1 1 Practice Final John H. Vande Vate Fall, 2002

2 2 Question In the retailer game, we discovered that a discount of increases average weekly sales by  10% 31%  20% 73%  40% 181%  Any goods we don’t sell can be salvaged for $25/unit Accepting these values as given, should we ever sell at 40% off? If so, provide a clear description of when we would sell at 40% off. If not, provide a clear and compelling argument showing why we should not sell at 40% off.

3 3 Question 2 Provide a brief, well-organized description of how you would implement a pricing strategy based on the optimization we developed in class for the Retail Game. Your description need not cover the issues of collecting or maintaining the data required to build the model. Instead, it should focus on the question of how to use the model in a retail business to manage the discount policy for a given item.

4 4 Question 3 In class we described how a model used by financial managers can be adapted to manage the supply chain impacts of release variability. This model proposed holding inventory of the part to buffer the supply chain from variations in customer demand.

5 5 Question 3 Under the model, the supplier normally ships the same quantity every day. When inventory rises to T, the model recommends curtailing shipments until it falls to t and when inventory falls to 0, the model recommends expediting shipments or sending an unusually large shipment to bring inventory levels back up to b.

6 6 Question 3 Consider the special case in which: a. there is no inventory holding cost (H = 0) b. because of space limitations, the maximum inventory level T for the part cannot exceed a given level M c. the cost of expediting and the cost of curtailing shipments are equal (E = C > 0) d. there is no bound on y, the difference between T and t (i.e., a = infinity)

7 7 Question 3 What is the optimal strategy in this special case? T = t = b =

8 8 Question 4 A clothing retailer believes demand for an item over the course of the Spring season is uniformly distributed between 1,000 and 2,000 units. The total landed cost of getting the product to the store is $10/unit. The product sells for $30/unit and any unsold units can be liquidated for $5/unit. The retailer must sign a single contract for the full season’s order.

9 9 Question 4 A. How large should the retailer’s order be? B. What are the retailer’s expected net profits from selling the item in the Spring?

10 Retailer’s Order Find Q, the order quantity Risk: $5/item unsold Reward: $20/item sold P = Probability demand is less than Q 20(1-P) = 5P P = 20/25 That happens at Q = 1,800

11 Retailer’s Profit

12 Question 5 We are developing a Location/Allocation approach to building Milk Runs between suppliers and an auto assembly plant. Our approach, however takes into account the variance in the daily volumes assigned to the routes. Specifically, we are given the means variances and covariances of the daily volumes suppliers ship along with the capacity of the trucks. We locate R route “anchors” in the Location Step.

13 Question 5 In the Allocation Step we wish to assign each supplier to a single route so as to  Minimize the total distance between the route anchors and the suppliers assigned to them (We are given the distances between supplier j and route r)  Ensure that the average volume assigned to a route does not exceed the capacity of the truck  Ensure that the variance in the volume assigned to each route does not exceed a given parameter.

14 Question 5 Formulate a linear mixed-integer programming model for the Allocation Step. Use the following parameters and sets: Set R; /* The set of Routes */ Set S; /* The set of Suppliers */ Param Avg{S}; /* The average daily cubic feet of product each supplier ships to the plant */ Param Sigma2{S}; /* The variance in the daily cubic feet of product each supplier ships to the plant */ Param Cap; /* The cubic capacity of each truck */ Param MaxVar; /* The maximum variance in the daily volume assigned to any route. By this we mean that the variance of the sum of the volumes assigned to the route may not exceed MaxVar */

15 Question 5 Param Cov{S,S}; /* The covariances in the daily cubic feet of product suppliers ship to the plant */ Param Dist{S, R}; /* The distance between each supplier and each route */

16 Solution Var Assign{S, R} binary; /*Assign supplier to route */ Var CoAssign{s in S, t in S, R: s < t} binary; /* Assign j and k to route r */ Minimize Distance: sum{s in S, r in R}Dist[s,r]*Assign[s,r]; S.t. VehicleCap{r in R}: Sum{s in S} Avg[s]*Assign[s,r] <= Cap;

17 Solution S.t. VarianceCap{r in R}: Sum{s in S}Sigma2[s]*Assign[s,r] + 2*Sum{s in S, t in S: s < t}Cov[s,t]*CoAssign[s,t,r] <= MaxVar; S.t. AssignOne{s in S}: Sum{r in R} Assign[s,r] = 1; S.t. DefineCoAssign1[s in S, t in S, r in R: s < t}: CoAssign[s,t,r] >= Assign[s, r] + Assign[t, r] –1; S.t. DefineCoAssign2{s in S, t in S, r in R: s < t }: CoAssign[s, t, r] <= Assign[s, r]; S.t. DefineCoAssign3{s in S, t in S, r in R: s < t }: CoAssign[s, t, r] <= Assign[t, r];

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