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A new approximate algorithm for solving multiple objective linear programming problems with fuzzy parameters
Applied Mathematics and Computation 174 (2006) 524–544 www.elsevier.com/locate/amc
A new approximate algorithm for solving multiple objective linear programming problems with fuzzy parameters Fengjie Wu *, Jie Lu, Guangquan Zhang Faculty of Information Technology, University of Technology, Sydney, P.O. Box 123, Broadway, NSW 2007, Australia
Abstract Many business decision problems involve multiple objectives and can thus be described by multiple objective linear programming (MOLP) models. When a MOLP problem is being formulated, the parameters of objective functions and constraints are normally assigned by experts. In most real situations, the possible values of these parameters are imprecisely or ambiguously known to the experts. Therefore, it would be more appropriate for these parameters to be represented as fuzzy numerical data that can be represented by fuzzy numbers. In this paper, a new approximate algorithm is developed for solving fuzzy multiple objective linear programming (FMOLP) problems involving fuzzy parameters in any form of membership functions in both objective functions and constraints. A detailed description and analysis of the algorithm are supplied. In addition, an example is given to illustrate the approximate algorithm. 2005 Elsevier Inc. All rights reserved. Keywords: Optimization; Decision support technology; Fuzzy multiple objective linear programming; Approximate algorithm
*
Corresponding author. E-mail address: [email protected] (F. Wu).
0096-3003/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.04.106
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1. Introduction Decision-making is a process for selecting a possible course of actions from all the available alternatives. In almost all such selecting problems, the decision maker (DM) wants to attain more than one objective or goal in selecting the course of actions while satisfying the constraints dictated by environment, processes, and resources. The main feature of these problems is that the objectives are apparently noncommensurable, and conflict with each other [3]. Multiple objective linear programming (MOLP) is one of the popular methods to deal with such complex and ill-structured decision problems. When formulating an MOLP problem, various factors of the real world system should be reflected in the description of the objective functions and the constraints. Naturally, these objective functions and constraints involve many parameters which possible values may be assigned by the experts. Normally, such parameters are set at some values in an experimental or subjective manner through the expertsÕ understanding of the nature for the parameters [9,10]. With this observation, it is natural to recognize that the possible values of these parameters are often imprecisely or ambiguously known to the experts. In this case, it may be more appropriate to interpret the expertsÕ understanding of the parameters as fuzzy numerical data that can be represented by fuzzy numbers. The fuzzy multiple objective linear programming (FMOLP) problems involving fuzzy parameters would be viewed as a more realistic version than the conventional one [9,10]. Various kinds of FMOLP models have been proposed to deal with different decision-making situations that involve fuzzy values in objective function parameters, constraints parameters, or goals. Transforming a FMOLP problem into a crisp programming one is still employed by researchers [4]. Tanaka and Asai [11] formulated FMOLP with triangular fuzzy numbers, and the nonlinear programming problem obtained was solved by using mix–min operator. Lai and Hwang [5] resolved imprecise objectives with triangle fuzzy numbers with maximizing the most possible value, minimizing risk of obtaining lower profit and maximizing possibilities of obtaining higher profit, and used fuzzy ranking concept to resolve imprecise constraints. Luhandjula [6] proposed the concept of a-possible feasibility and b-possible efficiency, and resolved imprecise objectives and constraints with fuzzy numbers by solving an auxiliary crisp MODM problem derived by using the extension principle and a and b-level cuts. Although various approaches have been made to solve FMOLP problems with fuzzy parameter, most of the works listed above are dealt with the fuzzy parameters only with symmetric triangular, trapezoidal, or some certain kind of membership functions. In order to deal with the situation in which the membership functions of parameters can be in any form, Zhang et al. [15,16] proposed two methods: one method is to solve a fuzzy linear programming
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(FLP) problem by transforming it into a corresponding four-objective constrained optimization problem, in which the fuzzy parameters are only included in the objective functions; another method is to formulate linear programming problems with fuzzy equality and inequality constraints. Based on these methods, some further researches have been done [13,14]. In this paper, an approximate algorithm for solving the proposed FMOLP problems with fuzzy parameters in any form of membership functions in both objective functions and constraints is developed. This paper is organized in six sections. Related concepts about fuzzy set theory are introduced in Section 2. A review of related definitions and theorems for solving FMOLP problems with fuzzy parameters in objective functions and constraints is described in Section 3. The core idea of the approximate algorithm proposed for solving FMOLP problems is described in Section 4. An illustrative example is presented in Section 5 to demonstrate the approximate algorithm developed. Some conclusions about the potentialities of the algorithm are drawn in Section 6.
2. Preliminaries In this section, we review some fundamentals of basic fuzzy set theory, which will be used through the remainder of this paper. Starting with several basic definitions about fuzzy sets, the extension principle is presented. It provides a general method for extending nonfuzzy mathematical concepts to the fuzzy framework. With the extension principle, operations on fuzzy sets, especially on fuzzy numbers, are systematically developed. ~ of X is Definition 2.1. Let X denote a universal set. Then a fuzzy sub-set A defined by its membership function lA~ : X ! ½0; 1;
ð1Þ
which assigns to each element x 2 X a real number lA~ ðxÞ in the interval [0, 1] ~ where the value of lA~ ðxÞ at x represents the grade of membership of x in A [7]. ~ is convex if and only if Definition 2.2. A fuzzy set A lA~ ðkx1 þ ð1 kÞx2 Þ P minðlA~ ðx1 Þ; lA~ ðx2 ÞÞ
ð2Þ
for every x1, x2 2 X and k 2 [0, 1] [7]. Among fuzzy sets, numbers such as ‘‘approximately m’’ or ‘‘about n’’ can be defined as fuzzy sets of the real lines R1. Such fuzzy numbers are formally defined as follows.
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Definition 2.3. Let R1 be the set of all real numbers. Then a real fuzzy number ~ a is defined by its membership function l~a ðxÞ that satisfies: (1) (2) (3) (4) (5) (6)
A continuous mapping from R1 to the closed internal [0, 1]. l~a ðxÞ ¼ 0 for all x 2 (1, c]. Strictly increasing and continuous on [c, a]. l~a ðxÞ ¼ 1 for all x 2 [a, b]. Strictly decreasing and continuous on [b, d]. l~a ðxÞ ¼ 0 for all x 2 [d, +1) [2].
The concept of a-level sets serves as an important transfer between ordinary sets and fuzzy sets. It also plays an important role in the construction of a fuzzy set by a series of ordinary sets. ~ is defined as an ordinary set Aa Definition 2.4. The a-level set of a fuzzy set A for which the degree of its membership function exceeds the level a [7] Aa ¼ fxjlA~ ðxÞ P ag; a 2 ½0; 1.
ð3Þ
It is clear that the following property holds for the a-level sets: a1 6 a2 () Aa1 Aa2 .
ð4Þ
Using the concept of a-level sets, the relationships between ordinary sets and fuzzy sets can be featured by the following decomposition theorem. Theorem 2.1 (Decomposition theorem). A fuzzy sets can be represented by [ ~¼ aAa ; ð5Þ A a2½0;1
where aAa denotes the algebraic product of a scalar a with the a-level set Aa [7]. According to Decomposition theorem, Theorem 2.1, a fuzzy number a~ can be represented as follows: [ ~ ð6Þ k aLk ; aRk . a¼ k2½0;1
As the above discussion, both the a-level sets and the decomposition theorem are used in relating fuzzy sets to ordinary sets and vice versa. The extension principle that will be defined in Definition 2.5 is to provide a general method for extending nonfuzzy mathematical concepts to the fuzzy framework.
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Definition 2.5. Let f:X ! Y be a mapping from a set X to a set Y. Then the extension principle allows defining the fuzzy set B in Y induced by the fuzzy set A in X through f as follows: B ¼ fðy; lB ðy ÞÞjy ¼ f ðxÞ; x 2 X g with lB ðy Þ,lf ðAÞ ðy Þ ¼
ð7Þ
8 < sup lA ðxÞ;
f 1 ðyÞ 6¼ /;
:
f 1 ðy Þ ¼ /;
y¼f ðxÞ
0;
ð8Þ
where f1(y) is the inverse image of y [7]. By the application of the extension principle in Definition 2.5, we have the following theorem. ~ be two fuzzy numbers with the membership functions l~ ðxÞ ~; b Theorem 2.2. Let a a ~ and lb~ ðxÞ, respectively, and 0 6 k 2 R, the addition of two fuzzy numbers ~a þ b and the scalar product of k and ~ a are defined by the membership functions n o l~aþb~ ðzÞ ¼ sup min l~a ðxÞ; lb~ ðy Þ ; ð9Þ z¼xþy
lk~a ðzÞ ¼ sup l~a ðxÞ.
ð10Þ
z¼kx
Also from Decomposition theorem and Theorem 2.2, we have the following theorem. ~ be two fuzzy numbers with the membership functions l~ ðxÞ Theorem 2.3. Let a~; b a and lb~ ðxÞ, respectively, and 0 6 a 2 R, 06b 2 R, we have [ ~¼ ~ ð11Þ k aLk þ bLk ; aRk þ bRk ; aþb k2½0;1
a~ a¼
[
k aaLk ; aaRk ;
ð12Þ
k2½0;1
~¼ a~ a þ bb
[
k aaLk þ bbLk ; aaRk þ bbRk .
