Posynomial geometric programming problem subject to max–min fuzzy relation equations

Information Sciences 328 (2016) 15–25

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Posynomial geometric programming problem subject to max–min fuzzy relation equations Xue-Gang Zhou a,b, Xiao-Peng Yang a,c, Bing-Yuan Cao a,d,∗ a School of Mathematics and Information Science, Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong, Higher Education Institutes, Guangzhou University, Guangzhou, Guangdong, 510006, China b Department of Applied Mathematics, Guangdong University of Finance, Guangzhou, Guangdong, 510521, China c Department of Mathematics and Statistics, Hanshan Normal University, Chaozhou, Guangdong, China d Guangzhou Vocational College of Science and Technology, Guangzhou, Guangdong, 510550, China

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 22 July 2014 Revised 16 July 2015 Accepted 29 July 2015 Available online 24 August 2015

We discuss a class of posynomial geometric programming problem(PGPF), aimed at minimizing a posynomial subject to fuzzy relational equations with max–min composition. By introducing auxiliary variables, we convert the PGPF into an equivalent programming problem whose objective function is a non-decreasing function with an auxiliary variable. We show that an optimal solution consists of a maximum feasible solution and one of the minimal feasible solutions by an equivalent programming problem. In addition, we introduce some rules for simplifying the problem. Then by using a branch and bound method and fuzzy relational equations (FRE) path, we present an algorithm to obtain an optimal solution to the PGPF. Finally, numerical examples are provided to illustrate the steps of the procedure.

Keywords: Posynomial geometric programming Fuzzy relation equation Max–min composition

© 2015 Elsevier Inc. All rights reserved.

1. Introduction In this study, we consider a certain type of posynomial geometric programming problems subject to max–min fuzzy relation equations described as follows:

(PGPF) min z(x) =

p  k=1

ck

n  γk xj j j=1

s.t.A ◦ x = b,

(1)

where A = (ai j )m×n , x = (x j )n×1 , b = (bi )m×1 , aij , xj , bi ∈ [0, 1], ck , γk j ∈ R, ck > 0, i ∈ I = {1, 2, . . . , m}, j ∈ J = {1, 2, . . . , n}, k ∈ K =

{1, 2, . . . , p}, and for given j ∈ J, γk j (k ∈ K ) are either all non-positive real numbers or all non-negative real numbers. Without loss of generality, we assume that problem (1) satisfies the following inequalities:

1  b1  b2  · · ·  bm  0. Otherwise, rearrange the components of b in decreasing order and adjust the rows of A accordingly b. Fuzzy relational equations have played an important role in fuzzy set theory and fuzzy logic systems, and many researchers have discussed fuzzy relation equations based on different compositions of fuzzy relations, including max–min fuzzy relational ∗

Corresponding author. Tel.: +8615876504405. E-mail address: [email protected], [email protected] (B.-Y. Cao).

http://dx.doi.org/10.1016/j.ins.2015.07.058 0020-0255/© 2015 Elsevier Inc. All rights reserved.

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X.-G. Zhou et al. / Information Sciences 328 (2016) 15–25

equations [29,30], max-product fuzzy relational equations [1], continuous t-norm fuzzy relational equations [5], interval-valued fuzzy relation equations [16], and so on. It is worth noting that the complete solution set of continuous t-norm fuzzy relational equations can be completely determined by a unique maximum solution and a finite number of minimal solutions [4,5,14,16]. Computing the maximum solution is easy, but finding all the potential minimal solutions of fuzzy relational equations is an NP-hard problem [3,25]. Recently, studies on fuzzy relational equations have been extended to the setting of interval-valued equations with max-t-norm composition. Wang et al. [35] have proposed three types of solution sets for this type of equation. Li and Fang [18] presented the detailed analysis of the classification and the solvability of general fuzzy relational equations with various compositions. Applications of fuzzy relational equations can be found in [5,12,26,27]. Wang et al. [34] first studied a class of latticized linear programming subject to max–min fuzzy relation inequalities. Li and Fang [19] considered the latticized linear optimization problem and its variants, which are a special class of optimization problems constrained by fuzzy relational equations or inequalities. Li and Wang [17] investigated the latticized linear programming that is subject to the fuzzy-relation inequality constraints with the max–min composition by using a semi-tensor product method, and proposed a matrix approach to this problem. Fang and Li [6] discussed an optimization model with a linear objective function subject to max–min fuzzy relational equations. They converted such an optimization model into a 0–1 integer programming problem and solved it by the branch-and-bound method. Wu and Guu [38] proposed a necessary condition for an optimal solution to explore the same optimization problem with positive cost coefficients in the objective function. Loetamonphong and Fang [22] considered a minimization problem with a linear objective function and a max-product fuzzy relational equations constraint. By using the nonnegative and negative coefficients in the objective function, this optimization problem is separated into two subproblems. The maximum solution of max-product fuzzy relational equations is an optimal solution to a subproblem formed by negative coefficients. In addition, the subproblem formed by the nonnegative coefficients can be converted into a 0–1 integer programming problem which can be solved by the branch-and-bound method. Guu and Wu [13] provided a necessary condition for an optimal solution in terms of the maximum solution derived from the fuzzy relational equations. We employed this necessary condition to provide an efficient procedure for solving the minimization problem. Wu and Guu [37] extended the fuzzy relational constraints with a max-product composition to the situation with max-strict t-norm composition. They proved that the necessary condition for an optimal solution in terms of the maximum solution also can be applied to the situation of max-strict t-norm composition. Peeva [28] found all the minimal solutions and compared their corresponding objective function values in order to obtain optimal solutions. Wang [33] investigated multiple linear objective functions subject to max–min fuzzy relational equations. Lu et al. [24] proposed a genetic algorithm for the problems where the objective function is a single nonlinear objective function and the constraints are max–min fuzzy relational inequalities. Loetamonphong et al. [23] presented the nonlinear multi-objective optimization problem with a fuzzy relational equation constraint. Ghodousian and Khorram [8] focused on subset of these problems where the solutions are fuzzy relational equations with max-prod composition and the objective function is linear. Khorram and Ghodousian [15] presented an optimization model with a linear objective function subject to a system of the fuzzy relational equations with max-av composition. Ghodousian and Khorram [9] investigated linear programming problem with the convex combination of the max–min and the max-average fuzzy relational equations. Li and Fang [20] considered the problem of minimizing a linear fractional function subject to a system of sup-T equations, where T means a continuous Archimedean triangular norm. Li and Fang [21] considered the detailed analysis of the resolution and optimization of a system of sup-T equations. Shieh [31] examined the feasibility of minimizing a linear objective function subject to a max-t fuzzy relation equation constraint, where t is a continuous/Archimedean t-norm. Ghodousian and Khorram [10] discussed linear optimization with an arbitrary fuzzy relational inequality. They proved that its optimal solution can be obtained if the problem is defined by a non-decreasing or non-increasing function. Freson et al. [7] considered a generalization of the linear optimization problem with fuzzy relational (in)equality constraints by allowing for bipolar max–min constraints, i.e. constraints in which not only the independent variables but also their negations occur. Guo et al. [11], Chang and Shieh [2] also proposed some rules to reduce linear optimization subject to the fuzzy (in)equality constraints. (c ∧ Yang and Cao [39], as well as Wu [36] considered the problem where the objective function in (1) becomes Z = ∨m i=1 i r

