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1. Consider the transportation problem where costs are to be minimized from locations 1, 2 and 3 to destinations A, B, C, and D according to the following cost and requirements table, where M is a very large number: Destination A B C D Supply Location 1 6 3 M 7 8 2 5 2 5 8 9 3 3 6 7 M 5 Demand: 4 6 6 6 (a) Solve for the optimal solution using the transportation simplex method and the northwest corner rule to obtain an initial basic feasible solution. Be sure to identify the shipping assignments and compute total shipping costs for your solution. 6 3 M 7 5 2 5 8 3 6 7 M 6 3 M 7 5 2 5 8 3 6 7 M 6 3 M 7 5 2 5 8 3 6 7 M 6 3 M 7 5 2 5 8 3 6 7 M 6 3 M 7 5 2 5 8 3 6 7 M 6 3 M 7 5 2 5 8 3 6 7 M 6 3 M 7 5 2 5 8 3 6 7 M (b) What would be the initial basic feasible solution for this problem using Vogel’s Approximation? Show your work. 6 3 M 7 5 2 5 8 3 6 7 M (c) What would be the initial basic feasible solution for this problem using Russell’s Approximation? Show your work. 6 3 M 7 5 2 5 8 3 6 7 M (d)What would be the initial basic feasible solution for this problem using the minimum cost technique? Show your work. 6 3 M 7 5 2 5 8 3 6 7 M 2. Solve one iteration of the following problem using the Interior Point Algorithm, with an initial starting solution of (12, 2) and α = 0.8. Max Z = x1 + 4x2 s.t. x1 + x2 = 14 x1, x2 ≥ 0 Initial solution: x1 = 12, x2 = 2, Z = . . Solution after one iteration: x1 = . ., x2 = . ., Z = . . 3. Four new products are being considered for production. The start-up costs and marginal revenues are shown below. Product 1 2 3 4 Start-up costs $15,000 $20,000 $25,000 $30,000 Marginal revenue $25 $30 $45 $40 Let x1, x2, x3, and x4 denote the continuous variables associated with the production levels of the four products. Formulate this problem as a mathematical programming problem, including the following additional constraints. • No more than three of the products can be produced • Product 1 is produced only if products 3 and 4 are produced • Only one of the following constraints holds, not both: 3x1 + 2x2 + x3 + 4x4 ≤ 2000 2x1 + 3x2 + x3 + 3x4 ≤ 2500 4. Consider the following network problem: 6 6 7 5 8 4 5 S 7 9 E 5 3 4 6 6 2 4 a. If the arc values represent distances, find the shortest path from S to E. Display the final solution and compute the shortest path. 6 6 7 5 8 4 5 S 7 9 E 5 3 4 6 6 2 4 b. If the arc values represent distances, find the minimal spanning tree that would connect all the nodes. (Disregard the arc direction.) Display the final solution and compute the distance of the minimal spanning tree. 6 6 7 5 8 4 5 S 7 9 E 5 3 4 6 6 2 4 c. If the arc values represent maximum capacities, find the maximum flow from S to E. Display the final solution and compute the maximum flow through the network. 6 6 7 5 8 4 5 S 7 9 E 5 3 4 6 6 2 4

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