Definitions [7]
An optimisation problem is a problem in which the value of one quantity has to be made as large as possible or as small as possible under given restrictions. If the quantity and restrictions are linear, the problem becomes a Linear Programming Problem (LPP).
A Linear Programming Problem (LPP) is a problem in which a linear objective function is to be maximised or minimised subject to a set of linear constraints and non-negative conditions on the variables.
A region is said to be convex if the line segment joining any two points in the region lies entirely within the region.
The common region satisfying all the given inequalities is called the solution set.
The equation ax + by = c is called the associated equation of the inequality.
The linear function whose maximum or minimum value is to be determined is called the objective function.
To optimise means to maximise or minimise.
Theorems and Laws [1]
Statement:
If a linear objective function has a maximum or minimum value over a feasible region, then the maximum or minimum occurs at one of the corner points of the feasible region.
Key Points
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Linear Programming is a method of optimisation under linear constraints.
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The quantity to be optimised is called the objective function.
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The unknown quantities are called decision variables.
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Restrictions are called constraints.
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Non-negative restrictions must always be included.
| Condition | Region represented |
|---|---|
| ( x > 0 ) | Right of the y-axis |
| ( x < 0 ) | Left of the y-axis |
| ( y > 0 ) | Above x-axis |
| ( y < 0 ) | Below x-axis |
| ( x ≥ 0 ) | Includes y-axis |
| ( y ≥ 0 ) | Includes x-axis |