xi i ) and the constraint parts are fuzzy relational inequalities with max–min composition. Where the objective function in (1) r

(c · xi i ) and the constraint parts are fuzzy relational inequalities with max-product composition, the results becomes Z = ∨m i=1 i by Zhou and Ahat [41] inspired the current paper. Yang and Cao [40] discussed the following monomial geometric programming with fuzzy relation equation constraints: min

z=c

n 

r

x jj

j=1

s.t. A ◦ x = b. Shivanian and Khorram [32] proposed monomial geometric programming subject to fuzzy relation inequalities with maxproduct composition. We aim to establish an optimization management model in a BitTorrent-like Peer-to-Peer (P2P) resource sharing system. Suppose there exist n terminals in the system, denoted by Aj , j ∈ J = {1, 2, . . . , n}. According to the BitTorrent-like P2P transmission protocol, every terminal shares an owner resource to the other ones and meanwhile it can download resource from any other terminals. Let the quality level when the jth terminal sends the resource data to the other terminals be xj , j ∈ J. The bandwidth between Ai and Aj is aij . Due to the bandwidth limitation, the network traffic that Ai receives from Aj is actually aij ∧xj , where i ∈ I, j ∈ J and i = j. When i = j, we always assume that aii = 0 since Ai does not need to download resource from itself. Furthermore,

X.-G. Zhou et al. / Information Sciences 328 (2016) 15–25

17

the highest quality of the network traffic Ai obtained from other terminals is

ai1 ∧ x1 ∨ ai2 ∧ x2 ∨ . . . ∨ ain ∧ xn . Suppose that the highest quality requirement of download traffic of Ai is bi (bi ≥ 0) where i ∈ I⊆J. Without loss of generality, we assume that I = {1, 2, . . . , m} at m ≤ n. Consequently, the BitTorrent-like Peer-to-Peer (P2P) system can be reduced into the following system of max–min fuzzy relation equations:

⎧ ⎪ ⎨a11 ∧ x1 ∨ a12 ∧ x2 ∨ . . . ∨ a1n ∧ xn = b1 ,

a21 ∧ x1 ∨ a22 ∧ x2 ∨ . . . ∨ a2n ∧ xn = b2 , . ⎪ ⎩ ...................................... am1 ∧ x1 ∨ am2 ∧ x2 ∨ . . . ∨ amn ∧ xn = bm . Next, we consider the dissatisfaction degree of the terminals. Assume that there exist p factors which would influence the disγk  satisfaction degree. Moreover, the dissatisfaction degree of the terminals caused by the kth factor is ck nj=1 x j j where ck is the γk  p weight coefficient. Hence, the total dissatisfaction degree of the terminals is z(x) = k=1 ck nj=1 x j j . In general, an optimization management objective in the BitTorrent-like Peer-to-Peer (P2P) system is to minimize the total dissatisfaction degree of the terminals. Additionally, in order to distinguish the positive and negative of the exponent γk j , we divide the terminals into two classes. One class is called the server and the other one is called the users. The server terminals have significantly more resources than the users. The major task of the server terminals is to send out resource data to the others, while that of the users is to receive resource data. Based on the discussion above, we establish optimization model (1). If γk j > 0 for any k ∈ K and j ∈ J, we

believe that the optimal solution indicates the dissatisfaction degree of the terminals in the worst case. If γk j < 0 for any k ∈

K and j ∈ J, we believe that the optimal solution illustrates dissatisfaction degree of the terminals in the best case. When the optimal solution is closer to 0, It implies that there is a higher degree of dissatisfaction. In this paper, we develop an algorithm to deal with the PGPF based on a special structure of a solution set to the fuzzy relation equations. The material is arranged as follows. In Section 2, we summarize some concepts and important properties related to max–min fuzzy relational equations. In Section 3, we present some main conclusions for developing the algorithm in order to solve the PGPF. We introduce three rules to simplify the PGPF. We give a step-by-step algorithm to the PGPF in Section 4. In Section 5, we use some example to illustrate how the algorithm works. Section 6 consists of our concluding remarks.

