Probing Line in Linear Program

The probing line is a line used to investigate the optimum value obtained from the objective function.

The optimum value of the objective form of the set of inequality system resolutions other than by using the corner point method can also be solved by using the probing line.

Steps to Find the Optimum Value by The Probing Line

Here are the steps to find the optimum value by using the probing line method:

1. Make the line ax + by = k, where ax + by is the objective form that the optimum value is sought. To make it easier to take k = ab.
2. Make parallel lines ax + by = k, ie by taking different k or shifting ax + by = k line to the left or right.

• If ax + by = k₁ is the leftmost line in the settlement region through the point (x₁, y₁), then k₁ = ax₁ + bx₁ is the minimum value.
• If ax + by = k₂ is the rightmost line in the settlement region through the point (x₂, y₂), then k₂ = ax₂ + by₂ is the maximum value.

Example:
Using the probing line method, determine the maximum and minimum values of the objective function Z = 2x + 3y in the feasible region in the figure below:

Answer:
To determine the maximum and minimum of the first done is to make the line equation of the known objective function is 2x + 3y = 6 = k, and named with the line g.

Look at the picture above. Scroll the line g so as to cut the feasble region at the leftmost point, ie the line g₁ which is the line parallel to the line g and right through the point (1, 2). Thus the minimum value Z is k₁ = 2 (1) + 3 (2) = 8. While g₂ is the most right and right line through the point (5, 4). Thus the maximum value of Z is k₂ = 2 (5) + 3 (4) = 22.

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Referensi :

• To'Ali's book math group accounting and sales

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