ð13Þ
k2½0;1
Definition 2.6. Let ~ ai be fuzzy number with the membership function l~ai ðxi Þ, i = 1, . . ., n, we define ~ a ¼ ð~ a1 ; ~ a2 . . . ~ an Þ, whose membership function is defined as l~a ðxÞ ¼ min l~a1 ðx1 Þ; l~a2 ðx2 Þ; . . . ; l~an ðxn Þ ;
ð14Þ
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where x = (x1, x2, . . ., xn) is n-dimensional vector. Then a~ is called an n-dimensional fuzzy number on Rn. In order to deal with FMOLP problems with fuzzy parameters in this paper, we define the ranking way between two n-dimensional fuzzy numbers as follows. ~ be two n-dimensional fuzzy numbers, we define Definition 2.7. Let ~ a; b ~ iff aL =bL and aR =bR , i = 1, . . ., n "k 2 [0, 1], ~b (1) a ik ik ik ik ~ iff aL P bL and aR P bR , i = 1, . . ., n "k 2 [0, 1], (2) ~ ab ik ik ik ik ~ iff aL > bL and aR > bR , i = 1, . . ., n "k 2 [0, 1] [16]. (3) ~ ab ik ik ik ik 3. Fuzzy multiple objective linear programming problem and its related definitions and theorems 3.1. Multiple objective linear programming In classical mathematical programming, multiple objective problems are concerned with the optimization of multiple, conflicting, and noncommensurable objective functions subject to constraints representing the availability of limited resources and requirements. The MOLP problem is specified by linear functions which are to be maximized subject to a set of linear constraints. The standard form of MOLP can be written as follows: ðMOLPÞ
max
f ðxÞ ¼ Cx;
s:t:
x 2 X ¼ fx 2 Rn jAx 6 b; x P 0g;
ð15Þ
where C is an k · n objective function matrix, A is an m · n constraint matrix, ~ b is an m-vector of right-hand side, and x is an n-vector of decision variables. If we directly apply the notion of optimality of single objective linear programming (SOLP) to MOLP, we can get the following notion of a complete optimal solution. Definition 3.1. x* is said to be a complete optimal solution, if and only if there exists x* 2 X such that fi(x*) P fi(x), i = 1, . . ., k for all x 2 X [8]. However, in general, such a complete optimal solution that simultaneously maximizes all objective functions does not always exist when the objective functions conflict with each other. Thus, Pareto-optimal solution is introduced into MOLP.
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Definition 3.2. x* is said to be a Pareto optimal solution, if and only if there does not exist another x 2 X such that fi (x) P fi(x*) for all i and fj(x) 5 fj(x*) for at least one j [8]. In addition to Pareto optimality, the following weak Pareto optimal solution is defined as a slight weak solution concept than Pareto optimality. Definition 3.3. x* is said to be a weak Pareto optimal solution, if and only if there does not exist another x 2 X such that fi(x) > fi(x*), i = 1, . . . , k [8]. 3.2. Fuzzy multiple objective linear programming with fuzzy parameters In Section 3.1, we review some basic concepts, models and definitions about MOLP problems. However, when formulating an MOLP problem, various factors of the real world system should be reflected in the description of the objective functions and the constraints. Naturally, these objective functions and constraints involve many parameters of which possible values are assigned by the experts. But in most real situations, the possible values of these parameters are often imprecisely or ambiguously known to the experts. Therefore, it may be more appropriate for these parameters to be represented by fuzzy numbers. In this paper, we will consider the situation in which all coefficients of the objective functions and the constraints are fuzzy number parameters represented in any form of membership functions. Such FMOLP problems can be formulated as follows: ( ðFMOLPÞ
max
e f~ ðxÞ ¼ Cx
s:t:
~ ~b; x P 0g x 2 X ¼ fx 2 Rn jAx
ð16Þ
e is an k · n matrix, each element of which ~cij is a fuzzy number where C ~ is an m · n matrix, each element represented by membership function l~cij ðxÞ; A of which ~ aij is a fuzzy number represented by membership function l~aij ðxÞ; ~b is an m-vector, each element of which ~ bi is a fuzzy number represented by membership function l~bi ðxÞ, and x is an n-vector of decision variables, x 2 R n. Referring to Theorems 2.2 and 2.3, in the proposed FMOLP model (16), for each crisp value of x 2 X, each objective function f~ i ðxÞ is a fuzzy number. Because there is no universal concept of optimal solution to be accepted widely, based on Definition 2.7, which supplies a ranking way for any two ndimensional fuzzy numbers, we propose the following definitions about the optimal solutions for FMOLP problems with fuzzy parameters.
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Definition 3.4. x* is said to be a complete optimal solution to the FMOLP problems, if and only if there exists x* 2 X such that f~ i ðx Þ f~ i ðxÞ, i = 1, . . . , k for all x 2 X. Definition 3.5. x* is said to be a Pareto optimal solution to the FMOLP problems, if and only if there does not exist another x 2 X such that f~ i ðxÞ f~ i ðx Þ for all i. Definition 3.6. x* is said to be a weak Pareto optimal solution to the FMOLP problems, if and only if there does not exist another x 2 X such that f~ i ðxÞ f~ i ðx Þ, for all i. Associated with the FMOLP problems (16), let us consider the following multiple objective linear programming (MOLPk) problems: ! 8 > C Lk x > > 8k 2 ½0; 1 > < max C Rk x ð17Þ ðMOLPk Þ > > s:t: x 2 X ¼ fx 2 Rn jALk x 6 bLk ; ARk x 6 bRk ; x P 0 > > : 8k 2 ½0; 1g where 2 C Lk
cL12k
cL22k
.. .
..
cLk1k
cLk2k
aL11k
aL12k
aL22k
.. .
..
6 L 6 c21k 6 ¼6 . 6 . 4 . 2
ALk
cL11k
6 L 6 a21k 6 ¼6 . 6 . 4 . aLm1k
.
.
cL1nk
3
7 cL2nk 7 7 ; .. 7 7 . 5 cLknk aL1nk
3
7 aL2nk 7 7 ; .. 7 7 . 5
2 C Rk
cR11k
6 R 6 c21k 6 ¼6 . 6 . 4 .
cR12k
cR22k
.. .
..
.
cR1nk
3
7 cR2nk 7 7 ; .. 7 7 . 5
cRk1k cRk2k cRknk 3 2 R a11k aR12k aR1nk 7 6 R 6 a21k aR22k aR2nk 7 7 6 R Ak ¼ 6 . ; .. 7 .. .. 7 6 . . 5 . . 4 .
aLm2k
aRm1k aRm2k aRmnk aLmnk T T bLk ¼ bL1k ; bL2k ; . . . ; bLmk ; bRk ¼ bR1k ; bR2k ; . . . ; bRmk . The following theorem shows the relationship between FMOLP problem and the MOLPk problem: Theorem 3.1. Let x* 2 X be a feasible solution to the FMOLP problem. Then 1. x* is a complete optimal solution to the FMOLP problem, if and only if x* is a complete optimal solution to the MOLPk problem.
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2. x* is a Pareto optimal solution to the FMOLP problem, if and only if x* is a Pareto optimal solution to the MOLPk problem. 3. x* is a weak Pareto optimal solution to the FMOLP problem, if and only if x* is a weak Pareto optimal solution to the MOLPk problem. Proof. The proof is obvious from Definition 2.7.
h
In the next section, we will propose an approximation algorithm to solve the MOLPk problem. According to Theorem 3.1, the solution of MOLPk problem is also the solution of the FMOLP problem. 4. An approximation algorithm for solving FMOLP problems Refer to the description of MOLPk problem (17) in Section 3, there are an infinite number of objective functions and an infinite number of constraints included in MOLPk problem. In order to fix the problem, in this paper, an approximation algorithm is proposed as following for solving MOLPk problem, the solution of which is equally the solution of FMOLP problem. For the simplicity in presentation, we define X k ¼ x 2 Rn jALk x 6 bLk ; ARk x 6 bRk ; x P 0 8 k 2 ½0; 1.
ð18Þ
The main steps of the approximation algorithm are described as follows: Let the interval [0,1] be decomposed into l mean sub-intervals with (l + 1) nodes ki (i = 0, . . ., l) which are arranged in the order of 0 = k0 < k1 < < klT= 1. Based l on the current decomposing, we define the constraint X l ¼ i X ki , and denote
ðMOLPk Þl
8 > > > > < max > > > > :
s:t:
0 @
cLikj x cRikj x
1 A;
i ¼ 1; . . . ; k; j ¼ 0; . . . ; l ð19Þ
x 2 Xl
Step 1. Set l = 1, then solve (MOLPk)l with (x)l, where (x)l = (x1, x2, . . ., xn)l, and the solution obtained is subject to constraint x 2 Xl. Step 2. Solve (MOLPk)2l with (x)2l, and the solution obtained is subject to constraint x 2 X2l. Step 3. If k(x)2l (x)lk < e, then the solution x* of MOLPk problem is (x)2l. Otherwise, update l to 2l and go to Step 2.
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In Step 1, initially the interval [0, 1] is not split, and only k0 = 0 and k1 = 1 are considered. Then, each fuzzy objective function f~ i ðxÞ ¼ ~ci x is converted into four corresponding nonfuzzy objective functions 0 L 1 ci0 x B cL x C B i1 C ð20Þ B R C; i ¼ 1; . . . ; k. @ ci1 x A cRi0 x This conversion is also applied on constraints in the same way, and each fuzzy constraint ~ ai x ~ bi is converted into four corresponding nonfuzzy constraints, which are as follows: 0 L 1 0 L1 ai0 x bi0 B aL x C B bL C B i1 C B i1 C ð21Þ B R C 6 B R C; i ¼ 1; . . . ; m @ ai1 x A @ bi1 A aRi0 x
bRl0
The solution (x)l is based on new nonfuzzy objective functions (20) and nonfuzzy constraints (21). In Step 2, the interval [0, 1] is split further. At current step, suppose there are (l + 1) nodes ki (i = 0, 2, 4, . . . , 2l) in the interval [0, 1], and l new nodes ki (i = 1, 3, . . ., 2l 1) are inserted. The relationship between the new inserted nodes and the previous nodes are: k2iþ1 ¼
k2i þ k2iþ2 ; 2
i ¼ 0; 1; . . . ; l 1.
