a. Formulate an integer programming model that could be used to find the minimum number of locations necessary to provide coverage to all areas. b. Solve the problem in part (a).

8 Use the simplex method to find the optimal solution to the following LP: max z = 5x1 + x2 s.t. 2x, + x2 5 6 X1 - X2 50 X1, X2 2 )

A health Centre will be built to serve 7 communities. The geographical location of the communities and their population is shown in the table below.     Communities A,  X =2.5 (km),       Y=4.5(km),           Population( 000's)=2 Communities B,  X =2.5 (km),       Y=2.5(km),           Population( 000's)=5    Communities C, X =5.5 (km),       Y=4.5(km),           Population( 000's)=10     Communities D, X =5 (km),           Y=2(km),               Population( 000's)=7  Communities E, X =8 (km),           Y=5(km),               Population( 000's)=10    Communities F, X =7 (km),           Y=2(km),               Population( 000's)=20                  Communities G, X =9 (km),          Y=2.5(km),               Population( 000's)=14                There is a possibility of building the Centre in two communities, community C and F. Based on the Demand-Distance criterion what is your recommendation for the site of the Centre. The distances to be measured based on Rectilinear method. Based on…

5. Consider the transportation problem having the following parameter table: Destination 1 Destination 2 Destination 3 Source 1 Source 2 Demand The initial BF solution given by Source 1 Source 2 Demand Vj 3 2.9 3 X11 = 3, X12 = 2, X22= 2, X23 = 2. Interactively apply the transportation simplex method to obtain an optimal solution. The initial transportation simplex tableau is Destination 1 Destination 2 3 2.9 3 3 2.7 2.8 4 2.7 2.8 4 2 2 0 0 0 2 Destination 3 supply 5 0 2 supply 5 4 2 4 Ui For each iteration tableau, clearly mark: U₁, V₁ values, č¡¡ + or negative values, the loop, the entering BV, the leaving BV. At the end of each iteration, also write out clearly what the new BF solution, and the new objection value.

3 A product is produced at three plants and shipped to three warehouses (the transportation costs per unit are shown in the following table). Warehouse Plant W, W2 W3 Plant Capacity P, P2 P3 20 16 24 300 10 10 500 12 18 10 100 Warehouse demand 200 400 300 a. Show a network representation of the problem. b. Develop a linear programming model for minimizing transportation costs; solve this model to determine the minimum cost solution. c. Suppose that the entries in the table represent profit per unit produced at plant i and sold to warehouse j. How does the model change from that in part (b)?

I. The Childfair Company has three plants producing child push chairs that are to be shipped to four distribution centers. Plants 1, 2, and 3 produce 12, 17, and 11 shipments per month, respectively. Each distribution center needs to receive 10 shipments per month. The distance from each plant to the respective distributing centers is given to the right: Distance Distribution Center 1 2 4 1 800 1300 400 700 2 3 1100 1400 600 1000 600 1200 800 900 The freight cost for each shipment is $100 plus 50 cents per mile. How much should be shipped from each plant to each of the distribution centers to minimize the total shipping cost? (a) Formulate this problem as a transportation problem by constructing the appropriate parameter table. (b) Draw the network representation of this problem. (c) Obtain an optimal solution.

A manufacturing company has 4 factories and 3 warehouses. The costs for transporting its products from factories to ware houses, the supply capacity of factories and demands in warehouses are given respectively by 6 4 81 70 170 4 3 2 C = 130 S = D= 200 2 4 7 100 130 1 5 8 200 a. Use minimum cost method to find an initial feasible solution. b. Determine the amount of products to be transported from factories to warehouses to minimize the cost.

6 see pictures to solve

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