Arbitrary Processing Times Research Articles

Job Shop Scheduling under a Non-Renewable Resource Constraint

In this paper we consider the job shop scheduling problem under a discrete non-renewable resource constraint. We assume that jobs have arbitrary processing times and resource requirements and there is a unit supply of the resource at each time period. We develop an approximation algorithm for this problem and empirically test its effectiveness in finding the minimum makespan schedules. Most of the research done in the area of scheduling deals with the allocation of a single scarce resource over time to perform a collection of tasks. In this study, the scheduling environment is extended to include an additional non-renewable resource. The term non-renewable implies that the resource is actually being consumed by the jobs competing for it. Financial constraints are typical examples of such constraints and for this reason non-renewable resource constraints are often called financial constraints1. Materials shared by different products can also be regarded as a non-renewable resource in manufacturing environments. Considering these constraints explicitly is likely to yield more realistic schedules. A limited amount of research has been done in the area of non-renewable resource constrained scheduling. There are some polynomially bounded solution algorithms for precedence constrained scheduling problems2 and for pre-emptive scheduling of independent jobs on parallel machines1. In the case of arbitrary resource requirements and availabilities, Slowinski' states that Carlier has shown that even the non-preemptive single machine scheduling problem is NP-complete if job processing times are different from unity. In Toker et al.3, it is shown that when the amount of resource available at each time period is constant, the single machine non-renewable resource constrained problem is equivalent to a resource- free, two-machine flowshop problem. Hence it is solvable in polynomial time. They also extended their results to the m-machine case and showed how to transform different types of resource constrained problems into equivalent, unconstrained problems. The scheduling environment we consider is a job shop. A single non-renewable resource becomes available over time in equal quantities (say unity) and there is a set of jobs to be processed. Each operation requires an arbitrary amount of the non-renewable resource which must be available at the start of that operation and which is consumed during its processing. We used makespan as the performance measure. Optimization algorithms for the general job shop scheduling problem are restricted to implicit enumeration techniques, such as branch-and-bound based procedures4-7. On the other hand, approximation algorithms usually use a dispatching rule to give priorities to operations to be scheduled8. The non-renewable resource constrained job shop scheduling problem is NP-complete as, when all resource requirements are zero, it reduces to the unconstrained job shop problem which is NP-complete9. This complexity result serves as a formal justification to use approximation algorithms for the constrained problem. We first describe the approximation algorithm developed and then introduce two lower bounds for the n-job, m-machine resource-constrained job shop scheduling problem. Next, we discuss the computational results. Finally, we present our conclusions.

Job Shop Scheduling Under a Non-renewable Resource Constraint

In this paper we consider the job shop scheduling problem under a discrete non-renewable resource constraint. We assume that jobs have arbitrary processing times and resource requirements and there is a unit supply of the resource at each time period. We develop an approximation algorithm for this problem and empirically test its effectiveness in finding the minimum makespan schedules.

On Optimal Permutation Scheduling in Stochastic Proportionate Flowshops

Consider m machines in series with unlimited intermediate buffers and n jobs available at time zero. The processing times of job j on all m machines are equal to a random variable Xj with distribution Fj. Various cost functions are analyzed using stochastic order relationships. First, we focus on minimizing where cj is the weight (holding cost) and Tj the completion time of job j. We establish that if are in a class of distributions we define as SIFR, and and are increasing sequences of likelihood ratio-ordered and stochastic-ordered random variables, respectively, the job sequence [1, 2, … n ] is optimal among all static permutation schedules. Second, for arbitrary processing time distributions, if is an increasing sequence of likelihood ratio-ordered (hazard rate-ordered) random variables and the costs are nonincreasing, then a general cost function is minimized by the job sequence [1,2,…, n] in the stochastic ordering (increasing convex ordering) sense.

Analysis of two-stage transfer-line production system subject to inter-stage inspection and arbitrary arrival and processing time distributions

The purpose of this paper is to study the transient and steady-state behaviour of a two-stage transfer-line production system subject to attribute type of instantaneous inter-stage inspection. A stochastic model is developed to describe the system and analytical expressions have been obtained for certain measures of performance with arbitrary distributions of arrival and processing time at both the stages of production by using the state-space method. The results obtained by Gopalan and Ramesh [ Int. J. Management Syst. 4(1), 50–66 (1988)] and Gopalan and Anantharaman [ Ind. Engng J. XIX(3), 17–23 (March, 1990)] have been shown to be particular cases.

Scheduling under a Non-renewable Resource Constraint

This paper examines a single-machine, non-renewable-resource-constrained scheduling problem where jobs have arbitrary processing times and resource requirements. Unit supply of a resource is assumed at each time period. Performance criterion is makespan. It is proved that this problem is identical to the two-machine flowshop problem, enabling the use of Johnson's algorithm. Immediate extensions of this result are presented.

Random Review Production/Inventory Systems with Compound Poisson Demands and Arbitrary Processing Times

A production/inventory system is considered in which a single production facility produces items of a given type. The demand for the item is assumed to arrive according to a compound Poisson process. The time required to produce an item is assumed to follow an arbitrary distribution. An (s, S) policy is considered in which production stops at the instant that the inventory level is raised to S and production resumes at an inspection point when the inventory level is observed to have dropped to or below s for the first time. The time interval between two successive inspection points during a nonproduction period is a random variable which follows an arbitrary distribution. Under a cost structure which includes a set-up cost, a linear holding cost and a linear backorder cost, an expression for the expected cost per unit time is obtained for given control values. The operating cost during a cycle is shown to be convex in S for a given value of S-s. For the continuous review simple Poisson demand system, as well as some other special cases of the system considered here, it is shown that the expected cost per unit time is unimodal. Based on these properties, a procedure to find the optimal (s, S) policy is presented.

Scheduling the Open Shop to Minimize Mean Flow Time

It is shown that the problem of scheduling a two-processor n-job open shop nonpreemptively in order to minimize mean flow time is NP-complete even if input length is measured by the sum of the task lengths. The proof is similar in approach to that used by Garey, Johnson and Sethi to show NP-completeness of the two-processor flow shop mean flow problem. We assume previous results from their paper where possible and concentrate on those elements of the proof that are distinct from theirs.In addition, bounds are derived for the mean flow times of arbitrary and shortest processing time (SPT) first schedules for m-processor n-job systems in terms of the mean flow time of an optimal schedule.