ð22Þ
Therefore, each fuzzy objective function f~ i ðxÞ ¼ ~ci x is converted into 2*(2l + 1) corresponding nonfuzzy objective functions, and the same conversion happens on the constraints. Suppose that the number of fuzzy objective functions and fuzzy constraints are k and m respectively, the total number of nonfuzzy objective functions and nonfuzzy constraints of (19) are 2*k*(2l + 1) and 2*m*(2l + 1), respectively. The solution (x)2l is also based on new nonfuzzy objective functions and nonfuzzy constraints. In Step 3, if the difference between (x)l and (x)2l in two consecutive steps is within the preset tolerance e, the solution of current step is the final result; otherwise, the algorithm needs further iteration. The flow chart of the algorithm is listed in Fig. 1. In each step of the algorithm, after the original FMOLP problem is converted into nonfuzzy (MOLPk)l problem, the feasible solution to (MOLPk)l problem can be formed and solved in the following maxmini sense. Zimmermann [17] extended his fuzzy linear programming approach into MOLP problem. For each objective function fi(x) = cix of MOLP (15), the corresponding linear membership function l(fi (x)) is defined as
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Fig. 1. Flow chart for the approximate algorithm.
lðfi ðxÞÞ ¼
8 0; > > < > > :
fi ðxÞfi0 fi1 fi0
1;
fi ðxÞ 6 fi0 ; fi0 6 fi ðxÞ 6 fi1 fi ðxÞ P fi1
ð23Þ
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where fi0 or fi1 denotes the value of the objective function fi(x) = cix such that the degree of membership function is 0 or 1, respectively. Using the linear membership function l(fi(x)) (23) and following the fuzzy decision concept of Bellman and Zadeh [1], the original MOLP can be interpreted as following: max min lðfi ðxÞÞ ¼ max min i¼1;...;k
s:t:
i¼1;...;k
fi ðxÞfi0 fi1 fi0
ð24Þ
x 2 X ¼ fx 2 Rn jAx 6 b; x P 0g.
By introducing the auxiliary value c, it can be reduced into the following conventional linear programming problem: max
c
s: t:
lðfi ðxÞÞ ¼
fi ðxÞfi0 fi1 fi0
P c;
i ¼ 1; 2; . . . ; k;
ð25Þ
Ax 6 b; x P 0 By assuming the optimal solution xi of the individual objective function fi(x) = cix under the constraints is defined by max fi ðxÞ ¼ ci x;
i ¼ 1; . . . ; k;
x2X
ð26Þ
Zimmermann [17] suggested a way to determine the linear membership function l(fi(x)). To be more specific, using the individual maximum fimax ¼ f xi ¼ max fi ðxÞ ¼ ci x; i ¼ 1; . . . ; k ð27Þ x2X
together with fimin ¼ min
j¼1;...;k j6¼i
fi xj ;
i ¼ 1; . . . ; k
ð28Þ
the membership function as in (23) is defined by choosing fi1 ¼ fimax and fi0 ¼ fimin . Then the linear programming problem (25) is transferred into as following: max
c
s:t:
lðfi ðxÞÞ ¼ f imax fimin P c;
f ðxÞf min i
Ax 6 b;
i ¼ 1; . . . ; k;
ð29Þ
i
x P 0.
Following the approach described above, from the idea of (24), the (MOLPk)l problem can be interpreted as 0 cL xf L min 1T ikj
ikj
B fikLjmax fikLjmin C C max min B @ cRik xfikR min A ; j j fikR max fikR min j j
s:t:
x 2 Xl
i ¼ 1; . . . ; k; j ¼ 0; . . . ; l
ð30Þ
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L For formula (30), by assuming the optimal solution xL ikj of cikj x under the constraints is defined by
max fikLj ðxÞ ¼ cLikj x; x2X l
i ¼ 1; . . . ; k; j ¼ 0; . . . ; l
ð31Þ
R and the optimal solution xR ikj of cikj x under the constraints is defined by
max fikRj ðxÞ ¼ cRikj x; x2X l
i ¼ 1; . . . ; k; j ¼ 0; . . . ; l.
ð32Þ
Then L L fikLjmax ¼ fikLj xL ikj ¼ cikj xikj ; R R fikRjmax ¼ fikRj xR ikj ¼ cikj xikj ;
i ¼ 1; . . . ; k; j ¼ 0; . . . ; l;
ð33Þ
i ¼ 1; . . . ; k; j ¼ 0; . . . ; l
ð34Þ
together with fikLjmin ¼ min
s¼1;...;k t¼0;...;l s6¼i;t6¼j
L R L L L R ; f fikLj xL x c x ; c x ¼ min skt ikj skt ikj skt ikj skt ; s¼1;...;k t¼0;...;l s6¼i;t6¼j
i ¼ 1; . . . ; k; j ¼ 0; . . . ; l R R R L R R ; f x c x ; c x ¼ min fikRjmin ¼ min fikRj xL skt ikj skt ikj skt ikj skt ; s¼1;...;k t¼0;...;l s6¼i;t6¼j
ð35Þ
s¼1;...;k t¼0;...;l s6¼i;t6¼j
i ¼ 1; . . . ; k; j ¼ 0; . . . ; l.
ð36Þ
By introducing the auxiliary variable c, from the formula (30), the (MOLPk)l problem can finally be reduced to the following SOLP problem: max
s:t:
c 8 cL xf L min ikj ikj > > P c; > fikL max fikL min > > j j > > > > < cRikj xfikRjmin P c; fikR max fikR min j j > > > > > aLskj x 6 bLskj ; > > > > : R askj x 6 bRskj ;
i ¼ 1; . . . ; k; j ¼ 0; . . . ; l; s ¼ 1; 2; . . . m;
where fikLjmax ; fikRjmax ; fikLjmin and fikRjmax are in (33)–(36), respectively. And the above SOLP problem can be solved by normal Simplex algorithm.
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5. An illustrative example 5.1. The description of an example of FMOLP To illustrate the approximate algorithm developed in Section 4, let us consider the following FMOLP problem with two fuzzy objective functions and four fuzzy constraints: ! ! ~c11 x1 þ ~c12 x2 f~ 1 ðxÞ max f~ ðxÞ ¼ max ð37Þ ¼ max ~c21 x1 þ ~c22 x2 f~ 2 ðxÞ 8 ~ a12 x2 ~ b1 ; a11 x1 þ ~ > > > > > > <~ a22 x2 ~ b2 ; a21 x1 þ ~ ð38Þ s:t: > > ~ a32 x2 ~ b3 ; a31 x1 þ ~ > > > > : ~ a42 x2 ~ b4 ; a41 x1 þ ~ where the membership functions of the coefficients of the objective functions and the constraints are as following: 8 0; x < 1; > > > > > > 1 6 x < 2; ðx2 1Þ=3; > > < l~c11 ðxÞ ¼ 1; 2 6 x 6 3; > > > > ð256 x2 Þ=247; 3 < x 6 16; > > > > : 0; 16 < x; 8 0; x < 0; > > > > > 2 > 0 6 x < 1; x; > > < l~c12 ðxÞ ¼ 1; 1 6 x 6 2; > > > 2 > ð576 x Þ=572; 2 < x 6 24; > > > > : 0; 24 < x; 8 0; x < 2; > > > > > > 2 6 x < 1; ð4 x2 Þ=3; > > < l~c21 ðxÞ ¼ 1; 1 6 x 6 0; > > > > ð13 xÞ=13; 0 < x 6 13; > > > > : 0; 13 < x;
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8 0; x < 1; > > > > 2 > ðx 1Þ=3; 1 6 x < 2; > > < 2 6 x 6 3; l~c22 ðxÞ ¼ 1; > > > > ð25 xÞ=22; 3 < x 6 25; > > > : 0; 25 < x; 8 0; x < 2; > > > > > 2x þ 4; 2 6 x < 1.5; > > < l~a11 ðxÞ ¼ 1; 1.5 6 x 6 0.5; > > > 2 > 4x ; 0.5 < x 6 0; > > > : 0; 0 < x; 8 0; x < 0; > > > > > x=2; 0 6 x < 2; > > < l~a12 ðxÞ ¼ 1; 2 6 x 6 4; > > > > ðe12 ex Þ=ðe12 e4 Þ; 4 < x 6 12; > > > : 0; 12 < x; 8 0; x < 0; > > > > > 2x; 0 6 x < 0.5; > > < l~a21 ðxÞ ¼ 1; 0.5 6 x 6 1.5; > > > 10 x 10 1.5 > ðe e Þ=ðe e Þ; 1.5 < x 6 10; > > > : 0; 10 < x; 8 0; x < 0; > > > > > ðex 1Þ=ðe2 1Þ; 0 6 x < 2; > > < l~a22 ðxÞ ¼ 1; 2 6 x 6 4; > > > 18 x 18 4 > ðe e Þ=ðe e Þ; 4 < x 6 18; > > > : 0; 18 < x; 8 0; x < 1; > > > > x 1 3 1 > ðe e Þ=ðe e Þ; 1 6 x < 3; > > < l~a31 ðxÞ ¼ 1; 3 6 x 6 5; > > > > ð18 xÞ=13; 5 < x 6 18; > > > : 0; 18 < x;
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8 0; x < 0; > > > 2 > x =4; 0 6 x < 2; > < 2 6 x 6 4; l~a32 ðxÞ ¼ 1; > > > ð10 xÞ=6; 4 < x 6 10; > > : 0; 10 < x; 8 0; x < 0; > > > > 06x<2 > < x=2; l~a41 ðxÞ ¼ 1; 2 6 x 6 4; > > 10 x 10 4 > ðe e Þ=ðe e Þ; 4 < x 6 10; > > : 0; 10 < x; 8 0; x < 0; > > > > 0 6 x < 0.5; > < 2x; l~a42 ðxÞ ¼ 1; 0.5 6 x 6 1.5; > > 3 > ð1000 x Þ=996.625; 1.5 < x 6 10; > > : 0; 10 < x; 8 0; x < 18; > > > x 18 20 18 > > < ðe e Þ=ðe e Þ; 18 6 x < 20; l~b1 ðxÞ ¼ 1; 20 6 x 6 22; > > > ð27000 x3 Þ=16; 352; 22 < x 6 30; > > : 0; 30 < x; 8 0; x < 24; > > > x 24 26 24 > ðe e Þ=ðe e Þ; 24 6 x < 26; > < l~b2 ðxÞ ¼ 1; 26 6 x 6 28; > > > ð40 xÞ=12; 28 < x 6 40; > > : 0; 40 < x; 8 0; x < 42; > > > > ðx 42Þ=2; 42 6 x < 44; > < 60 x 60 46 l~b3 ðxÞ ¼ ðe e Þ=ðe e Þ; 44 6 x 6 46; > > > 1; 46 < x 6 60; > > : 0; 60 < x; 8 0; x < 27; > > > 2 > x ð 729 Þ=112; 27 6 x < 29; > < l~b1 ðxÞ ¼ 1; 29 6 x 6 31; > > 3 > ð64; 000 x Þ=34; 209; 31 < x 6 40; > > : 0; 40 < x;