2. Preliminaries In this section, we recall some basic concepts and important properties associated with fuzzy relational equations with max– min composition. However, all proofs are omitted to keep the paper succinct and readable. The readers may refer to [1,5,7,11,16,34] for a complete overview of the following max–min fuzzy relational equations:

A ◦ x = b,

(2)

xi ∈ [0, 1], i ∈ I.

For x, y ∈ Rn , we write x ≤ y, if xj ≤ yj holds for all j ∈ {1, 2, . . . , n}, and x < y, if x ≤ y and x = y. It is easy to verify that ≤ is a partial order relation over Rn . We denote the solution set of (1) by X ∗ = {x ∈ [0, 1]n |A ◦ x = b}. The system (2) is said to be consistent if and only if X∗ = ∅. A solution xˆ ∈ X ∗ is said to be a maximum solution if and only if x  xˆ for all x ∈ X∗ . A solution xˇ ∈ X ∗ is said to be a minimal solution if and only if x  xˇ implies x = xˇ for any x ∈ X∗ . Theorem 1 ([34]). The system (2) is consistent if and only if xˆ = (xˆ1 , xˆ2 , . . . , xˆn )T is the solution of (2), where xˆ j is defined as:



xˆ j =

min{bi |ai j > bi } if {bi |ai j > bi } = ∅, 1

otherwise.

(3)

Moreover, xˆ is the maximum solution. If (2) is consistent, the maximum solution exists. Furthermore, it is unique. But, unlike the uniqueness of the maximum ˇ solution, there may be many minimal solutions of (2). We denote the set of all minimal solutions by X. Theorem 2 ([34]). If the system (2) is consistent, then solution set X∗ is fully characterized by one maximum solution and a finite number of minimal solutions, i.e.

X∗ =



{x ∈ [0, 1]n |xˇ  x  xˆ}.

ˇ Xˇ x∈

From this theorem we know that the solution set of X∗ , when it is non-empty, is a non-convex set in general. So the traditional convex programming method is useless. Next we will present the conservative path method for obtaining all the minimal solutions of (2). Readers may refer to [11,34] and references cited therein.

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X.-G. Zhou et al. / Information Sciences 328 (2016) 15–25

For any i ∈ I, define index sets Ji = { j ∈ J| min{(bi j , xˆ j } = bi } and  = J1 × J2 × . . . × Jm . A vector q = (q1 , q2 , . . . , qm ) ∈  if and only if qi ∈ Ji , ∀i ∈ I. For any q ∈ , define

Iqj = {i ∈ I | qi = j}, j ∈ J.

(4)

and F :  −→ Rn such that,



Fj (q) =

max bi

if Iqj = ∅,

0

if Iqj = ∅,

i∈Jqj

∀ j ∈ J.

(5)

Obviously, a vector q ∈  is called a conservative path or C-path [14,34] of (2). Let the set of all the C-paths of (2) be CP. Definition 1 ([11,34]). A vector q = (q1 , . . . , qm ) is called a fuzzy relational equalities(FRE) path of (2) if it satisfies that

⎧ = 1, ⎨∈ J1 , k qi ∈ Ji , Ji {q1 , q2 , . . . , qi−1 } = ∅; i ∈ I, ⎩ = 0,

otherwise.

Denote the set of all FRE paths of (2) by FREP. Definition 2 ([11]). Let q ∈ CP. A solution xq = (x1 , x2 , . . . , xn ) is called a quasi-minimal solution corresponding to C-path q, q where, for any j ∈ J, x j = Fj (q). q is called an corresponding C-path of xq . q

q

q

x, q ∈ CP } = {x ∈ X | xq  x   x, q ∈ F REP}. Theorem 3 ([11]). If X∗ = ∅, then X ∗ = {x ∈ X | xq  x   Theorem 4 ([16]). Let X∗ = ∅. The next results hold: (1) If p ∈ , then F(p) ∈ X∗ . (2) For any x ∈ X∗ , there exists p ∈  such that F(p) ≤ x. Theorem 5 ([11]). Suppose xˇ is a minimal solution of system (2). There must exist a FRE path q ∈ FREP such that xˇ = xq , where q x j = Fj (q) for each j ∈ J. Furthermore, if system (2) satisfies d1 > d2 >  > dm , then for any given FRE path q, xq is a minimal solution of Eq. (2). It follows from Theorem 5 that we can obtain all minimal solutions of system (2) by only using the FRE paths. 3. Main results In this paper, we always assume that 0−1 = +∞. In the following, we firstly convert the PGPF into an equivalent programming problem showing that the optimal solution consists of the maximum feasible solution and one of the minimal feasible solution by introducing auxiliary variables. Then, we introduce some rules for simplifying the problem. Suppose the solution set of A ◦ x = b is

