Transportation problem formulation

3. Three depots, F, G and H, supply petrol to three service stations, S, T and U. The table gives the cost, in pounds, of transporting 1000 litres of petrol from each depot to each service station. F, G and H have stocks of 540000,789000 and 673000 litres respectively.
S, T and U require 257000,348000 and 412000 litres respectively. The total cost of transporting the petrol is to be minimised. Formulate this problem as a linear programming problem. Make clear your decision variables, objective function and constraints.

5. The manager of a car hire firm has to arrange to move cars from three garages \(A , B\) and \(C\) to three airports \(D , E\) and \(F\) so that customers can collect them. The table below shows the transportation cost of moving one car from each garage to each airport. It also shows the number of cars available in each garage and the number of cars required at each airport. The total number of cars available is equal to the total number required.

1. Use the North-West corner rule to obtain a possible pattern of distribution and find its cost.
   (3)
2. Calculate shadow costs for this pattern and hence obtain improvement indices for each route.
3. Use the stepping-stone method to obtain an optimal solution and state its cost.

3. Three warehouses \(W , X\) and \(Y\) supply televisions to three supermarkets \(J , K\) and \(L\). The table gives the cost, in pounds, of transporting a television from each warehouse to each supermarket. The warehouses have stocks of 34, 57 and 25 televisions respectively, and the supermarkets require 20, 56 and 40 televisions respectively. The total cost of transporting the televisions is to be minimised. Formulate this transportation problem as a linear programming problem. Make clear your decision variables, objective function and constraints.
(Total 7 marks)

1. A glazing company runs a promotion for a special type of window. As a result of this the company receives orders for 30 of these windows from business \(B _ { 1 } , 18\) from business \(B _ { 2 }\) and 22 from business \(B _ { 3 }\). The company has stocks of 20 of these windows at factory \(F _ { 1 } , 35\) at factory \(F _ { 2 }\) and 15 at factory \(F _ { 3 }\). The table below shows the profit, in pounds, that the company will make for each window it sells according to which factory supplies each business.

The glazing company wishes to supply the windows so that the total profit is a maximum.
Formulate this information as a linear programming problem.

1. State your decision variables.
2. Write down the objective function in terms of your decision variables.
3. Write down the constraints and state what each one represents.

1. A team of gardeners is called in to attend to the grounds of a stately home. The three gardeners will each be assigned to one of three areas, the lawns, the hedgerows and the flower beds. The table below shows the estimated time, in hours, it will take each gardener to do each job.

The team wishes to complete the tasks in the least total time.
Formulate this information as a linear programming problem.

1. State your decision variables.
2. Write down the objective function in terms of your decision variables.
3. Write down the constraints and explain what each one represents.

1. Four workers, A, B, C and D, are to be assigned to three tasks, 1, 2 and 3 . Each task must be assigned to just one worker and each worker can do one task only.

Worker A cannot do task 2 and worker D cannot do task 3
The cost of assigning each worker to each task is shown in the table below.
The total cost is to be minimised. Formulate the above situation as a linear programming problem. You must define your decision variables and make the objective function and constraints clear.
(6) \section*{(Total for Question 1 is 6 marks)}