Scheduling identical parallel machines with fixed delivery dates to minimize total tardiness

Highlights

• We consider identical parallel machine scheduling to minimize total tardiness.

• Jobs can only be delivered at exogenously given fixed delivery dates.

• Mathematical programming and a branch-and-bound algorithm solve small problems.

• A tabu search and a hybrid genetic algorithm are developed for larger problems.

Abstract

This paper addresses the problem of minimizing the total tardiness of a set of jobs to be scheduled on identical parallel machines where jobs can only be delivered at certain fixed delivery dates. Scheduling problems with fixed delivery dates are frequent in industry, for example when a manufacturer has to rely on the timetable of a logistics provider to ship its products to customers. We develop and empirically evaluate both optimal and heuristic solution procedures to solve the problem. As the problem is NP-hard, only relatively small instances can be optimally solved in reasonable computational time using either an efficient mathematical programming formulation or a branch-and-bound algorithm. Consequently, we develop a tabu search and a hybrid genetic algorithm to quickly find good approximate solutions for larger instances.

Introduction

This paper addresses the problem of scheduling jobs on identical parallel machines to minimize total tardiness, where jobs can only be delivered at certain fixed delivery dates.

More precisely, a set of jobs, J, where each job j = 1, …, n has a given integer processing time p_j and due date d_j is to be scheduled non-preemptively on m identical parallel machines, where each machine i = 1, …, m is ready at time zero and can process all jobs. The due dates are the times when a customer asks to receive a job and hence a job has to be both, processed and delivered before or at that due date to avoid tardiness penalties. There is a set K of exogenously given fixed delivery dates k = 1, …, s, where deliveries occur at times Δ₁ < ⋅⋅⋅ < Δ_s. The delivery capacity at each delivery date is assumed to be infinite, and times for delivery are neglected.

Each job has a completion time C_j and a delivery time D_j where D_j = min _{k ∈ K}{Δ_k|Δ_k ≥ C_j}, i.e. a job is delivered at the first delivery date after its completion, and its tardiness is defined as T_j = max {0, D_j − d_j}. In the standard three-field notation we can state this problem as $Pm|s=\overline{s}|{\sum }_j{T}_j,$ where $\overline{s}$ implies that the number of delivery dates is known and fixed in advance.

Scheduling problems with fixed delivery dates are often found in industry, for example when a manufacturing company relies on the timetables of shipping or airfreight companies to deliver its products to customers. The practical relevance of such problems is underlined by various papers addressing real-world situations with fixed delivery dates. For instance, Wang, Batta, and Szczerba (2005) consider a scheduling problem arising at the U.S. Postal Service, where mail processing operations are scheduled subject to an outbound truck delivery schedule. Scheduling of order consignment tasks at a logistics center subject to inbound and outbound truck arrivals and departures with different objective functions is studied by Carrera, Ramdane-Cherif, and Portmann (2010a); 2010b). Further practical examples of scheduling problems with fixed delivery dates originate in the consumer electronics industry. Li, Ganesan, and Sivakumar (2005) investigate how a computer manufacturer can synchronize its production schedule to airfreight timetables. Computer manufacturer Dell has the completed orders picked up by a logistics provider once a day, and Dell can reduce transportation cost by avoiding costly express services if jobs are scheduled sufficiently early (Stecke and Zhao, 2007). Ma, Chan, and Chung (2012) integrate sea freight schedules and production schedules for a manufacturer of electronic household appliances which essentially results in a single machine earliness-tardiness scheduling problem with additional consideration of different shipping times and costs.

Several authors conduct more theoretically oriented studies of scheduling problems with fixed delivery dates. Hall, Lesaoana, and Potts (2001) provide complexity results and algorithms for a variety of problems. Among others, they prove that the problem $Pm|s=\overline{s}|{\sum }_j{T}_j$ is NP-hard in the ordinary sense and propose a pseudopolynomial dynamic programming algorithm to solve it. Single machine problems with fixed delivery dates are studied by Matsuo (1988) who addresses a weighted total tardiness problem with the possibility of using overtime to expand capacity, by Yang (2000) who studies a problem with earliness penalties, and by Seddik, Gonzales, and Kedad-Sidhoum (2013) who employ a payoff function that aims at maximizing the cumulative number of jobs processed before each delivery date subject to job release dates. Liu (2012) investigates a job shop with job release dates, equidistant fixed delivery dates and the objective to minimize the sum of weighted flow times and storage times.