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As shown above, the fuzzy parameters of the objective functions and the constraints in the illustrative example can be represented in different kind of forms of membership functions, such as linear, quadratic, cubic, exponent, etc. In the approximate algorithm, for each step, the FMOLP problem will be converted into corresponding nonfuzzy MOLPk problem. Associated with the FMOLP problem of the illustrative example, the corresponding MOLPk problem in the illustrative example is listed as follows: pffiffiffi 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 3k þ 1 k 6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7" # 6 256 247k x1 576 572k 7 max f ðxÞ ¼ max 6 ð39Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 x 4 3k 3k þ 1 5 2 4 2
13 13k ðk 4Þ=2 pffiffiffi k=2
6 6 6 6 k=2 6 6 6 ln ðe10 ðe10 e1.5 ÞkÞ 6 s:t: 6 6 ln ððe3 e1 Þk þ e1 Þ 6 6 6 18 13k 6 6 2k 4
25 22k 2k
3
7 ln ðe12 ðe12 e4 ÞkÞ 7 7 ln ððe2 1Þk þ 1Þ 7 7 7" # 18 18 4 ln ðe ðe e ÞkÞ 7 7 x1 pffiffiffi 7 7 x1 2 k 7 7 7 10 6k 7 7 k=2 5 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 3 1000 996.625k 3
ln ðe10 ðe10 e4 ÞkÞ 2 ln ððe20 e18 Þk þ e18 Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6 p 6 3 27000 16352k 7 7 6 6 ln ððe26 e24 Þk þ e24 Þ 7 7 6 7 6 7 6 40 12k 7 6 66 7; 7 6 42 þ 2k 7 6 7 6 6 ln ðe60 ðe60 e46 ÞkÞ 7 7 6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6 729 þ 112k 5 4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 3 64; 000 34; 209k
ð40Þ
where "k 2 [0, 1]. 5.2. Solving MOLPk problem by the approximate algorithm Referring to MOLPk problem, in Step 1, the interval [0, 1] is not split, and only k0 = 0 and k1 = 1 are considered, then totally 8 nonfuzzy objective functions and 16 nonfuzzy constraints are generated. The result of conversion is listed as follows:
F. Wu et al. / Appl. Math. Comput. 174 (2006) 524–544
2
1
6 2 6 6 6 16 6 6 3 6 max f ðxÞ ¼ max 6 6 2 6 6 1 6 6 4 13 0 s:t:
2
2 6 1.5 6 6 6 0 6 6 0.5 6 6 6 0 6 6 0.5 6 6 6 10 6 6 1.5 6 6 6 1 6 6 3 6 6 6 18 6 6 5 6 6 6 0 6 6 2 6 6 4 10 4
0
541
3
1 7 7 7 24 7 7 2 7 7 x1 7 1 7 7 x2 2 7 7 7 25 5
ð41Þ
3
3 2 3 0 18 7 7 6 2 7 6 20 7 7 6 7 6 30 7 12 7 7 6 7 7 6 22 7 4 7 6 7 7 6 7 6 24 7 0 7 7 6 7 7 6 26 7 2 7 6 7 7 6 7 6 40 7 18 7 7 6 7 6 28 7 x 4 7 7 1 6 7 6 6 7. 7 6 42 7 0 7 x 7 1 6 7 7 6 44 7 2 7 6 7 7 6 7 7 6 60 7 10 7 6 7 7 6 46 7 4 7 6 7 7 6 7 6 27 7 0 7 7 6 7 7 6 29 7 0.5 7 6 7 7 6 7 5 4 40 5 10 1.5 31
ð42Þ
By the maxmini method that is described in Section 4, the solution to the problem in Step 1 is that the decision variables are x1 ¼ 1.5179;
x2 ¼ 1.3790
and the fuzzy objective functions are ( f~ 1 x1 ; x2 ¼ 1.5179~c11 þ 1.3790~c12 ; f~ x ; x ¼ 1.5179~c21 þ 1.3790~c22 . 2
1
ð43Þ
ð44Þ
2
In Step 2, one more node is inserted into the interval [0,1·], and three nodes are considered in this step, they are k0 = 0, k1 = 0.5 and k2 = 1. Then totally 12
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nonfuzzy objective functions and 24 nonfuzzy constraints are generated. By the maxmini method, the solution in Step 2 is that the decision variables are x1 ¼ 1.5985;
x2 ¼ 1.1049;
ð45Þ
and the fuzzy objective functions are ( f~ 1 x1 ; x2 ¼ 1.5985~c11 þ 1.1049~c12 ; f~ x ; x ¼ 1.5985~c21 þ 1.1049~c22 . 2
1
ð46Þ
2
More detailed information about further steps is listed in Table 1. Before the algorithm starts running, the tolerance e in the algorithm needs to be preset. Different tolerance will let the approximate algorithm stop at different step. From Table 1, we can find that if the tolerance e = 102, the approximate algorithm stops at step 5, if the tolerance e = 104, the approximate algorithm will stop at step 8, and the number of nonfuzzy objective functions and constraints will increase dramatically. Suppose we select the tolerance e = 104, then the final solution for the FMOLP problem for the illustrative example is that the decision variables are x1 ¼ 1.6166;
x2 ¼ 1.0164
ð47Þ
and the fuzzy objective functions are ( f~ 1 x1 ; x2 ¼ 1.6166~c11 þ 1.0164~c12 ; f~ x ; x ¼ 1.6166~c21 þ 1.0164~c22 . 2
1
ð48Þ
2
When choose different w1 and w2 for the objective functions, the different results are summarized in Table 2. be found from Table 2, when w1 = 1 and w2 = 0, the solution It can x1 ; x2 ¼ ð3.2260; 0.1932Þ is only concerned about the first objective function. When w1 = 0 and w2 = 1, the solution x1 ; x2 ¼ ð0.0; 1.8506Þ is only concerned about the second objective function. When w1 decreases from 1 to 0 and w2
Table 1 Summary of the running solution Step
x1
x2
1 2 3 4 5 6 7 8
1.51786 1.59854 1.61299 1.61614 1.61629 1.61666 1.61659 1.61655
1.37897 1.10480 1.02958 1.02795 1.02788 1.02634 1.02638 1.02640
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Table 2 Summary of the running solution by setting different weights of objective functions w2 x1 x2 f~ 2 x1 ; x2 f~ 1 x1 ; x2 w1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
3.2260 2.9141 2.5954 2.2721 1.9454 1.6166 1.2872 0.9589 0.6337 0.3134 0
0.1932 0.3543 0.5201 0.6878 0.8568 1.0264 1.1957 1.3637 1.5296 1.6922 1.8506
3.2260~c11 þ 0.1932~c12 2.9141~c11 þ 0.3543~c12 2.5954~c11 þ 0.5201~c12 2.2721~c11 þ 0.6878~c12 1.9454~c11 þ 0.8568~c12 1.6166~c11 þ 1.0264~c12 1.2872~c11 þ 1.1957~c12 0.9589~c11 þ 1.3637~c12 0.6337~c11 þ 1.5296~c12 0.3134~c11 þ 1.6922~c12 1.8506~c12
3.2260~c21 þ 0.1932~c22 2.9141~c21 þ 0.3543~c22 2.5954~c21 þ 0.5201~c22 2.2721~c21 þ 0.6878~c22 1.9454~c21 þ 0.8568~c22 1.6166~c21 þ 1.0264~c22 1.2872~c21 þ 1.1957~c22 0.9589~c21 þ 1.3637~c22 0.6337~c21 þ 1.5296~c22 0.3134~c21 þ 1.6922~c22 1.8506~c22
increases from 0 to 1 simultaneously, the solution will move from (3.2260,0.1932) to (0.0,1.8506) gradually.
6. Conclusion Decisions to be made in complex contexts, characterized by the presence of multiple evaluation aspects, are normally affected by uncertainty, which is essentially due to the insufficient and/or imprecise nature of input data as well as the subjective and evaluative preferences of the decision maker. In this paper, a new approximate algorithm for solving FMOLP problems with fuzzy parameters in any form of membership function in both objective functions and constraints has been developed. And a detailed description and analysis of the algorithm have been supplied. Based on the algorithms, a decision support system for fuzzy multiple objective linear programming with fuzzy parameters is being developed [12]. Research is currently underway to extend the idea to multiple objective decision-making problems with fuzzy goals and fuzzy parameters both of which are represented by fuzzy numbers.
References [1] R.E. Bellman, L.A. Zadeh, Decision-making in a fuzzy environment, Management Science 17 (1970) 141–164. [2] D. Dubois, H. Prade, Operations on fuzzy number, International Journal of Systems Science 9 (1978) 613–626. [3] C.L. Hwang, K. Yoon, Multiple Attribute Decision Making: Methods and Applications—a State-of-the-art Survey, Springer, Berlin, 1981.
