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Simplex Method Example 4.2 # 17 Produced by E. Gretchen Gascon

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The Problem ChurchLaborMax Writing2216 Follow-up1312 Money Raised Let x1 be the # of church groups, x2 be the # of labor unions Add the slack variable:

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Set Up the Initial Simplex Tableau Find the Most Negative Indicator Decide on the pivot element by finding the smallest non-negative quotient in the column 16/2 = 8 or 12/3 = 4 The winner is….

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Perform the row operations ~ to Gauss-Jordan for that pivot column times r1 minus 2 times r2 r1 3(r1c1) – 2(r2c1) 3(2) – 2(1) 3(r1c2) – 2(r2c2) 3(2) – 2(3) 3(r1c3) – 2(r2c3) 3(1) – 2(0) 3(r1c4) – 2(r2c4) 3(0) – 2(1) 3(r1c5) – 2(r2c5) 3(0) – 2(0) 3(r1c6) – 2(r2c6) 3(16) – 2(12) times r times r2 r3 3(r3c1) + 200(r2c1) 3(-100) + 200(1) 3(r3c2) + 200(r2c2) 3(-200) + 200(3) 3(r3c3) + 200(r2c3) 3(0) + 200(0) 3(r3c4) + 200(r2c4) 3(0) + 200(1) 3(r3c5) + 200(r2c5) 3(1) + 200(0) 3(r3c6) + 200(r2c6) 3(0) +200(12)

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Must repeat process as the bottom row has negative numbers still Pivot element is 4 because 24/4 = 6, and 12/1 = 12, 4 produces the smallest quotient.

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Perform the row operations ~ to Gauss-Jordan for that pivot column times r2 minus r1 r2 4(r2c1) – r1c1 4(1) – 4 4(r2c2) – r1c2 4(3) – 0 4(r2c3) – r1c3 4(0) – 3 4(r2c4) – r1c4 4(1) – (-2) 4(r2c5) – r1c5 4(0) – 0 4(r2c6) – r1c6 4(12) – r times r1 r3 r3c1 + 25(r1c1) (4) r3c2 + 25(r1c2) 0 +25(0) r3c3 +25(r1c3) 0 +25(3) r3c4 + 25(r1c4) (-2) r3c5 + 25(r1c5) 3 +25(0) r3c6 + 25(r1c6) (24)

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Solution x1x2s1s2z Read the solutions from the columns x1, x2 and z 6 Churches, 2 Labor Unions, for a maximum return of $1000 How do I know that I am done with row operations? Because column x1, x2 and z all have only one element and the rest are all 0’s in those columns

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Same Problem with EXCEL Solver 1. Start by entering the following into an Excel Spreadsheet. You can find a tutorial document in course material on using the Solver. 2. Make cell C3 the active cell, and select the Solver (this will be different depending upon which version of Excel you have) (You may need to add in the solver if you do not already have it.) Your spread sheet should look like

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Solver window Be sure to check the solver options:

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Solution By Solver: By Matrix Algebra (slide # 7) The answer: 6 Churches, 2 Labor Unions, for a maximum return of $1000

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Review Two ways were shown how to complete the simplex method of solving simultaneous inequalities. Please post comments, questions, regarding this slide presentation in the Main forum

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