Scheduling problems with two competing agents to minimized weighted earliness–tardiness

doi:10.1016/j.cor.2012.05.019

Computers & Operations Research

Volume 40, Issue 1, January 2013, Pages 109-116

https://doi.org/10.1016/j.cor.2012.05.019 Get rights and content

Abstract

We study scheduling problems with two competing agents, sharing the same machines. All the jobs of both agents have identical processing times and a common due date. Each agent needs to process a set of jobs, and has his own objective function. The objective of the first agent is total weighted earliness–tardiness, whereas the objective of the second agent is maximum weighted deviation from the common due date. Our goal is to minimize the objective of the first agent, subject to an upper bound on the objective value of the second agent. We consider a single machine, and parallel (both identical and uniform) machine settings. An optimal solution in all cases is shown to be obtained in polynomial time by solving a number of linear assignment problems. We show that the running times of the single and the parallel identical machine algorithms are O(n^m+3), where n is the number of jobs and m is the number of machines. The algorithm for solving the problem on parallel uniform machine requires O(n^m+3m³) time, and under very reasonable assumptions on the machine speeds, is reduced to O(n^m+3). Since the number of machines is given, these running times are polynomial in the number of jobs.

Introduction

In classical Just-in-Time (JIT) scheduling problems, the objective is minimum earliness–tardiness cost of the job completion times from their due dates. Most of the early studies (see e.g. Baker and Scudder [1]) focused on scheduling problems where all the jobs share a common due date. Among these studies, some considered minsum objectives (where the scheduler goal is to minimize the total cost incurred by all the jobs). Others researchers focused on minmax objectives (where the goal is to minimize the cost of the worst scheduled job). In their seminal paper, Hall and Posner [2] proved that the single machine problem to minimize the weighted deviation of the jobs completion times from a common due date is NP-hard. This minsum version is known as the Weighted Earliness–Tardiness (WET) problem. Cheng and Li [3] proved that the problem of minimizing the maximum weighted deviations among the job completion times from a common due date is NP-hard. This minmax version is known as the Minimum Weighted Absolute Lateness (MWAL) problem.

In recent years, scheduling researchers have focused on a setting of two competing agents. In this setting, two agents who need to process their own sets of jobs, compete on the use of a common resource. Each one of the two agents has his own objective function, and the goal is to find the joint schedule that minimizes the value of the objective function of one agent, subject to an upper bound on the value of the objective function of the second agent. Baker and Cole Smith [4] introduced the first scheduling paper dealing with two agents sharing a single processor. They focused on minimizing makespan, maximum lateness and total weighted completion time. Agnetis et al. [5] extended significantly the list of scheduling measures and the machine settings, and were followed by: Ng et al. [6], Cheng et al. [7], Lee et al. [8], Agnetis et al. [9], Gawiejnowicz et al. [10], Leung et al. [11], Mor and Mosheiov [12], Lee et al. [13], Li and Hsu [14], Li and Yuan [15], and Mor and Mosheiov [16], among others.

In this paper we study a two-agent scheduling JIT problem, where the objective is to minimize the minsum measure of the first agent (WET), subject to an upper bound on the minmax measure of the second agent (MWAL). This setting of objective functions is a generalization of some of the settings introduced in [5], since the weighted earliness–tardiness objective (of the first agent) is clearly an extension of weighted tardiness or of total completion times studied in Agnetis et al., and maximum weighted deviation (of the second agent) is an extension of maximum weighted completion time assumed in Agnetis et al. The general problem (i.e. assuming general job-dependent processing times) is NP-hard, since, as mentioned above, even the single agent WET problem is NP-hard. (Note that WET is NP-hard even for symmetric earliness–tardiness cost, and for a non-restrictive, i.e. sufficiently large, common due date, see Hall and Posner [2]). We focus here on the important and extensively studied special case of identical jobs. Beyond its theoretical importance, this setting is known to have numerous applications, including the wide range of common production lines focusing on identical items. Scheduling with identical jobs has been studied extensively, under various objective functions and machine settings. The list of papers dealing with JIT scheduling with both identical jobs and a common due date contains (among others): Hall and Posner [2], Cheng and Chen [17], Mosheiov and Shadmon [18], Cheng et al. [19], Mosheiov and Yovel [20], Li et al. [21], Tuong and Soukhal [22], Mosheiov and Sarig [23], [24], Tuong and Soukhal [25], and Drobouchevitch and Sidney [26].

