Multi-Location Newsvendor: Consider a fashion retailer in Singapore that manages three retail stores at Jurong West, Orchard and Harbour Front respectively. The retailer faces the prob- lem of how to manage the inventories of its best-selling product at these three locations for the new selling season. The product of concern has a retail price of S$100 at all three locations and costs S$50 per unit to procure from the manufacturer. The demands for the product at location i, i ? {Jurong West, Orchard, Harbour Front} are assumed to be normal truncated above zero, i.e., max{N (?i, ?i2), 0} and the corresponding parameters, estimated from historical data, are summarized in the first two rows in the table below. The demands across different locations are assumed to be independent.

Table 3.1: Estimation of Demand Parameters and Inventory Position Jurong West Orchard Harbour Front Hi 300 500 500 Oi 20 20 40 Inventory 300 500 500(3) Compute the total expected prot aggregated over the three locations (the model of process exibility example discussed in class can be used to compute this with minor modication). (b) What is the total expected prot if there is no demand uncertainty at all (i.e., 0-,; = 0)? (c) To deal with the demand uncertainty, the management team is considering employing a logistic service company to help transship its products from one store to another. For example, after the selling season begins, the demand is realized to be 250, 550 and 450 units at Jurong West, Orchard and Harbour Front respectively. In this case, Orchard is running out of inventory but both J urong West and Harbour Front have some leftover. The logistic company can then transship the leftover at Jurong West or Harbour Front to satisfy the remaining demand at Orchard. The logistic company charges a xed transaction cost of S3320!) (which is incurred even if nothing is shipped) and a unit transshipment cost depending on the origin and destination summarized in the table below. For example, if 50 units of product were to be shipped from Harbour Front to Orchard, then the total cost to the retailer is S$200 + 7 x 50. Alternatively, if 20 and 30 units of product were shipped from Harbour Front and J urong West to Orchard respectively, then the total cost to the retailer is S$200 +7 X 20 + 22 X 30. Should the retailer adopt the service provided Table 3.2: Transshipment Cost (S$/ur1it) H Harbour Front J urong West 19 | Orchard 7 | Harbour Front 0 by the logistic company? (Note that your solution may be unstable if the sample size is too small. You can check the robustness of your answer by looking at different sample sizes and see whether the solution stabilize or not.) (d) Suppose the retailer adopts the transshipment service in part (c) (so the xed transaction cost S$200 becomes sunk cost). The retailer now would like to determine the optimal amount of inventory to procure from the manufacturer and the amount of inventory to place at each locations. Let (51,52,53) denote the inventory position at the three locations (so the total procurement quantity is simply 31 + 32 + 53). Formulate the problem of determining the optimal (.51, .52, .53) as a two-stage stochastic optimization problem and provide the corresponding sample average approximation formulation with K demand samples. (Solving your formulation using Gurobi is encouraged but not required.)