X (A, b) = {xˇ1  x  xˆ} ∪ {xˇ2  x  xˆ} ∪ · · · ∪ {xˇl  x  xˆ}, where x = (x1 , x2 , . . . , xn )T ,xˇt = (xˇt1 , xˇt2 , . . . , xˇtn )T ,xˆ = (xˆ1 , xˆ2 , . . . , xˆn )T ,t = 1, 2, . . . , l, and xˆ is the maximum solution, while xˇ1 , xˇ2 , . . . , xˇl are all of the minimal solutions of A ◦ x = b. Then the model (PGPF) can be written as:

(PGPF1) min z(x) =

p  k=1

ck

n  γk xj j j=1

(6)

s.t.x ∈ X (A, b). Proposition 1. Let K j = {k ∈ K |γk j = 0} ⊆ K, j = 1, 2, . . . , n. The following statements hold. (i) If K j = ∅, then γ1 j = γ2 j = . . . = γ p j = 0. γk1

γk2

j

j

(ii) If Kj = ∅, then ∀k1j , k2j ∈ K j , |γ j | = |γ j | = 1 or − 1. k1 k2 Proof. The proof of this proposition follows easily for given j ∈ J, γk j (k ∈ K ) which are either all non-positive real numbers or all non-negative real numbers.  γk

γk

Let J = { j ∈ J| |γ j | = 1, ∀k j ∈ K j }, J = { j ∈ J|K j = ∅ or |γ j | = −1, ∀k j ∈ K j }. Then J = J ∪ J . k k j

j

Now, we take some variable substitutions y = (y1 , y2 , . . . , yn )T where



yj =

xj x−1 j

if j ∈ J , if j ∈ J ,

for any j = 1, 2, . . . , n, k = 1, 2, . . . , p. Let γk = |γk j | for all j = 1, 2, . . . , n, k = 1, 2, . . . , p. j

X.-G. Zhou et al. / Information Sciences 328 (2016) 15–25

19

We construct the set Y(A, b) based on X(A, b) as follows. For any t ∈ {1, 2, . . . , l }, let



yˇtj

=

yˆtj =

xˇtj

if j ∈ J ,

xˆ j

if j ∈ J ,

(xˆ j )−1 if j ∈ J .

 (xˇtj )−1 if j ∈ J .

for each j = 1, 2, . . . , n. Let yˇt = (yˇt1 , yˇt2 , . . . , yˇtn )T , yˆt = (yˆt1 , yˆt2 , . . . , yˆtn )T . The set Y(A, b) is defined by

Y (A, b) = {yˇ1  y  yˆ1 } ∪ {yˇ2  y  yˆ2 } ∪ . . . ∪ {yˇl  y  yˆl }. Now, we define the following problem (PGPF2):

(PGPF2) min z (y) =

p 

ck

n  γk yj j j=1

k=1

s.t.y ∈ Y (A, b). Theorem 6. The problem (PGPF1) is equivalent to the problem (PGPF2) in the following sense: if y∗ = (y∗1 , y∗2 , . . . , y∗n )T is the optimal solution of PGPF2. Assume that



x∗j

=

y∗j

( )

y∗j −1

if j ∈ J , if j ∈ J ,

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

(7)

then x∗ = (x∗1 , x∗2 , . . . , x∗n )T is the optimal solution of PGPF1. Conversely, if x∗ = (x∗1 , x∗2 , . . . , x∗n )T is the optimal solution of PGPF1 and



y∗j

=

x∗j

( )

x∗j −1

if j ∈ J , if j ∈ J ,

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

(8)

then y∗ = (y∗1 , y∗2 , . . . , y∗n )T is the optimal solution of PGPF2. Proof. Let y∗ = (y∗1 , y∗2 , . . . , y∗n )T be an optimal solution of PGPF2 and



x∗j

=

y∗j

if j ∈ J ,

(y∗j )−1 if j ∈ J ,

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

Then y∗ ∈ Y(A, b) and there exists some t satisfying yˇt  y∗  yˆt , that is, for any j ∈ J, yˇtj  y∗j  yˆtj . If j ∈ J , then y∗j = x∗j , yˇtj = xˇtj , yˆtj = xˆ j . So we can get xˇtj ≤ x∗j ≤ xˆ j . Otherwise, j ∈ J , y∗j = (x∗j )−1 , yˇtj = (xˆ j )−1 , yˆtj = (xˇtj )−1 . It follows that xˇtj ≤ x∗j ≤ xˆ j . So. x∗ ∈ X(A, b)

and z(x∗ ) = z (y∗ ). Assume that x1 ∈ X(A, b) is the optimal solution of PGPF1 and z(x∗ ) > z(x1 ). It follows from the definition of y that there exists y1 such that



y1j

=

x1j

if j ∈ J ,

(x1j )−1 if j ∈ J ,

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

and z(x1 ) = z (y1 ). We know that y∗ is an optimal solution of PGPF2, which implies that z (y∗ ) ≤ z (y1 ). It follows that z (y∗ ) = z(x∗ ) > z(x1 ) = z (y1 )  z (y∗ ). This results in a contradiction. It implies that x∗ is the optimal solution of PGPF1. To show the converse statement, let x∗ ∈ X(A, b) be an optimal solution for problem (PGPF1) and



y∗j

=

x∗j

if j ∈ J ,

(x∗j )−1 if j ∈ J ,

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

It is obvious that y∗ ∈ Y(A, b) and z(x∗ ) = z (y∗ ). We suppose that y1 ∈ Y(A, b) is the optimal solution of PGPF2 and z (y∗ ) > z (y1 ).e Then, there exists some x1 ∈ X(A, b) satisfying z(x1 ) = z(y1 ) and



x1j =

y1j

( )

y1j −1

if j ∈ J , if j ∈ J ,

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

This implies that z(x∗ ) = z (y∗ ) > z (y1 ) = z(x1 )  z(x∗ ). It follows that y∗ is the optimal solution of PGPF2.



