) =

(

x∈_[0,1]n _{: max}n

j=1

n

T_Yp(aij, xj)

o

≤_b1

i

) .

Sim-ilarly, for eachi ∈_{I2, we defineS}

T_Yp(di, b2i) =

(

x∈_[0,1]n _{: max}n

j=1

n

T_Yp(dij, xj)

o

≥_b2

i

) .

According to Definition 3 and the constraints stated in (2), setsS_Tp Y(ai, b

1

i) andST_Yp(di, b2i)

actually denote the feasible solutions sets of the i’th inequality max

j∈_J

{_Tp

Y (aij, xj)} ≤b1_i

(i ∈_{I1) and max}

j∈_J

{_Tp

Definitions 1 and 3, it can be easily concluded that for a fixedi ∈_I1,S

T_Yp(ai, b1i),∅iff S_Tp

Y(aij,b

1

i),∅,

∀_j ∈_{J. On the other hand, by Corollary 1 we know that}

S_Tp Y(aij,b

1

i),∅,

∀_i ∈_I1

and∀_j∈_{J. As a result,S}

T_Yp(ai, bi1),∅for eachi ∈I1. However, in contrast toST_Yp(ai, bi1),

setS_Tp Y(di, b

2

i) may be empty. Actually, for a fixed i ∈I2,ST_Yp(di, b2_i) is nonempty if and

only if_S T_Yp(dij,b

2

i) is nonempty for at least some j

∈_{J. Additionally, for each i} ∈_{I2 and}

j∈_{Jwe have}

S_Tp Y

(dij,b2i),∅if and only ifdij≥b2i. These results have been summarized in the following lemma. Part (b) of the lemma gives a necessary and sufficient condition for the feasibility of setS_Tp

Y(di, b

2

i) (∀i ∈ I2). It is to be noted that the lemma 2 (part (b))

also provides a necessary condition for problem (1).

Lemma 2.(a)S_Tp Y(ai, b

1

i),∅,∀i∈I1. (b) For a fixedi∈I2,ST_Yp(di, b2i),∅iffSn

j=1ST_Yp(dij,b

2

i),∅. Additionally, for eachi ∈_I2andj∈_J,

S

T_Yp(dij,b

2

i),∅iffdij≥b2i.

Definition 4. For each i ∈ _{I2 and j} ∈ _J2

i, we define ST_Yp(di, bi2, j) = [0,1]×...×[0,1]×

[Lij,1]×[0,1]×...×[0,1], where [Lij,1] is in thej’th position.

In the following lemma, the feasible solutions set of thei’th fuzzy relational inequality is characterized.

Lemma 3. (a) S_Tp Y(ai, b

1

i) = [0, Ui1]×[0, Ui2]×...×[0, Uin], ∀i ∈ I1. (b) ST_Yp(di, b_i2) =

S

j∈_J2

i ST p Y(di, b

2

i, j),∀i ∈I2.

Proof. (a) Fixi ∈_{I1 and letx}∈_S

T_Yp(ai, b1i). By Definition 3,xj ∈[0,1] for eachj∈J, and n

max

j=1

n

T_Yp(aij, xj)

o

≤_b1

i. The latter inequality impliesT p

Y(aij, xj)≤b1_i,∀j∈J. Thus, by

Def-inition 1 and Corollary 3 we havexj ∈ST_Yp(aij, bi1) = [0, Uij],∀j ∈J, which necessitates

x∈_{[0, Ui1]}×_{[0, Ui2]}×_...×_{[0, Uin]. Conversely, suppose thatx}∈_{[0, Ui1]}×_{[0, Ui2]}×_...×

[0, Uin]. Then, by Corollary 3,xj∈[0, Uij] =S_Tp

Y(aij, b

1

i),∀j ∈J, which impliesxj ∈[0,1]

and T_Yp(aij, xj)≤ bi1, ∀j ∈ J. Thus, x ∈ [0,1]n and n

max

j=1

n

T_Yp(aij, xj)

o

≤ _b1

i. Therefore, by

Definition 3,x∈_S

T_Yp(ai, bi1).

(b) Fixi∈_{I2and letx}∈_S

T_Yp(di, bi2). By Definition 3,x∈[0,1]nand n

max

j=1

n

T_Yp(dij, xj)

o

≥_b2

i.

