Elsevier

Computers & Operations Research

Volume 109, September 2019, Pages 89-101
Computers & Operations Research

Heuristic methods for the capacitated stochastic lot-sizing problem under the static-dynamic uncertainty strategy

https://doi.org/10.1016/j.cor.2019.03.007Get rights and content

Highlights

  • We study a capacitated inventory system under static-dynamic uncertainty strategy.

  • The demand is stochastic and non-stationary.

  • Several heuristics developed produce high quality solutions and scale-up efficiently.

  • The effectiveness of the heuristics is tested statistically in an extensive numerical study.

Abstract

We consider a lot-sizing problem in a single-item single-stage production system facing non-stationary stochastic demand in a finite planning horizon. Motivated by common practice, the set-up times need to be determined and frozen once and for all at the beginning of the horizon while decisions on the exact lot sizes can be deferred until the setup epochs. This operating scheme is referred to as the static-dynamic uncertainty strategy in the literature. It has been shown that a modified base stock policy is optimal for a capacitated system with minimum lot size restrictions under the static-dynamic uncertainty strategy. However, the optimal policy parameters require an exhaustive search, for which the computational time grows exponentially in the number of periods in the planning horizon. In order to alleviate the computational burden for real-life size problems, we developed and tested seven different heuristics for computational efficiency and solution quality. Our extensive numerical experiments showed that average optimality gaps less than 0.1% and maximum optimality gaps below 4% can be attained in reasonable running times by using a combination of these heuristics.

Introduction

This paper considers a multi-period single-item stochastic capacitated lot-sizing problem. The costs are non-stationary and they consist of linear production, inventory holding and penalty costs, as well as a fixed set-up cost. Any unfulfilled demand is backlogged and satisfied as soon as possible. There are restrictions on the minimum and maximum number of products that can be produced in any period. The objective is to find the periods in which production will take place and the corresponding production quantities such that the total expected cost is minimized.

Contrary to its deterministic counterpart (i.e., deterministic lot-sizing problems), which has been studied extensively in the literature, the stochastic lot-sizing problem poses different alternatives depending on how production decisions are made and executed. Bookbinder and Tan (1988) mention three strategies: static uncertainty strategy, dynamic uncertainty strategy and static-dynamic uncertainty strategy. Under the static uncertainty strategy, the timing and quantities are fixed at the beginning of the planning horizon once and for all. The practical benefit of following this strategy is reduced system nervousness due to the fixed production plan. However, this strategy is inflexible and cannot respond to the demand realizations. A practical remedy is to solve the problem in a rolling horizon fashion in pre-set frequencies to partially alleviate this inflexibility without creating excessive nervousness in the system. Under the dynamic uncertainty strategy, neither the timing nor the quantities are fixed at the beginning of the planning horizon, but rather, are determined as time evolves. This strategy reduces inventory holding and backlog costs, and permits flexibility in production/inventory management by adjusting the inventory position at the beginning of each period. However, it is bound to create higher system nervousness. Finally, in the static-dynamic uncertainty strategy, the timing of the production is fixed at the beginning of the planning horizon but the exact quantities are determined later in the horizon. Bookbinder and Tan (1988) consider this strategy a more accurate representation of industrial practice since freezing schedules is a common practice to decrease system nervousness. Inderfurth (1994) notes that nervousness due to “deviations in delivery timing” requests is considered to be more severe in practice compared to “deviations in quantities”. Note that only “deviations in quantities” are present in the static-dynamic uncertainty strategy.