γ

 p k Lemma 1. z (y) = k=1 ck nj=1 y j j is a non-decreasing function with variable yj , j ∈ J. Proof. The proof of this proposition follows easily from the facts that c1 , c2 , . . . , c p are positive and, yi ∈ [0, 1] and γk = |γk j | are non-negative for any k, j; therefore, it is omitted.



j

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X.-G. Zhou et al. / Information Sciences 328 (2016) 15–25

Theorem 7. Suppose z∗ = min{z (yˇ1 ), z (yˇ2 ), . . . , z (yˇq )}. Then the optimal solution set of (PGPF2) is

Y ∗ = {yˇt |z (yˇt ) = z∗ , 1 ≤ t ≤ q}, and the optimal objective function value is z ∗ . Proof. It is clear that Y∗ ⊆Y(A, b) and z (y∗ ) = z∗ , ∀y∗ ∈ Y∗ . Next we need to prove z ∗ ≤ z (y), ∀y ∈ Y(A, b). Let y ∈ Y(A, b). According to the special structure of Y(A, b), we can find at least one t ∈ {1, 2, . . . , l } such that yˇt ≤ y ≤ yˆt . Since

z (y) =

p 

n  γk yj j

ck

j=1

k=1

is a non-decreasing function with variable yj , j ∈ J, so we get

z∗ = min{z (yˇ1 ), z (yˇ2 ), . . . , z (yˇq )} ≤ z (yˇt ) ≤ z (y).



Let y∗ = (y∗1 , y∗2 , . . . , y∗n )T be an optimal solution of PGPF2. It follows from Theorem 7 that there exists some t satisfying y∗ = yˇt , that is, for any j ∈ J, y∗j = yˇtj . If j ∈ J , y∗j = xˇtj , otherwise, y∗j = xˆ j . Then, we can easily obtain the following results. Lemma 2. If γk j > 0 for any k ∈ K and j ∈ J, that is, J = ∅, one of the minimum solutions xˇt is an optimal solution of problem (1). Lemma 3. If γk j < 0 for any k ∈ K and j ∈ J, that is, J = ∅, the optimal solution of problem (1) is xˆt . On the basis of the discussion above, we present some rules for reducing the size of problem (1). Rule 1. Define I0 := {i ∈ I|∃ ji ∈ J suchthat min{ai j1 , xˆ ji } = bi }. For any j ∈ J , set x j = xˆ j and delete column j of A; for any i ∈ I0 , delete the ith constraint. Then, problem (1) will be reduced to the following problem:

min z(x) =

p 

ck

 γk j  γk j xj xˆ j

k=1

j∈J

j∈J

s.t.A ◦ x = b ,

(9)

x j ∈ [0, 1], j ∈ J. where A and b are the updated fuzzy relational matrix and fuzzy vector, respectively. It follows from Lemma 2 that one of the minimum solutions xˇt is an optimal solution of problem (9). Theorem 8. Assume that xˇt is an optimal solution (not necessary unique) of problem (9). Then the optimal solution x∗ of problem (1) is defined as follows:



x∗j =

xˇtj xˆ j

if j ∈ J , if j ∈ J ,

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

In order to obtain the optimal solution x∗ of (1), it follows from Lemma 2 and Theorem 8 that we only need to solve problem (9). Now, we will mainly discuss how to solve problem (9). Rule 2 ([33,34]). If i ≥ k and Ji ⊇Jk , then deleting Ji from  does not effect on the set of all the minimal feasible solutions Xˇ of problem (9). Theorem 9. If there exists some j1 ∈ J satisfying γk j > 0 for all k and j1 ∈ Ji for all i ∈ I, then, the optimal value of problem (1) is zero 1

and the optimal solution x∗ satisfying x∗j = 0. 1

Proof. Assume that x∗ is an optimal solution of problem (1). We see that there exists some t such that x∗i = xˇti . q For any q ∈ , we can get qi = j1 for all i ∈ I, since j1 ∈ Ji for all i ∈ I. This implies that x j = 0. It follows from Theorem 3 that i

xˇtj = 0 for all t = 1, 2, . . . , l. So, x∗j = 0 and 1

1

z(x∗ ) =

p  k=1

ck

n 

γk j

(x∗j )γk j (x∗j1 )

1

= 0.

j=1, j= j1

 Theorem 10. If problem (1) satisfies the following conditions: (1) K = K  ∪ K  , K  ∩ K  = ∅, K  , K  ⊆ K; (2) K  := {k ∈ K |γk j = 0, ∀ j ∈ J };

ˆ / Ji , ∀i ∈ I}; then, the optimal solution to this problem is x. (3) K  := {k ∈ K |∃ j1 ∈ J , γk j = 0 and j1 ∈ 1

X.-G. Zhou et al. / Information Sciences 328 (2016) 15–25

21

Proof. From the proof of Theorem 9, we see that the optimal solution x∗ satisfying x∗j = 0. It follows that, for any k ∈ K , 1

ck

n 

(x∗j )γk j = 0.

