A joint optimisation model for inventory replenishment,product assortment,shelf space and display area allocation decisions

In this paper, we propose an optimisation model to determine the product assortment, inventory replenishment, display area and shelf space allocation decisions that jointly maximize the retailer’s profit under shelf space and backroom storage constraints. The variety of products to be displayed in the retail store, their display locations within the store, their ordering quantities, and the allocated shelf space in each display area are considered as decision variables to be determined by the proposed integrated model. In the model formulation, we include the inventory investment costs, which are proportional to the average inventory, and storage and display costs as components of the inventory costs and make a clear distinction between showroom and backroom inventories. We also consider the effect of the display area location on the item demand. The developed model is a mixed integer non-linear program that we solved using LINGO software. Numerical examples are used to illustrate the developed model.     

A heuristic for triggering emergency orders in an inventory system

This paper considers a single-echelon inventory system with a warehouse facing compound Poisson customer demand. Normally the warehouse replenishes from an outside supplier according to a continuous review reorder point policy. However, it is also possible to use emergency orders. Such orders incur additional costs but have a much shorter lead time. We consider standard holding and backorder costs as well as ordering costs. A heuristic decision rule for triggering emergency orders is suggested. The decision rule minimizes the expected costs under the assumption that there is only a single possibility for an emergency replenishment, but the rule is used repeatedly as a heuristic. Given a certain reorder point policy for normal replenishments, our decision rule will always reduce the expected costs. A simulation study illustrates that the suggested technique performs well under different conditions.     

A Note on the Heuristic for Replenishment of Trended Inventories Considering Shortages

We discuss the inventory replenishment policy for an item having a deterministic demand pattern with a linear (positive) trend and shortages. A heuristic is developed to determine the decision rule for selecting the times and sizes of replenishments over a finite time-horizon so as to keep the total costs minimum. The use of the heuristic is illustrated with a numerical example.     

A multiple replenishment contract with ARIMA demand processes

This paper is concerned with a multiple replenishment contract with a purchase price discount in a supply chain. The chain is composed of one supplier, one buyer and consumers for a product. The replenishment contract is based upon the well-known (s, Q) policy, but allows us to contract replenishments at a future time with a price discount. Owing to the larger forecast error of future demand, the buyer should keep a higher level of safety stock to provide the same level of service as the usual (s, Q) policy. However, the buyer can reduce his purchase cost by ordering a larger quantity at a discounted price. Hence, there exists a trade-off between the price discount and the inventory holding cost. For the ARIMA demand processes, we present a model for the contract and an algorithm to find the number of the future replenishments. Computational experiments show that the algorithm finds the global optimum solution very quickly.     

Temporary price increase during replenishment lead time

Pricing and inventory management make up together revenue management, which is a significant effort to boost revenues out of available resources. Firms use various forms of dynamic pricing, including personalized pricing, markdowns, promotions, coupons, discounts, and clearance sales, to respond to market fluctuations and demand uncertainty. In this paper, we study a temporary price increase policy, a form of dynamic pricing, for a non-perishable product, a practice used by several giant retailers such as Amazon, Walmart, and Apple. We develop a continuous review inventory model that allows for joint replenishment and pricing decisions, where the lead time is not zero. A replenishment decision controls supply, while a pricing decision controls demand. A manager exercises a temporary price increase to slow demand and avoid a stock-out situation while waiting for a shipment, which may not necessarily increase revenues, but decrease stock-out costs. The problem is to solve for the optimal replenishment and the pricing policy parameters that maximize the long-run expected profit. That is, when and how much to order and when to raise the price. In this paper, the inventory level and time trigger a price increase. We solve many numerical examples and perform extensive sensitivity analyses. Our results show that compared to a model that focuses on fixed pricing, our model brings an additional increase in profit of about 13%.     

