Computers ind. Engng Vol. 14, No. 4, pp. 387-393, 1988 0360-8352/88 $3.00+0.00 Printed in Great Britain. All rights reserved Copyright © 1988 Pergamon Press pie SINGLE FACILITY SCHEDULING WITH NONLINEAR PROCESSING TIMES JATINDERN. D. GUPTA 1 and SUSHIL K. GUPTA 2 1Department of Management Science, Ball State University, Muncie, IN 47303 and 2Department of Decision Sciences, Florida International University, Miami, FL 33199, U.S.A. (Received for publication 10 December 1987) Abstract--This paper considers the static single facility scheduling problem where the processing times of jobs are a monotonically increasing function of their starting (waiting) times and the objective is to minimize the total elapsed time (called the makespan) in which all jobs complete their processing. Based on the combinatorial analysis of the problem, an exact optimization algorithm is developed for the general processing time function which is then specialized for the linear case. In view of the excessive computational burden of the exact optimization algorithm for the nonlinear processing time functions, heuristic algorithms are proposed. The effectiveness of these proposed algorithms is empirically evaluated and found to indicate that these heuristic algorithms yield optimal or near optimal schedules in many cases. INTRODUCTION Consider the following static single facility problem: a set of n independent, single-operation jobs, is ready for being processed at time zero on a single machine. Neither job splitting nor machine idleness are allowed. The processing time of each job depends on its starting (or waiting) time in the sequence. It is desired to find that processing order (schedule) which minimizes the makespan, defined as the total elapsed time in which all jobs complete their processing. Such situations often occur in many chemical and metallurgical processes. For example, in steel rolling mills, ingots are heated to the required temperature before rolling. In this case, the furnace is the single facility and ingots are the independent jobs to be processed (heated). Heating time depends upon the ingot's current temperature, which depends upon the time it has been waiting. During the waiting period, the ingot cools down thus requiring more heating time in the furnace. It is desired to minimize the total time spent by all available ingots in the heating shop. For each job i, let Pi and ti be its processing and starting times respectively. Since the processing time of job i depends on its starting time, the relationship between Pi and ti can be represented as: Pi = f(ti). (1) The exact form of the function f in equation (1) depends upon the specific production process under consideration. Mathematically, the linear, the quadratic, and the general forms are expressed as follows: Pi = ai + biti (linear) (2) Pi = ai + biti + cit 2 (quadratic) (3) Pi = ai + biti + ci t2 + • • • + mit m (general) (4) where ai, bi, Ci, • • • , mi are non-negative constants. As stated before, the objective is to find the schedule (or an order) S = ([1], [2] .... [K] ..... [n]), where [K] is the job in Kth position that will minimize makespan, T(S), computed as follows: T(S) = Z Pi,I (5) i=1 387