Elsevier

Computers & Operations Research

Volume 65, January 2016, Pages 104-110
Computers & Operations Research

The stochastic lot sizing problem with piecewise linear concave ordering costs

https://doi.org/10.1016/j.cor.2015.07.004Get rights and content

Highlights

  • We address the stochastic lot sizing with piecewise linear concave ordering costs.

  • We introduce the generalized (R,S) policy and present a MIP formulation thereof.

  • The generalized (R,S) policy yields an optimality gap around 1%.

Abstract

We address the stochastic lot sizing problem with piecewise linear concave ordering costs. The problem is very common in practice since it relates to a variety of settings involving quantity discounts, economies of scales, and use of multiple suppliers. We herein focus on implementing the (R,S) policy for the problem under consideration. This policy is appealing from a practical point of view because it completely eliminates the setup-oriented nervousness – a pervasive issue in inventory control. In this paper, we first introduce a generalized version of the (R,S) policy that accounts for piecewise linear concave ordering costs and develop a mixed integer programming formulation thereof. Then, we conduct an extensive numerical study and compare the generalized (R,S) policy against the cost-optimal generalized (s,S) policy. The results of the numerical study reveal that the (R,S) policy performs very well – yielding an average optimality gap around 1%.

Introduction

The need for coordination and cooperation between supply chain players has dramatically increased due to rapid progress in globalization and competition. In this context, a common complaint raised by supply chain managers is that downstream players continually revise the timing and the size of their order requests [7]. It is commonly agreed that the revisions in replenishment schedules are particularly critical [4]. This issue is often referred to as the setup-oriented nervousness and it is prevalent in a variety of industrial settings involving managing joint replenishments [13], shipment consolidation in logistics [8], and buying raw materials in markets where price is fluctuating [6]. A practical approach towards offsetting setup-oriented nervousness is to employ an (R,S) policy [1]. Here, a replenishment schedule is fixed once and for all at the beginning of the planning horizon, but the size of replenishments are dynamically determined at the time of placing orders upon observing realized demands. This strategy has recently been subject to a detailed scrutiny due to its practical relevance, and applied to a variety of inventory control problems [1], [14], [15], [11] (see e.g.). All of these studies have analyzed the (R,S) policy under the assumption that ordering cost is comprised of a fixed and a linear component. In this study, we aim to extend this literature by presenting a mathematical programming model of the (R,S) policy for piecewise linear concave ordering costs. This cost structure is a class of decreasing average costs that captures the combination of linear and fixed ordering costs as a special case. It reflects many practical examples such as quantity discounts, effect of scale economies, and use of multiple suppliers (see e.g [9], [2], [17]).

The majority of research efforts on inventory problems with piecewise concave ordering costs has been concentrated on characterizing the structure of the optimal control policy. Porteus [9], [10] showed that a generalized (s,S) policy is optimal for inventory systems for a class of demand distributions. Fox et al. [2] considered a specific case of piecewise linear concave ordering costs, and proved that a generalized (s,S) policy is optimal for a larger class of demand distributions. Zhang et al. [18] addressed the same problem under limited order capacities. Yu and Benjafaar [17] extended earlier results by establishing the optimality of generalized (s,S) policies for general demand distributions. These research contributions have ultimately showed that generalized (s,S) policies are optimal under a large variety of settings. However, finding the optimal parameters of the generalized (s,S) policy is still a computational challenge. Also, an important drawback of the generalized (s,S) policy is that it does not provide the exact timing of replenishments in advance. As such, inventory systems controlled by the generalized (s,S) policy are exposed to a great deal of setup-oriented system nervousness which results in complications on the coordination between supply chain players (see e.g. [3], [4], [5], [7], [16].

In this paper, we adopt the (R,S) policy for the stochastic lot sizing problem with piecewise linear concave ordering costs, and show that it is a viable alternative to the cost optimal (s,S) policy. The contribution of the current study is two-fold. First, we introduce a generalized version of the (R,S) policy for the inventory problem with piecewise linear concave ordering costs, and present a mixed-integer programming (MIP) formulation thereof. Secondly, we conduct an experimental study that compares (s,S) and (R,S) policies, and thus reflect upon the trade-off between the setup-oriented system nervousness and cost optimality in case of piecewise concave ordering costs.

The remainder of this paper is organized as follows. Section 2 provides the problem definition. 3 The generalized (, 4 The generalized ( introduce solution methods for (s,S) and (R,S) policies, respectively. Section 5 provides an illustrative numerical example for two alternative policies. Section 6 is devoted to the design and findings of the computational experiments. Finally, Section 7 draws conclusions.

Section snippets

Problem definition and preliminaries

We consider a single product periodic-review finite-horizon inventory control problem. The planning horizon is comprised of T periods. The demand ξt in period t is an independent and normally distributed random variable with known parameters. The demand distribution may vary from period to period (i.e., demand is non-stationary). A holding cost h is incurred for each unit carried in inventory from one period to the next, and a shortage cost p is incurred for each unit of demand backordered. The

The generalized (s,S) policy

In this section, we take a closer look at the inventory problem under consideration, and examine the structure of the optimal control policy. The expected total cost of the inventory system is comprised of ordering, holding and penalty costs. If the inventory position at period t immediately after the delivery is y, then the sum of expected holding and penalty costs to be incurred during that period can be written asLt(y)=E{h(yξt)++p(yξt)}where x+=max{0,x}, and x=max{0,x}.

Then, given an

The generalized (R,S) policy

The (R,S) policy is an order-up-to policy whose essence lies in using a rigid replenishment schedule that is established at the very beginning of the planning horizon while allowing flexibility in the replenishment quantities. The order quantities are dynamically determined at replenishment epochs so as to raise the inventory position to prescribed order-up-to levels. Therefore, the policy specifies the replenishment periods and corresponding order-up-to levels minimizing the expected total

A numerical example

In this section, we provide a numerical example to illustrate generalized (s,S) and (R,S) policies. Henceforth, we drop the term “generalized” when referring to inventory policies for the sake of brevity. We consider a system with two suppliers. The variable unit and fixed ordering costs associated with these suppliers are: v1=2, K1=500, and v2=3, K2=200. The planning horizon is 12 periods, and average period demands μt are given in Table 1. We assume a constant coefficient of variation of 0.3

Numerical study

In this section, we investigate the cost performance of the (R,S) policy with respect to the cost-optimal (s,S) policy. In what follows, we first present the design of numerical experiments, and then provide the results and discuss our findings.

We employ demand patterns that reflect demand progressions common in practice. Here, STAT and RAND patterns provide the limit cases regarding stationarity and non-stationarity, respectively. Seasonal patterns SIN1 and SIN2 are sinusoidal waves with

Conclusion

In this paper, we addressed the stochastic lot sizing problem with piecewise linear concave ordering costs. This cost structure is common in many environments characterized by e.g. quantity discounts, economies of scales, and use of multiple suppliers. We focused on implementing the (R,S) policy as an alternative to the cost-optimal (s,S) policy. The (R,S) policy provides a rigid replenishment schedule, and thereby completely eliminates setup-oriented nervousness – an important practical issue

References (18)

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1

The author is supported by the Scientific and Technological Research Council of Turkey (TUBITAK) #110M500.

2

This publication has emanated from research supported in part by a research grant from Science Foundation Ireland (SFI) under Grant number SFI/12/RC/2289.

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