Long-term inventory policy-making through fuzzy decision-making models

The inventory policy, meant as a replenishment rule, has a considerable impact on most firms. The paper considers the determination of optimal inventory policy of firms from a global viewpoint of top management. The inventory is represented as a fuzzy system with the fuzzy inventory level as the output, the fuzzy replenishment as the input and fuzzy demand. The control problem is formulated in terms of decision-making in a fuzzy environment with fuzzy constraints imposed on replenishments, a fuzzy goal for preferable inventory levels to be attained and the fuzzy decision as the intersection of fuzzy constraints and the fuzzy goal at subsequent stages. The planning horizon is infinite. The problem is to find an optimal time-invariant strategy relating the optimal replenishments to the current inventory levels, maximizing the membership function of fuzzy decision. The existence of such a strategy is proved and an algorithm for its determination is given. The optimal time-invariant strategy obtained is represented as a fuzzy conditional statement equated with a fuzzy relation which is the firm's optimal fuzzy replenishment rule.     

Optimum inventory policies with discrete time proportional demand,discrete replenishment opportunities and different optimizing criteria

This paper extends the deterministic, single product, dynamic E0Q model to the case where demand increases linearly with time but at discrete time points and where the number of replenishments is also discrete. The problem is to find the number of orders and the replenishment schedule that will either maximize the return on the investment on inventory or minimize inventory costs. The proposed solution to either problem requires to first find the replenishment schedule that will minimize the total inventory throughout the planning horizon, for a given number of orders and then find the optimal number of replenishment points. The solution algorithms exploit the discrete nature of the demand and do not require the decomposability property of dynamic programming. This is particularly important in the return on investment case, where decomposability cannot be achieved.     

Deterministic lot-size inventory models with shortages and deterioration for fluctuating demand

We establish various inventory replenishment policies. We then analytically identify the best alternative among them based on the minimum total relevant costs. Finally, we prove that the relevant cost is convex with the number of replenishments. Consequently, the search for the optimal replenishment number is reduced to finding a local minimum.     

Separate vs. joint replenishment policies with maximum storage requirement costs

Multi-item inventory problems give rise to the possibility of time-phasing the replenishments of different items over the inventory cycle. Such a policy reduces the peak storage requirement, compared to a policy of simultaneous replenishment. This, in turn, increases the amount of warehouse space which is permanently available for leasing throughout the cycle. However, where cost savings may be achieved through combining setups of different items, as in the well known joint replenishment problem, such a time-phasing policy may increase total setup costs. This paper considers the two item joint replenishment problem, where a cost (equivalent to the opportunity cost of warehouse space) attaches to the peak storage requirement which occurs within the inventory cycle. Existing joint replenishment models do not consider such costs, but their consideration suggests that joint replenishment is not always optimal. We analyze possible policies under both joint and separate replenishment, and provide optimal closed form solutions. A numerical example to illustrate the tradeoff between joint and separate replenishment is provided.     

Integrating inventory control and a price change in the presence of reference price effects: a two-period model

Demand and procurement planning for consumer electronics products must cope with short life cycles, limited replenishment opportunities and a willingness to pay that is influenced by past prices and decreases over time. We therefore propose the use of an integrated pricing and inventory control model with a two-period linear demand model, in which demand also depends on the difference between a price-history-based reference price and the current price. For this model we prove that the optimal joint pricing/inventory policy for the replenishment opportunity after the first period is a base-stock list-price policy. That is, stock is either replenished up to a base-stock level and a list-price is charged, or it is not replenished and a discount is given that increases with the stock-level. Furthermore, we use real-world cell phone data to study the differences between an integrated policy and traditional sequential optimization, where prices are initially optimized based on the expected demand and ordering cost, and the resulting demand distribution is used to determine an optimal inventory policy. Finally, we discuss possible extensions of the model.     

Inventory and shelf-space optimization for fresh produce with expiration date under freshness-and-stock-dependent demand rate

It is well documented that the demand for fresh produce, to a great extent, depends on how fresh it is and an increase in shelf space for displayed stocks may induce more purchase of the produce. However, relatively little attention has been paid to the effect of expiration date despite the fact that produce deteriorates over time and expiration dates are often an important factor in consumers’ purchase decision. In this paper, we propose an economic order quantity model in which we explicitly specify the demand for fresh produce to be a function of its freshness-expiration date and displayed volume. With the demand being freshness-and-stock dependent, it may be profitable to maintain high stock level at the end of the replenishment cycle. Hence, we relax the traditional assumption of zero ending inventory to non-zero ending inventory. Consequently, the objective here is to determine the optimal level of shelf space size, replenishment cycle time, and/or ending inventory level in an effort of maximizing the total annual profit. We found that the total annual profit is strictly pseudo-concave with regard to the three decision variables, which simplifies the search for the global solution to a local optimal. Numerical examples are then presented to highlight the theoretical implications and managerial insights.     