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[4] Y.J. Lai, C.L. Hwang, Fuzzy Multiple Objective Decision Making: Methods and Applications, Springer-Verlag, Berlin, 1994. [5] Y.J. Lai, C.L. Hwang, A new approach to some possibilistic linear programming problems, Fuzzy Sets and Systems 49 (1992) 121–133. [6] M.K. Luhandjula, Multiple objective programming problems with possibilistic coefficients, Fuzzy Sets and Systems 21 (1987) 135–145. [7] M. Sakawa, Fundamentals of fuzzy set theory, in: Fuzzy Sets and Interactive Multiobjective Optimization, Plenum Press, New York, 1993. [8] M. Sakawa, Fuzzy linear programming, in: Fuzzy Sets and Interactive Multiobjective Optimization, Plenum Press, New York, 1993. [9] M. Sakawa, Interactive multiobjective linear programming with fuzzy parameters, in: Fuzzy Sets and Interactive Multiobjective Optimization, Plenum Press, New York, 1993. [10] M. Sakawa, Large Scale Interactive Fuzzy Multiobjective Programming, Physica-Verlag, Heidelberg, 2000. [11] H. Tanaka, K. Asai, Fuzzy linear programming problems with fuzzy numbers, Fuzzy Sets and Systems 13 (1984) 1–10. [12] F. Wu, J. Lu, G.Q. Zhang, A decision support system for multiple objective linear programming with fuzzy parameters, in: Proceedings of the 2004 IEEE International Conference on e-Technology, e-Commerce and e-Service (EEE-04), Taipei, Taiwan, 2004. [13] F. Wu, J. Lu, G.Q. Zhang, An extension of scalarization-based approach to fuzzy multiple objective linear programming with fuzzy parameters, in; Proceedings of the Second International Conference on Information and Management Sciences, Chengdu, China, 2003, pp. 420–426. [14] F. Wu, J. Lu, G.Q. Zhang, A new approximate algorithm for solving multiple objective linear programming with fuzzy parameters, in: Proceedings of The Third International Conference on Electronic Business (ICEB 2003), Singapore, 2003, pp. 532–534. [15] G.Q. Zhang, Y. Wu, M. Remias, J. Lu, An a-fuzzy max order and solution of linear constrained fuzzy optimization problems, East–West Journal of Mathematics P84 (2002), Special Volume. [16] G.Q. Zhang, Y.H. Wu, M. Remias, J. Lu, Formulation of fuzzy linear programming problems as four-objective constrained optimization problems, Applied Mathematics and Computation 39 (2003) 383–399. [17] H.J. Zimmermann, Fuzzy programming and linear programming with several objective functions, Fuzzy Sets and Systems 1 (1978) 45–55.
A new approximate algorithm for solving multiple objective linear programming problems with fuzzy parameters Fengjie Wu *, Jie Lu, Guangquan Zhang Faculty of Information Technology, University of Technology, Sydney, P.O. Box 123, Broadway, NSW 2007, Australia
Abstract Many business decision problems involve multiple objectives and can thus be described by multiple objective linear programming (MOLP) models. When a MOLP problem is being formulated, the parameters of objective functions and constraints are normally assigned by experts. In most real situations, the possible values of these parameters are imprecisely or ambiguously known to the experts. Therefore, it would be more appropriate for these parameters to be represented as fuzzy numerical data that can be represented by fuzzy numbers. In this paper, a new approximate algorithm is developed for solving fuzzy multiple objective linear programming (FMOLP) problems involving fuzzy parameters in any form of membership functions in both objective functions and constraints. A detailed description and analysis of the algorithm are supplied. In addition, an example is given to illustrate the approximate algorithm. 2005 Elsevier Inc. All rights reserved. Keywords: Optimization; Decision support technology; Fuzzy multiple objective linear programming; Approximate algorithm
*
Corresponding author. E-mail address: [email protected] (F. Wu).
0096-3003/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.04.106
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1. Introduction Decision-making is a process for selecting a possible course of actions from all the available alternatives. In almost all such selecting problems, the decision maker (DM) wants to attain more than one objective or goal in selecting the course of actions while satisfying the constraints dictated by environment, processes, and resources. The main feature of these problems is that the objectives are apparently noncommensurable, and conflict with each other [3]. Multiple objective linear programming (MOLP) is one of the popular methods to deal with such complex and ill-structured decision problems. When formulating an MOLP problem, various factors of the real world system should be reflected in the description of the objective functions and the constraints. Naturally, these objective functions and constraints involve many parameters which possible values may be assigned by the experts. Normally, such parameters are set at some values in an experimental or subjective manner through the expertsÕ understanding of the nature for the parameters [9,10]. With this observation, it is natural to recognize that the possible values of these parameters are often imprecisely or ambiguously known to the experts. In this case, it may be more appropriate to interpret the expertsÕ understanding of the parameters as fuzzy numerical data that can be represented by fuzzy numbers. The fuzzy multiple objective linear programming (FMOLP) problems involving fuzzy parameters would be viewed as a more realistic version than the conventional one [9,10]. Various kinds of FMOLP models have been proposed to deal with different decision-making situations that involve fuzzy values in objective function parameters, constraints parameters, or goals. Transforming a FMOLP problem into a crisp programming one is still employed by researchers [4]. Tanaka and Asai [11] formulated FMOLP with triangular fuzzy numbers, and the nonlinear programming problem obtained was solved by using mix–min operator. Lai and Hwang [5] resolved imprecise objectives with triangle fuzzy numbers with maximizing the most possible value, minimizing risk of obtaining lower profit and maximizing possibilities of obtaining higher profit, and used fuzzy ranking concept to resolve imprecise constraints. Luhandjula [6] proposed the concept of a-possible feasibility and b-possible efficiency, and resolved imprecise objectives and constraints with fuzzy numbers by solving an auxiliary crisp MODM problem derived by using the extension principle and a and b-level cuts. Although various approaches have been made to solve FMOLP problems with fuzzy parameter, most of the works listed above are dealt with the fuzzy parameters only with symmetric triangular, trapezoidal, or some certain kind of membership functions. In order to deal with the situation in which the membership functions of parameters can be in any form, Zhang et al. [15,16] proposed two methods: one method is to solve a fuzzy linear programming
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(FLP) problem by transforming it into a corresponding four-objective constrained optimization problem, in which the fuzzy parameters are only included in the objective functions; another method is to formulate linear programming problems with fuzzy equality and inequality constraints. Based on these methods, some further researches have been done [13,14]. In this paper, an approximate algorithm for solving the proposed FMOLP problems with fuzzy parameters in any form of membership functions in both objective functions and constraints is developed. This paper is organized in six sections. Related concepts about fuzzy set theory are introduced in Section 2. A review of related definitions and theorems for solving FMOLP problems with fuzzy parameters in objective functions and constraints is described in Section 3. The core idea of the approximate algorithm proposed for solving FMOLP problems is described in Section 4. An illustrative example is presented in Section 5 to demonstrate the approximate algorithm developed. Some conclusions about the potentialities of the algorithm are drawn in Section 6.
2. Preliminaries In this section, we review some fundamentals of basic fuzzy set theory, which will be used through the remainder of this paper. Starting with several basic definitions about fuzzy sets, the extension principle is presented. It provides a general method for extending nonfuzzy mathematical concepts to the fuzzy framework. With the extension principle, operations on fuzzy sets, especially on fuzzy numbers, are systematically developed. ~ of X is Definition 2.1. Let X denote a universal set. Then a fuzzy sub-set A defined by its membership function lA~ : X ! ½0; 1;
ð1Þ
which assigns to each element x 2 X a real number lA~ ðxÞ in the interval [0, 1] ~ where the value of lA~ ðxÞ at x represents the grade of membership of x in A [7]. ~ is convex if and only if Definition 2.2. A fuzzy set A lA~ ðkx1 þ ð1 kÞx2 Þ P minðlA~ ðx1 Þ; lA~ ðx2 ÞÞ
ð2Þ
for every x1, x2 2 X and k 2 [0, 1] [7]. Among fuzzy sets, numbers such as ‘‘approximately m’’ or ‘‘about n’’ can be defined as fuzzy sets of the real lines R1. Such fuzzy numbers are formally defined as follows.
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527
Definition 2.3. Let R1 be the set of all real numbers. Then a real fuzzy number ~ a is defined by its membership function l~a ðxÞ that satisfies: (1) (2) (3) (4) (5) (6)
A continuous mapping from R1 to the closed internal [0, 1]. l~a ðxÞ ¼ 0 for all x 2 (1, c]. Strictly increasing and continuous on [c, a]. l~a ðxÞ ¼ 1 for all x 2 [a, b]. Strictly decreasing and continuous on [b, d]. l~a ðxÞ ¼ 0 for all x 2 [d, +1) [2].
The concept of a-level sets serves as an important transfer between ordinary sets and fuzzy sets. It also plays an important role in the construction of a fuzzy set by a series of ordinary sets. ~ is defined as an ordinary set Aa Definition 2.4. The a-level set of a fuzzy set A for which the degree of its membership function exceeds the level a [7] Aa ¼ fxjlA~ ðxÞ P ag; a 2 ½0; 1.
ð3Þ
It is clear that the following property holds for the a-level sets: a1 6 a2 () Aa1 Aa2 .
ð4Þ
Using the concept of a-level sets, the relationships between ordinary sets and fuzzy sets can be featured by the following decomposition theorem. Theorem 2.1 (Decomposition theorem). A fuzzy sets can be represented by [ ~¼ aAa ; ð5Þ A a2½0;1
where aAa denotes the algebraic product of a scalar a with the a-level set Aa [7]. According to Decomposition theorem, Theorem 2.1, a fuzzy number a~ can be represented as follows: [ ~ ð6Þ k aLk ; aRk . a¼ k2½0;1
As the above discussion, both the a-level sets and the decomposition theorem are used in relating fuzzy sets to ordinary sets and vice versa. The extension principle that will be defined in Definition 2.5 is to provide a general method for extending nonfuzzy mathematical concepts to the fuzzy framework.
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Definition 2.5. Let f:X ! Y be a mapping from a set X to a set Y. Then the extension principle allows defining the fuzzy set B in Y induced by the fuzzy set A in X through f as follows: B ¼ fðy; lB ðy ÞÞjy ¼ f ðxÞ; x 2 X g with lB ðy Þ,lf ðAÞ ðy Þ ¼
ð7Þ
8 < sup lA ðxÞ;
f 1 ðyÞ 6¼ /;
:
f 1 ðy Þ ¼ /;
y¼f ðxÞ
0;
ð8Þ
where f1(y) is the inverse image of y [7]. By the application of the extension principle in Definition 2.5, we have the following theorem. ~ be two fuzzy numbers with the membership functions l~ ðxÞ ~; b Theorem 2.2. Let a a ~ and lb~ ðxÞ, respectively, and 0 6 k 2 R, the addition of two fuzzy numbers ~a þ b and the scalar product of k and ~ a are defined by the membership functions n o l~aþb~ ðzÞ ¼ sup min l~a ðxÞ; lb~ ðy Þ ; ð9Þ z¼xþy
lk~a ðzÞ ¼ sup l~a ðxÞ.
