Retail inventory management with lost sales

The inventory control problem of traditional store-based grocery retailers has several challenging features. Demand for products is stochastic, and is typically lost when no inventory is available on the shelves. As the consumer behavior studies reveal, only a small percentage of customers are willing to wait when confronted with an out-of-stock situation, whereas the remaining majority will either buy a different product, visit another store, or entirely drop their demand. A store orders inventory on a periodic basis, and receives replenishment according to a fixed schedule. The ordered stock is typically delivered before the next ordering moment, which results in lead times shorter than the review period length. Order sizes are often constrained to integer multiples of a fixed batch size, the case packs, generally dictated by the manufacturer. Upon order receipt at the store, the stock is manually stacked on the shelves, to serve customer demand. Shelf space allocation of many products is limited, dictated by marketing constraints. Hence, surplus stock, which does not fit on the regular shelf, is temporarily stored in the store’s backroom, often a small place, poorly organized. The focus of this dissertation is on developing quantitative models and designing solution approaches for managing the inventory of a single item, under periodic review, when some or all of the following characteristics are taken into account: ?? Lost sales. Demand that occurs when no inventory is available is lost, rather than backordered. ?? Fractional lead time. Time between order placement and order delivery is shorter than the review period length. ?? Batch ordering. Order sizes are constrained to integer multiples of a fixed batch size. ?? Limited shelf space. Shelf space allocation is predetermined. The retailer’s inventory is split between the sales floor and the backroom, which is used to temporarily store surplus inventory not accommodated by the regular shelves. We consider optimal, as well as easy-to-understand inventory replenishment policies, where the objective is to minimize the long-run average cost of the system. Two types of costs are primarily recognized in the inventory models developed in this dissertation: ?? inventory related costs: for ordering, for holding products on stock, and penalty costs for not being able to satisfy end-customer demand, and ?? handling related costs: for shelf stacking, and for handling backroom stock. Despite empirical evidence on the dominance of handling costs in the store, remarkably little is reported in the academic literature on how to manage inventory in the presence of handling costs. A reason for this is that formal models of handling operations are still scarce. In this dissertation, we first formalize a model of shelf stacking costs, using insights from an empirical study. Then, we extend the traditional single-item lost-sales periodic-review inventory control model with several realistic dimensions of the replenishment practices of grocery retailers: batch ordering, handling costs, shelf space and backroom operations. The models we consider are too complex to lend themselves to straightforward analytical tractability. As a result, numerical solution methods based on stochastic dynamic programming are proposed in this dissertation, and near-optimal alternative replenishment policies are investigated. Chapter 2 addresses operational concerns regarding the shelf stacking process in grocery retail stores, and the key factors that influence the execution time of this common store operation. Shelf stacking represents the regular store process of manually refilling the shelves with products from new deliveries, which is typically time consuming and costly. We focus on products that are replenished in pre-packed form but presented to the end-customer in individual units. A motion and time study is executed, and the complete shelf stacking process is broken down into several sub-activities. The main time drivers for each activity are identified, relationships are established, tested and validated using real-life data collected at two European grocery retailers. A simple prediction model of the total stacking time per order line is then inferred, in terms of the number of case packs and consumer units. The model can be applied to estimate the workload and potential time savings in the stacking process. Implications of our empirical findings for inventory replenishment decisions are illustrated by a lot-sizing analysis in Chapter 2, and further explored in Chapter 3. Chapter 3 defines a single item stochastic lost sales inventory control model under periodic review, which is designed to handle fractional lead times, batch ordering and handling costs. We study the settings in which replenishment costs reflect shelf stacking costs and have an additive form with fixed and linear components, depending on the number of batches and units in the replenishment order. We explore the structure of optimal policies under the long-run average cost criterion and propose a new policy, referred to as the (s;QjS; nq) policy, which partially captures the optimal policy structure and shows close-to-optimal performance in many settings. In a numerical study, we compare the performance of the policy against the best (s; Q; nq) and (s; S; nq) policies, and demonstrate the relative improvements. Sensitivity analyses illustrate the impact of the different problem parameters, in particular the batch size and the handling cost parameters, on the optimal solutions and associated average costs. Managerial insights into the effect of ignoring handling costs in the optimization of replenishment decisions are also discussed. Chapter 4 extends the retail setting from Chapter 3 to situations in which there is a limited shelf space to display goods on the sales floor, and the retailer uses the store’s backroom to temporarily store surplus stock. As a result, the back stock is regularly transferred from the backroom to the sales floor to satisfy end-customer demand, which results in additional handling costs for the retailer. We investigate the effect of using the backroom on the inventory system performance, where performance is measured with respect to the optimal ordering decisions, and the long-run average cost of ordering, holding, lost-sales and merchandise handling. Two extensions of the inventory model with ample shelf space are proposed in Chapter 4, which include a (i) linear or (ii) fixed cost structure for additional handling operations. In a numerical study, we discuss several qualitative properties of the optimal solutions, illustrate the additional complexities of the second model, and compare the findings with those of the previous chapter. Furthermore, we build several managerial insights into the effect of problem parameters, in particular the shelf space capacity, on the system’s performance. Finally, we quantify the expected cost penalty the retailer may face by ignoring the additional handling costs in the optimization of inventory decisions, and illustrate the trade-off between the different cost components. Chapter 5 studies a variant of the traditional infinite-horizon, periodic-review, singleitem inventory system with random demands and lost sales, where we assume fractional lead times and batch ordering, and allow for ??xed non-negative ordering costs. We present a comparison of four situations: zero vs. positive setup costs, and unit vs. non-unit batch sizes. For all cases, the optimal policy structure is only partially known in general. We show in a numerical study that the optimal policy structure of the most general model is usually more complex than that of the models with positive setup cost, or batch ordering only. Based on the gained insights, we further test the performance of the near-optimal (s;QjS; nq) heuristic policy in the different cases, and demonstrate its effectiveness. Also, well-known inventory control policies of base-stock, or (s; S) type are extended to the case of batch ordering and studied in comparison with the new heuristic under several conditions.