Fuzzy job-shop scheduling based on ranking level (λ, 1) interval-valued fuzzy numbers

Job-shop scheduling is a difficult problem, both theoretically and practically. The theoretical problems stem from the search for optimal schedules subject to a limited number of constraints, while the complexity of practical problems is due to the number and variety of constraints that are not rigid in the actual situations. Indeed, in real world descriptions there are many vaguely formulated relations and imprecise data. Although the job-shop scheduling problem has often been investigated, very little of this research is concerned with the uncertainty characterized by the imprecision in problem variables. In this paper, we consider a fuzzy job-shop scheduling problem with imprecise processing times. We use fuzzy numbers and level (lambda, 1) interval-valued fuzzy numbers for the representation of vague processing times. This problem is similar to a fuzzy multiple criteria optimization problem. The primary results obtained from this research are: 1) signed distance ranking fuzzy numbers used to obtain Property 5, a job-shop scheduling problem in the fuzzy sense and 2) signed distance ranking level (lambda, 1) interval-valued fuzzy numbers used to obtain Property 6, another job-shop scheduling problem in the fuzzy sense. We conclude that 1) the schedules obtained from Properties 5 and 6 are the same type as the crisp case, and 2) Property 5 is a special case of Property 6.