j=1

Then, the problem (1) can be simplified to

min

 k∈K 

ck

 γk j xj j∈J

s.t.A ◦ x = b, x j ∈ [0, 1], j ∈ J. It follows from Lemma 3 that xˆ is the optimal solution to problem (1).  Rule 3. Define J0 := {j ∈ J |j ∈ Ji , ∀i ∈ I}. For all j ∈ J0 , set K 0j := {k ∈ K |γk j > 0}. If k ∈ K 0j , then there exists some j1 ∈ J such that 1

γk j > 0 and j1 ∈ Ji for all i ∈ I. So, for each t = 1, 2, . . . , l, xˇtj = 0 and, for any feasible solution x ∈ X(A, b), 1

1

ck

n 

(x j )γk j  0 = ck

j=1

n 

(x∗j )γk j ,

j=1

since xj ≥ 0 for all j ∈ J. Therefore, if k ∈ K0 , it has no effect on the optimal value of the problem removing the kth factor   γk γk ck nj=1 (x j ) j from objective function. Then, if j1 ∈ J0 and k ∈ K 0j , set x j1 = 0 and eliminate the kth factor ck nj=1 (x j ) j of 1

z(x).

4. An algorithm and its computational complexity Based on the theorems presented, we first develop the following algorithm to solve the model (PGPF). Then, we discuss the complexity of the proposed method. Step 1. Solve the maximum feasible solution xˆ of problem (1). If A ◦ xˆ = b, then goto Step 2. Otherwise, problem (1) is not feasible, stop. Step 2. If j ∈ J then set x j = xˆ j and convert problem (1) into problem (9) by Rule 1. Step 3. For all i ∈ I\I0 , calculate index sets Ji by (4). Step 4. Reduce the problem (9) by Rule 2. γk  Step 5. If j1 ∈ J0 and k ∈ K 0j , set x j1 = 0 and remove ck nj=1 (x) j j from z(x) by Rule 3. 1

Step 6. Generate the solution tree by FRE path and calculate an optimal solution of problem (9) by the branch and bound method. Then, compute the optimal solution xˇ∗ and the optimal value z(x∗ ) of problem (1). The computational complexity of the proposed algorithm is proposed as follows. Notation: m: row number of A; n: column number of A or component number of x; p: element number of K; |J |: element number of J ; |J |: element number of J ; |J0 |: element number of J0 ; |Ji |: element number of Ji ;  0 k: element number of Kj ; j∈J0

|I0 |: element number of I0 ; Computational complexity of reducing rules: Rule 1: |J | × m + p × |J |2 times; Rule 2: 12 m(m − 1)n times; Rule 3: |J |(m − |I0 |) + pk times; Finding the maximum solution costs m2 × n operations. Judging the feasibility of problem (1) costs m × (n + 1) operations. Computing index sets Ji (i ∈ I\I0 ) costs 2(n − |J |) × (m − |I0 |) operations. Obtaining all minimal solutions by by branch and bound  method and FRE-path costs |Ji | + |Ji |( p − k) including comparison, addition and multiplication. Therefore, we obtain the i∈I\I0

i∈I\I0

22

X.-G. Zhou et al. / Information Sciences 328 (2016) 15–25

following computational complexity of the presented algorithm.

m(mn + n + 1) + 2(n − |J |) × (m − |I0 |) +

 i∈I\I0

+|J | × m + p × |J |2 +

|Ji | +



|Ji |( p − k)

i∈I\I0

1 m(m − 1)n + |J |(m − |I0 |) + pk. 2

5. Numerical examples In this section, we provide an numerical example to illustrate the proposed method. Example 1. Consider the following problem:

min

−1 3 −1.5 −1.5 2.5 −2 −4 −1 −3 3 −2 −1 z(x) = 3x−2 + x−1 x4 x5 + 2x−3 1 x3 x4 x6 1 x2 2 x3 x5 x6 + 5x4 x5 x6

A ◦ x = b,

s.t.

(10)

x ∈ [0, 1]6 where

⎡0.4 0.7 1 0.7 0.8 0.9⎤ ⎢0.5 0.8 0.7 0.3 0.6 0.6⎥ A = ⎢0.4 0.6 0.3 0.5 0.2 0.9⎥, ⎣ ⎦ 0.2 0.7 0.2 0.1 0.2 0.4 0.1 0.2 0.1 0.1 0.2 0.1

b = [0.7, 0.6, 0.5, 0.2, 0.2]T . Step 1. The maximum feasible solution xˆ of problem (10) is xˆ = (1, 0.2, 0.6, 1, 0.7, 0.2)T . And A ◦ xˆ = b, then goto Step 2. Step 2. It is obvious that J = {4} J = {1, 2, 3, 5, 6}. So, set x1 = 1, x2 = 0.2, x3 = 0.6, x5 = 0.7, x6 = 0.2 and delete columns 1, 2, 3, 5, 6 of A. Since I0 = {1, 2, 4, 5}, we remove constraints 1, 2, 4, 5 of problem (10). Then, problem (10) is converted into the following problem:

min

−3 z(x) = 3 · 0.6−1 · 0.2−1.5 x34 + 0.2−1.5 · 0.7−2 x2.5 · 0.6−4 · 0.7−1 · 0.2−3 4 + 2 · 0.2

+ 5 · 0.7−2 · 0.2−1 · x34 = 106.9221x34 + 22.817x2.5 4 + 344470 s.t.

