Approximate Reasoning for Solving Fuzzy Linear Programming Problems

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Approximate Reasoning for Solving Fuzzy

Linear Programming Problems

Robert Full´er

Department of Computer Science, E¨otv¨os Lor´and University, P.O.Box 157, H-1502 Budapest 112, Hungary

Hans-J ¨urgen Zimmermann

Department of Operations Research, RWTH Aachen Templergraben 64, W-5100 Aachen, Germany

Abstract

We interpret fuzzy linear programming (FLP) problems (where some or all coefficients can be fuzzy sets and the inequality relations between fuzzy sets can be given by a certain fuzzy relation) as multiple fuzzy reasoning schemes (MFR), where the antecedents of the scheme correspond to the con- straints of the FLP problem and the fact of the scheme is the objective of the FLP problem. Then the solution process consists of two steps: first, for every decision variablex∈ Rn, we compute the maximizing fuzzy set, M AX(x), via sup-min convolution of the antecedents/constraints and the fact/objective, then an (optimal) solution to FLP problem is any point which produces a maximal element of the set{MAX(x) | x ∈ Rn} (in the sense of the given inequality relation). We show that our solution process for a classi- cal (crisp) LP problem results in a solution in the classical sense, and (under well-chosen inequality relations and objective function) coincides with those suggested by [Buc88, Del87, Ram85, Ver82, Zim76]. Furthermore, we show how to extend the proposed solution principle to non-linear programming problems with fuzzy coefficients.

Keywords: Compositional rule of inference, multiple fuzzy reasoning, fuzzy mathematical programming, possibilistic mathematical programming.

Published in: Proceedings of 2nd International Workshop on Current Issues in Fuzzy Technolo- gies, Trento, May 28-30, 1992, University of Trento, 1993 45-54.

Partially supported by the German Academic Exchange Service (DAAD) and Hungarian Re- search Foundation OTKA under contracts T 4281, I/3-2152 and T 7598.

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1 Statement of LP problems with fuzzy coefficients

We consider LP problems, in which all of the coefficients are fuzzy quantities (i.e.

fuzzy sets of the real line R), of the form maximize c˜1x1+· · · + ˜cnxn

subject to a˜i1x1+· · · + ˜ainxn <

∼ ˜bi, i = 1, . . . , m, (1) wherex∈ Rnis the vector of decision variables,˜aij, ˜biand˜cjare fuzzy quantities, the operations addition and multiplication by a real number of fuzzy quantities are defined by Zadeh’s extension principle [Zad75], the inequality relation∼ is given<

by a certain fuzzy relation and the objective function is to be maximized in the sense of a given crisp inequality relation≤ between fuzzy quantities.

The FLP problem (1) can be stated as follows: Findx ∈ Rnsuch that

˜

c1x1+· · · + ˜cnxn≤ ˜c1x1+· · · + ˜cnxn (2)

˜

ai1x1+· · · + ˜ainxn <

∼ ˜bi, i = 1, . . . , m,

i.e. we search for an alternative,x, which maximizes the objective function sub- ject to constraints.

We recall now some definitions needed in the following.

A fuzzy number˜a is a fuzzy quantity with a normalized (i.e. there exists a unique a∈ R, such that µ˜a(a) = 1), continuous, fuzzy convex membership function of bounded support. The real numbera which maximizes the membership function of the fuzzy number˜a called the peak of ˜a, and we write peak(˜a) = a.

A symmetric triangular fuzzy number˜a denoted by (a, α) is defined as µ˜a(t) = 1− |a − t|/α if |a − t| ≤ α and µ˜a(t) = 0 otherwise, where a∈ R is the peak and 2α > 0 is the spread of ˜a.

We shall use the following crisp inequality relations between two fuzzy quantities

˜ a and ˜b

˜

a≤ ˜b iff max{˜a, ˜b} = ˜b˜ (3)

˜

a≤ ˜b iff ˜a ⊆ ˜b (4)

wheremax is defined by Zadeh’s extension principle.˜

The compositional rule of inference scheme with several relations (called Multiple Fuzzy Reasoning Scheme) [Zad73] has the general form

Fact X has property P

Relation 1: X and Y are in relation W1

. . . . . .

