A Fuzzy Approach for Minimizing Machine Rental Cost for a Specially-Structured Three-Stages Flow-Shop Scheduling Problem in a Fuzzy Environment

1. Introduction

Scheduling problem is a problem concerned with searching for an optimal or near optimal schedule subject to a number of constraints. Neuromas approaches have been investigated to solve the problem of scheduling with an aim to determine the sequence of processing jobs on a given set of machines. It's important and relevance to industry has promoted. Scheduling requires a variety of activities to accomplish a particular purpose with time and budget. Flow shop is the most classical studies manufacturing environment in scheduling literature. One of the earliest results in flow shop scheduling theory is an algorithm given by Johnson's (1954) for scheduling jobs in a two, three machine flow shop to minimize the time at which all jobs are completed. Gupta (1975) gave an algorithm to find the optimal schedule for specially structured flow shop scheduling. The work developed by Ignall and Scharge (1965), Bagga (1969), Szware (1977), Maggu and Das (1981), Singh et al. (2005), Gupta (1988) have an important consideration. Narain and Bagga (2005) studied a problem for obtaining a sequence gives minimum possible rental cost while minimize total elapsed time constrained by pre- defined rental policy. Schulz et al. (2020) studied a hybrid flow shop scheduling problem with variable discrete production speed level. Gheisariha et al. (2021) solved flexible flow shop scheduling problem with sequence based setup time, transportation time, and probable network using an enhanced multi- objective harmony search algorithm and Gaussian mutation.

In many scientific areas, such as system analysis and operators research, a model has to be setup-using data, which is only approximately known. Fuzzy sets theory, introduced by Zadeh (1965) makes this possible. Fuzzy numerical data can be represented by means of fuzzy subsets of the real line, known as fuzzy numbers. Dubois and Prade (1980) extended the use of algebraic operations on real numbers to fuzzy numbers by use of a fuzzification principle. In spite of having a vast decision making experience, the decision maker cannot always articulate the goals precisely. Decision-making in a fuzzy environment, developed by Bellman and Zadeh (1970) improved and a great help in the management decision problems. Zimmermann (1978) proposed the fuzzy set theory and its applications. Petrovic and Song (2006) studied the job sequence problem in a two- machine flow shop with the processing times of the jobs are characterized by triangular fuzzy set. Saracoglu and Suer (2018) proposed a methodology for multi- product parallel multistage cellular manufacturing company, where they applied on a case study from a shoe manufacturing plant for producing the product based on their due dates. Pang et al. (2020) introduced the flow shop scheduling problem and hybrid flow shop scheduling together aim to find the best scheduling strategy for making production plants. Shao et al. (2020) investigated a distributed fuzzy blocking flow- shop scheduling problem with processing times characterized by fuzzy numbers aim to minimize the fuzzy makespan among all factors.

In this paper, a special structured n- job of three machines flow shop scheduling with piecewise quadratic fuzzy processing times. (PFN) is studied. A heuristic approach for obtaining the sequence of the jobs, which minimize the rental cost of the machines, is proposed.