Dynamic lot-sizing model under perishability, substitution, and limited storage capacity
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 ≤ t ≤ T) at unit substitution cost st. The unit ordering cost at the beginning of period t is (resp. ) for product 1 (resp. 2). The unit holding cost in period t is (resp. ) for product 1 (resp.
Two Properties and DP Algorithm
Prior to establishing the two structural proprieties, we define
We can easily determine that
In addition, if , then
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 units of products n in period i and carry units of inventory in each subsequent period k, where i ≤ k ≤ t − 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 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
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.
References (70)
- et al.
The multi-item capacitated lot-sizing problem with safety stocks and demand shortage costs
Computers & Operations Research
(2009) - et al.
Impact of carbon emissions in a sustainable supply chain design for a second generation biofuel
Journal of Cleaner Production
(2018) - et al.
An O(n2) algorithm for lot sizing with inventory bounds and fixed costs
Operations Research Letters
(2008) - et al.
A polynomial algorithm for a lot-sizing problem with backlogging, outsourcing and limited inventory
Computers & Industrial Engineering
(2013) - et al.
Discrete forecast horizons for two-product variants of the dynamic lot-size problem
International Journal of Production Economics
(2009) Solving large-scale requirements planning problems with component substitution options
Computers & Industrial Engineering
(2003)- et al.
A polynomial algorithm for the production/ordering planning problem with limited storage
Computers & Operations Research
(2007) - et al.
An efficient approach for solving the lot-sizing problem with time-varying storage capacities
European Journal of Operational Research
(2008) - et al.
Effective replenishment policies for the multi-item dynamic lot-sizing problem with storage capacities
Computers & Operations Research
(2013) - et al.
Warehouse space capacity and delivery time window considerations in dynamic lot-sizing for a simple supply chain
International Journal of Production Economics
(2004)
Forecast horizon for dynamic lot sizing model under product substitution and perishable inventories
Computers & Operations Research
Fix-and-Optimize heuristic for capacitated lot-sizing with sequence-dependent setups and substitutions
European Journal of Operational Research
Economic lot sizing problem with inventory bounds
European Journal of Operational Research
Production planning with limited inventory capacity and allowed stockout
International Journal of Production Economics
A comparison of simple heuristics for multi-product dynamic demand lot-sizing with limited warehouse capacity
International Journal of Production Economics
A note on “The economic lot sizing problem with inventory bounds”
European Journal of Operational Research
Two-level lot-sizing with inventory bounds
Discrete optimization
The economic lot-sizing problem with remanufacturing and one-way substitution
International Journal of Production Economics
Note on the economic lot-sizing problem with remanufacturing and one-way substitution
International Journal of Production Economics
Dynamic economic lot size model with perishable inventory and capacity constraints
Applied Mathematical Modelling
How does an industry manage the optimum cash flow within a smart production system with the carbon footprint and carbon emission under logistics framework?
International Journal of Production Economics
Optimal production delivery policies for supplier and manufacturer in a constrained closed-loop supply chain for returnable transport packaging through metaheuristic approach
Computers & Industrial Engineering
Environmental and economic assessment of closed-loop supply chain with remanufacturing and returnable transport items
Computers & Industrial Engineering
Recovery-channel selection in a hybrid manufacturing- remanufacturing production model with RFID and product quality
International Journal of Production Economics
Polyhedral analysis for the two item uncapacitated lot-sizing problem with one-way substitution
Discrete Applied Mathematics
Erratum to: Polyhedral analysis for the two item uncapacitated lot-sizing problem with one-way substitution
Discrete Applied Mathematics
Multi-item uncapacitated lot sizing problem with inventory bounds
Optimization Letters
Capacitated lot sizing problems with inventory bounds
Annals of Operations Research
Exploring the potential and the boundaries of the rolling horizon Technique for the management of reservoir systems with one-year behaviour
Water Resources Management
Lot sizing with inventory bounds and fixed costs: polyhedral study and computation
Operations Research
Requirements planning with substitutions: exploiting bill-of-materials flexibility in production planning
Manufacturing and Service Operations Management
Forecast and rolling horizons under demand substitution and production changeovers: analysis and insights
IIE Transactions
Models and Lagrangian heuristics for a two-level lot-sizing problem with bounded inventory
OR Spectrum
A dynamic lot sizing problem with multiple customers: customer-specific shipping and backlogging costs
IIE Transactions
Forecast, solution, and rolling horizons in operations management problems: a classified bibliography
Manufacturing and Service Operations Management
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2022, Omega (United Kingdom)Citation Excerpt :They used the Branch-and-Cut algorithms to solve this problem and tested the performance by conducting numerical experiments. Jing and Mu [37,38] investigated two-item DLS model with perishable products. They developed DP algorithms to solve these models and obtained the planning horizons.