ð10Þ
z¼kx
Also from Decomposition theorem and Theorem 2.2, we have the following theorem. ~ be two fuzzy numbers with the membership functions l~ ðxÞ Theorem 2.3. Let a~; b a and lb~ ðxÞ, respectively, and 0 6 a 2 R, 06b 2 R, we have [ ~¼ ~ ð11Þ k aLk þ bLk ; aRk þ bRk ; aþb k2½0;1
a~ a¼
[
k aaLk ; aaRk ;
ð12Þ
k2½0;1
~¼ a~ a þ bb
[
k aaLk þ bbLk ; aaRk þ bbRk .
ð13Þ
k2½0;1
Definition 2.6. Let ~ ai be fuzzy number with the membership function l~ai ðxi Þ, i = 1, . . ., n, we define ~ a ¼ ð~ a1 ; ~ a2 . . . ~ an Þ, whose membership function is defined as l~a ðxÞ ¼ min l~a1 ðx1 Þ; l~a2 ðx2 Þ; . . . ; l~an ðxn Þ ;
ð14Þ
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529
where x = (x1, x2, . . ., xn) is n-dimensional vector. Then a~ is called an n-dimensional fuzzy number on Rn. In order to deal with FMOLP problems with fuzzy parameters in this paper, we define the ranking way between two n-dimensional fuzzy numbers as follows. ~ be two n-dimensional fuzzy numbers, we define Definition 2.7. Let ~ a; b ~ iff aL =bL and aR =bR , i = 1, . . ., n "k 2 [0, 1], ~b (1) a ik ik ik ik ~ iff aL P bL and aR P bR , i = 1, . . ., n "k 2 [0, 1], (2) ~ ab ik ik ik ik ~ iff aL > bL and aR > bR , i = 1, . . ., n "k 2 [0, 1] [16]. (3) ~ ab ik ik ik ik 3. Fuzzy multiple objective linear programming problem and its related definitions and theorems 3.1. Multiple objective linear programming In classical mathematical programming, multiple objective problems are concerned with the optimization of multiple, conflicting, and noncommensurable objective functions subject to constraints representing the availability of limited resources and requirements. The MOLP problem is specified by linear functions which are to be maximized subject to a set of linear constraints. The standard form of MOLP can be written as follows: ðMOLPÞ
max
f ðxÞ ¼ Cx;
s:t:
x 2 X ¼ fx 2 Rn jAx 6 b; x P 0g;
ð15Þ
where C is an k · n objective function matrix, A is an m · n constraint matrix, ~ b is an m-vector of right-hand side, and x is an n-vector of decision variables. If we directly apply the notion of optimality of single objective linear programming (SOLP) to MOLP, we can get the following notion of a complete optimal solution. Definition 3.1. x* is said to be a complete optimal solution, if and only if there exists x* 2 X such that fi(x*) P fi(x), i = 1, . . ., k for all x 2 X [8]. However, in general, such a complete optimal solution that simultaneously maximizes all objective functions does not always exist when the objective functions conflict with each other. Thus, Pareto-optimal solution is introduced into MOLP.
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Definition 3.2. x* is said to be a Pareto optimal solution, if and only if there does not exist another x 2 X such that fi (x) P fi(x*) for all i and fj(x) 5 fj(x*) for at least one j [8]. In addition to Pareto optimality, the following weak Pareto optimal solution is defined as a slight weak solution concept than Pareto optimality. Definition 3.3. x* is said to be a weak Pareto optimal solution, if and only if there does not exist another x 2 X such that fi(x) > fi(x*), i = 1, . . . , k [8]. 3.2. Fuzzy multiple objective linear programming with fuzzy parameters In Section 3.1, we review some basic concepts, models and definitions about MOLP problems. However, when formulating an MOLP problem, various factors of the real world system should be reflected in the description of the objective functions and the constraints. Naturally, these objective functions and constraints involve many parameters of which possible values are assigned by the experts. But in most real situations, the possible values of these parameters are often imprecisely or ambiguously known to the experts. Therefore, it may be more appropriate for these parameters to be represented by fuzzy numbers. In this paper, we will consider the situation in which all coefficients of the objective functions and the constraints are fuzzy number parameters represented in any form of membership functions. Such FMOLP problems can be formulated as follows: ( ðFMOLPÞ
max
e f~ ðxÞ ¼ Cx
s:t:
~ ~b; x P 0g x 2 X ¼ fx 2 Rn jAx
ð16Þ
e is an k · n matrix, each element of which ~cij is a fuzzy number where C ~ is an m · n matrix, each element represented by membership function l~cij ðxÞ; A of which ~ aij is a fuzzy number represented by membership function l~aij ðxÞ; ~b is an m-vector, each element of which ~ bi is a fuzzy number represented by membership function l~bi ðxÞ, and x is an n-vector of decision variables, x 2 R n. Referring to Theorems 2.2 and 2.3, in the proposed FMOLP model (16), for each crisp value of x 2 X, each objective function f~ i ðxÞ is a fuzzy number. Because there is no universal concept of optimal solution to be accepted widely, based on Definition 2.7, which supplies a ranking way for any two ndimensional fuzzy numbers, we propose the following definitions about the optimal solutions for FMOLP problems with fuzzy parameters.
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Definition 3.4. x* is said to be a complete optimal solution to the FMOLP problems, if and only if there exists x* 2 X such that f~ i ðx Þ f~ i ðxÞ, i = 1, . . . , k for all x 2 X. Definition 3.5. x* is said to be a Pareto optimal solution to the FMOLP problems, if and only if there does not exist another x 2 X such that f~ i ðxÞ f~ i ðx Þ for all i. Definition 3.6. x* is said to be a weak Pareto optimal solution to the FMOLP problems, if and only if there does not exist another x 2 X such that f~ i ðxÞ f~ i ðx Þ, for all i. Associated with the FMOLP problems (16), let us consider the following multiple objective linear programming (MOLPk) problems: ! 8 > C Lk x > > 8k 2 ½0; 1 > < max C Rk x ð17Þ ðMOLPk Þ > > s:t: x 2 X ¼ fx 2 Rn jALk x 6 bLk ; ARk x 6 bRk ; x P 0 > > : 8k 2 ½0; 1g where 2 C Lk
cL12k
cL22k
.. .
..
cLk1k
cLk2k
aL11k
aL12k
aL22k
.. .
..
6 L 6 c21k 6 ¼6 . 6 . 4 . 2
ALk
cL11k
6 L 6 a21k 6 ¼6 . 6 . 4 . aLm1k
.
.
cL1nk
3
7 cL2nk 7 7 ; .. 7 7 . 5 cLknk aL1nk
3
7 aL2nk 7 7 ; .. 7 7 . 5
2 C Rk
cR11k
6 R 6 c21k 6 ¼6 . 6 . 4 .
cR12k
cR22k
.. .
..
.
cR1nk
3
7 cR2nk 7 7 ; .. 7 7 . 5
cRk1k cRk2k cRknk 3 2 R a11k aR12k aR1nk 7 6 R 6 a21k aR22k aR2nk 7 7 6 R Ak ¼ 6 . ; .. 7 .. .. 7 6 . . 5 . . 4 .
aLm2k
aRm1k aRm2k aRmnk aLmnk T T bLk ¼ bL1k ; bL2k ; . . . ; bLmk ; bRk ¼ bR1k ; bR2k ; . . . ; bRmk . The following theorem shows the relationship between FMOLP problem and the MOLPk problem: Theorem 3.1. Let x* 2 X be a feasible solution to the FMOLP problem. Then 1. x* is a complete optimal solution to the FMOLP problem, if and only if x* is a complete optimal solution to the MOLPk problem.
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2. x* is a Pareto optimal solution to the FMOLP problem, if and only if x* is a Pareto optimal solution to the MOLPk problem. 3. x* is a weak Pareto optimal solution to the FMOLP problem, if and only if x* is a weak Pareto optimal solution to the MOLPk problem. Proof. The proof is obvious from Definition 2.7.
h
In the next section, we will propose an approximation algorithm to solve the MOLPk problem. According to Theorem 3.1, the solution of MOLPk problem is also the solution of the FMOLP problem. 4. An approximation algorithm for solving FMOLP problems Refer to the description of MOLPk problem (17) in Section 3, there are an infinite number of objective functions and an infinite number of constraints included in MOLPk problem. In order to fix the problem, in this paper, an approximation algorithm is proposed as following for solving MOLPk problem, the solution of which is equally the solution of FMOLP problem. For the simplicity in presentation, we define X k ¼ x 2 Rn jALk x 6 bLk ; ARk x 6 bRk ; x P 0 8 k 2 ½0; 1.
ð18Þ
The main steps of the approximation algorithm are described as follows: Let the interval [0,1] be decomposed into l mean sub-intervals with (l + 1) nodes ki (i = 0, . . ., l) which are arranged in the order of 0 = k0 < k1 < < klT= 1. Based l on the current decomposing, we define the constraint X l ¼ i X ki , and denote
ðMOLPk Þl
8 > > > > < max > > > > :
s:t:
0 @
cLikj x cRikj x
1 A;
i ¼ 1; . . . ; k; j ¼ 0; . . . ; l ð19Þ
x 2 Xl
Step 1. Set l = 1, then solve (MOLPk)l with (x)l, where (x)l = (x1, x2, . . ., xn)l, and the solution obtained is subject to constraint x 2 Xl. Step 2. Solve (MOLPk)2l with (x)2l, and the solution obtained is subject to constraint x 2 X2l. Step 3. If k(x)2l (x)lk < e, then the solution x* of MOLPk problem is (x)2l. Otherwise, update l to 2l and go to Step 2.