Then there exists somej0∈J_i2 such thatT

p

Y(dij0, xj0)≥b

2

i. Therefore, from Definition 1

and Corollary 3, it is concluded thatxj0∈ST_Yp(dij0, b

2

i) = [Lij0,1]. Now, from Definition

4 we have x∈_S

T_Yp(di, bi2, j0). Thus, x∈Sj∈J_i2ST_Yp(di, b2i, j). Conversely, suppose thatx∈

S

j∈_J2

i ST p Y(di, b

2

i, j). Then there exists somej0∈Ji2such thatx∈ST_Yp(di, b2i, j0). Therefore,

by Definition 4,x∈_[0,1]n_andxj

0 ∈ST_Yp(dij0, b

2

i) = [Lij0,1], which impliesT

p

Y(dij0, xj0)≥

b2_i. Thus,x∈_[0,1]n_andmaxn

j=1

n

T_Yp(dij, xj)

o

≥_b2

i, which requiresx∈ST_Yp(di, b2i).

∀_i ∈_{I2 and}∀_j∈_J2

i, where

X(i, j)k =

(

Lij k=j

0 k,j

Lemma 3 together with Definitions 4 and 5, results in Theorem 1, which completely determines the feasible region for thei’th relational inequality.

Theorem 1. (a) S_Tp Y(ai, b

1

i) = [0, X(i)], ∀i ∈ I1. (b)ST_Yp(di, b2_i) =Sj∈_J2

i[X(i, j),1],

∀_i ∈_I2,

where0and1aren–dimensional vectors with each component equal to zero and one, respectively.

Theorem 1 gives the upper and lower bounds for the feasible solutions set of the i’th relational inequality. Actually, for eachi ∈_{I1, vectors0and X(i) are the unique}

min-imum and the unique maxmin-imum of set S_Tp Y(ai, b

1

i). In addition, for each i ∈ I2, set

S_Tp Y(di, b

2

i) has the unique maximum (i.e., vector1), but the finite number of minimal

solutionsX(i, j)(∀_j∈_J2

i). Furthermore, part (b) of Theorem 1 presents another feasible

necessary condition for problem (1) as stated in the following corollary.

Corollary 4.IfS_Tp

Y(A, D, b

1_{, b}2₎

,∅, then1∈S_Tp Y(di, b

2

i),∀i ∈I2(i.e.,1∈Ti∈_I2S_Tp Y

(di, bi2) =

S_Tp Y(D, b

2_)).

Proof. Let S_Tp

Y(A, D, b

1_{, b}2₎

, ∅. Then, S_Tp Y(D, b

2₎

, ∅, and therefore, S_Tp Y

(di, b2_i) , ∅,

∀_i ∈_{I2. Now, Theorem 1 (part (b)) implies1}∈_S

T_Yp(di, b2i),∀i ∈I2.

Lemma 4 describes the shape of the feasible solutions set for the fuzzy relational in-equalitiesAϕx≤_b1 _andDϕx≥_b2_{, separately.}

Lemma 4. (a)S_Tp Y(A, b

1_{) =}T

i∈_I1[0, Ui1]×Ti∈_I1[0, Ui2]×...×Ti∈_I1[0, Uin]. (b)ST_Yp(D, b2) =

T

i∈_I2

S

j∈_J2

i ST p Y(di, b

2

i, j).

Proof.The proof is obtained from Lemma 3 and equationsS_Tp Y(A, b

1_{) =}T

i∈_I1S_Tp Y

(ai, b1i)

andS_Tp Y(D, b

2_{) =}T

i∈_I2S

T_Yp(di, b

2

i).

Definition 6. Lete :I2 → J_i2 so thate(i) = j ∈J_i2, ∀i ∈ I2, and let ED be the set of all

vectorse. For the sake of convenience, we represent eache∈_{ED as anm2–dimensional}

vectore= [j1, j2, ..., jm2] in whichjk =e(k),k= 1,2, ..., m2.

Definition 7. Let e = [j1, j2, ..., jm2] ∈ ED. We define X = min_i∈I 1

n

X(i)o, that is, Xj =

min

i∈_I1

n X(i)j

o

,∀_j ∈_{J. Moreover, letX(e) = [X(e)1, X(e)2, ..., X(e)n], whereX(e)j= max}

i∈_I2

n

X(i, e(i))j

o =

max

i∈I2

n

X(i, ji)j

o

,∀_j ∈_J.