(11)

0.5 ∧ x4 = 0.5, x4 ∈ [0, 1]

Step 3. By virtue of (11), we calculate index set J3 = {4}. Step 4. It is obvious that the optimal solution of (11) is xˇ14 = 0.5 and the optimal solution of (10) is

x∗ = (1, 0.2, 0.6, 0.5, 0.7, 0.2)T . Example 2. Consider the following problem:

min

0.6 0.3 0.8 −0.8 −0.2 0.5 0.3 z(x) = 0.3x0.6 x5 x8 x9 + 0.2x2−0.3 x30.4 x−0.1 x70.3 x90.2 + 0.3x−0.4 x−0.2 x0.3 1 x3 x4 x6 + 0.2x2 6 5 2 5

s.t.

A ◦ x = b,

(12)

9

x ∈ [0, 1] , where

⎡

1 0.8 0.4 ⎢ 0.9 0.8 0.4 ⎢ 0.8 0.95 0.1 ⎢ ⎢ 0.8 0.8 0.5 ⎢ 0.6 0.7 A = ⎢ 0.1 ⎢ 0.6 0.5 0.6 ⎢ ⎢ 0.4 0.5 0.2 ⎣ 0.4 0.1 0.15 0.25 0.12 0.2

⎤

1 1 0.5 1 0.6 0.9 0.9 0.4 0.98 0.75 0.9 0.95⎥ 0.3 0.9 0.9 0.8 0.5 0.9 ⎥ ⎥ 0.8 0.2 0.8 0.8 0.85 0.7 ⎥ ⎥ 0.5 0.6 0.7 0.2 0.7 0.65⎥, 0.2 0.6 ⎥ 0.6 0.6 0.2 0.6 ⎥ 0.5 0.5 0.4 0.5 0.45 0.5 ⎥ ⎦ 0.2 0.14 0.4 0.25 0.4 0.35 0.15 0.1 0.2 0.15 0.2 0.15

b = [1, 0.95, 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.2]T . Step 1. The maximum feasible solution xˆ of problem (12) is xˆ = (0.2, 0.9, 1, 1, 1, 0.95, 1, 0.8, 1)T . And A ◦ xˆ = b, then goto Step 2.

X.-G. Zhou et al. / Information Sciences 328 (2016) 15–25

0 x9 = 0.95

x6 = 0.95 2

0 1 x8 = 0.7 x3 = 0.7 x6 = 0.7 0 3 x6 = 0.4 6

23

0.3062/stop

5

4

0.2794/stop 0.1793/stop

x8 = 0.4 7

0.2362/stop 0.1355/stop Fig. 1. Solution tree generated by the algorithm proposed in the paper.

Step 2. Obviously J = {1, 3, 4, 6, 7, 8, 9}, J = {2, 5}. Therefore, we can set x2 = 0.9, x5 = 1 and delete columns 2, 5 of A. Since I0 = {1, 3, 4, 6, 7}, we eliminate constraints 1,3,4,6,7 of problem (12). Then, problem (12) is converted into the following one:

min

0.8 −0.8 −0.2 0.5 0.3 −0.1 0.3 0.2 z(x) = 0.3x10.6 x30.6 x0.3 1 x8 x9 + 0.2 · 0.9−0.3 x0.4 x7 x9 4 x6 + 0.2 · 0.9 3 1

+ 0.3 · 0.9−0.4 1−0.2 x0.3 6 0.6 0.3 0.8 0.5 0.3 0.4 0.3 0.2 0.3 = 0.3x0.6 1 x3 x4 x6 + 0.2176x8 x9 + 0.2064x3 x7 x9 + 0.3109x6

(13)

A ◦ x = b , x ∈ [0, 1]7 ,

s.t. where

⎡

⎤

0.9 0.4 0.9 0.98 0.75 0.9 0.95 ⎢ 0.1 0.7 0.5 0.7 0.2 0.7 0.65⎥  A =⎣ , 0.4 0.15 0.2 0.4 0.25 0.4 0.35⎦ 0.25 0.2 0.15 0.2 0.15 0.2 0.15 b = (0.95, 0.7, 0.4, 0.2)T , x = (x1 , x3 , x4 , x6 , x7 , x8 , x9 )T . Step 3. By virtue of (4), we calculate index sets Ji (i = 2, 5, 8, 9) as follows:

J2 = {6, 9}, J5 = {3, 6, 8}, J8 = {6, 8}, J9 = {1, 3, 6, 8}. Step 4. Since J9 ⊇ J5 and J9 ⊇ J8 , then delete J9 by Rule 2. Therefore,  = J2 × J5 × J8 . Step 5. Since J0 = {1, 4, 7}, K10 = {1}, K40 = {1} and K70 = {3}, then, we can set x1 = 0, x4 = 0, x7 = 0 and remove the first and third factors of z(x). So, the objective function z(x) is reduced into z(x) = 0.2176x80.5 x90.3 + 0.3109x60.3 . Step 6. We now generate the solution tree by the branch and bound method for obtaining the optimal solution(see Fig. 1. for the details). Then, the optimal solution of (13) is xˇ1 = (0, 0, 0, 0.95, 0, 0, 0)T and the optimal solution of (12) is x∗ = (0, 0.9, 0, 0, 1, 0, 0, 0.4, 0.95)T . The optimal value is z(x∗ ) = 0.1355. Example 3. Consider the following problem:

min

0.3 0.8 0.8 0.2 0.5 0.3 0.8 0.2 0.5 0.5 0.2 0.4 0.2 0.3 0.3 0.4 z(x) = 0.4x10.6 x0.7 3 x4 x6 + 0.5x2 x5 x8 x9 + 0.8x1 x3 x6 + 0.4x8 x9 + 0.9x2 x5 x6 + 0.4x7 x9

s.t.

