Exercise 1 (30 percent) Gram A/S is a Danish appliance manufacturer headquartered in Kolding producing environmentally-friendly and functional appliances. The company plans to launch a new model of refrigerator to the European market in early 2019. Gram has 5 factories producing this new model, which have production capacities listed in Table 1.1 below: Factory 1 850 Factory 2 500 Factory 3 700 Table 1.1 Factory 4 650 Factory 5 800 Client 1 and 2 require exact quantities 420 and 565units respectively to be delivered to them, no more and no less. The other clients require a minimum quantity to be delivered. The exact order quantities for client 1 and 2 as well as the minimum order quantities for the other clients are listed in Table 1.2 below: Client 1 Client 2 Client 3 Client 4 Client 5 Client 6 420 565 840 790 520 200 Table 1.2 Suppose that there are unit transportation costs induced by shipping refrigerators from different factories to different clients. The unit costs of transportation are listed in Table 1.3 below: Client 1 Client 3 Client 4 Client 6 Client 2 67 Client 5 59 66 56 76 40 89 81 54 73 62 34 Factory 1 Factory 2 Factory 3 Factory 4 Factory 5 67 45 30 90 50 56 54 49 62 74 62 53 45 54 54 45 77 71 42 Table 1.3 The unit costs above are given in DKK. 1. Formulate a transportation problem for Gram A/S, where you minimize the total transportation cost. How many decision variables are there in the model? How many constraints are there in the model? 2. Formulate and solve the problem by using Excel and the Problem Solver. Use the Problem Solver to conduct a sensitivity analysis. 3. How many units of product should be transported from factory 4 to client 5? 4. How much cost does Gram A/S need to pay at the optimal decision you obtained in Question 2? Based on the sensitivity report from Question 2 answer the following Questions 5-9 5. How much more does Gram A/S need to pay if the unit cost of transporting from factory 1 to client 5 increases from 59 DKK to 61 DKK? 6. Assume that the unit cost of transporting from factory 1 to client 5 is 59 DDK. What are the shadow prices of the capacities of the five factories? Explain their meanings. 7. Assume that the unit cost of transporting from factory 1 to client 5 is 59 DDK. What are the shadow prices of the demands from the six clients? Explain their meanings. 8. Assume that the unit cost of transporting from factory 1 to client 5 is 59 DDK. Assume that Gram 1 A/S can increase the capacity of any one of the plants by one unit at the cost of 15 DKK. Should Gram A/S buy additional capacity - if yes - why? If yes, at what plant should Gram A/S buy additional capacity? Exercise 1 (30 percent) Gram A/S is a Danish appliance manufacturer headquartered in Kolding producing environmentally-friendly and functional appliances. The company plans to launch a new model of refrigerator to the European market in early 2019. Gram has 5 factories producing this new model, which have production capacities listed in Table 1.1 below: Factory 1 850 Factory 2 500 Factory 3 700 Table 1.1 Factory 4 650 Factory 5 800 Client 1 and 2 require exact quantities 420 and 565units respectively to be delivered to them, no more and no less. The other clients require a minimum quantity to be delivered. The exact order quantities for client 1 and 2 as well as the minimum order quantities for the other clients are listed in Table 1.2 below: Client 1 Client 2 Client 3 Client 4 Client 5 Client 6 420 565 840 790 520 200 Table 1.2 Suppose that there are unit transportation costs induced by shipping refrigerators from different factories to different clients. The unit costs of transportation are listed in Table 1.3 below: Client 1 Client 3 Client 4 Client 6 Client 2 67 Client 5 59 66 56 76 40 89 81 54 73 62 34 Factory 1 Factory 2 Factory 3 Factory 4 Factory 5 67 45 30 90 50 56 54 49 62 74 62 53 45 54 54 45 77 71 42 Table 1.3 The unit costs above are given in DKK. 1. Formulate a transportation problem for Gram A/S, where you minimize the total transportation cost. How many decision variables are there in the model? How many constraints are there in the model? 2. Formulate and solve the problem by using Excel and the Problem Solver. Use the Problem Solver to conduct a sensitivity analysis. 3. How many units of product should be transported from factory 4 to client 5? 4. How much cost does Gram A/S need to pay at the optimal decision you obtained in Question 2? Based on the sensitivity report from Question 2 answer the following Questions 5-9 5. How much more does Gram A/S need to pay if the unit cost of transporting from factory 1 to client 5 increases from 59 DKK to 61 DKK? 6. Assume that the unit cost of transporting from factory 1 to client 5 is 59 DDK. What are the shadow prices of the capacities of the five factories? Explain their meanings. 7. Assume that the unit cost of transporting from factory 1 to client 5 is 59 DDK. What are the shadow prices of the demands from the six clients? Explain their meanings. 8. Assume that the unit cost of transporting from factory 1 to client 5 is 59 DDK. Assume that Gram 1 A/S can increase the capacity of any one of the plants by one unit at the cost of 15 DKK. Should Gram A/S buy additional capacity - if yes - why? If yes, at what plant should Gram A/S buy additional capacity