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In Step 1, initially the interval [0, 1] is not split, and only k0 = 0 and k1 = 1 are considered. Then, each fuzzy objective function f~ i ðxÞ ¼ ~ci x is converted into four corresponding nonfuzzy objective functions 0 L 1 ci0 x B cL x C B i1 C ð20Þ B R C; i ¼ 1; . . . ; k. @ ci1 x A cRi0 x This conversion is also applied on constraints in the same way, and each fuzzy constraint ~ ai x ~ bi is converted into four corresponding nonfuzzy constraints, which are as follows: 0 L 1 0 L1 ai0 x bi0 B aL x C B bL C B i1 C B i1 C ð21Þ B R C 6 B R C; i ¼ 1; . . . ; m @ ai1 x A @ bi1 A aRi0 x
bRl0
The solution (x)l is based on new nonfuzzy objective functions (20) and nonfuzzy constraints (21). In Step 2, the interval [0, 1] is split further. At current step, suppose there are (l + 1) nodes ki (i = 0, 2, 4, . . . , 2l) in the interval [0, 1], and l new nodes ki (i = 1, 3, . . ., 2l 1) are inserted. The relationship between the new inserted nodes and the previous nodes are: k2iþ1 ¼
k2i þ k2iþ2 ; 2
i ¼ 0; 1; . . . ; l 1.
ð22Þ
Therefore, each fuzzy objective function f~ i ðxÞ ¼ ~ci x is converted into 2*(2l + 1) corresponding nonfuzzy objective functions, and the same conversion happens on the constraints. Suppose that the number of fuzzy objective functions and fuzzy constraints are k and m respectively, the total number of nonfuzzy objective functions and nonfuzzy constraints of (19) are 2*k*(2l + 1) and 2*m*(2l + 1), respectively. The solution (x)2l is also based on new nonfuzzy objective functions and nonfuzzy constraints. In Step 3, if the difference between (x)l and (x)2l in two consecutive steps is within the preset tolerance e, the solution of current step is the final result; otherwise, the algorithm needs further iteration. The flow chart of the algorithm is listed in Fig. 1. In each step of the algorithm, after the original FMOLP problem is converted into nonfuzzy (MOLPk)l problem, the feasible solution to (MOLPk)l problem can be formed and solved in the following maxmini sense. Zimmermann [17] extended his fuzzy linear programming approach into MOLP problem. For each objective function fi(x) = cix of MOLP (15), the corresponding linear membership function l(fi (x)) is defined as
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Fig. 1. Flow chart for the approximate algorithm.
lðfi ðxÞÞ ¼
8 0; > > < > > :
fi ðxÞfi0 fi1 fi0
1;
fi ðxÞ 6 fi0 ; fi0 6 fi ðxÞ 6 fi1 fi ðxÞ P fi1
ð23Þ
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535
where fi0 or fi1 denotes the value of the objective function fi(x) = cix such that the degree of membership function is 0 or 1, respectively. Using the linear membership function l(fi(x)) (23) and following the fuzzy decision concept of Bellman and Zadeh [1], the original MOLP can be interpreted as following: max min lðfi ðxÞÞ ¼ max min i¼1;...;k
s:t:
i¼1;...;k
fi ðxÞfi0 fi1 fi0
ð24Þ
x 2 X ¼ fx 2 Rn jAx 6 b; x P 0g.
By introducing the auxiliary value c, it can be reduced into the following conventional linear programming problem: max
c
s: t:
lðfi ðxÞÞ ¼
fi ðxÞfi0 fi1 fi0
P c;
i ¼ 1; 2; . . . ; k;
ð25Þ
Ax 6 b; x P 0 By assuming the optimal solution xi of the individual objective function fi(x) = cix under the constraints is defined by max fi ðxÞ ¼ ci x;
i ¼ 1; . . . ; k;
x2X
ð26Þ
Zimmermann [17] suggested a way to determine the linear membership function l(fi(x)). To be more specific, using the individual maximum fimax ¼ f xi ¼ max fi ðxÞ ¼ ci x; i ¼ 1; . . . ; k ð27Þ x2X
together with fimin ¼ min
j¼1;...;k j6¼i
fi xj ;
i ¼ 1; . . . ; k
ð28Þ
the membership function as in (23) is defined by choosing fi1 ¼ fimax and fi0 ¼ fimin . Then the linear programming problem (25) is transferred into as following: max
c
s:t:
lðfi ðxÞÞ ¼ f imax fimin P c;
f ðxÞf min i
Ax 6 b;
i ¼ 1; . . . ; k;
ð29Þ
i
x P 0.
Following the approach described above, from the idea of (24), the (MOLPk)l problem can be interpreted as 0 cL xf L min 1T ikj
ikj
B fikLjmax fikLjmin C C max min B @ cRik xfikR min A ; j j fikR max fikR min j j
s:t:
x 2 Xl
i ¼ 1; . . . ; k; j ¼ 0; . . . ; l
ð30Þ
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L For formula (30), by assuming the optimal solution xL ikj of cikj x under the constraints is defined by
max fikLj ðxÞ ¼ cLikj x; x2X l
i ¼ 1; . . . ; k; j ¼ 0; . . . ; l
ð31Þ
R and the optimal solution xR ikj of cikj x under the constraints is defined by
max fikRj ðxÞ ¼ cRikj x; x2X l
i ¼ 1; . . . ; k; j ¼ 0; . . . ; l.
ð32Þ
Then L L fikLjmax ¼ fikLj xL ikj ¼ cikj xikj ; R R fikRjmax ¼ fikRj xR ikj ¼ cikj xikj ;
i ¼ 1; . . . ; k; j ¼ 0; . . . ; l;
ð33Þ
i ¼ 1; . . . ; k; j ¼ 0; . . . ; l
ð34Þ
together with fikLjmin ¼ min
s¼1;...;k t¼0;...;l s6¼i;t6¼j
L R L L L R ; f fikLj xL x c x ; c x ¼ min skt ikj skt ikj skt ikj skt ; s¼1;...;k t¼0;...;l s6¼i;t6¼j
i ¼ 1; . . . ; k; j ¼ 0; . . . ; l R R R L R R ; f x c x ; c x ¼ min fikRjmin ¼ min fikRj xL skt ikj skt ikj skt ikj skt ; s¼1;...;k t¼0;...;l s6¼i;t6¼j
ð35Þ
s¼1;...;k t¼0;...;l s6¼i;t6¼j
i ¼ 1; . . . ; k; j ¼ 0; . . . ; l.
ð36Þ
By introducing the auxiliary variable c, from the formula (30), the (MOLPk)l problem can finally be reduced to the following SOLP problem: max
s:t:
c 8 cL xf L min ikj ikj > > P c; > fikL max fikL min > > j j > > > > < cRikj xfikRjmin P c; fikR max fikR min j j > > > > > aLskj x 6 bLskj ; > > > > : R askj x 6 bRskj ;
i ¼ 1; . . . ; k; j ¼ 0; . . . ; l; s ¼ 1; 2; . . . m;
where fikLjmax ; fikRjmax ; fikLjmin and fikRjmax are in (33)–(36), respectively. And the above SOLP problem can be solved by normal Simplex algorithm.
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537
5. An illustrative example 5.1. The description of an example of FMOLP To illustrate the approximate algorithm developed in Section 4, let us consider the following FMOLP problem with two fuzzy objective functions and four fuzzy constraints: ! ! ~c11 x1 þ ~c12 x2 f~ 1 ðxÞ max f~ ðxÞ ¼ max ð37Þ ¼ max ~c21 x1 þ ~c22 x2 f~ 2 ðxÞ 8 ~ a12 x2 ~ b1 ; a11 x1 þ ~ > > > > > > <~ a22 x2 ~ b2 ; a21 x1 þ ~ ð38Þ s:t: > > ~ a32 x2 ~ b3 ; a31 x1 þ ~ > > > > : ~ a42 x2 ~ b4 ; a41 x1 þ ~ where the membership functions of the coefficients of the objective functions and the constraints are as following: 8 0; x < 1; > > > > > > 1 6 x < 2; ðx2 1Þ=3; > > < l~c11 ðxÞ ¼ 1; 2 6 x 6 3; > > > > ð256 x2 Þ=247; 3 < x 6 16; > > > > : 0; 16 < x; 8 0; x < 0; > > > > > 2 > 0 6 x < 1; x; > > < l~c12 ðxÞ ¼ 1; 1 6 x 6 2; > > > 2 > ð576 x Þ=572; 2 < x 6 24; > > > > : 0; 24 < x; 8 0; x < 2; > > > > > > 2 6 x < 1; ð4 x2 Þ=3; > > < l~c21 ðxÞ ¼ 1; 1 6 x 6 0; > > > > ð13 xÞ=13; 0 < x 6 13; > > > > : 0; 13 < x;
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8 0; x < 1; > > > > 2 > ðx 1Þ=3; 1 6 x < 2; > > < 2 6 x 6 3; l~c22 ðxÞ ¼ 1; > > > > ð25 xÞ=22; 3 < x 6 25; > > > : 0; 25 < x; 8 0; x < 2; > > > > > 2x þ 4; 2 6 x < 1.5; > > < l~a11 ðxÞ ¼ 1; 1.5 6 x 6 0.5; > > > 2 > 4x ; 0.5 < x 6 0; > > > : 0; 0 < x; 8 0; x < 0; > > > > > x=2; 0 6 x < 2; > > < l~a12 ðxÞ ¼ 1; 2 6 x 6 4; > > > > ðe12 ex Þ=ðe12 e4 Þ; 4 < x 6 12; > > > : 0; 12 < x; 8 0; x < 0; > > > > > 2x; 0 6 x < 0.5; > > < l~a21 ðxÞ ¼ 1; 0.5 6 x 6 1.5; > > > 10 x 10 1.5 > ðe e Þ=ðe e Þ; 1.5 < x 6 10; > > > : 0; 10 < x; 8 0; x < 0; > > > > > ðex 1Þ=ðe2 1Þ; 0 6 x < 2; > > < l~a22 ðxÞ ¼ 1; 2 6 x 6 4; > > > 18 x 18 4 > ðe e Þ=ðe e Þ; 4 < x 6 18; > > > : 0; 18 < x; 8 0; x < 1; > > > > x 1 3 1 > ðe e Þ=ðe e Þ; 1 6 x < 3; > > < l~a31 ðxÞ ¼ 1; 3 6 x 6 5; > > > > ð18 xÞ=13; 5 < x 6 18; > > > : 0; 18 < x;
F. Wu et al. / Appl. Math. Comput. 174 (2006) 524–544
