Linear Programming Results
The Simplex algorithm will find the solution whenever the feasible region is non-empty but limited in size. There is no solution if the feasible region is empty (the constraints are such that no point satisfies all constraints) and when some variable can increase without limit.
The solution from a commercial package for the LP problem in the previous section is shown below.
The output has three sections: Solution; Rows (constraints); and Columns (decision variables). The maximum profit is £424.15 by making 16.65 of item A and 83.16 of item B. The maximum profit is in the 'Solution' section and the values for the decision variables are in the 'Value' row of the 'Columns Section'.
The results from the LP solution provide other information which is very useful for understanding the origin of profitability in a complex organisation. These are explained in the following table.
| Column | Description |
|---|---|
| At | Indicates if a constraint or variable is at a limit (LL or UL for upper and lower limits) or has freedom (BS). Limits on decision variables are specified as 'bounds' on the variable rather than constraints for computational efficiency. |
| Value | Gives the total left hand side of constraints (total quantity used) and the value of the decision variables. |
| Slack Value | The spare capacity in a constraint (resource). |
| Dual Value | The rate of increased profit/cost if the constraint can be relaxed. This is the marginal profit/cost of the resource. |
| Input Cost | Simply the coefficient in the cost function. |
| Reduced Cost | The extent by which the profit/cost has to change before the decision variable will become a critical variable. |
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