Punctuality and idleness in just-in-time scheduling

Abstract

In scheduling problems with irregular cost functions, such as deviation functions in earliness–tardiness scheduling problems, optimal solutions usually contain some idleness periods during which no activity is processed. Then, minimizing the penalties for not delivering on time and minimizing the idleness cost are two complementary criteria for a schedule. A Dynamic Programming procedure, in which the states are represented by continuous two-dimensional piecewise linear functions, is proposed to compute the cost of the Pareto optimal schedules.

Introduction

Just-in-time (JIT) scheduling problems form a well studied class of multicriteria scheduling problems. Indeed, these models in which earliness is penalized are very useful to represent practical problems where perishable goods have to be delivered or where storage costs cannot be disregarded.

Most of these problems are based on the two following criteria that are related to the presence of a due date d_i for each task i of the instance:

• the tardiness T_i = max(0, C_i − d_i) where C_i denotes the completion time of i,

• the earliness E_i = max(0, d_i − C_i).

Clearly, a task cannot simultaneously have a positive tardiness and a positive earliness: a task is either tardy or early. Moreover, earliness and tardiness are often comparable in terms of costs induced by a delivery that is not on time. Therefore, these two criteria are often aggregated into a single criterion f_i(C_i) = α_iE_i + β_iT_i, which is calles the weighted deviation of the task with respect to its due date. Interestingly, this deviation function can be generalized to express more complex combinations of contradicting criteria and soft or hard constraints on the date a customer should be delivered. For example, in addition to a global due date at which the customer would like to be delivered, there can be time windows modeling the time periods during which the customer’s shop is closed or time periods the customer cannot be present for any reason. Delivery inside such time windows must be more or less severely penalized. Fig. 1 illustrates the form of a cost function in the presence of time windows. We observe that the function is not convex but is piecewise linear with 17 segments. In this paper, this generalization of deviation functions will be called punctuality functions and we will assume that the punctuality functions f_i are piecewise linear, which is the case in most scheduling models. The complexity of the algorithms will of course depend of the number of segments of the functions. ∥f_i∥ will denote the number of segments of f_i.

While JIT scheduling problems have been widely studied in the case where the earliness and tardiness criteria are aggregated into a single criterion, T’kindt and Billaut [22, Chapter 6] observed that the multicriteria JIT scheduling problems with more than two criteria have not been well studied in the literature. In this paper, we will consider a JIT scheduling problem in which we will consider the punctuality costs (subsuming earliness criterion and tardiness criterion) and the idleness criterion. In a feasible schedule, idleness periods are time intervals in which a worker or a machine processes no activity. The presence of idleness periods is a well-known consequence of the fact that the punctuality cost functions are not non-decreasing, or, according to the scheduling theory terminology, the criteria are non-regular. For example, in a classical earliness–tardiness problem with two tasks i and j with unit processing times and due dates such that d_j > d_i + 1, the minimization of the criterion E_i + T_i leads to a schedule in which both tasks complete at their due date. Therefore, the time interval [d_i, d_j − 1) is idle since no task is processed in between d_i and d_j − 1. In practical situations, workers may be assigned other tasks to perform during the idleness periods but generally, the presence of—too much—idleness in a schedule is negatively perceived since it often means that some non-multifunctional workers are being paid but are producing nothing. A possible approach in order to minimize the idleness in a schedule, is to introduce a regular secondary criterion such as the makespan C_max = max_iC_i or the flowtime ${\sum }_i{C}_i$. For example, Fry et al. [5] consider the minimization of a linear combination of the earliness, tardiness and flowtime. This modeling approach may be unsatisfying because it tends to favor earliest schedules. On the contrary, when the workday is defined as the time interval starting with the first task and completing with the completion time of the last task, a good policy may be to postpone the start of the workday in order to decrease the workday length and to reduce the idle time. In our approach in this paper, we will explicitly consider each possible idleness period between consecutive tasks.

The problem of finding an optimal sequence for the earliness/tardiness problem is NP-complete even for a common due date [8], [10]. There are however some special cases where the sequence can be found in polynomial cases. They are listed in the book of T’kindt and Billaut [22, Chapter 6]. In most of these problems, the due date is common for all the tasks—see also the survey of Gordon et al. [7]. Polynomial cases for problems with distinct due date (but equal processing times) were studied by Garey et al. [6] and by Verma and Dessouky [24]. The recent paper of Hassin and Shani [9] presents some generalization of these polynomial algorithms for early/tardy scheduling. Other just-in-time scheduling problems that model assembly lines [18] have also been presented and can efficiently be solved as assignment problems [17], [15], [16].

Much research effort was devoted to the case where the tasks form a chain, that is they are already sequenced in order to be processed (on a single machine). This paper also focuses on this special problem. This problem typically models a situation in which a tour to deliver customers has been build and the best timing for the deliveries is searched for. More generally, this problem is very important because most scheduling algorithms first rank the tasks by the mean of either a (meta)heuristic or an enumeration scheme arid next determine the optimal—if possible—timing for the sequenced tasks. For the single machine problem with earliness/tardiness penalties, we can refer to several branch-and-bound algorithms by Kim and Yano [13], Hoogeveen and van de Velde [11] and Sourd and Kedad-Sidhoum [20], heuristic algorithms by James and Buchanan [12] and Wan and Yen [25]. Also, Ventura and Radhakrishnan [23] and Sourd and Kedad-Sidhoum [20] present algorithms to make feasible some unfeasible solutions of relaxed problems.

This problem in which only the punctuality criterion is considered has been already studied in the literature. When the punctuality cost is equal to the weighted deviation α_iE_i + β_iT_i (α_i, β_i ⩾ 0), the problem can be formulated as a linear program [4] and this formulation is still valid while all the punctuality cost functions are convex and piecewise linear. However, this problem can be more efficiently solved by a direct combinatorial algorithm based on the blocks of adjacent tasks in the earliness–tardiness case [6], [3], [21] and in the more general case where the cost functions are convex [2]. More recently, Sourd [19] has shown that, by a dynamic programming approach, the problem is still polynomial when the punctuality cost functions are not assumed to be convex (but piecewise linear). The complexity of this algorithm depends on the number of segments.

The present paper is a continuation of the work presented in [19]. In contrast to it, the approach adopted in this paper is multicriteria in the sense that the sum of the punctuality costs on the one hand and the sum of the idleness costs on the other hand are separately considered. The algorithm proposed here computes the costs of all the Pareto optimal schedules by a dynamic programming approach. Dynamic programming (DP) has already been used to compute the set of Pareto optimal solutions. For example, Klamroth and Wiecek [14] survey several DP approaches to compute the set of Pareto optimal solutions for the Knapsack problem. However, to the best of our knowledge, our approach is original in a methodological view since the proposed recursive scheme deals with an infinite number of states that can be compactly represented by two-dimensional piecewise linear functions. Similarly, the returned set of Pareto optimal schedules contains an infinite number of schedules that can be represented by a one-dimensional piecewise linear function with a finite number of segments. Under some conditions, it is shown that the number of segments is polynomial.

Section 2 introduces the problem and the notations. In Section 3, the DP algorithm is presented to solve the general problem. Section 4 focuses on the special case where we consider the idleness length instead on idleness costs: a special data structure is described in order to have a polynomial algorithm and a polynomial output. An example is given in Section 5, with some remarks in order to have an efficient implementation.