In a recent study, Özen et al. (2012) introduced an exact formulation for the problem under the static-dynamic uncertainty strategy and showed that the optimal policy is a modified based stock policy. They proposed two dynamic programming based heuristics that are shown to perform well for uncapacitated problems. In this paper, we focused on the capacitated version of the problem with minimum production quantities and capacity limits. We developed two groups of heuristic solutions. The first group is the modifications of the dynamic programming based (DP-based) heuristics by Özen et al. (2012), while the second group is search based heuristics. We evaluated all of our heuristic procedures in terms of the optimality gap and computation time. Interestingly, our extensive numerical experiments demonstrate that these two types of heuristics show complementary performance. DP-based heuristics perform better in the scenarios where capacity constraints are not restrictive (i.e., a high maximum lot size regime) whereas search-based heuristics achieve better results when capacity restrictions are tight (i.e., a low maximum lot size regime). We statistically tested which combinations of these two groups of heuristics dominate the others. The best performing combinations can achieve a less than 1% optimality gap under certain parameter settings. We also observed that the computation time of our heuristics grows polynomially in the number of periods.

The remainder of the paper is organized as follows. In Section 2, we provide a literature review. Section 3 introduces the model in detail and the notation used. Section 4 discusses the optimal policy and how optimal policy parameters can be calculated. Sections 5 and 6 are dedicated to our dynamic programming and search-based heuristics, respectively. We present the numerical study in Section 7 with a brief discussion of the findings. Finally, we conclude in Section 8. An additional discussion on the lower bounds and detailed results of the numerical study are presented in the Online Appendix.

Section snippets

Literature review

The literature on the deterministic lot-sizing problem is very extensive. We refer to Brahimi et al. (2006) for a comprehensive review on the single-item lot-sizing problem. In this section, we review the stochastic lot-sizing literature in three groups based on the strategy used (i.e., static uncertainty, dynamic uncertainty and static-dynamic uncertainty).

Regarding the static uncertainty strategy, Vargas (2009) studies the stochastic version of the classical lot-sizing problem by Wagner and

Model and notation

In this section, we introduce the model and the basic assumptions. We follow the same notation used in Özen et al. (2012).

Consider a single-stage, single-item production system in a finite horizon under periodic review. The planning horizon consists of equal length time periods (days, weeks, etc.). Let t{1,2,,N} be the index for periods with N being the last period in the planning horizon. Let Dt denote non-negative random demand in period t, which is independent and not necessarily identical

Optimal policy for Problem P

Consider Problem P under a given production schedule (m,r). For a given starting inventory level Ii, let yi*(Ii) be the optimal inventory level just after production in cycle i for i=1,,m. The optimal inventory levels (y1*(I1),,ym*(Im)) can be found by the solving a backward dynamic program with m stages where a stage is a production cycle. The recursive function is given byfm+1(Im+1)=cNIm+1fi(Ii)=minyi:Ii+oriyiIi+uriGi(yi)for i=1,,m where fi(Ii) is the minimum expected cost of the system

Dynamic programming-based heuristics

This section presents the heuristic procedures that are based on dynamic programming. We start with introducing a heuristic procedure by Özen et al. (2012), which simultaneously determines the production schedule and base stock levels. However, the performance of this heuristic deteriorates with tighter capacity and minimum order quantity constraints. Therefore, we introduce two new modified heuristics to improve the performance.

Search based heuristics

This section introduces our new heuristic procedures for solving Problem P. The basic idea of these heuristics is to perform a search on production schedules in a greedy fashion by considering three operations: merging, dividing and switching.

  • 1.

    Merging: In a given production schedule, a merging operation combines two consecutive production cycles and forms a production schedule with one setup less by simply removing one setup from the schedule.

  • 2.

    Dividing: In contrast to merging, a dividing

Numerical study

In this section, we evaluate the performance of the heuristics developed in Sections 5 and 6. We first show the effectiveness of the lower bounds used in the procedure described in 4.1. Next, we test the performance of the heuristics in terms of the quality of the solution (optimality gap) and the computation time.

We start our experiments with stationary cost parameters and lot size restrictions, i.e., values do not change from period to period, and create the test bed as follows. For each

Conclusion

We consider the stochastic lot-sizing problem under the static-dynamic uncertainty strategy. Özen et al. (2012) have shown that a modified base stock policy is optimal, but finding an optimal solution requires an exhaustive search that grows exponentially with the length of the planning horizon. They proposed dynamic programming (DP) based heuristics for the uncapacitated problem and provided numerical evidence of their good performance. In this paper, we study the capacitated problem and

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