Inventory Management with Advance Demand Information and Flexible Delivery

Tong Wang, Beril L. Toktay
2008 Management science  
This paper considers inventory models with advance demand information and flexible delivery. Customers place their orders in advance, and delivery is flexible in the sense that early shipment is allowed. Specifically, an order placed at time t by a customer with demand leadtime T should be fulfilled by period t + T ; failure to fulfill it within the time window [t, t + T ] is penalized. We consider two situations: (1) customer demand leadtimes are homogeneous and demand arriving in period t is
more » ... scalar dt to be satisfied within T periods. We show that state-dependent (s, S) policies are optimal, where the state represents advance demands outside the supply leadtime horizon. We find that increasing the demand leadtime is more beneficial than decreasing the supply leadtime. (2) Customers are heterogeneous in their demand leadtimes. In this case, demands are vectors and may exhibit crossover, necessitating an allocation decision in addition to the ordering decision. We develop a lower-bound approximation based on an allocation assumption, and propose protection level heuristics that yield upper bounds on the optimal cost. Numerical analysis quantifies the optimality gaps of the heuristics (2% on average for the best heuristic) and the benefit of delivery flexibility (14% on average using the best heuristic), and provides insights into when the heuristics perform the best and when flexibility is most beneficial. before the quoted due dates. Flexible delivery arrangements where the customer accepts delivery of the product at any time before its quoted due date are prevalent in many instances where companies sell directly to consumers. In contrast, the inventory management literature on advance demand information (ADI) focuses almost exclusively on exact delivery: When customers place an order with a given due date, it is assumed that they will not accept early delivery. In other words, early shipment is forbidden. At the same time, delayed shipment is penalized. Hariharan and Zipkin (1995) , one of the first papers to incorporate advance demand information into inventory management, provides the following justification: "This assumption is realistic in many though not all situations. The costs to customers of early deliveries are now widely appreciated, partly due to the JIT movement." The authors argue that early payment associated with early delivery is a deterrent. The additional inventory cost borne by the customer, and the uncertainty in delivery timing may also make flexible delivery unappealing to the customer. Much of the literature following has taken the exact delivery assumption for granted. However, in many situations where companies interact with end users directly (e.g. online retailing, services), it is customary for firms to tell their customers that the product/service will be delivered by a particular due date. It is easy to see why this is acceptable: If the product is for use or consumption, customers would typically prefer receiving their goods earlier rather than later. In this case, early delivery offers firms a powerful mechanism to reduce their inventory costs by transforming the firm's inventory cost into the customers' utility. Some firms recognize the variety in customer preferences and offer a range of options. For example, Dell's Intelligent Fulfillment program includes both delivery within five days and delivery on an exact date in its delivery options (Özer and Wei 2004). In this paper, we analyze inventory management with advance demand information and the possibility of early shipment, which we call flexible delivery. The model we use is closely related to the discrete-time, uncapacitated, advance demand information model of Gallego andÖzer (2001) , except that we allow for delivery flexibility. We first consider a model where customers are homogeneous in that they all have an identical demand leadtime T : Demand d i observed in period i needs Wang and Toktay: Advance Demand Information and Flexible Delivery 3 to be satisfied on or before period i + T . The supply leadtime is L. Flexible delivery introduces a nonlinearity into the system evolution equations. Nevertheless, we show that the structure of the optimal solution parallels that of Gallego andÖzer (2001) : If T ≤ L + 1, the system reduces to the traditional model by replacing the inventory position with the modified inventory position, and a modified (s, S) policy is optimal; if T > L + 1, a state-dependent (s(V ), S(V )) policy is optimal, where the stateV represents information about advance demands beyond the supply leadtime. We next turn to the more general model where customers are heterogeneous in their demand leadtimes: There are T + 1 categories of customers, with demand leadtimes ranging from 0 to T . Demand in period i is now a vector (d , where d j i stands for orders received in period i and to be satisfied by the end of period j. Unlike the homogeneous demand case, it is no longer optimal to satisfy orders as early as possible since future orders may be due earlier (called "demand cross-over") with T ≥ 2. Fulfilling observed advance orders early reduces holding cost, but at the same time, increases the probability of shortage as unobserved urgent orders may arrive in the future. Besides choosing when and how much to order (ordering decision), now inventory managers have to decide when and by how much to fulfill advance orders (allocation decision). As the analysis becomes intractable in this case, we develop an approximation that relaxes the nonnegativity constraints on delivery quantities. This is equivalent to allowing the firm to take previously "misallocated" units back and to reuse them to satisfy urgent demands. Imposing such a relaxation helps bypass the allocation decisions, and ensures that myopic allocation is optimal for the relaxed problem. This approximation yields a lower bound on the optimal objective function value. We then propose three protection level heuristics (PL(0), PL(σ), and PL(Σ)) that use different levels ("zero," "optimal," and "maximal") of stock to protect against shortages due to mis-allocation. These heuristics yield upper bounds on the optimal cost. We benchmark their performance by determining the optimality gap between the upper bounds they yield and the lower bound obtained from the relaxation. Numerical experiments yield structural results concerning the state-dependent (s(V ), S(V )) policies, some of which we prove for a special case. These experiments quantify the performance of the Wang and Toktay: Advance Demand Information and Flexible Delivery 4 heuristic -over an experiment with 540 instances, we find the average optimality gap obtained from the best heuristic PL(σ) to be 2.08%; and identify the cost benefit of advance demand information and delivery flexibility -on average a 14.06% cost reduction was achieved in our experiments by introducing flexible delivery to an ADI system. An interesting finding is that increasing the demand leadtime by one period has a higher benefit than shortening the supply leadtime by one period. This is in contrast to previous research (Hariharan and Zipkin 1995) showing that the two are equivalent for systems with ADI but no delivery flexibility. We show that delivery flexibility and ADI are complements: The benefit of delivery flexibility is higher when there are higher degrees of advance demand availability. The remainder of this paper is organized as follows. Section 2 positions our work in the context of the advance demand information literature. In Sections 3 and 4, we develop and analyze models with homogeneous and heterogeneous customers, respectively. Each section includes numerical analysis followed by structural and managerial insights obtained from them. Concluding remarks are presented in Section 5. All the proofs can be found in the e-companion to this paper, unless otherwise noted. Literature Review Our model directly contributes to the stream of research that analyzes uncapacitated inventory systems (where the supply leadtime is exogenous) with advance demand information and exact delivery. In addition to Hariharan and Zipkin (1995)'s continuous-review model discussed above, Gallego andÖzer (2001) study a periodic-review model with heterogeneous advance demand information. They show that it is optimal to adopt a modified (s, S) policy, where replenishments are made to raise the modified inventory position (=inventory position minus advance demands, hereafter MIP) to S whenever MIP reaches or drops below s. Gallego andÖzer (2003) andÖzer (2003) extend this analysis to multi-echelon models, and distribution systems, respectively. Other related models that all demonstrate the benefits of ADI are Bourland et al. (1996) in a two-stage supply system, Güllü (1997) in a two-echelon, single-depot, multiple-retailer problem, Decroix and Wang and Toktay: Advance Demand Information and Flexible Delivery
doi:10.1287/mnsc.1070.0831 fatcat:yudrzl6bn5arhf7rfp3jrmyfpu