Based on Theorem 1 and the above definition, we have the following theorem charac-terizing the feasible regions of the general inequalitiesAϕx≤_b1 _andDϕx≥_b2 _{in the}

most familiar way.

Theorem 2.(a)S_Tp Y(A, b

1_{) = [0, X],∀i∈I}

1. (b)ST_Yp(D, b2) =

S

e∈_ED[X(e),1].

Proof. (a) By considering Definitions 5 and 7, for each j ∈ _{J we have} T

" 0,min

i∈_I1

n Uij

o #

= "

0,min

i∈_I1

n X(i)j

o #

= [0, Xj]. Therefore, part (a) of lemma 4 can be

rewrit-ten asS_Tp Y(A, b

1_{) =}h 0, X1

i

×h_{0, X2}i×_...×h_{0, Xn}i_{= [0, X], where0is the zero vector. This}

proves part (a).

(b) From part (b) of lemma 4,S_Tp Y(D, b

2_{) =}T

i∈_I2

S

j∈_J2

i ST p Y(di, b

2

i, j). Now, by Definitions

4 and 5, we have

S_Tp Y(D, b

2_{) =}\

i∈_I2

[

j∈_J2

i

[0,1]×_...×_[0,1]×_[Lij,1]×_[0,1]×_...×_{[0,1] =}\

i∈_I2

[

j∈_J2

i

[X(i, j),1].

Therefore, from Definitions 6 and 7 we have

S_Tp Y(D, b

2_{) =}T

i∈_I2

S

e∈_ED[X(i, e(i)),1] =

S

e∈_ED

T

i∈_I2[X(i, e(i)),1] =

S

e∈_ED

" max

i∈I2

{_{X(i, e(i))}}_,1

# =

S

e∈_ED[X(e),1]where, the last equality is resulted from Definition 7. This completes the

proof.

Corollary 5.Assume thatS_Tp

Y(A, D, b

1_{, b}2₎

,∅. Then, there exists somee∈ED such that

[0, X]T

[X(e),1],∅.

Corollary 6.Assume thatS_Tp

Y(A, D, b

1_{, b}2₎

,∅. Then,X∈S_Tp Y(D, b

2_).

Proof. Let S_Tp

Y(A, D, b

1_{, b}2₎

,∅. By Corollary 5, [0, X]T[X(e0),1],∅ for somee0 ∈ED.

Thus, X ∈ _[X(e0_{),1] that means X} ∈S

e∈_ED[X(e),1]. Therefore, from Theorem 2 (part

(b)),X∈_S

T_Yp(D, b2).

3. Feasible solutions set and simplification operations

In this section, two operations are presented to simplify the matrices AandD, and a necessary and sufficient condition is derived to determine the feasibility of the main problem. At first, we give a theorem in which the bounds of the feasible solutions set of problem (1) are attained. As is shown in the following theorem, by using these bounds, the feasible region is completely found.

Theorem 3.Suppose thatS_Tp

Y(A, D, b

1_{, b}2₎

,∅. ThenST_Yp(A, D, b1, b2) =

S

e∈_ED[X(e), X].

Proof.SinceS_Tp

Y(A, D, b

1_{, b}2_{) =S}

T_Yp(A, b1)

T S_Tp

Y(D, b

2_{), then by Theorem 2,S}

T_Yp(A, D, b1, b2) =

[0, X]T (S

e∈_ED[X(e),1] ) and the statement is established.

In practice, there are often some components of matricesAandD, which have no effect on the solutions to problem (1). Therefore, we can simplify the problem by changing the values of these components to zeros. We refer the interesting reader to [13] where a brief review of such these processes is given. Here, we present two simplification techniques based on the Yager family of t-norms.

Definition 8. If a value changing in an element, say aij, of a given fuzzy relation

matrixAhas no effect on the solutions of problem (1), this value changing is said to be an equivalence operation.

Corollary 7. Suppose that i ∈ _{I1 and T}p

Y(aij0, xj0) < bi,∀x ∈ ST_Yp(A, b

is obvious that maxn

j=1

n

T_Yp(aij, xj)

o

≤_b1

i is equivalent to n

max j = 1 j ,j0

n

T_Yp(aij, xj)

o

≤ _b1

i, that is,

“resetting aij0 to zero” has no effect on the solutions of problem (1) (since component

aij0 only appears in the i‘th constraint of problem (1)). Therefore, if T

p

Y(aij0, xj0) <

b1_i,∀_x∈_S

T_Yp(A, b1), then “resettingaij0 to zero” is an equivalence operation.