Relation m: X and Y are in relation Wm

Consequence: Y has property Q

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whereX and Y are linguistic variables taking their values from fuzzy sets in clas- sical setsU and V , respectively, P and Q are unary fuzzy predicates in U and V , respectively,Wiis a binary fuzzy relation inU × V , i=1,. . . ,m.

The consequenceQ is determined by [Zad73]

Q = P

m i=1

Wi

or in detail,

µQ(y) = sup

x∈Umin{µP(x), µW1(x, y), . . . , µW1(x, y)}.

2 Multiply fuzzy reasoning for solving FLP problems

We consider FLP problems as MFR schemes, where the antecedents of the scheme correspond to the constraints of the FLP problem and the fact of the scheme is interpreted as the objective function of the FLP problem. Then the solution pro- cess consists of two steps: first, for every decision variable x ∈ Rn, we com- pute the maximizing fuzzy set, M AX(x), via sup-min convolution of the an- tecedents/constraints and the fact/objective, then an (optimal) solution to the FLP problem is any point which produces a maximal element of the set{MAX(x) | x∈ Rn} (in the sense of the given inequality relation).

We interpret the FLP problem

maximize c˜1x1+· · · + ˜cnxn

subject to a˜i1x1+· · · + ˜ainxn <

∼ ˜bi, i = 1, . . . , m, (5) as Multiple Fuzzy Reasoning schemes of the form

Antecedent 1 Constraint1(x) := ˜a11x1+· · · + ˜a1nxn <

∼ ˜b1

. . . . . .

Antecedent m Constraintm(x) := ˜am1x1+· · · + ˜amnxn <

∼ ˜bm Fact Goal(x) := ˜c1x1+· · · + ˜cnxn

Consequence M AX(x)

wherex ∈ Rn and the consequence (i.e. the maximizing fuzzy set)M AX(x) is computed as follows

M AX(x) = Goal(x)◦

m i=1

Constrainti(x),

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i.e.

µM AX(x)(v) = sup

u min{µGoal(x)(u), µConstraint1(x)(u, v), . . . , µConstraintm(u, v)}.

We regard M AX(x) as the (fuzzy) value of the objective function at the point x∈ Rn. Then an optimal value of the objective function of problem (5), denoted byM , is defined as

M := sup{MAX(x)|x ∈ Rn}, (6)

wheresup is understood in the sense of the given inequality relation≤. Finally, a solutionx ∈ Rnto problem (5) is obtained by solving the equation

M AX(x) = M.

The set of solutions of problem (5) is non-empty iff the set of maximizing elements of

{MAX(x)|x ∈ Rn} (7)

is non-empty.

Remark 2.1 To determine the maximum of the set (7) with respect to the in- equality relation≤ is usually a very complicated process. However, this problem can lead to a crisp LP problem [Zim78, Buc89], crisp multiple criteria parametric linear programming problem (see e.g. [Del87, Del88, Ver82, Ver84]) or nonlinear mathematical programming problem (see e.g. [Zim86]). If the inequality relation for the objective function is not crisp, then we somehow have to find an element from the set (7) which can be considered as a best choice in the sense of the given fuzzy inequality relation [Ovc89, Orl80, Ram85, Rom89, Rou85, Tan84].

3 Extension to FMP problems with fuzzy coefficients

In this section we show how the proposed approach can be extended to non-linear FMP problems with fuzzy coefficients. Generalizing the classical MP problem

maximize g(c, x)

subject to fi(ai, x)≤ bi, i = 1, . . . , m,

where x ∈ Rn, c = (ci, . . . , ck) and ai = (ai1, . . . , ail) are vectors of crisp coefficients, we consider the following FMP problem

maximize g(˜c1, . . . ˜ck, x)

subject to fiai1, . . . ˜ail, x)∼ ˜b< i, i = 1, . . . , m,

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wherex∈ Rn,c˜h,h = 1, . . . , k, ˜ais,s = 1, . . . , l, and ˜biare fuzzy quantities, the functionsg(˜c, x) and fiai, x) are defined by Zadeh’s extension principle, and the inequality relation∼ is defined by a certain fuzzy relation. We interpret the above<

FMP problem as MFR schemes of the form

Antecedent 1: Constraint1(x) := fia11, . . . ˜a1l, x)∼ ˜b< 1

. . . . . .

