Theory and Methodology
Optimal fuzzy counterparts of scheduling rules

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Abstract

The optimality of a fuzzy logic alternative to the usual treatment of uncertainties in a scheduling system using probability theory is examined formally. Fuzzy scheduling techniques proposed in the literature either fuzzify directly the existing scheduling rules, or solve mathematical programming problems to determine the optimal schedules. In the former method, the fuzzy optimality for the optimal scheduling rules is usually not justified but still assumed. In this paper, the necessary conditions for fuzzy optimality are defined, and fuzzy counterparts of some of the well-known scheduling rules such as shortest processing time (SPT) and earliest due date (EDD) are developed.

Section snippets

Introduction and scope

The purpose of this paper is to investigate the necessary conditions for optimality of fuzzy counterparts of classical optimal scheduling rules, such as shortest processing time (SPT) and earliest due date (EDD). Essentially, any element of a scheduling problem may be uncertain. Rather than following the usual procedure of fuzzifying a crisp optimum scheduling rule, we examine the optimization of fuzzy schedules by using precise definitions of fuzzy ranking and dominance.

Fuzzy logic, which was

Fuzzy ranking and dominance

In all fuzzy scheduling problems, a fuzzy dominance relation needs to be defined in order to obtain a sequence of jobs. In mathematical programming solutions, this dominance relation is embedded in the problem formulation, but most of the fuzzy scheduling procedures developed to date simply fuzzify the crisp optimal rules and use a fuzzy dominance relation to rank the schedulable jobs, without justifying that fuzzy optimality has been reached. In this section, we will define the concept of

Fuzzy scheduling

Here, we assume that every element of a scheduling system may be fuzzy. Let j=1,…,J denote the jobs to be scheduled, t̂j the fuzzy processing times, d̂j the fuzzy due dates, Ĉj the fuzzy completion times, where Ĉ[j]=∑ji=1t̂[i] and subscript [j] is used to denote the job sequenced in the jth place, L̂j the fuzzy lateness, where L̂j=Ĉjd̂j, L̂max the maximum fuzzy lateness, where L̂max=wmaxj{Ĉjd̂j}, T̂j the fuzzy tardiness, where T̂j=wmax{0,Ĉjd̂j}, and T̂max the maximum fuzzy tardiness,

Numerical example

In this section, we use a GLRFN [3] with L(z)=1−zpL and R(z)=1−zpR. Thus, a GLRFN of this type can be represented by the six-tuple (pL,pR,a1(0),a1(1),a2(1),a2(0)). Table 2 shows the fuzzy data for eight jobs to be sequenced. Since the data involve GLRFN, the fuzzy mean will not necessarily satisfy consistency-I, therefore it will result in a heuristic approximation for FWSCPT, CWSFPT, SFPT, EFDD, LNFDD rules; on the other hand, for this case, F0 i.e. F1 with p1(α)=p2(α)=12 will yield fuzzy

Summary and conclusions

After defining necessary conditions, namely linearity, invertibility, multiplicity, additivity, consistency I and -II for the fuzzy optimality of scheduling rules, properties of AC family of ranking functions are investigated for the purpose of fuzzy scheduling. Two scenarios, weak and strong dominance, are distinguished and it is shown that independent of the shape of the membership function, under strong dominance, the AC family satisfies consistency-I. In the weak dominance case, which is

Acknowledgements

The research in this paper has been supported in part by US National Science Foundation. The authors would like to thank to the referees and Ágnes Galambosi for their constructive suggestions.

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