Transportation problems

The table below shows the cost, in pounds, of transporting one unit of stock from each of four supply points, A, B, C and D, to three demand points, P, Q and R. It also shows the stock held at each supply point and the number of units required at each demand point.

A minimum cost solution is required.

Use the north-west corner method to obtain an initial solution to this transportation problem.

Perform one iteration of the stepping-stone method to obtain an improved solution. You must make your method clear by showing the route and stating the

• shadow costs

• improvement indices

• entering cell and exiting cell

Formulate the transportation problem as a linear programming problem. You must define your decision variables and make the objective function and constraints clear.

A standard transportation problem is described in the linear programming formulation below.

Let x_ij be the number of units transported from i to j

where i i {A, B, C, D}

Minimise P x_AR + 17x_AS + 24x_AT + 15x_BR + 29x_BS + 32x_BT

subject to

Given that the problem is balanced,

state the value of k.

transportation problem.

Use the north‑west corner method to obtain the cost of an initial solution to this transportation problem.

Perform one iteration of the stepping‑stone method to obtain an improved solution. You must make your method clear by showing the route and the

• shadow costs

• improvement indices

• entering cell and exiting cell.

The table below shows the cost, in pounds, of transporting one unit of stock from each of four supply points, A, B, C and D, to four sales points, P, Q, R and S. It also shows the number of units held at each supply point and the number of units required at each sales point.

A minimum cost solution is required.

Use the north‑west corner method to obtain an initial solution.

Taking AS as the entering cell, use the stepping‑stone method to find an improved solution. Make your method clear.

Perform one further iteration of the stepping‑stone method to obtain an improved solution. You must make your method clear by showing the route and stating the

• shadow costs

• improvement indices

• entering cell and exiting cell

State the cost of the solution found in (c).

Determine whether the solution obtained in (c) is optimal, giving a reason for your answer.

Table 1 shows the cost, in pounds, of transporting one unit of stock from each of four supply points, A, B, C and D, to three sales points, P, Q and R. It also shows the number of units held at each supply point and the number of units required at each sales point. A minimum cost solution is required.

Table 2 shows an initial solution given by the north-west corner method.

Taking AR as the entering cell, use the stepping-stone method to find an improved solution. Make your method clear.

Perform one further iteration of the stepping-stone method to obtain an improved solution. You must make your method clear by stating ●shadow costs ●improvement indices ●route ●entering cell and exiting cell.

Determine whether the solution obtained from this second iteration is optimal, giving the reason for your answer.

Formulate this situation as a linear programming problem. You must define your decision variables and make the objective function and constraints clear.

Explain why the Simplex algorithm cannot be used to solve transportation linear programming problems such as that formulated in (d).

Table 1 shows the cost, in pounds, of transporting one unit of stock from each of four supply points, A, B, C and D, to each of four demand points, P, Q, R and S. It also shows the stock held at each supply point and the stock required at each demand point. A minimum cost solution is required.

Table 2 shows an initial solution given by the north-west corner method.

Taking DQ as the entering cell, use the stepping-stone method to find an improved solution. Make your method clear.

Perform one further iteration of the stepping-stone method to obtain an improved solution. You must make your method clear by stating the

entering cell and exiting cell.

Determine whether the solution obtained from this second iteration is optimal, giving a reason for your answer.

State the cost of the solution found in (b).