Dynamic lot-sizing model under perishability, substitution, and limited storage capacity

https://doi.org/10.1016/j.cor.2020.104978Get rights and content

Highlights

  • We use two structural properties to develop a dynamic programming algorithm that can solve the dynamic lot-sizing problem with perishable inventory and demand substitution under storage capacity constraints.

  • We obtain the forecast horizons for the general problem and the case with constant unit ordering costs by using the marginal analysis method and establishing the monotonicity of the regeneration points of two products, respectively.

  • We determine the effects of storage capacity, product lifetime, joint setup, and inventory costs on the length of the forecast horizon and the total costs by using a detailed test bed of instances.

Abstract

Various commodities including blood, pharmaceutical, and agricultural products are perishable in nature and require special storage conditions, such as controlled temperature. However, expanding the space of cold storage is expensive. Therefore, storage capacity frequently restricts the operational efficiency of enterprises in the perishable product industry. Numerous enterprises have adopted the policies of product substitution and multiple item joint procurement to achieve an efficient inventory management and reduce their operation costs. This paper considers a two-product dynamic lot-sizing problem for perishable inventory under product substitution and limited storage capacity. This study aims to identify the ordering, inventory, and substitution decisions over a planning horizon under the criteria that (i) the inventory holding costs and deterioration rates are age dependent, (ii) a one-direction product substitution and joint ordering of two products are possible, and (iii) the storage capacity has an upper bound that limits the inventory quantities. The contributions of this study are summarized in three points. First, we develop a dynamic programming algorithm by using two structural properties to solve the dynamic lot-sizing problem with perishable inventory and demand substitution under storage capacity constraints. Second, we obtain the forecast horizons for the general problem by using the marginal analysis method and for the case with constant unit ordering costs through establishing the monotonicity of the regeneration points of two products. Third, we determine the effects of storage capacity, product lifetime, joint setup, and inventory costs on the length of the forecast horizon and the total costs by using a detailed test bed of instances. The major findings are also discussed in three aspects. First, the single-period satisfaction property fails to hold under limited storage capacity and time-varying unit ordering costs. Second, the forecast horizon inconsistently increases with joint setup costs or storage capacity. Third, the total costs fail to decrease with increasing storage capacity or product lifetime.

Introduction

Perishable products, such as vegetables, fruits, meat, seafood, and aquatic and dairy products, are indispensable and ubiquitous in our lives and account for the majority of supermarket or grocery sales. The Food Market Institute of America reports that these products account for 53.4% of the 501.35 billion supermarket sales in 2016.1 Perishable products also include pharmaceutical products (e.g., blood products and drugs), which comprise a billion-dollar industry. However, the mismanagement of perishable products increases costs for companies. In China, fruits and vegetables, meat, and aquatic products that are circulated in the market have inventory deterioration (breakage) rates of 30%, 12%, and 15%, respectively; all of these products exceed the 5% level reported in developed countries.2 In practice, perishable products require special storage conditions, such as clean rooms or controlled temperature. Storage capacity is typically a scarce resource for the majority of the manufacturers and retailers of perishable products. In the fresh product industry (e.g., Shuanghui Group, the largest meat processor in China), operational efficiency is frequently restricted by the space of cold storage, that is, by the inventory or storage capacity. When the setup costs are extremely high, an enterprise must enhance its production in one operation to reduce its average costs. However, if the space of cold storage is insufficient, then an enterprise suffers from a limited production. Thus, restrictions due to storage capacity have become increasingly critical. Managing the inventory of perishable products introduces storage capacity constraint problems that motivate us to study the trade-off between high setup costs and limited storage capacity. Varying grades of perishable goods, such as fresh and frozen meat, also have a limited storage capacity. Fresh meat can be used to fulfill the demand for frozen meat to improve operational efficiency.

Under multiperiod decision making, a strong decision process requires managers to determine the role of future business data (e.g., costs and demands) in their current decisions. In production and operation decisions, managers predict the first few periods and subsequently use these predictions to influence their present decisions. Although inaccuracies may exist, the information gained from forecast data can be habitually integrated into the current decisions of enterprises and thus cannot be disregarded. A key question on how future data affect current decisions arise. Studies on the concepts of forecast and decision horizons have been conducted in the operation management field. Enterprises in the perishable product industry require a short horizon and accurate information about their product features. Defining short problems requires the acquisition of data for short periods in the future, thereby incurring low forecast costs. Solving an optimization problem with a short horizon may also considerably reduce the required computational burden. However, short forecast information corresponds to imprecise decisions on the first few periods, thereby increasing costs. Therefore, a problem horizon that most efficiently balances this trade-off must be determined.

In this paper, we present a two-perishable-item dynamic lot-sizing (DLS) problem with limited storage capacity and one-direction substitution, which is also referred to as downward substitution. Perishable products can be classified into high- and low-grade products; the former can be used to satisfy the demand for the latter. These types share the same limited storage capacity. We develop a polynomial time dynamic programming (DP) algorithm to solve the problem. Subsequently, we obtain forecast horizon results for the scenario with the problem on constant unit ordering costs. We also describe the application of the marginal analysis method to establish forecast horizon results for the general problem with time-varying unit ordering costs. The basic idea is described as follows. First, we define the smallest marginal cost for t-period problem. Second, we let the last ordering period be equal to the period with the smallest marginal cost and subsequently obtain the monotonicity of the last ordering period. Several studies have examined the forecast horizon under the case of two products. For instance, Dawande et al. (2009) consider a two-product variant of the DLS model with inventory capacity constraints and compute discrete forecast horizons via integer programming. Bardhan et al. (2013) investigate a two-item DLS problem with product substitution and production changeovers and develop a DP algorithm to determine an approximate solution and a forecast horizon. Jing and Mu (2019) examine a two-item DLS problem with perishable inventory and product substitution while assuming an infinite storage capacity. Table 1 highlights the differences between our model and those of Dawande et al. (2009), Bardhan et al. (2013), and Jing and Mu (2019), characterizes our problem, and summarizes the contributions of our work.

