C ompanies such as Zara and World Co. have recently implemented novel product development processes and supply chain architectures enabling them to make more product design and assortment decisions during the selling season, when actual demand information becomes available. How should such retail firms modify their product assortment over time in order to maximize overall profits for a given selling season? Focusing on a stylized version of this problem, we study a finite horizon multiarmed bandit model with several plays per stage and Bayesian learning. Our analysis involves the Lagrangian relaxation of weakly coupled dynamic programs (DPs), results contributing to the emerging theory of DP duality, and various approximations. It yields a closed-form dynamic index policy capturing the key exploration versus exploitation trade-off and associated suboptimality bounds. In numerical experiments its performance proves comparable to that of other closedform heuristics described in the literature, but this policy is particularly easy to implement and interpret. This last feature enables extensions to more realistic versions of the motivating dynamic assortment problem that include implementation delays, switching costs, and demand substitution effects.
C arbon footprinting is a tool for firms to determine the total greenhouse gas (GHG) emissions associated with their supply chain or with a unit of final product or service. Carbon footprinting typically aims to identify where best to invest in emission reduction efforts, and/or to determine the proportion of total emissions that an individual firm is accountable for, whether financially and/or operationally. A major and underrecognized challenge in determining the appropriate allocation stems from the high degree to which GHG emissions are the result of joint efforts by multiple firms. We introduce a simple but general model of joint production of GHG emissions in general supply chains, decomposing the total footprint into processes, each of which can be influenced by any combination of firms. We analyze two main scenarios. In the first scenario, the social planner allocates emissions to individual firms and imposes a cost on them (such as a carbon tax) in proportion to the emissions allocated. In the second scenario, a carbon leader voluntarily agrees to offset all emissions in the entire supply chain and then contracts with individual firms to recoup (part of) the costs of those offsets. In both cases, we find that, to induce the optimal effort levels, the emissions need to be overallocated, even if the carbon tax is the true social cost of carbon. This is in contrast to the usual focus in the life-cycle assessment (LCA) and carbon footprinting literatures on avoiding double counting. Our work aims to lay the foundation for a framework to integrate the economics-and LCA-based perspectives on supply chain carbon footprinting.
Working in collaboration with Spain-based retailer Zara, we address the problem of distributing, over time, a limited amount of inventory across all the stores in a fast-fashion retail network. Challenges specific to that environment include very short product life cycles, and store policies whereby an article is removed from display whenever one of its key sizes stocks out. To solve this problem, we first formulate and analyze a stochastic model predicting the sales of an article in a single store during a replenishment period as a function of demand forecasts, the inventory of each size initially available, and the store inventory management policy just stated. We then formulate a mixed-integer program embedding a piecewise-linear approximation of the first model applied to every store in the network, allowing us to compute store shipment quantities maximizing overall predicted sales, subject to inventory availability and other constraints. We report the implementation of this optimization model by Zara to support its inventory distribution process, and the ensuing controlled pilot experiment performed to assess the model's impact relative to the prior procedure used to determine weekly shipment quantities. The results of that experiment suggest that the new allocation process increases sales by 3% to 4%, which is equivalent to $275 M in additional revenues for 2007, reduces transshipments, and increases the proportion of time that Zara's products spend on display within their life cycle. Zara is currently using this process for all of its products worldwide.
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