A ssigning products to and retrieving them from proper storage locations are crucial decisions in minimizing the operating cost of a unit-load warehouse. The problem becomes intractable when the warehouse faces variable supply and uncertain demand in a multiperiod setting. We assume a factor-based demand model in which demand for each product in each period is affinely dependent on some uncertain factors. The distributions of these factors are only partially characterized. We introduce a robust optimization model that minimizes the worst-case expected total travel in the warehouse with distributional ambiguity of demand. Under a linear decision rule, we obtain a storage and retrieval policy by solving a moderate-size linear optimization problem. Surprisingly, despite imprecise specification of demand distributions, our computational studies suggest that the linear policy achieves close to the expected value given perfect information and significantly outperforms existing heuristics in the literature.
A classical result in risk measure theory states that every coherent risk measure has a dual representation as the supremum of certain expected value over a risk envelope. We study this topic in more detail. The related issues include: 1. Set operations of risk envelopes and how they change the risk measures, 2. The structure of risk envelopes of popular risk measures, 3. Aversity of risk measures and its impact to risk envelopes, and 4. A connection between risk measures in stochastic optimization and uncertainty sets in robust optimization.2. We present independent proofs in Subsections 3.1-3.5 for the correspondence between several popular risk measures and their risk envelopes.3. We study sufficient and necessary conditions on the risk envelope that guarantee the aversity of the corresponding risk measure (See Propositions 4.2-4.5).4. We indicate a connection between the so-called uncertainty sets in robust optimization and the dual representation of risk measures (See Propositions 5.1-5.2specify and Theorem 5.1 for details).
Problem definition: In each period of a planning horizon, an online retailer decides how much to replenish each product and how to allocate its inventory to fulfillment centers (FCs) before demand is known. After the demand in the period is realized, the retailer decides on which FCs to fulfill it. It is crucial to optimize the replenishment, allocation, and fulfillment decisions jointly such that the expected total operating cost is minimized. The problem is challenging because the replenishment allocation is done in an anticipative manner under a push strategy, but the fulfillment is executed in a reactive way under a pull strategy. We propose a multiperiod stochastic optimization model to delicately integrate the anticipative replenishment allocation decisions with the reactive fulfillment decisions such that they are determined seamlessly as the demands are realized over time. Academic/practical relevance: The aggressive expansion in e-commerce sales significantly escalates online retailers’ operating costs. Our methodology helps boost their competency in this cutthroat industry. Methodology: We develop a two-phase approach based on robust optimization to solve the problem. The first phase decides whether the products should be replenished in each period (binary decisions). We fix these binary decisions in the second phase, in which we determine the replenishment, allocation, and fulfillment quantities. Results: Numerical experiments suggest that our approach outperforms existing methods from the literature in solution quality and computational time and performs within 7% of a benchmark with perfect information. A study using real data from a major fashion online retailer in Asia suggests that the two-phase approach can potentially reduce the retailer’s cumulative cost significantly. Managerial implications: By decoupling the binary decisions from the continuous decisions, our methodology can solve large problem instances (up to 1,200 products). The integration, robustness, and adaptability of the decisions under our approach create significant value.
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