An increasing number of products are becoming perishable. The perishable products that require controlled temperature and humidity may have a limited warehouse capacity. The DLS problem with perishable products under storage capacity constraints is a realistic problem. However, DLS studies on perishable products disregards the limited storage capacity of these products. In theory, the recognized zero-inventory property (ZIP, which posits that if a positive production exists in period t, then the inventory at the end of period t − 1 is 0) and the single-period satisfaction property (SPSP, which posits that the demands in a period are satisfied entirely by production in exactly one of the periods) fail to cover the scenario under a limited storage capacity. Thus, we must explore other structural properties to devise a new DP algorithm that can solve the DLS problem under a limited storage capacity. The major contributions of our paper are summarized as follows:

  • (1)

    We use two structural properties to develop a DP algorithm that can solve the DLS problem with perishable inventory and demand substitution under storage capacity constraints.

  • (2)

    We obtain the forecast horizons for the general problem and the case with constant unit ordering costs by using the marginal analysis method and establishing the monotonicity of the regeneration points of two products, respectively.

  • (3)

    We determine the effects of storage capacity, product lifetime, joint setup, and inventory costs on the length of the forecast horizon and the total costs by using a detailed test bed of instances.

The rest of this paper is organized as follows. Section 2 reviews the literature. Section 3 describes the DLS problem for two perishable products with inventory bounds and one-direction substitution. Section 4 explores two properties and devises a DP algorithm to address the problem. Section 5 presents a sufficient condition to obtain the forecast and decision horizons. Section 6 discusses the computational experiences and insights. Section 7 concludes the paper and presents future research directions.

Section snippets

Literature Review

This study is related to the DLS problem and forecast horizon. In this section, we present a detailed discussion of the DLS models and forecast horizons.

Model Formulation

The cases proposed by Dawande et al. (2009) and Jing and Mu (2019) are appropriate for studying the DLS problem with perishable inventory and downward substitution under limited storage capacity. Product 1 can be used to satisfy the demand for product 2 at the beginning of period t (1 ≤ tT) at unit substitution cost st. The unit ordering cost at the beginning of period t is ct1 (resp. ct2) for product 1 (resp. 2). The unit holding cost in period t is hit1 (resp. hit2) for product 1 (resp.

Two Properties and DP Algorithm

Prior to establishing the two structural proprieties, we defineAiini1andAktni=1l=kt1(1αiln)forn=1,2;1ik<tT.

We can easily determine thatAktni=AkqniAqtniforn=1,2;k<q<t.

In addition, if αitnαjtn, thenAktniAktnjforn=1,2;1i<jk<tT.

Remark 1

Given the stock deterioration, to satisfy one unit demand of product n (n = 1, 2) in period t by ordering in period i (i < t), we must order Aitni units of products n in period i and carry Aktni units of inventory in each subsequent period k, where ikt − 1.

Forecast Horizon

Under variable unit ordering costs, we cannot directly obtain the monotonicity of the ordering and regeneration points. For obtaining the forecast horizon under this general case, applying the marginal analysis method proposed by Eppen et al. (1969) may be appropriate. To apply this method, we must modify the DP algorithm proposed in Section 4. Let V(t) denote the optimal costs for t-period problem. Let Vt(i1t1,i2t1,...,iRt1;i1t2,i2t2,...,iSt2) denote the optimal costs for t-period problem, in

Computational Results and Managerial Insights

Our computational study aims to (i) analyze the behavior of the forecast horizon under varying storage capacities, product lifetimes, and cost parameters and (ii) evaluate the effects of storage capacity and product lifetime on the total costs. To address these issues, we consider the following test beds, which are similar to those presented in Dawande et al. (2006, 2007, 2009, 2010), Bardhan et al. (2013), and Jing and Mu (2019).

Test 1. The demands for both products are assumed to be normally

Conclusion and Suggestions for Future Research

This paper investigates a two-perishable-item DLS problem with one-direction product substitution and limited storage capacity. We explore two properties in an optimal solution and use them to devise a DP method to solve the aforementioned problem. We also consider the case with constant unit ordering costs and establish forecast and decision horizon results for this case. We generate beneficial managerial insights from our computational study and present that the restrictions on storage

Endnotes

1. https://www.fmi.org/docs/default-source/default-document-library/supermarket-sales-by-department-2017.pdf?sfvrsn=9946796e_0

2. https://www.qianzhan.com/analyst/detail/220/170622-973ef7dc.html

CRediT authorship contribution statement

Fuying Jing: Conceptualization, Methodology, Formal analysis, Writing - original draft, Writing - review & editing. Yinping Mu: Writing - review & editing, Software, Funding acquisition, Supervision.

Acknowledgments

This research is supported by the National Natural Science Foundation of China, under grants 71772025, 71531003 and 71772026. We thank the Editor-in-chief, Prof. Francisco Saldanha da Gama, and the Area Editor, for their constructive comments. We also thank the anonymous reviewers for their insightful comments.

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