Antecedent m: Constraintm(x) := fmam1, . . . ˜aml, x)∼ ˜b< m Fact: Goal(x) := g(˜c1, . . . , ˜ck, x)

Consequence M AX(x)

Then the solution process is carried out analogously to the linear case, i.e an opti- mal value of the objective function,M , is defined by (6), and a solution x ∈ Rn is obtained by solving the equation

M AX(x) = M.

4 Relation to classical LP problems

In this section we show that our solution process for classical LP problems results in a solution in the classical sense. A classical LP problem can be stated as follows

Axmax≤b< c, x > (8)

Let X be the set of optimal solutions to (8), and if X is not empty, then let v =< c, x> denote the optimal value of its objective function.

In the following¯a denotes the characteristic function of the singleton a ∈ R, i.e.

µ¯a(t) = 1 if t = a and µ¯a(t) = 0 otherwise.

Consider now the crisp problem (8) with fuzzy singletons and crisp inequality re- lations

max¯

Ax≤¯b< ¯c, x > (9)

where ¯A = [¯ai,j], ¯b = (¯bi),¯c = (¯cj), and

¯

ai1x1+· · · + ¯ainxn≤ ¯biiffai1x1+· · · + ainxn≤ bi, i.e.

µ¯ai1x1+···+¯ainxn≤¯bi(u, v) =

1 if u = v and < ai, x >≤ bi

0 if u = v and < ai, x > > bi

0 if u = v

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The following theorem can be proved directly by using the definition of the in- equality relation (10).

Theorem 4.1 The sets of solutions of LP problem (8) and FLP problem (9) are equal, and the optimal value of the FLP problem is the characteristic function of the optimal value of the LP problem.

5 Crisp objective and fuzzy constraints

FLP problems with crisp inequality relations in fuzzy constraints and crisp objec- tive function can be formulated as follows (see e.g. Negoita’s robust programming [Neg81], Ramik and Rimanek’s approach [Ram85])

max < c, x >

s.t. ˜ai1x1+· · · + ˜ainxn≤ ˜bi, i = 1, . . . , m. (11) It is easy to see that problem (11) is equivalent to the crisp MP:

maxx∈X < c, x > (12)

where,

X ={x ∈ Rn| ˜ai1x1+· · · + ˜ainxn≤ ˜bi, i = 1, . . . , m}.

Now we show that our approach leads to the same crisp MP problem (12). Consider problem (11) with fuzzy singletons in the objective function

max < ¯c, x >

s.t. ˜ai1x1+· · · + ˜ainxn≤ ˜bi, i = 1, . . . , m.

where the inequality relation≤ is defined by

µ˜ai1x1+···+˜ainxn≤˜bi(u, v) =

1 if u = v and < ˜ai, x >≤ ˜bi

0 if u = v and < ˜ai, x > > ˜bi

0 if u = v

Then we have

µM AX(x)(v) =

 1 if v =< c, x > and x∈ X 0 otherwise

Thus, to find a maximizing element of the set{MAX(x) | x ∈ Rn} in the sense of the given inequality relation we have to solve the crisp problem (12).

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6 Fuzzy objective and crisp constraints

Consider the FLP problem with fuzzy coefficients in the objective function and fuzzy singletons in the constraints

maximize ˜c1x1+· · · + ˜cnxn

subject to ¯ai1x1+· · · + ¯ainxn≤ ¯bi, i = 1, . . . , m, (13) where the inequality relation for constraints is defined by (10) and the objective function is to be maximized in the sense of the inequality relation (3), i.e.

< ˜c, x >≤ < ˜c, x > iff ˜max{< ˜c, x >, < ˜c, x >} =< ˜c, x>

ThenµM AX(x)(v), ∀v ∈ R, is the optimal value of the following crisp MP prob- lem

maximize µ˜c1x1+···+˜cnxn(v) subject to Ax≤ b,

and the problem of computing a solution to FLP problem (13) leads to the same crisp multiple objective parametric linear programming problem obtained by Del- gado et al. [Del87, Del88] and Verdegay [Ver82, Ver84].

7 Relation to Buckley’s possibilistic LP

We show that when the inequality relations in an FLP problem are defined in a possibilistic sense then the optimal value of the objective function is equal to the possibility distribution of the objective function defined by Buckley [Buc88].