Leung and Chen (2013) and Fu, Huo, and Zhao (2012) consider single machine problems with fixed delivery dates and limited delivery capacity. In particular, Leung and Chen (2013) develop polynomial-time algorithms for minimizing the maximum lateness and the number of vehicles used. Hence, they coordinate production scheduling and delivery decisions. Integrated production and delivery scheduling which may also include delivery date setting has received very strong interest recently. Chen (2010) gives an overview of this research.

While very few researchers have addressed the parallel machine scheduling problem with fixed delivery dates, many insights are available on the correspondent scheduling problem without delivery dates, Pm||∑_jT_j. This problem is also NP-hard in the ordinary sense since 1||∑_jT_j has been shown to be NP-hard in the ordinary sense by Du and Leung (1990). However, various efficient branch-and-bound algorithms (e.g., Azizoglu, Kirca, 1998, Schaller, 2009, Shim, 2009, Shim, Kim, 2007, Tanaka, Araki, 2008, Yalaoui, Chu, 2002) and heuristics (e.g., Biskup, Herrmann, and Gupta, 2008) have been developed for Pm||∑_jT_j. These findings can be used to develop sequence-oriented solution procedures for the problem with fixed delivery dates. In a sequence-oriented solution approach, the focus is on finding a good schedule where each job is delivered at the first delivery date at or after its completion. Thus, the delivery dates of the jobs depend on the schedule. A schedule is infeasible if some job is completed after the last delivery date. Obviously, the last delivery date should be large enough so that there exists at least one schedule such that Δ_s ≥ C_j ∀ j. Throughout this paper, we assume Δ_s ≥ min {C_max}, i.e., the last delivery date is at least as large as the minimum makespan. Details are provided in Section 4.

Apart from the scheduling literature, the problem $Pm|s=\overline{s}|{\sum }_j{T}_j$ is also related to the generalized assignment problem which consists of assigning agents with limited capacity to a set of tasks. To illustrate this point, let us say that a job is in block k if Δ_{k − 1} < C_j ≤ Δ_k and hence D_j = Δ_k. The following property holds:

Property 1 Hall et al., 2001

In an optimal solution of $Pm|s=\overline{s}|{\sum }_j{T}_j,$ all jobs assigned to one block on one machine can be processed in arbitrary sequence.

Consequently, it is sufficient to identify an optimal assignment of jobs to blocks. This may be achieved with assignment-oriented solution approaches that focus on assigning each job to a block with a certain delivery date. An assignment is infeasible if the processing time assigned to a block exceeds the capacity. Both the sequence- and the assignment-oriented approaches appear to have their particular merits and drawbacks. For example, the sequence-oriented approaches appear less likely to generate infeasible solutions because infeasibility can only occur with respect to the last delivery date, but they may add unnecessary complexity. In this paper, we consider both avenues. However, we understand that our discussion of exemplary solution procedures cannot provide a conclusive result if one approach outperforms the other.

The remainder of the paper is structured as follows: In Section 2, we present a mathematical programming formulation and a branch-and-bound algorithm to find optimal solutions. Since the problem is NP-hard in the ordinary sense, the optimal solution procedures require much computational time for larger instances. Consequently, we propose an assignment-based tabu search and a more sequence-oriented hybrid genetic algorithm in Section 3. In Section 4, we present numerical results which suggest that the optimal solution procedures can solve instances with up to 20 jobs in a reasonable computational time while both heuristics yield good approximate solutions for instances with up to 50 jobs. Section 5 summarizes the findings and suggests some fruitful directions for future research.