Lemma 5 (simplification of matrix A).Suppose that matrix ˜A= ( ˜aij)m1×n is resulted

from matrixAas follows:

˜ aij=

(

0 aij< b1i

aij aij≥bi1

for eachi ∈_{I1 andj} ∈_{J. Then,S}

T_Yp(A, b1) =ST_Yp( ˜A, b1).

Proof. From corollary 7, it is sufficient to show thatT_Yp(aij0, xj0)< b

1

i,∀x ∈ST_Yp(A, b1).

But, from the monotonicity and identity laws ofT_Yp, we haveT_Yp(aij0, xj0)≤T

p

Y(aij0,1) =

aij0< b

1

i,∀xj0∈[0,1]. Thus,T

p

Y(aij0, xj0)< b

1

i,∀x∈ST_Yp(A, b1).

Lemma 5 gives a condition to reduce the matrixA. In this lemma, ˜Adenote the sim-plified matrix resulted fromAafter applying the simplification process. Based on this notation, we define ˜J_i1=nj∈_{J : S}

T_Yp( ˜aij, b1i),∅

o

(∀_i ∈_{I1) where ˜aij denotes (i, j)‘th}

com-ponent of matrix ˜A. So, from Corollary 1 and Remark 2, it is clear that ˜J_i1 = J_i1 = J. Moreover, since S_Tp

Y(A, D, b

1_{, b}2_{) = S}

T_Yp(A, b1)

T S_Tp

Y(D, b

2_{), from Lemma 5 we can also}

conclude thatS_Tp

Y(A, D, b

1_{, b}2_{) =S}

T_Yp( ˜A, D, b1, b2).

By considering a fixed vectore ∈_{ED in Theorem 3, interval [X(e), X] is meaningful i}ff

X(e)≤_{X. Therefore, by deleting infeasible intervals [X(e), X] in which X(e)}6≤_{X, the}

feasible solutions set of problem (1) stays unchanged. In order to remove such infea-sible intervals from the feainfea-sible region, it is sufficient to neglect vectors e generating infeasible solutions X(e) (i.e., solutions X(e) such that X(e) 6≤ _{X). These}

considera-tions lead us to introduce a new set E0_D =ne∈_{ED : X(e)}≤_Xo _{to strengthen Theorem}

3. By this new set, Theorem 3 can be written as S_Tp

Y(A, D, b

1_{, b}2_{) =} S

e∈E0

D[X(e), X], if S_Tp

Y(A, D, b

1_{, b}2₎

,∅.

Lemma 6.LetIj(e) ={i∈I2 : e(i) =j}andJ(e) =

n

j ∈_{J : Ij(e),}∅o_,∀_e∈_{ED. Then,}

X(e)j=

      

max

i∈_Ij(e)

n Li e(i)

o

j∈_J(e)

0 j<J(e)

Proof. From Definition 7, X(e)j = max i∈_I2

n

X(i, e(i))j

o

,∀_j ∈_{J. On the other hand, by}

Defi-nition 5, we have

X(i, e(i))j=

(

Li e(i) j=e(i)

Now, the result follows by combining these two equations.

Corollary 8.e∈_E0

D if and only ifLi e(i)≤Xe(i),∀i∈I2.

Proof.Firstly, from the definition of setE_D0 , we note thate∈_E0

D if and only ifX(e)j≤Xj,

∀_j ∈ _{J. Now, let e} ∈ _E0

D and by contradiction, suppose that Li0e(i0) > Xe(i0) for some

i0 ∈ I2. So, by setting e(i0) = j0, we have j0 ∈ J(e), and therefore lemma 6 implies

X(e)j0 = max

i∈_Ij 0(e)

n Li e(i)

o

≥_Li

0e(i0)> Xe(i0). Thus,X(e)j0> Xe(i0) that contradictse∈E 0

D. The

converse statement is easily proved by Lemma 6.

As mentioned before, to accelerate identification of the meaningful solutionsX(e), we reduce our search to set E_D0 instead of set ED. As a result from Corollary 8, we can

confine set J_i2 by removing each j ∈ _J2

i such that Lij > Xj before selecting the vectors

e to construct solutions X(e). However, lemma 7 below shows that this purpose can be accomplished by resetting some components of matrixD to zeros. Before formally presenting the lemma, some useful notations are introduced.

