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# FM Optimal Solution

## 2.5 Integer Solutions for Linear Programming Problems

Note: if you cannot remember how to analyse linear programming problems using graphical methods, revise notes for 2.4 Graphical Method for Solving Linear Programming Problems.

### Integer Solutions for Linear Programming Problems

• When using graphical methods for solving linear programming problems, we are presented with a region which encompasses all possible solutions. If the decision variables are continuous any point within this region it corresponds to a feasible solution. However, if we are dealing with discrete variables, only a finite number of points within this region will correspond to values the variables can actually take on.
• If the region is sufficiently small, it is possible for us to find all possible solutions for a linear programming problem with integer solutions.
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## 2.4 Graphical Method for Solving Linear Programming Problems

Note: if you cannot remember how to setup a linear programming problem, revise notes for 2.3 Linear Programming.

### Sliding Line Method

• By drawing lines on a graph representing the minimum/maximum conditions as allowed by the systems inequalities, we bound a region. This region is known as the feasible region.
• Any point within the feasible region represents a possible solution to this system. However, in general, we do not want just any possible solution, we want the “best” solution, known as the optimal solution. Often this is the combination of decision variable values that produces a maximum or minimum allowed value from the objective function.
• If we substitute a point from the feasible region into the objective function we get a possible value for the quantity being optimised. If we then equate the objective function to this value, we produce a line of the form where a, b and c are constants and x and y are the decision variables. If we do this for another point, we will produce another equation which is parallel to the one just derived with the only difference being a different c value shifting the line upwards or downwards.
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