Simplex Method (Maximization).pptx - The Simplex Method(Maximization The geometric method of solving linear programming problems presented before The

Simplex Method (Maximization).pptx - The Simplex...

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The Simplex Method (Maximization) The geometric method of solving linear programming problems presented before. The graphical method is useful only for problems involving two decision variables and relatively few problem constraints. What happens when we need more decision variables and more problem constraints? We use an algebraic method called the simplex method, which was developed by George B. DANTZIG (1914-2005) in 1947 while on assignment with the U.S. Department of the air force.
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Standard Maximization Problems in Standard Form A linear programming problem is said to be a standard maximization problem in standard form if its mathematical model is of the following form: Maximize the objective function Subject to problem constraints of the form With non-negative constraints max 1 1 2 2 ... n n Z P c x c x c x 1 1 2 2 ... , 0 n n a x a x a x b b 1 2 , ,..., 0 n x x x
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Slack Variables “A mathematical representation of surplus resources.” In real life problems, it’s unlikely that all resources will be used completely, so there usually are unused resources. Slack variables represent the unused resources between the left-hand side and right-hand side of each inequality.
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Basic and Nonbasic Variables Basic variables are selected arbitrarily with the restriction that there be as many basic variables as there are equations. The remaining variables are non-basic variables. This system has two equations, we can select any two of the four variables as basic variables. The remaining two variables are then non-basic variables. A solution found by setting the two non-basic variables equal to 0 and solving for the two basic variables is a basic solution . If a basic solution has no negative values, it is a basic feasible solution . 1 2 1 1 2 2 2 32 3 4 84 x x s x x s
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SIMPLEX METHOD Step-1 Write the standard maximization problem in standard form, introduce slack variables to form the initial system, and write the initial tableau. Step-3 Select the pivot column Step-5 Select the pivot element and perform the pivot operatio n STOP The optimal solution has been found. STOP The linear programming problem has no optimal solution Step 2 Are there any negative indicators in the bottom row? Step 4 Are there any positive elements in the pivot column above the dashed line? Simplex algorithm for standard maximization problems
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To solve a linear programming problem in standard form, use the following steps.
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