A ◦ x = b, x ∈ [0, 1]9 ,

(14)

24

X.-G. Zhou et al. / Information Sciences 328 (2016) 15–25

0 1 x9 = 0.72 x7 = 0.56 x8 = 0.56 3

0.2947 2

x1 = 0.45

0.3058/stop

x3 = 0.45

0.2947 4

0.2947 5

6

7

0.2947/stop 0.2947/stop Fig. 2. Solution tree generated by the algorithm proposed in the paper.

where

⎡

0.65 0.92 0.72 0.61 0.53 ⎢0.75 0.9 0.76 0.32 0.95 ⎢0.82 0.61 0.67 0.65 0.8 ⎢ ⎢0.43 0.56 0.56 0.57 0.81 ⎢ A = ⎢0.23 0.56 0.71 0.62 0.8 ⎢ 0.7 0.72 0.45 0.54 0.7 ⎢ ⎢0.35 0.68 0.43 0.7 0.40 ⎣ 0.45 0.46 0.48 0.42 0.38 0.42 0.43 0.40 0.20 0.42

⎤

0.78 0.82 0.62 0.73 0.61 0.49 0.64 0.7 ⎥ 0.63 0.54 0.76 0.64⎥ ⎥ 0.59 0.8 0.56 0.47⎥ ⎥ 0.93 0.55 0.55 0.38⎥, 0.9 0.34 0.52 0.52⎥ ⎥ 0.55 0.45 0.25 0.48⎥ ⎦ 0.45 0.43 0.32 0.22 0.8 0.33 0.42 0.26

b = [0.72, 0.70, 0.64, 0.56, 0.55, 0.52, 0.48, 0.45, 0.42]T . Step 1. The maximum feasible solution xˆ of problem (14) is xˆ = (0.52, 0.42, 0.45, 0.48, 0.52, 0.42, 0.56, 0.64, 0.72)T . And A ◦ xˆ = b, then goto Step 2. Step 2. It is obvious that J = {1, 2, 3, 4, 5, 6, 7, 8, 9}, J = ∅. Then, Rule 1 cannot be applied. Step 3. By virtue of (4), we calculate index sets Ji as follows:

J1 = {9}, J2 = {9}, J3 = {8, 9}, J4 = {7, 8}, J5 = {7, 8}, J6 = {1, 5, 8, 9}, J7 = {4, 9}, J8 = {1, 3}, J9 = {1, 2, 6, 8}. Step 4. Since J2 ⊇ J1 , J3 ⊇ J1 , J6 ⊇ J1 , J7 ⊇ J1 and J5 ⊇ J4 , then delete J2 , J3 , J5 , J6 , J7 by Rule 2. Therefore,  = J1 × J4 × J8 × J9 = {9} × {7, 8} × {1, 3} × {1, 2, 6, 8}. Step 5. Since J0 = {4, 5}, K40 = {1} and K70 = {2, 5}, then, we can set x4 = 0, x5 = 0 and remove the first, second and fifth factors of z(x). So, the objective function z(x) is reduced into z(x) = 0.8x10.8 x30.2 x60.5 + 0.4x80.5 x90.2 + 0.4x70.3 x90.4 . Step 6. We now generate the solution tree by the branch and bound method for obtaining the optimal solution(see Fig. 2. for the details). Then, the optimal solutions of (14) are x∗1 = (0.45, 0, 0, 0, 0, 0, 0, 0.56, 0.72)T and x∗2 = (0, 0, 0.45, 0, 0, 0, 0, 0.56, 0.72)T the optimal value is z(x∗ ) = 0.2947. 6. Conclusions In this paper, we have proposed a posynomial geometric programming problem subject to max–min fuzzy relation equations. By introducing auxiliary variables, we converted the PGPF into an equivalent programming problem where the objective function is a non-decreasing function with the auxiliary variable. Additionally, we showed that an optimal solution consists of the maximum feasible solution and one of the minimal feasible solution by the equivalent programming problem. Then we introduce some rules for simplifying the problem. By using the branch and bound method and fuzzy relational equalities(FRE) paths, we presented an algorithm to obtain an optimal solution to the PGPF. It remains a challenging problem in how to extend the work to

X.-G. Zhou et al. / Information Sciences 328 (2016) 15–25

25

handle posynomial objective functions and how to solve the fuzzy relation geometric programming problem under more general circumstances. When γk j ∈ R for any k ∈ K and j ∈ J, the proposed method does not produce a solution to the PGPF. Finally, we also discussed how to solve multi-objective posynomial geometric programming problem subject to fuzzy relation equations. Acknowledgments The author is very grateful to the anonymous referees for their comments and suggestions, which have been very helpful in improving the paper. Thanks to the support by the PhD Start-up Fund of Natural Science Foundation of Guangdong Province, China (no. S2013040012506), the China Postdoctoral Science Foundation funded project (2014M562152), the innovation capability of independent innovation to enhance the class of building strong school projects of colleges of Guangdong Province (20140207) and the GuangZhou Postdoctoral Science Foundation funded project (gzhubsh2013006). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41]

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