Definition 9 (simplification of matrix D).Let ˜D = ( ˜dij)m2×n denote a matrix resulted

fromD as follows:

˜ dij=

(

0 j∈_J2

i and Lij> Xj

dij otherwise

Also, similar to Definition 1, assume that ˜J_i2 =nj ∈_{J : S}

T_Yp( ˜dij, b_i2),∅

o

(∀_i ∈ _{I2) where}

˜

dij denotes (i, j)‘th components of matrix ˜D.

According to the above definition, it is easy to verify that ˜J_i2⊆_J2

i ,∀i∈I2. Furthermore,

the following lemma demonstrates that the infeasible solutionsX(e) are not generated, if we only consider those vectors e generated by the components of the matrix ˜D, or equivalently vectorse generated based on the set ˜J_i2instead ofJ_i2.

Lemma 7. E_D˜ =E_D0 , where E_D˜ is the set of all functionse :I2 →J˜i2 so thate(i) =j ∈J˜i2,

∀_i ∈_I2.

Proof.Lete∈_E0

D. Then, by Corollary 8,Li e(i)≤Xe(i),∀i∈I2. Therefore, we have ˜di e(i) =

di e(i),∀i∈I2, that necessitates ˜Ji2=Ji2,∀i ∈I2. Hence,e(i)∈J˜i2,∀i ∈I2, and thene∈E_D˜.

Conversely, lete∈_{ED˜. Therefore,e(i)}∈_J˜2

i,∀i ∈I2. Since ˜Ji2 ⊆Ji2,∀i ∈I2, thene(i)∈Ji2,

∀_i ∈_{I2, and thereforee} ∈_{ED. By contradiction, suppose that e <E}0

D. So, by Corollary

8, there is somei0∈I2 such thatLi0e(i0) > Xe(i0). Hence, ˜di0e(i0) = 0 (sincee(i0)∈J

2

i0 and

Li0e(i0) > Xe(i0)) and Li0e(i0) >0. The latter inequality together with Definition 2 and

Remark 3 impliesb_i2

0 >0. But in this case,T

p

Y( ˜di0e(i0), x) =T

p

Y(0, x) = 0< b2i0,

∀_x∈_[0,1],

that contradictse(i0)∈J_i2₀.

By Lemma 7, we always have X(e)≤ _{X for each vector e, which is selected based on}

the components of matrix ˜D. Actually, matrix ˜D as a reduced version of matrix D, removes all the infeasible intervals from the feasible region by neglecting those vec-tors e generating the infeasible solutions X(e). Also, similar to Lemma 5 we have S_Tp

Y(A, D, b

1_{, b}2_{) = S}

S_Tp

Y(A, D, b

1_{, b}2_{) =S}

T_Yp( ˜A,D, b˜ 1, b2).

Definition 10. LetL= (Lij)m2×n be a matrix whose (i, j)’th component is equal toLij.

We define the modified matrixL∗= (L∗_ij)m2×nfrom the matrixLas follows:

L∗_ij= (

+∞ _{Lij> Xj}

Lij otherwise

As will be shown in the following theorem, matrixL∗ is useful for deriving a necessary and sufficient condition for the feasibility of problem (1) and accelerating identifica-tion of the setS_Tp

Y(A, D, b

1_{, b}2_).

Theorem 4. S_Tp

Y(A, D, b

1_{, b}2₎

,∅iff there exists at least somej ∈J_i2 such thatL∗_ij ,+∞,

∀_i ∈_I2.

Proof. Letx∈_S

T_Yp(A, D, b1, b2). Then, from Corollary 5, there exists somee

0_∈

ED such

that [X(e0), X],∅. Therefore, X(e0)≤X that impliese0 ∈ED0 . Now, by Corollary 8, we

haveLi e0_(i) ≤X_e0_(i),∀i ∈ I₂. Hence, by considering Definition 10, L∗

i e0_(i) ,+∞ ,∀i ∈I₂.

