Math 1324 Finite Mathematics Section 6.4 The Simplex Method: Standard Maximization Problems

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1 Finite Mathematics Section 6.4 The Simplex Method: Standard Maximization Problems A linear programming problem consists of a linear objective function to be maximized or minimized subject to certain constraints in the form of linear equations or inequalities. The Simplex Method is a method of finding the corner points for a linear programming problem with n variables algebraically. In this section, we will only cover standard maximization problems. These problems meet the following conditions: 1. The objective function is maximized 2. All variables in the problem are non-negative. 3. Each constraint can be written so that the expression containing the variables is less than or equal to a non-negative constant. Here is the Simplex Method: 1. Set up the initial simplex tableau: (a) Create slack variables. (b) Rewrite the objective function so that the coefficient of P is 1. all variables and P are on the same side of the equal sign. (c) Place the constraints and the objective function in the initial simplex tableau. 2. Determine whether or not the optimal solution has been reached: (a) The optimal solution has been reached if all entries in the last column above the horizontal line and all entries in the last row to the left of the vertical line are non-negative. (b) If an optimal solution has been reached, skip to step 4. (c) If an optimal solution has not been reached, go to step Perform pivot operations: (a) Locate pivot element: pivot column: (a). Are there negatives in the constants column (above the horizontal line)? If no, skip to b. If yes, pick any negative in that row. The column for that entry is the pivot column. (b). The column with the most negative entry in the last row to the left of the vertical line pivot row: Divide each entry in the pivot column into the corresponding entry in the constants column. The pivot row is the row with the smallest NON-NEGATIVE such ratio. (Cannot divide by 0) pivot element: The element in both the pivot column and pivot row. (b) Convert pivot element to 1 by dividing all elements in the pivot row by the pivot element. (c) Use row operations to convert the pivot column into a unit column. (add multiples of pivot row to other rows as needed). (d) Return to Step Determine the solution: (a) The value of the variable heading each unit column is given by the entry lying in the column of constants in the row containing the 1. (b) Variables heading columns not in unit form are assigned the value of 0.

2 Ex: Solve the linear programming problem using the Simplex Method. Maximize P = x + 3y Subject to: x + y 80 3x 90 x 0 y 0 2

3 Ex: A company manufactures two models of fax machines, model A and model B. Each model A costs $100 to make, and each model B costs $10. The profits are $30 for each model A and $40 for each model B machine. The total number of fax machines demanded does not exceed 200 and the company has earmarked no more than $600,000 for manufacturing costs. (a) How many units of each model should the company make to maximize its profits? (b) What is the largest monthly profit the company can make? (c) Are there any resources left over? 3

4 A linear programming problem will have no solution: When the simplex method breaks down (i.e. can find no non-negative ratios). x y s 1 s 2 P constant Infinitely many solutions: When the last row to the left of the vertical line has a 0 in a column that is not a unit column. x y z s 1 s 2 P constant Ex: CalJuice Company has decided to introduce three fruit juices made from blending two or more concentrates. These juices will be packaged in 2-qt. (64-oz) cartons. One carton of pineapple-orange juice requires 8 oz each of pineapple and orange juice concentrates. One carton of orange-banana juice requires 12 oz of orange juice concentrate and 4 oz of banana pulp concentrate. Finally, one carton of pineapple-orange-banana juice requires 4 oz of pineapple juice concentrate, 8 oz of orange juice concentrate, and 4 oz of banana pulp. The company has decided to allot 16,000 oz of pineapple juice concentrate, 24,000 oz of orange juice concentrate, and 000 oz of banana pulp concentrate for the initial production run. The company has also stipulated that the production of pineapple-orange-banana juice should not exceed 800 cartons. Its profit on one carton of pineapple-orange juice is $1.00, its profit on one carton of orange-banana juice is $.80, and its profit on one carton of pineapple-orange-banana is $.90. To realize a maximum profit, how many cartons of each blend should the company produce? What is the largest profit it can realize? Are there any concentrates left over? 4

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