Coordinating ordering and pricing decisions in a two-stage distribution system with price-sensitive demand through short-term discounting

We consider a short-term discounting model in which the distributor offers a discounted price for the retailers’ orders placed at the beginning of its replenishment cycle, in a non-cooperative distribution system with one distributor and multiple retailers, each facing price-sensitive demand. We examine the value of the price discount strategy as a mechanism for the distributor to coordinate the retailers’ ordering and pricing decisions under two common types of demand, linear demand in price and constant elasticity demand in price. Our numerical study reveals that, in the presence of homogeneous retailers (namely, retailers with identical demand rates), the distributor’s profit improvement due to coordination generally decreases as the number of retailers or the inventory holding cost rate increases, but increases as price elasticity increases. Although an increase in the inventory holding cost rate has a negative effect on the distributor’s profit, it may have a positive effect on the retailers’ profits. We further find that with heterogeneous retailers (namely, retailers with different demand rates), offering a discounted price under linear demand benefits the distributor when both the inventory holding cost rate and the variation in demand are either small or large. This cross effect, however, is absent under constant elasticity demand.     

Determining optimal selling price and lot size with a varying rate of deterioration and exponential partial backlogging

In this paper, a deterministic inventory model for deteriorating items with price-dependent demand is developed. The demand and deterioration rates are continuous and differentiable function of price and time, respectively. In addition, we allow for shortages and the unsatisfied demand is partially backlogged at a negative exponential rate with the waiting time. Under these assumptions, for any given selling price, we first develop the criterion for the optimal solution for the replenishment schedule, and prove that the optimal replenishment policy not only exists but also is unique. If the criterion is not satisfied, the inventory system should not be operated. Next, we show that the total profit per unit time is a concave function of price when the replenishment schedule is given. We then provide a simple algorithm to find the optimal selling price and replenishment schedule for the proposed model. Finally, we use numerical examples to illustrate the algorithm.     

Optimal ordering policies for periodic-review systems with replenishment cycles

In this paper, we consider inventory models for periodic-review systems with replenishment cycles, which consist of a number of periods. By replenishment cycles, we mean that an order is always placed at the beginning of a cycle. We use dynamic programming to formulate both the backorder and lost-sales models, and propose to charge the holding and shortage costs based on the ending inventory of periods (rather than only on the ending inventory of cycles). Since periods can be made any time units to suit the needs of an application, this approach in fact computes the holding cost based on the average inventory of a cycle and the shortage cost in proportion to the duration of shortage (for the backorder model), and remedies the shortcomings of the heuristic or approximate treatment of such systems (Hadley and Whitin, Analysis of Inventory Systems, Prentice-Hall, Englewood Cliffs, NJ, 1963). We show that a base-stock policy is optimal for the backorder model, while the optimal order quantity is a function of the on-hand inventory for the lost-sales model. Moreover, for the backorder model, we develop a simple expression for computing the optimal base-stock level; for the lost-sales model, we derive convergence conditions for obtaining the optimal operational parameters.     

The inventory replenishment planning and staggering problem: a bi-objective approach

To the best of our knowledge, this paper is the first one to suggest formulating the inventory replenishment problem as a bi-objective decision problem where, in addition to minimizing the sum of order and inventory holding costs, we should minimize the required storage space. Also, it develops two solution methods, called the exploratory method (EM) and the two-population evolutionary algorithm (TPEA), to solve the problem. The proposed methods generate a near-Pareto front of solutions with respect to the considered objectives. As the inventory replenishment problem have never been formulated as a bi-objective problem and as the literature does not provide any method to solve the considered bi-objective problem, we compared the results of the EM to three versions of the TPEA. The results obtained suggest that although the TPEA produces good near-Pareto solutions, the decision maker can apply a combination of both methods and choose among all the obtained solutions.     