Conversely, suppose thatL∗_ij i ,+

∞ _{for someji} ∈_J2

i ,∀i ∈I2. Then, from Definition 10

we have

Liji ≤Xji,∀i∈I2 (3) Consider vectore0 = [j1, j2, ..., jm]∈ED. So, by noting Lemma 6,X(e0)ji = max_i∈I

j(e0) n

Li e0_(i)

o =

max

i∈_Ij(e0₎

n Liji

o

, ∀_i ∈ _{I2, and X(e}0_{)j = 0 for each j} ∈ _J − {_{j1, j2, ..., jm}}_{. These equations}

to-gether with (3) implyX(e0)≤_{Xthat means [X(e}0_{), X],}∅_{. Now, the result follows from}

Corollary 5.

4. Optimization of the problem

According to the well-known schemes used for optimization of linear problems such as (1) [9,13,16,28], problem (1) is converted to the following two sub-problems:

min Z1=Pnj=1c+jxj

Aϕx≤_b1

Dϕx≥_b2

x∈_[0,1]n

(4)

and

min Z2=Pnj=1c

−

jxj

Aϕx≤_b1

Dϕx≥_b2

x∈_[0,1]n

(5)

Wherec+_j = max{_cj,0} _andc−

j = min{cj,0}for j = 1,2, ..., n. It is easy to prove thatX is

the optimal solution of (5), and the optimal solution of (4) isX(e0) for somee0∈_E0

Theorem 5. Suppose thatS_Tp

Y(A, D, b

1_{, b}2₎

,∅, and X and X(e∗) are the optimal

solu-tions of sub-problems (5) and (4), respectively. Then cTx∗ is the lower bound of the optimal objective function in (1), wherex∗= [x∗₁, x∗₂, ..., x∗n] is defined as follows:

x∗_j= (

Xj cj<0

X(e∗)j cj ≥0

(6)

forj= 1,2, ..., n.

Proof. Let x ∈ _S

T_Yp(A, D, b1, b2). Then, from Theorem 3 we have x ∈

S

e∈_ED[X(e), X].

Therefore, for each j ∈ _{J such that cj} ≥ _{0, inequality x}∗

j ≤ xj implies c+jx

∗

j ≤ cj+xj. In

addition, for eachj ∈_{J such thatcj <0, inequalityx}∗

j ≥xj impliesc

−

jx

∗

j ≤c

−

jxj. Hence,

Pn

j=1cjx∗_j≤Pnj=1cjxj.

Corollary 9. Suppose that S_Tp

Y(A, D, b

1_{, b}2₎

, ∅. Then, x∗ = [x1∗, x

∗

2, ..., x

∗

n] as defined in

(6), is the optimal solution of problem (1).

Proof. As in the poof of Theorem 5, cTx∗ is the lower bound of the optimal objective function. According to the definition of vector x∗, we have X(e∗)j ≤ xj∗ ≤ Xj, ∀j ∈ J,

which impliesx∗∈S

e∈_ED[X(e), X] =S_Tp

Y(A, D, b

1_{, b}2_).

We now summarize the preceding discussion as an algorithm.

Algorithm 1 (solution of problem (1))

Given problem (1):

1. ComputeUij (∀i ∈I1 and∀j∈J) andLij (∀i∈I2and∀j ∈J) by Definition 2.

2. If 1∈_S

T_Yp(D, b2), then continue; otherwise, stop, the problem is infeasible (Corollary

4).

3. Compute vectorsX(i) (∀_i∈_{I1) from Definition 5, and then vectorXfrom Definition}

7.

4. IfX∈_S

T_Yp(D, b2), then continue; otherwise, stop, the problem is infeasible (Corollary

6).

5. Compute simplified matrices ˜Aand ˜D from Lemma 5 and Definition 9, respectively.

6. Compute modified matrixL∗ from Definition 10.

7. For eachi ∈_{I2, if there exists at least somej} ∈_J2

i such thatL

∗

ij,+∞, then continue;

otherwise, stop, the problem is infeasible (Theorem 4).

8. Find the optimal solutionX(e∗) for the sub-problem (4) by considering vectorse∈_ED˜

and set ˜J_i2,∀_i∈_{I2( Lemma 7).}

9. Find the optimal solutionx∗= [x∗₁, x∗₂, ..., x∗_n] for the problem (1) by (6) (Corollary 9).

It should be noted that there is no polynomial time algorithm for complete solution of FRIs with the expectationN ,N P. Hence, the problem of solving FRIs is an NP-hard

problem in terms of computational complexity [2].

