Approximate Reasoning for Solving Fuzzy Linear Programming

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We interpret fuzzy linear programming (FLP) problems (where some or all coeffi- ..... a generally non-convex and non-differentiable mathematical programming.
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 constraints 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 variable x ∈ 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 {M AX(x) | x ∈ Rn } (in the sense of the given inequality relation). We show that our solution process for a classical (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.

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˜1 x1 + · · · + c˜n xn < ˜ subject to a ˜i1 x1 + · · · + a ˜in xn ∼ bi , i = 1, . . . , m,

(1)

where x ∈ Rn is the vector of decision variables, a ˜ij , ˜bi and c˜j are fuzzy quantities, the operations addition and multiplication by a real number of fuzzy quantities are defined < is given by a certain by Zadeh’s extension principle [Zad75], the inequality relation ∼ fuzzy relation and the objective function is to be maximized in the sense of a given crisp ∗

Published in: Proceedings of 2nd International Workshop on Current Issues in Fuzzy Technologies, Trento, May 28-30, 1992, University of Trento, 1993 45-54. † Partially supported by the German Academic Exchange Service (DAAD) and Hungarian Research Foundation OTKA under contracts T 4281, I/3-2152 and T 7598.

1

inequality relation ≤ between fuzzy quantities. The FLP problem (1) can be stated as follows: Find x∗ ∈ Rn such that c˜1 x1 + · · · + c˜n xn ≤ c˜1 x∗1 + · · · + c˜n x∗n

(2)

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 a ˜ ≤ ˜b iff a ˜ ⊆ ˜b

(3) (4)

where max ˜ 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 Relation 1: ... Relation m:

X has property P X and Y are in relation W1 ... X and Y are in relation Wm

Consequence: Y has property Q where X and Y are linguistic variables taking their values from fuzzy sets in classical sets U and V , respectively, P and Q are unary fuzzy predicates in U and V , respectively, Wi is a binary fuzzy relation in U × V , i=1,. . . ,m. The consequence Q is determined by [Zad73] Q=P ◦

m \

Wi

i=1

or in detail, µQ (y) = sup min{µP (x), µW1 (x, y), . . . , µW1 (x, y)}. x∈U

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 process consists of two steps: first, for every decision variable x ∈ 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 the FLP problem is any point which produces a maximal 2

element of the set {M AX(x) | x ∈ Rn } (in the sense of the given inequality relation). We interpret the FLP problem maximize c˜1 x1 + · · · + c˜n xn < ˜ subject to a ˜i1 x1 + · · · + a ˜in xn ∼ bi , i = 1, . . . , m,

(5)

as Multiple Fuzzy Reasoning schemes of the form Antecedent 1 ... Antecedent m

< ˜ Constraint1 (x) := a ˜11 x1 + · · · + a ˜1n xn ∼ b1 ... < ˜ Constraintm (x) := a ˜m1 x1 + · · · + a ˜mn xn ∼ bm

Fact

Goal(x) := c˜1 x1 + · · · + c˜n xn

Consequence

M AX(x)

where x ∈ Rn and the consequence (i.e. the maximizing fuzzy set) M AX(x) is computed as follows m M AX(x) = Goal(x) ◦

\

Constrainti (x),

i=1

i.e.

µM AX(x) (v) = sup min{µGoal(x) (u), µConstraint1 (x) (u, v), . . . , µConstraintm (u, v)}. u

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 by M , is defined as M := sup{M AX(x)|x ∈ Rn }, (6) where sup is understood in the sense of the given inequality relation ≤. Finally, a solution x∗ ∈ Rn to 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 {M AX(x)|x ∈ Rn }

(7)

is non-empty. Remark 2.1 To determine the maximum of the set (7) with respect to the inequality 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, 3

where x ∈ Rn , c = (ci , . . . , ck ) and ai = (ai1 , . . . , ail ) are vectors of crisp coefficients, we consider the following FMP problem maximize g(˜ c1 , . . . c˜k , x) < ˜ subject to fi (˜ ai1 , . . . a ˜il , x) ∼ bi , i = 1, . . . , m, where x ∈ Rn , c˜h , h = 1, . . . , k, a ˜is , s = 1, . . . , l, and ˜bi are fuzzy quantities, the functions g(˜ c, x) and fi (˜ ai , 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: ... Antecedent m:

