3. Table 1 shows the cost, in pounds, of transporting one unit of stock from each of four supply points, \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D , to three sales points, \(\mathrm { P } , \mathrm { Q }\) 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.
\begin{table}[h]
| P | Q | R | Supply |
| A | 25 | 24 | 17 | 42 |
| B | 7 | 12 | 14 | 68 |
| C | 13 | 11 | 20 | 25 |
| D | 16 | 15 | 13 | 40 |
| Demand | 59 | 72 | 44 | |
\captionsetup{labelformat=empty}
\caption{Table 1}
\end{table}
Table 2 shows an initial solution given by the north-west corner method.
\begin{table}[h]
\captionsetup{labelformat=empty}
\caption{Table 2}
\end{table}
- 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).