Production, Manufacturing, Transportation and Logistics
Efficient algorithms for the joint replenishment problem with minimum order quantities

https://doi.org/10.1016/j.ejor.2021.07.025Get rights and content

Highlights

  • Structural results and new algorithms under minimum order quantities.

  • ZIO policies are not optimal for joint ordering with MOQ restrictions.

  • Optimal inventory policy requires non-constant order quantities and intervals.

  • Delivery time constraints increase the relative gains of the proposed policies.

  • Empty replenishments lead to shorter fixed intervals and reduced cost.

Abstract

Suppliers often impose a minimum order quantity (MOQ) to ensure that production runs and shipping quantities of each specific item are economically viable. Additional economies of scale may arise as various items share high joint costs (e.g. the cost of an overseas container shipment) and require coordination of their replenishment policies. The buyer needs to find the joint ordering interval and replenishment policies for each of the individual items to minimize the system-wide ordering and inventory costs while satisfying the quantity restrictions. We focus on the case of constant demand without backlogging. Given a fixed joint reorder interval, we characterize the optimal inventory ordering strategy for each item, which we refer to as Multi-Replenishment (MR) inventory ordering policy, and derive a closed-form expression for the optimal average inventory costs per unit of time. In contrast to the previous literature, the MR inventory ordering policy does not follow the zero-inventory-ordering (ZIO) rule and allows for orders of different sizes over time to optimally accommodate the MOQ restriction. A numerical approach is used to determine the optimal joint reorder interval, and is later extended to account for the presence of empty replenishments. An extensive computational study shows 1) the total inventory and setup cost reduction associated with the MR inventory policies and empty replenishments; 2) the impact of various parameters on cost and policy performance; and 3) the loss associated with discretizing time at different levels of granularity (weeks, days, hours, minutes, seconds) to accommodate restrictions in the timing of item deliveries.

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 T.

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 T 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 N items that are jointly replenished, incurring a fixed cost of A for every order. Each item i,i=1,2,,N,is characterized by a constant demand rate Di per unit of time, a holding cost hi per unit per unit of time, and a minimum order quantity Mi. 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 NT (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 AT is the true ordering cost per unit of time in the system. When NT=, 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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