< ˜ Constraint1 (x) := fi (˜ a11 , . . . a ˜1l , x) ∼ b1 ... < ˜ Constraintm (x) := fm (˜ am1 , . . . a ˜ml , x) ∼ bm

Fact:

Goal(x) := g(˜ c1 , . . . , c˜k , x)

Consequence

M AX(x)

Then the solution process is carried out analogously to the linear case, i.e an optimal 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 max < c, x >

Ax≤b

(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. µa¯ (t) = 1 if t = a and µa¯ (t) = 0 otherwise. Consider now the crisp problem (8) with fuzzy singletons and crisp inequality relations max < c¯, x >

¯ ¯b Ax≤

(9)

where A¯ = [¯ ai,j ], ¯b = (¯bi ), c¯ = (¯ cj ), and a ¯i1 x1 + · · · + a ¯in xn ≤ ¯bi iff ai1 x1 + · · · + ain xn ≤ bi , i.e.

   1

if u = v and < ai , x > ≤ bi 0 if u = v and < ai , x > > bi µa¯i1 x1 +···+¯ain xn ≤¯bi (u, v) =   0 if u 6= v

(10)

The following theorem can be proved directly by using the definition of the inequality 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. 4

5

Crisp objective and fuzzy constraints

FLP problems with crisp inequality relations in fuzzy constraints and crisp objective 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. a ˜i1 x1 + · · · + a ˜in xn ≤ ˜bi , i = 1, . . . , m.

(11)

It is easy to see that problem (11) is equivalent to the crisp MP: max < c, x > x∈X

where,

(12)

X = {x ∈ Rn | a ˜i1 x1 + · · · + a ˜in xn ≤ ˜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. a ˜i1 x1 + · · · + a ˜in xn ≤ ˜bi , i = 1, . . . , m. where the inequality relation ≤ is defined by if u = v and < a˜i , x > ≤ b˜i µa˜i1 x1 +···+˜ain xn ≤˜bi (u, v) = 0 if u = v and < a˜i , x > > b˜i   0 if u 6= v    1

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 {M AX(x) | x ∈ Rn } in the sense of the given inequality relation we have to solve the crisp problem (12).

6

Fuzzy objective and crisp constraints

Consider the FLP problem with fuzzy coefficients in the objective function and fuzzy singletons in the constraints maximize c˜1 x1 + · · · + c˜n xn subject to a ¯i1 x1 + · · · + a ¯in xn ≤ ¯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′ > Then µM AX(x) (v), ∀v ∈ R, is the optimal value of the following crisp MP problem maximize µc˜1 x1 +···+˜cn xn (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 Delgado et al. [Del87, Del88] and Verdegay [Ver82, Ver84]. 5

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 := c˜1 x1 + · · · + c˜n xn subject to a ˜i1 x1 + · · · + a ˜in xn ≤ ˜bi , i = 1, . . . , m.

(14)

The possibility distribution of the objective function Z, denoted by P oss[Z = z], is defined by [Buc88] P oss[Z = z] = sup min{P oss[Z = z | x], P oss[< a ˜1 , x >≤ ˜b1 ], . . . , P oss[< a ˜m , x >≤ ˜bm ]}, x

where P oss[Z = z | x], the conditional possibility that Z = z given x, is defined by P oss[Z = z | x] = µc˜1 x1 +···+˜cn xn (z). The following theorem can be proved directly by using the definitions of P oss[Z = z] and µM (v). Theorem 7.1 For the FLP problem maximize c˜1 x1 + · · · + c˜n xn < ˜ subject to a ˜i1 x1 + · · · + a ˜in xn ∼ bi , i = 1, . . . , m.

(15)

< is defined by where the inequality relation ∼

µa˜

< ˜b (u, v) ain xn ∼ i1 x1 +···+˜ i

=

(

P oss[< a ˜i , 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 generalization of Zimmermann’s method [Zim76] for solving LP problems with soft constraints.

8

Example

We illustrate our approach by the following FLP problem maximize c˜x