8 0; x < 0; > > > 2 > x =4; 0 6 x < 2; > < 2 6 x 6 4; l~a32 ðxÞ ¼ 1; > > > ð10 xÞ=6; 4 < x 6 10; > > : 0; 10 < x; 8 0; x < 0; > > > > 06x<2 > < x=2; l~a41 ðxÞ ¼ 1; 2 6 x 6 4; > > 10 x 10 4 > ðe e Þ=ðe e Þ; 4 < x 6 10; > > : 0; 10 < x; 8 0; x < 0; > > > > 0 6 x < 0.5; > < 2x; l~a42 ðxÞ ¼ 1; 0.5 6 x 6 1.5; > > 3 > ð1000 x Þ=996.625; 1.5 < x 6 10; > > : 0; 10 < x; 8 0; x < 18; > > > x 18 20 18 > > < ðe e Þ=ðe e Þ; 18 6 x < 20; l~b1 ðxÞ ¼ 1; 20 6 x 6 22; > > > ð27000 x3 Þ=16; 352; 22 < x 6 30; > > : 0; 30 < x; 8 0; x < 24; > > > x 24 26 24 > ðe e Þ=ðe e Þ; 24 6 x < 26; > < l~b2 ðxÞ ¼ 1; 26 6 x 6 28; > > > ð40 xÞ=12; 28 < x 6 40; > > : 0; 40 < x; 8 0; x < 42; > > > > ðx 42Þ=2; 42 6 x < 44; > < 60 x 60 46 l~b3 ðxÞ ¼ ðe e Þ=ðe e Þ; 44 6 x 6 46; > > > 1; 46 < x 6 60; > > : 0; 60 < x; 8 0; x < 27; > > > 2 > x ð 729 Þ=112; 27 6 x < 29; > < l~b1 ðxÞ ¼ 1; 29 6 x 6 31; > > 3 > ð64; 000 x Þ=34; 209; 31 < x 6 40; > > : 0; 40 < x;
539
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As shown above, the fuzzy parameters of the objective functions and the constraints in the illustrative example can be represented in different kind of forms of membership functions, such as linear, quadratic, cubic, exponent, etc. In the approximate algorithm, for each step, the FMOLP problem will be converted into corresponding nonfuzzy MOLPk problem. Associated with the FMOLP problem of the illustrative example, the corresponding MOLPk problem in the illustrative example is listed as follows: pffiffiffi 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 3k þ 1 k 6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7" # 6 256 247k x1 576 572k 7 max f ðxÞ ¼ max 6 ð39Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 x 4 3k 3k þ 1 5 2 4 2
13 13k ðk 4Þ=2 pffiffiffi k=2
6 6 6 6 k=2 6 6 6 ln ðe10 ðe10 e1.5 ÞkÞ 6 s:t: 6 6 ln ððe3 e1 Þk þ e1 Þ 6 6 6 18 13k 6 6 2k 4
25 22k 2k
3
7 ln ðe12 ðe12 e4 ÞkÞ 7 7 ln ððe2 1Þk þ 1Þ 7 7 7" # 18 18 4 ln ðe ðe e ÞkÞ 7 7 x1 pffiffiffi 7 7 x1 2 k 7 7 7 10 6k 7 7 k=2 5 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 3 1000 996.625k 3
ln ðe10 ðe10 e4 ÞkÞ 2 ln ððe20 e18 Þk þ e18 Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6 p 6 3 27000 16352k 7 7 6 6 ln ððe26 e24 Þk þ e24 Þ 7 7 6 7 6 7 6 40 12k 7 6 66 7; 7 6 42 þ 2k 7 6 7 6 6 ln ðe60 ðe60 e46 ÞkÞ 7 7 6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6 729 þ 112k 5 4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 3 64; 000 34; 209k
ð40Þ
where "k 2 [0, 1]. 5.2. Solving MOLPk problem by the approximate algorithm Referring to MOLPk problem, in Step 1, the interval [0, 1] is not split, and only k0 = 0 and k1 = 1 are considered, then totally 8 nonfuzzy objective functions and 16 nonfuzzy constraints are generated. The result of conversion is listed as follows:
F. Wu et al. / Appl. Math. Comput. 174 (2006) 524–544
2
1
6 2 6 6 6 16 6 6 3 6 max f ðxÞ ¼ max 6 6 2 6 6 1 6 6 4 13 0 s:t:
2
2 6 1.5 6 6 6 0 6 6 0.5 6 6 6 0 6 6 0.5 6 6 6 10 6 6 1.5 6 6 6 1 6 6 3 6 6 6 18 6 6 5 6 6 6 0 6 6 2 6 6 4 10 4
0
541
3
1 7 7 7 24 7 7 2 7 7 x1 7 1 7 7 x2 2 7 7 7 25 5
ð41Þ
3
3 2 3 0 18 7 7 6 2 7 6 20 7 7 6 7 6 30 7 12 7 7 6 7 7 6 22 7 4 7 6 7 7 6 7 6 24 7 0 7 7 6 7 7 6 26 7 2 7 6 7 7 6 7 6 40 7 18 7 7 6 7 6 28 7 x 4 7 7 1 6 7 6 6 7. 7 6 42 7 0 7 x 7 1 6 7 7 6 44 7 2 7 6 7 7 6 7 7 6 60 7 10 7 6 7 7 6 46 7 4 7 6 7 7 6 7 6 27 7 0 7 7 6 7 7 6 29 7 0.5 7 6 7 7 6 7 5 4 40 5 10 1.5 31
ð42Þ
By the maxmini method that is described in Section 4, the solution to the problem in Step 1 is that the decision variables are x1 ¼ 1.5179;
x2 ¼ 1.3790
and the fuzzy objective functions are ( f~ 1 x1 ; x2 ¼ 1.5179~c11 þ 1.3790~c12 ; f~ x ; x ¼ 1.5179~c21 þ 1.3790~c22 . 2
1
ð43Þ
ð44Þ
2
In Step 2, one more node is inserted into the interval [0,1·], and three nodes are considered in this step, they are k0 = 0, k1 = 0.5 and k2 = 1. Then totally 12
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nonfuzzy objective functions and 24 nonfuzzy constraints are generated. By the maxmini method, the solution in Step 2 is that the decision variables are x1 ¼ 1.5985;
x2 ¼ 1.1049;
ð45Þ
and the fuzzy objective functions are ( f~ 1 x1 ; x2 ¼ 1.5985~c11 þ 1.1049~c12 ; f~ x ; x ¼ 1.5985~c21 þ 1.1049~c22 . 2
1
ð46Þ
2
More detailed information about further steps is listed in Table 1. Before the algorithm starts running, the tolerance e in the algorithm needs to be preset. Different tolerance will let the approximate algorithm stop at different step. From Table 1, we can find that if the tolerance e = 102, the approximate algorithm stops at step 5, if the tolerance e = 104, the approximate algorithm will stop at step 8, and the number of nonfuzzy objective functions and constraints will increase dramatically. Suppose we select the tolerance e = 104, then the final solution for the FMOLP problem for the illustrative example is that the decision variables are x1 ¼ 1.6166;
x2 ¼ 1.0164
ð47Þ
and the fuzzy objective functions are ( f~ 1 x1 ; x2 ¼ 1.6166~c11 þ 1.0164~c12 ; f~ x ; x ¼ 1.6166~c21 þ 1.0164~c22 . 2
1
ð48Þ
2
When choose different w1 and w2 for the objective functions, the different results are summarized in Table 2. be found from Table 2, when w1 = 1 and w2 = 0, the solution It can x1 ; x2 ¼ ð3.2260; 0.1932Þ is only concerned about the first objective function. When w1 = 0 and w2 = 1, the solution x1 ; x2 ¼ ð0.0; 1.8506Þ is only concerned about the second objective function. When w1 decreases from 1 to 0 and w2
Table 1 Summary of the running solution Step
x1
x2
1 2 3 4 5 6 7 8
1.51786 1.59854 1.61299 1.61614 1.61629 1.61666 1.61659 1.61655
1.37897 1.10480 1.02958 1.02795 1.02788 1.02634 1.02638 1.02640
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Table 2 Summary of the running solution by setting different weights of objective functions w2 x1 x2 f~ 2 x1 ; x2 f~ 1 x1 ; x2 w1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
3.2260 2.9141 2.5954 2.2721 1.9454 1.6166 1.2872 0.9589 0.6337 0.3134 0
0.1932 0.3543 0.5201 0.6878 0.8568 1.0264 1.1957 1.3637 1.5296 1.6922 1.8506
3.2260~c11 þ 0.1932~c12 2.9141~c11 þ 0.3543~c12 2.5954~c11 þ 0.5201~c12 2.2721~c11 þ 0.6878~c12 1.9454~c11 þ 0.8568~c12 1.6166~c11 þ 1.0264~c12 1.2872~c11 þ 1.1957~c12 0.9589~c11 þ 1.3637~c12 0.6337~c11 þ 1.5296~c12 0.3134~c11 þ 1.6922~c12 1.8506~c12
3.2260~c21 þ 0.1932~c22 2.9141~c21 þ 0.3543~c22 2.5954~c21 þ 0.5201~c22 2.2721~c21 þ 0.6878~c22 1.9454~c21 þ 0.8568~c22 1.6166~c21 þ 1.0264~c22 1.2872~c21 þ 1.1957~c22 0.9589~c21 þ 1.3637~c22 0.6337~c21 þ 1.5296~c22 0.3134~c21 þ 1.6922~c22 1.8506~c22
increases from 0 to 1 simultaneously, the solution will move from (3.2260,0.1932) to (0.0,1.8506) gradually.
6. Conclusion Decisions to be made in complex contexts, characterized by the presence of multiple evaluation aspects, are normally affected by uncertainty, which is essentially due to the insufficient and/or imprecise nature of input data as well as the subjective and evaluative preferences of the decision maker. In this paper, a new approximate algorithm for solving FMOLP problems with fuzzy parameters in any form of membership function in both objective functions and constraints has been developed. And a detailed description and analysis of the algorithm have been supplied. Based on the algorithms, a decision support system for fuzzy multiple objective linear programming with fuzzy parameters is being developed [12]. Research is currently underway to extend the idea to multiple objective decision-making problems with fuzzy goals and fuzzy parameters both of which are represented by fuzzy numbers.
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