Network Analysis of Fuzzy Bi-serial and Parallel Servers with a Multistage Flow Shop Model

Queuing theory is the one of the classical mathematical tool for study waiting lines or queues. In queuing theory a mathematical model is constructed to find various queue characteristics like average waiting time, mean queue length and number of the customers etc. Infact, waiting for service in queues has become an integral part of our daily life. Queues are often undesirable as they cost time, money and resources. Generally, queues come into play when the service resources are not sufficient to satisfy the demand. Most of the queuing optimization problems are static in nature in which the system characteristics are constant with respect to time or deterministic. In real life situations, one has to move from descriptive where all parameters are given to the prescriptive or normative where parameter changes with time and may be uncertain or vague in nature. In present scenario, it is difficult to find an exact distribution of arrival, service rate for the queuing applications. To express these rates mostly the linguistic expressions such as ‘service is fast’, ‘service is slow’ or ‘service speed is moderate’ are used. Therefore, it is more realistic to say the service rate, arrival rate etc are more possiblistic rather than probabilistic. Under these circumstances the fuzzy queuing models are more realistic approach than classical queuing theory model. From many decades, job scheduling has been a constructive area of research for deciding the order in which the jobs are to be processed. Proper scheduling of jobs not only save time but also increase the efficiency of the system. Flow shop scheduling models are effective tools for the optimal allocation of scare resources with objective of optimizing one or several criterions. In this paper a linkage between network of queues consisting of bi-serial and parallel servers linked to a common serve (phase I) with a multistage flow shop scheduling model (Phase II) is considered to deal with real world fuzzy network problem. Fuzzy sets are used to model uncertain arrival and service rates. The objective considered is average waiting time, mean queue length and total completion time. The triangular fuzzy membership function, α -cut approach and various fuzzy arithmetic operations are considered to estimate the uncertainty associated with input parameters. As problem related to completion time is NP-hard, a heuristic algorithm is developed to find the solution with maximum grades for objective. The developed algorithm is tested with numerical computation. The results of proposed model is useful and significant for system designers and practitioners. The model finds its applications in manufacturing or assembling line process in which units processed through a series of stations, each performing a given task. Practical situation can be observed in a registration process (vehicle registration) where the registrants have to visit a series of desks (advisor, department chairperson, cashier, etc.), or in a clinical physical test procedure where the patients have to pass through a series of stages (lab tests, electrocardiogram, chest X-ray etc.).