In this section, we present a method to generate random feasible regions formed as the intersection of two fuzzy inequalities with Yager family of t-norms. In section 5.1, we prove that the max-Yager fuzzy relational inequalities constructed by the introduced method are actually feasible. In section 5.2, the method is used to generate a ran-dom test problem for problem (1), and then the test problem is solved by Algorithm 1 presented in section 4.

5.1. Construction of test problems

There are several ways to generate a feasible FRI defined with max-Yager composition. In what follows, we present a procedure to generate random feasible max-Yager fuzzy relational inequalities:

Algorithm 2 (construction of feasible Max-Yager FRI)

1.Generate randon scalarsaij∈[0,1], i= 1,2, ...,m1andj = 1,2, ...,n, andb1i ∈[0,1], i= 1,2, ...,m1.

2.ComputeXby Definition 7.

2.Randomly selectm2columns{j1, j2, ..., jm2}fromJ={1,2, ...,n}.

2.Fori ∈ {_{1,2, ...,m2}}_{,assign a random number from [0, Xj}

i] tob

2

i.

3.Fori ∈ {_{1,2, ...,m2}}_,ifb2

i ,0,then

Assign a random number from interval

max

b2_i,1− p

q (1−_b2

i)p−(1−Xji)

p_{,1 tod} iji.

End

4.Fori ∈ {_{1,2, ...,m2}}

For eachk∈ {_{1,2, ...,m2}} − {_i}

Assign a random number from [0,1] todk ji. End

End

5.For eachi ∈ {_{1,2, ...,m2}} _{and each j <}{_{j1, j2, ..., jm}

2}

Assign a random number from [0,1] todij.

End

By the following theorem, it is proved that Algorithm 2 always generates random fea-sible max-Yager fuzzy relational inequalities.

Theorem 6. Problem (1) with feasible region constructed by Algorithm (2) has the nonempty feasible solutions set (i.e.,S_Tp

Y(A, D, b

1_{, b}2₎

,∅). Proof. By considering the columns {_{j1, j2, ..., jm}

2} selected by Algorithm 2, let e 0

= [j1, j2, ..., jm2]. We show thate

0_∈

ED andX(e0)≤X. Then, the result follows from

Corol-lary 5. From Algorithm 2, the following inequalities are resulted for eachi ∈_I2:

1. b2_i ≤_Xj

i.

2. b2_i ≤_dij

3. 1− p

q (1−_b2

i)p−(1−Xji)

p≤_dij

i.

By (I), we have 1− p

q (1−_b2

i)p−(1−Xji)

p ≤_{1. This inequality together withb}2

i ∈[0,1],

∀_i ∈_{I2, implies that the interval}

max

b_i2,1− p

q (1−_b2

i)p−(1−Xji)

p_{,1is meaningful.}

Also, by (II), e0(i) = ji ∈ Ji2, ∀i ∈ I2. Therefore, e0 ∈ED. Moreover, since the columns

{_{j1, j2, ..., jm}

2}are distinct, setsIji(e

0

) (i∈_{I2) are all singleton, i.e.,}

Iji(e

0

) ={_i}_,∀_i ∈_{I2 (7)}

As a result, we also haveJ(e0) ={_{j1, j2, ..., jm}

2}andIj(e 0

) =∅_{for each j <} {_{j1, j2, ..., jm}

2}.

On the other hand, from Definition 5, we haveX(i, e0(i))e0_(i)=X(i, j_i)_j

i =LijiandX(i, e

0

(i))j=

0 for each j <J − {ji}. This fact together with (7) and Lemma 6 implies X(e0)ji =Li ji,

∀_i ∈_{I2, andX(e}0_{)j = 0 forj <}{_{j1, j2, ..., jm}

2}. So, in order to proveX(e 0

)≤_{X, it is su}ffi

-cient to show thatX(e0)ji ≤Xji,∀i∈I2. But, from Definition 2 and Remark 3,

X(e0)ji =Li ji =       

0 b2_i = 0 1− p

q (1−_b2

i)p−(1−diji)

p _b2

i ,0

(8)

Now, inequality (III) implies

1− p

q (1−_b2

i)p−(1−diji)

p_≤X

ji (9)

Therefore, by relations (8) and (9), we haveX(e0)ji ≤ Xji,∀i ∈ I2. This completes the proof.

5.2. Numerical example