Possible optimality of solutions in a single machine

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Possible optimality of solutions in a single machine

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'A fuzzy set A is normal if and only if there exists x E R such that pA(x) = 1. ... S(p) = 0 then we assume that opts(p) = 0. Let us notice that .... jobs i E pl, that can be moved after job j in p ... the problem includes processing times tl, .. ., t,, due dates ...

Possible optimality of solutions in a single machine scheduling problem with fuzzy parameters Stefan Chanas Inst. of Industrial Eng. and Man. Wroclaw University of Technology Smoluchowskiego 25, 50-370 Wroclaw, POLAND [email protected]

Abstract A single machine scheduling problem with parameters given in the form of fuzzy numbers is considered. It is assumed that the optimal schedule in such a problem cannot be determined precisely (since the parameters are not known a priori). In the paper the concept of degree of possible optimality of a given schedule is introduced. An algorithm of calculating the degree of o p timality in the general case is presented and then it is applied to two particular cases of the problem.

Keywords: Scheduling; Single machine; f i z z y number; Degree of optimality

Adam Kasperski Inst. of Industrial Eng. and Man. Wroclaw University of Technology Smoluchowskiego 25, 50-370 Wroclaw, POLAND [email protected]

It follows from the properties of the membership function of a fuzzy number A that each its A-cut A X , A E (O,l], is a closed interval. We denote it by AX = [&(A), a(A)].

Definition 3 A fuzzy number A is called interval number if its membership function is of the following form: 1 for for

o where -m

x~[a,Z], x E (-03, a) u (z,m), (2)