2. Consider the data for a transportation problem in the matrix below. The matrix shows the cost of transporting 100 kg of the metal between the three sources and the four destinations. Total for Question 2: 32 marks (a) Represent this transportation problem using a directed graph. Indicate the transportation cost (2marks) on each arc.

Destination Source DI D2 D3 D4 Supplies 15 10 Si 10 10 6 S2 5 15 10 12 15 11 8 7 21 8 S3 Demands 5 3 8 17 (b) Formulate a linear programming problem which will enable to satisfy supply and demand requirements, and to minimize the total monthly transportation cost. All supply need to be used. Note: you can use appropriately defined vectors and matrices in your formulation. (c) Use the minimum cost method to find an initial feasible solution. (d) Use the transportation simplex method to find an optimal solution. (e) Write several sentences explaining the transportation plan for this transportation problem. (f) Add another source to the problem with a supply of five units and shipping costs to the four destinations of $4, $9, $7, and $13, respectively. Find a new optimal solution. Not all supplies need to be used. Provide the mathematical formulation of this problem and then use SAS to solve this problem. Ensure that your formulation is a linear or integer linear programming problem. Write several sentences to explain the optimal transportation plan (g) Continuing the transportation problem in part (f), suppose there are two plants, Pl and P2, which produce the metal. The P1 and P2 can supply 2000kg metal and 1500kg metal, respectively. Transportation costs per 100kg from the plant to the four sources (S1, S2, S3 and S4) are listed below, Source Plant si S2 S3 P1 3 5 2 S4 Supplies 1 20 5 15 P2 6 4 7 The transportation paths are from plants to sources, then to destinations (D1, D2, D3 and D4). Not all supplies need to be used. Provide the mathematical formulation of this problem and then use SAS to solve this problem. Ensure that your formulation is a linear or integer linear programming problem. Write several sentences to explain the optimal transportation plan. Destination Source DI D2 D3 D4 Supplies 15 10 Si 10 10 6 S2 5 15 10 12 15 11 8 7 21 8 S3 Demands 5 3 8 17 (b) Formulate a linear programming problem which will enable to satisfy supply and demand requirements, and to minimize the total monthly transportation cost. All supply need to be used. Note: you can use appropriately defined vectors and matrices in your formulation. (c) Use the minimum cost method to find an initial feasible solution. (d) Use the transportation simplex method to find an optimal solution. (e) Write several sentences explaining the transportation plan for this transportation problem. (f) Add another source to the problem with a supply of five units and shipping costs to the four destinations of $4, $9, $7, and $13, respectively. Find a new optimal solution. Not all supplies need to be used. Provide the mathematical formulation of this problem and then use SAS to solve this problem. Ensure that your formulation is a linear or integer linear programming problem. Write several sentences to explain the optimal transportation plan (g) Continuing the transportation problem in part (f), suppose there are two plants, Pl and P2, which produce the metal. The P1 and P2 can supply 2000kg metal and 1500kg metal, respectively. Transportation costs per 100kg from the plant to the four sources (S1, S2, S3 and S4) are listed below, Source Plant si S2 S3 P1 3 5 2 S4 Supplies 1 20 5 15 P2 6 4 7 The transportation paths are from plants to sources, then to destinations (D1, D2, D3 and D4). Not all supplies need to be used. Provide the mathematical formulation of this problem and then use SAS to solve this problem. Ensure that your formulation is a linear or integer linear programming problem. Write several sentences to explain the optimal transportation plan