2. (15') 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 E {Jurong West, Orchard, Harbour Front) are assumed to be normal truncated above zero, i.e., max{N (1.0),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 Table 3.1: Estimation of Demand Parameters and Inventory Position Jurong West Orchard Harbour Front Hi 0i Inventory 300 20 300 500 20 500 500 40 500 independent. The management team decides to procure in total 1300 products from the manufacturer 3-1 ssignment 3: Due on Oct 26, 2021 3-2 a and put 300, 500 and 500 units at Jurong West, Orchard and Harbour Front respectively (see last row in Table 3.1). That is, the inventory at a particular location is matched with the mean demand (before truncation). No inventory can be replenished from the manufacturer during the selling season and unsold products are worthless after the selling season. (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 Jurong 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 fixed transaction cost of S$200 (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 Jurong West to Orchard respectively, then the total cost to the retailer is S$200+7 x 20+22 x 30. Should the retailer adopt the service provided Table 3.2: Transshipment Cost (S$/unit) Jurong West Orchard Harbour Front Jurong West 0 22 Orchard 22 0 7 Harbour Front 19 7 0 19 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 (e) (so the fixed 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 (31,82,83) denote the inventory position at the three locations (so the total procurement quantity is simply 51 +52 +53). Formulate the problem of determining the optimal (81, 82, 83) 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.) 2. (15') 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 E {Jurong West, Orchard, Harbour Front) are assumed to be normal truncated above zero, i.e., max{N (1.0),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 Table 3.1: Estimation of Demand Parameters and Inventory Position Jurong West Orchard Harbour Front Hi 0i Inventory 300 20 300 500 20 500 500 40 500 independent. The management team decides to procure in total 1300 products from the manufacturer 3-1 ssignment 3: Due on Oct 26, 2021 3-2 a and put 300, 500 and 500 units at Jurong West, Orchard and Harbour Front respectively (see last row in Table 3.1). That is, the inventory at a particular location is matched with the mean demand (before truncation). No inventory can be replenished from the manufacturer during the selling season and unsold products are worthless after the selling season. (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 Jurong 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 fixed transaction cost of S$200 (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 Jurong West to Orchard respectively, then the total cost to the retailer is S$200+7 x 20+22 x 30. Should the retailer adopt the service provided Table 3.2: Transshipment Cost (S$/unit) Jurong West Orchard Harbour Front Jurong West 0 22 Orchard 22 0 7 Harbour Front 19 7 0 19 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 (e) (so the fixed 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 (31,82,83) denote the inventory position at the three locations (so the total procurement quantity is simply 51 +52 +53). Formulate the problem of determining the optimal (81, 82, 83) 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.)