Stochastic Single Machine JIT Scheduling with Geometric Processing Times and Due Dates

Yuncheng Luo,Frederic Kratz

doi:10.1155/2021/8874445

Abstract

In this paper, we investigate a static stochastic single machine JIT scheduling problem in which the jobs’ processing times are stochastically independent and follow geometric distributions whose mean is provided, due dates are geometrically distributed with a common mean, and both the unit penalty of earliness/tardiness and the fixed penalty of earliness/tardiness are deterministic and different. The objective is to minimize the expected total penalties for quadratic earliness, quadratic tardiness, and early and tardy jobs. We prove that the optimal schedule to minimize this problem is V-shaped with respect to the ratio of mean processing time to unit tardiness penalty under the specific condition. Also, we show a special case and two theorems related to this JIT scheduling problem under specific situations where the optimal solutions exist. Finally, based on the V-shaped characteristic, a dynamic programming algorithm is designed to achieve an optimal V-shaped schedule in pseudopolynomial time.

Highlights

In just-in-time (JIT) scheduling, the decision maker intends to find a rational scheduling scheme so that all jobs will not be completed either too early or too late
We show that the optimal solution of this problem has V-shaped characteristic with respect to the ratio of mean processing time to unit tardiness penalty, that is, the schedule will first arrange jobs in nonincreasing order of the ratio of mean processing time to unit tardiness penalty and arrange jobs in nondecreasing order of the ratio of mean processing time to unit tardiness penalty
We study a static stochastic single machine JIT scheduling problem in which the processing times are stochastically independent and geometrically distributed with distinct parameters, the due dates are geometrically distributed with a common parameter, and both the unit penalty of earliness/tardiness and the fixed penalty of earliness/tardiness are certain and distinct, where the JIT scheduling concept means that all the jobs are scheduled to complete as close to their due dates as possible. e objective is to find a schedule of jobs that minimizes the expected total penalties for quadratic earliness, quadratic tardiness, and early and tardy jobs

Summary

Introduction

In just-in-time (JIT) scheduling, the decision maker intends to find a rational scheduling scheme so that all jobs will not be completed either too early or too late. Soroush and Fredendall [28] analyzed the single machine scheduling problem with normally distributed processing times and deterministic due dates to minimize the total expected earliness and tardiness penalties. We, under stochastic scenario, address a static single machine JIT scheduling problem with the objective of minimizing the expected total penalties for quadratic earliness, quadratic tardiness, and early and tardy jobs. In this scheduling problem, it is assumed that processing times follow geometric distributions with distinct parameters and due dates follow geometric distributions with a common parameter.

Problem Definition and Formulation

The V-Shaped Characteristic of the Optimal Schedule

Dynamic Programming Algorithm

Conclusions