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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 100200 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 2210016 1301012 -100-2000010 Find the Most Negative Indicator 2210016 1301012 -100-2000010 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 2210016 1301012 -100-2000010 3 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) 6 - 246 -603 - 030 - 2-20 - 0048 - 2424 3 times r3 + 200 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) -300+200-100-600+60000 + 000 + 2002003 + 03 0 + 2400 2400

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Must repeat process as the bottom row has negative numbers still 403-2024 1301012 -1000020032400 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 403-2024 1301012 -1000020032400 4 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) – 24 4 - 4012 - 0120 - 3-3 4 +2 60 - 0048 -2424 r3 + 25 times r1 r3 r3c1 + 25(r1c1) -100 +25(4) r3c2 + 25(r1c2) 0 +25(0) r3c3 +25(r1c3) 0 +25(3) r3c4 + 25(r1c4) 200 +25(-2) r3c5 + 25(r1c5) 3 +25(0) r3c6 + 25(r1c6) 2400 +25(24) -100+100 00 + 000 +7575 200 - 50 1503 + 03 2400 +600 3000

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Solution 403-2024 012-36024 007515033000 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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5.5 Dual problem: minimization with problem constraints of the form Associated with each minimization problem with constraints is a maximization problem.

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