Production, Manufacturing, Transportation and LogisticsEfficient algorithms for the joint replenishment problem with minimum order quantities
Introduction
We study the Joint Replenishment Problem with Minimum Order Quantity constraints in a constant demand setting. The Joint Replenishment Problem (JRP) arises when shared fixed costs, such as those associated with the dispatch of a truck or the shipment of a full container on an ocean-liner, are incurred regardless of the mix or quantities of the items ordered. The coordination of the replenishment of the items involved is then necessary in order to strike the right balance of fixed ordering and inventory costs.
The JRP has been well studied in the literature, including not only the fixed shared costs but also separate fixed costs associated with the ordering of each individual product. These separate costs reflect the practical fact that there is a fixed cost related to initiating production, and/or providing transportation and handling for this particular product. We refer the reader to Goyal and Satir (1989), and Khouja and Goyal (2008) for thorough reviews. In the mosaic manufacturing industry that motivated this study, however, the manufacturer does not face any fixed costs for each individual item ordered. Instead, suppliers impose a minimum order quantity for each of the hundreds of tile color, size and finish varieties to ensure orders are sufficiently large to cover the costs associated with starting a production run. The requirement of minimum order quantities is common in practice; see Fisher and Raman (1996), Porras and Dekker (2005), Zhao and Katehakis (2006), Zhu, Liu and Chen (2015), and Noh, Kim and Sarkar (2019) for examples ranging from minimum order quantities for the production of families of items at an assembly plant due to component lot sizes, to Sports Obermeyer requiring minimum order quantities from buyers, to Home Depot, Walmart and Alibaba adhering to minimum quantities imposed by their suppliers. Slim profit margins in low-cost countries make the use of MOQ requirements necessary to ensure large production quantities that allow manufacturers to break even (Shen, Tian & Zhu, 2019). While it may be easy to identify the shared cost (e.g. the cost of shipping a container from Istanbul to Boston) for the entire multi-item order, the specific fixed costs associated with each individual item are hard to estimate. As a result, firms seek economies of scale through the MOQ requirement instead. The use and appeal of MOQs has recently been highlighted in the media (CSCMP Supply Chain SmartBrief, 2021; MacLean, 2021): “Companies with deep analytical capabilities are seeing massive gains from MOQ implementation. These initiatives reduce small orders, increase warehouse efficiency, reduce inventory write-offs, make money-losing small accounts into money makers, increase sales and significantly boost profits.”
Despite their widespread use, MOQ restrictions have not received sufficient attention in the literature (Noh et al., 2019). To our knowledge, the only work to date that addresses the Joint Replenishment Problem with MOQ constraints (JRP_MOQ) is that of Porras and Dekker (2005, 2006) which addresses the case of constant demand for each of the items, and that of Noh et al. (2019) which considers stochastic demand and the added complexity of quantity discounts.
Inventory in this previous work is modeled assuming an underlying zero-inventory-ordering (ZIO) policy, where inventory of any particular item is always ordered in the same quantities and only replenished when the current stock hits zero (or a reorder point in the case of stochastic demand). These policies are optimal in the unconstrained case. As we shall show, however, the ZIO property is no longer optimal under the MOQ restriction. We characterize the optimal inventory policy, and derive a closed form expression for the associated average cost per unit of time for any given constant joint reorder interval .
In calculating the total costs, we initially assume that an order is placed (and the joint setup cost charged) at every joint reorder interval. This is, however, not always the case under MOQ constraints, as shown in Porras and Dekker (2006). The large orders required may cover demand for each of the items over more than one reorder interval, and result in some reorder opportunities where all items have sufficient inventory to last until the next ordering point. These “empty” replenishments will not incur any fixed costs and thus result in lower annual ordering costs, which makes shorter reorder intervals more attractive. We adapt the calculation of the total cost associated with any given joint reorder interval to account for the presence of empty replenishments and explore the impact on the optimal solution.
The paper is organized as follows. Section 2 briefly introduces the previous literature and highlights the contribution of the paper. In Section 3, we present the model formulation, define the new multi-replenishment inventory ordering policy, derive a closed-form expression for the average inventory per unit of time and propose a numerical approach to determine the optimal joint replenishment interval. Section 4 highlights the importance of empty replenishments and extends the numerical approach to account for them. In Section 5, we present numerical experiments to show the practical value of the new policy, the impact of various parameters, and the effect of restricting the joint replenishment interval to accommodate delivery time requirements. Section 6 concludes the study and proposes directions for future work.
Section snippets
Literature review
The work most closely related is that of Porras and Dekker (2006). The authors consider a deterministic JRP_MOQ setting identical to that of the current paper and develop effective approaches to calculate the joint reorder interval, first assuming the fixed cost is charged every interval and then under the more realistic scenario that allows for empty replenishments. The current paper follows a very similar pattern but with some major differences: 1) We consider a more general set of inventory
Modeling approach
Consider items that are jointly replenished, incurring a fixed cost of for every order. Each item is characterized by a constant demand rate per unit of time, a holding cost per unit per unit of time, and a minimum order quantity . Individual item setup costs are not present; instead, the MOQ ensures the order covers any existing fixed costs. Demand must be satisfied without backlogging. We assume instantaneous replenishment of each of the products without loss of
Empty replenishments
As long as there is at least one item in set (i.e., one item whose demand over the reorder interval is greater than or equal to its MOQ) then positive orders will be placed on each replenishment interval and is the true ordering cost per unit of time in the system. When , however, some ordering intervals may have “empty” replenishments; this happens when all the items have sufficient inventory to cover their demand for one more interval without receiving any goods at that ordering
Numerical experiments
The examples presented above demonstrate that there are specific settings where allowing for multiple replenishments within an item's regeneration cycle leads to significant improvements in system performance. To explore the practical impact of the multi-replenishment inventory ordering strategy relative to the previously proposed policies that implicitly assume a ZIO policy, we replicate the extensive numerical experiments presented in Porras and Dekker (2006), which we will refer to as P&D,
Conclusion & future work
The Joint Replenishment Problem with Minimum Order Quantities has received little attention in the literature. Previous models assume each item follows a ZIO policy. In this study, we identify the optimal inventory policy, which allows for multiple replenishments for each item during its regeneration cycle and leads to substantially lower inventory costs. We derive a closed form expression for the inventory costs resulting from the optimal multi-replenishment inventory ordering policy, and use
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2024, European Journal of Operational ResearchReplenishment and transshipment in periodic-review systems with a fixed order cost
2023, European Journal of Operational ResearchCitation Excerpt :However, in the presence of transshipment, little is known about how to order new supplies. Nevertheless, the fixed cost of placing an order for small businesses is high due to the lack of an advanced ordering system, and optimal ordering is essential for effective inventory management (e.g., Chapman, Gatewood, Arnold, & Clive, 2016; Creemers & Boute, 2022; Muriel, Chugh, & Prokle, 2022). There are generally two types of transshipment, reactive transshipment and proactive transshipment, and they differ in the trigger mechanism.
Optimizing a multi-echelon location-inventory problem with joint replenishment: A Lipschitz ϵ-optimal approach using Lagrangian relaxation
2023, Computers and Operations ResearchCitation Excerpt :In such cases, JR policy creates an opportunity for cross-border e-commerce companies to reduce their logistic costs. Although applications of JR policy can be easily found in retail or manufacturing enterprises that adopt a centralized procurement strategy (Nilsson and Silver, 2008; Porras and Dekker, 2008; Hsu, 2009; Moon et al., 2011; Liu et al., 2018; Muriel et al., 2022). The consideration of integrated optimization of location and inventory is rare in JRPs.