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Simplex Method

Simplex Method. Example 4.2 # 17 Produced by E. Gretchen Gascon. The Problem. Let x1 be the # of church groups, x2 be the # of labor unions . Add the slack variable:. Set Up the Initial Simplex Tableau. Find the Most Negative Indicator.

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Simplex Method

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

  2. The Problem Let x1 be the # of church groups, x2 be the # of labor unions Add the slack variable:

  3. 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….

  4. Perform the row operations ~ to Gauss-Jordan for that pivot column 6 - 2 4 6 -6 0 3 - 0 3 0 - 2 -2 0 - 0 0 48 - 24 24 -300+200 -100 0 -600+600 0 + 0 0 0 + 200 200 3 + 0 3 0 + 2400 2400 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) 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)

  5. 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.

  6. Perform the row operations ~ to Gauss-Jordan for that pivot column 0 4 - 4 12 12 - 0 0 - 3 -3 6 0 0 - 0 48 -24 24 4 +2 0 0 + 0 0 0 +75 75 150 3 + 0 3 3000 -100+100 200 - 50 2400 +600 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 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)

  7. Solution x1 x2 s1 s2 z 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

  8. Same Problem with EXCEL Solver 1. Start by entering the following into an Excel Spreadsheet. 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.) You can find a tutorial document in course material on using the Solver. Your spread sheet should look like 

  9. Solver window Be sure to check the solver options:

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

  11. 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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