Optimal solution:

Consider the following linear programing problem:

min z= -x₁-2x₂

Subject to the constraints:

2x₁+x₂ ≤ 5

x₁+x₂ ≤ 3

x₁,x₂≥ 0

Convert the minimization into maximization, then the result is:

max z = -min(-z)

max z = -min z^*

Therefore, the objective function is, max z^* = x₁+2x₂

Consider s₁ and s₂ are the two slack variables.

Add each of the slack variables to each constraint when they are less than or equal to type.

Here, the given two constraints are less than or equal to type, then these two slack variables are added to make it as standard form and at the same time add these variables to the objective function with 0 quantity weight.

Therefore, the standard form of the LP is as follows:

min z= -x₁-2x₂-0×s₂

Subject to the constraints:

2x₁+x₂ +s₁= 5

x₁+x₂+s₂ = 3

The matrix form is as follows:

max z^*=cX

Subject to the constraints:

AX=b

The initial simplex table is shown below:

In the initial simplex table, the first row is filled with objective function coefficients, second and third columns are filled with matrix A values.

Basic variable column is filled with matrix b...

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Chapter 4 Solutions

Introduction to mathematical programming

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Solution Summary: The author analyzes the linear programing problem by converting the minimization into maximization and adding slack variables to each constraint.

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