The input for the problems studied here consists of (i) a list of earliness and tardiness weights for the first agent, (ii) a list of earliness and tardiness weights for the second agent, (iii) an upper bound on the maximum permitted earliness/tardiness value of the second agent, and (iv) the common due date (which may be either large or small, i.e. non-restrictive or restrictive). First we solve the single machine problem. We introduce an O(n⁴) solution procedure (where n is the number of jobs of both agents). Then we extend this solution procedure to the setting of parallel machines. We consider first parallel identical machines, and introduce an O(n^m+3) solution algorithm (where m is the number of machines). Then we study parallel uniform machines, and an O(n^m+3m³) time algorithm is introduced for the general case. An improved O(n^m+3) algorithm is proposed under a very reasonable assumption on the machine speed factors. Since m is given (i.e. m is not part of the input), the above running times for parallel machines settings are polynomial in n.

The paper is organized as follows. In Section 2 we introduce the notation and the formulation of the problem. Section 3 focuses on the single machine case. 4 Parallel identical machines, 5 Uniform machines present the solutions for parallel identical and uniform machines, respectively.

Section snippets

Formulation

Two agents, denoted by X and Y, respectively, need to process their jobs on a single processor. Agent X processes n^X jobs, and agent Y processes n^Y jobs. n=n^X+n^Y denote the total number of jobs. All the jobs are available at time zero, and preemption is not allowed. $p_{j}^{Z}$ denotes the processing time of job j of agent Z, $j = 1, \dots, n^{Z}, Z = X, Y$ . We assume that the job processing times are identical, and after appropriate scaling (i.e. without loss of generality) they are assumed to be unit time jobs, i.e. $p$

The single machine case

In this section we introduce a polynomial time solution for problem P1, where both agents share a single machine. We first claim that an optimal schedule exists with no idle time between consecutive jobs. [If an idle time exists between two early jobs, close this idle time by delaying the early jobs. The resulting schedule is clearly feasible since the cost of the early Y-jobs is reduced, and the objective function value (based on the X-jobs) is smaller. Similarly, if an idle time exists

Parallel identical machines

We start this section with an extension of the solution procedure introduced above to parallel identical machines (problem P2). The no idle-time property on each machine clearly holds. Property 1 remains valid as well for each one of the machines, i.e.:

Property 1

(parallel machines): An optimal schedule exists such that on each machine, either (i) the first scheduled job starts at time zero, or (ii) at least one Y-job has cost value of Q, or (iii) an X-job is completed exactly at the due date.

Proof

The proof is

Uniform machines

The above solution procedure is further extended in this section to a setting of parallel uniform machines (problem P3). We consider m uniform machines, i.e. machine-dependent speed factors are assumed. The speed factor of machine i is $σ_{i}, i = 1, \dots, m$ . For convenience we assume σ₁=1. Given the assumption of identical-time jobs, the processing time of job j on machine i is given by $P_{i j} = 1 / σ_{i}, i = 1, \dots, m$ . Without loss of generality we assume that the machines are sorted in a non-decreasing speed factor,

Conclusion

We solved scheduling problems with two competing agents on a single-machine and on parallel-machines (identical and uniform). The goal is to minimize the total weighted earliness–tardiness from a common due date of the first agent, subject to a maximum weighted deviation of the second agent. We focus on the setting of identical job processing times, and job-dependent earliness and tardiness weights. Several properties of an optimal solution are presented, and consequently, special structured

Acknowledgement

The second author is the Charles Rosen Chair of Management, The School of Business Administration, The Hebrew University. This paper was supported in part by the Recanati Fund of The School of Business Administration, The Hebrew University, Jerusalem, Israel.

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Scheduling problems with two competing agents to minimized weighted earliness–tardiness

Abstract

Introduction

Section snippets

Formulation

The single machine case

Parallel identical machines

Uniform machines

Conclusion

Acknowledgement

European Journal of Operational Research

Information Processing Letters

European Journal of Operational Research

Computers & Operations Research

European Journal of Operational Research

European Journal of Operational Research

Computers & Operations Research

European Journal of Operational Research

Computers & Operations Research

European Journal of Operational Research

Computers & Operations Research

Sequencing with earliness and tardiness penalties: a review

Operations Research Letters

Earliness–tardiness scheduling problems, I: weighted deviation of completion times about a common due date

Operations Research