Capacitated replenishment and disposal planning for multiple products with resalable returns

We consider a replenishment and disposal planning problem (RDPP) that arises in settings where customer returns are in as-good-as-new condition. These returns can be placed into inventory to satisfy future demand or can be disposed of, in case they lead to excess inventory. Our focus is on a multi-product setting with dynamic demands and returns over a finite planning horizon with explicit replenishment and disposal capacities. The problem is to determine the timing of replenishment and disposal setups, along with the associated quantities for the products, so as to minimize the total costs of replenishment, disposal, and inventory holding throughout the planning horizon. We examine two variants of the RDPP of interest both of which are specifically motivated by a spare part kitting application. In one variant, the replenishment capacity is shared among multiple products while the disposal capacity is product specific. In the other variant, both the replenishment and disposal capacities are shared among the products. We propose a Lagrangian Relaxation approach that relies on the relaxation of the capacity constraints and develop a smoothing heuristic that uses the solution of the Lagrangian problem to obtain near-optimal solutions. Our computational results demonstrate that the proposed approach is very effective in obtaining high-quality solutions with a reasonable computational effort.     

The optimality of myopic stocking policies for systems with decreasing purchasing prices

This paper studies the stocking/replenishment decisions for inventory systems where the purchasing price of an item decreases overtime. In a periodic review setting with stochastic demands, we model the purchasing prices of successive periods as a stochastic and decreasing sequence. To minimize the expected total discounted costs (purchasing, inventory holding and shortage penalty) for systems with backlogging and lost sales, we derive conditions, regarding the cost parameters, under which myopic stocking policies are optimal.     

A Simple Inventory Replenishment Decision Rule for a Linear Trend in Demand

We consider the situation of a deterministic demand pattern having a linear trend. The problem is to select the timing and sizes of replenishments so as to keep the total of replenishment and carrying costs as low as possible. An earlier developed heuristic for the general case of a deterministic, time-varying, demand pattern is specialized to the case of a linear trend. The simple decision rule is shown to lead to small cost penalties in two examples that have been exactly analyzed in an earlier article in this journal.     

Joint replenishment model with substitution

When products are coupled to the same cycle, the joint replenishment model (JRM) is used to determine optimal inventory levels, where the amount to order (for each item) is designed to minimize the joint holding and ordering costs based on a given demand. JRM studies assume that there is no substitution between items. However, this assumption is unrealistic in some settings where substitution cannot be ignored. This paper combines the separate works on substitution and joint replenishment and proposes a solution procedure for solving the joint replenishment model with substitution (JRMS) for two products within the framework of the classical economic order quantity model. We determine the optimal order quantities for each product taking into consideration substitution between them so that demand is partially met and the total cost associated with the delivery, holding, and shortage of the products is minimized. We also provide an extensive scenario analysis and draw insights. In particular, we shed some light on the role of substitution in reducing the fixed cost. We show that JRMS can result in substantial cost savings compared to the ordinary JRM.     

Economic order quantity under advance payment

Though advance payment is widely used in practice, its influences on buyer’s inventory policy are rarely discussed. This paper investigates the buyer’s inventory policy under advance payment, including all payment in advance and partial-advanced–partial-delayed payment. The buyer’s ordering policy is derived by minimizing his total inventory costs including inventory holding cost, ordering cost, and interest cost caused by advance payment or delayed payment. The conclusions show that when all the payment is paid in advance, the buyer’s optimal replenishment cycle is influenced only by the price discount associated with advance payment, and the length of advance payment has no effect. For the partial-advanced–partial-delayed payment case, the buyer’s replenishment cycle is also not influenced by the length of advance period. However, in this situation, the delayed period and the price discount may have impacts on the inventory policy. We also use discounted cash flow (DCF) model to derive the buyer’s replenishment cycle and show that the replenishment cycle is negatively related to the length of advance period. Numerical examples are presented to illustrate the results.