Consider a possibilistic LP

maximize Z := ˜c1x1+· · · + ˜cnxn

subject to ˜ai1x1+· · · + ˜ainxn≤ ˜bi, i = 1, . . . , m. (14) The possibility distribution of the objective functionZ, denoted by P oss[Z = z], is defined by [Buc88]

P oss[Z = z] = sup

x min{P oss[Z = z | x], P oss[< ˜a1, x >≤ ˜b1], . . . , P oss[< ˜am, x >≤ ˜bm]}, whereP oss[Z = z | x], the conditional possibility that Z = z given x, is defined

by

P oss[Z = z| x] = µ˜c1x1+···+˜cnxn(z).

The following theorem can be proved directly by using the definitions ofP oss[Z = z] and µM(v).

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Theorem 7.1 For the FLP problem maximize c˜1x1+· · · + ˜cnxn

subject to a˜i1x1+· · · + ˜ainxn <

∼ ˜bi, i = 1, . . . , m. (15) where the inequality relation∼ is defined by<

µ˜a

i1x1+···+˜ainxn<

˜bi(u, v) =



P oss[< ˜ai, x >≤ ˜bi] if u = v

0 otherwise

and the objective function is to be maximized in the sense of the inequality relation (4), the following equality holds

µM(v) = P oss[Z = v],∀v ∈ Rn.

So, if the inequality relations for constraints are defined in a possibilistic sense and the objective function is to be maximized in relation (4) then the optimal value of the objective function of problem (15) is equal to the possibility distribution of the objective function of possibilistic LP (14).

In a similar manner it can be swhon that the proposed solution concept is a gen- eralization of Zimmermann’s method [Zim76] for solving LP problems with soft constraints.

8 Example

We illustrate our approach by the following FLP problem maximize cx˜

subject to ax˜ ∼ ˜b, x ≥ 0,< (16) where ˜a = (1, 1), ˜b = (2, 1) and ˜c = (3, 1) are fuzzy numbers of symmetric triangular form, the inequality relation∼ is defined in a possibilistic sense, i.e.<

µ˜ax <

˜b(u, v) =



P oss[˜ax≤ ˜b] if u = v,

0 otherwise

and the inequality relation for the values of the objective function is defined by

˜

cx≤ ˜cx iffpeak(˜cx)≤ peak(˜cx).

It is easy to compute that

µM(v) = µM AX(2)(v) =

4− v/2 if v/x ∈ [3, 4]

v/x− 2 if v/x ∈ [2, 3]

0 otherwise

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4 6 8 1

so, the unique solution to (16) isx = 2. This result can be interpreted as: the solution to problem (16) is equal to the solution to the following crisp LP problem

max 3x

s.t. x≤ 2, x ≥ 0,

where the coefficients are the peaks of the fuzzy coefficients of problem (16), and the optimal value of problem (16), which can be called ”v is approximately equal to 6”, can be considered as an approximation of the optimal value of the crisp problemv= 6.

Figure 1. The membership function of the optimal value of problem (16).

Concluding remarks. We have interpreted FLP problems with fuzzy coeffi- cients as MFR schemes, where the antecedents of the scheme correspond to the constraints of the FLP problem and the fact of the scheme is the objective of the FLP problem. Using the compositional rule of inference, we have shown a method for finding an optimal value of the objective function and an optimal solution. In the general case the computerized implementation of the proposed solution prin- ciple is not easy. To computeM AX(x) we have to solve a generally non-convex and non-differentiable mathematical programming problem. However, the stability property of the consequence in MFR schemes under small changes of the member- ship function of the antecedents [Ful91] guarantees that small rounding errors of digital computation and small errors of measurement in membership functions of the coefficients of the FLP problem can cause only a small deviation in the member- ship function of the consequence,M AX(x), i.e. every successive approximation method can be applied to the computation of the linguistic approximation of the exactM AX(x). As was pointed out in Section 2, to find an optimal value of the objective function,M , from the equation (6) can be a very complicated process (for related works see e.g. [Car82, Car87, Neg81, Orl80, Ver82, Ver84, Zim85, Zim86) and very often we have to put up with a compromise solution [Rom89].

An efficient fuzzy-reasoning-based method is needed for the exact computation ofM . Our solution principle can be applied to multiple criteria mathematical pro-

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gramming problems with fuzzy coefficients. This topic will be the subject of future research.

Acknowledgements

The authors are indebted to Brigitte Werners of RWTH Aachen and the par- ticipants of the Workshop on Current Issues on Fuzzy Technologies 1992 for their valuable comments and helpful suggestions.

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