How to Solve a Standard Maximization Problem Using the Simplex Method and the Rowops Program

Transcription

1 How to Solve a Standard Maximization Problem Using the Simplex Method and the Rowops Program Problem: Maximize z = x + 0x subject to x + x 6 x + x 00 with x 0 y 0 I. Setting Up the Problem. Rewrite each structural constraint as an equation by adding a different slack variable and replacing the inequality symbol with an equal sign. x + x 6 x + x + s = 6 x + x 00 x + x + s = 00. Rewrite the objective function so that all the variables are on the left and the constant is on the right. When you do this, the coefficient of the variable to be optimized must be positive. correct: x 0x + z = 0 incorrect: x + 0x z = 0 Record the rewritten objective function below the constraint equations. Line the variables up vertically. x + x + s = 6 x + x + s = 00 x 0x + z = 0. Record the initial simplex tableau. Label each column. x x s s z Enter your initial simplex tableau into your calculator as matrix A. a. Hit MATRIX or nd MATRIX b. Hit to go to the Edit Menu c. Hit to select matrix A

2 d. Hit ENTER 6 ENTER to enter the number of rows and columns in the matrix. The calculator will then create an empty matrix with those dimensions. e. Enter the elements of the matrix, one number at a time, working from left to right. ALWAYS check every entry before you leave this screen. You can scroll left & right, up & down, to check the entire matrix. II. Using the ROWOPS Program. Enter the ROWOPS program. a. Hit the PRGM key. Select ROWOPS. b. You will see the matrix you just entered as Matrix A. Hit ENTER again. You will see a menu with four choices. The ROWOPS Menu. SWAP ROWS We never swap rows when using the simplex method.. MULTIPLY To get a one in a pivot position we multiply the row by the reciprocal of the pivot number. For example, if the pivot position is R C and it is occupied by, we would multiply row two by. To do this in the Rowops program we would select To Multiply, hit Enter, type the row number, hit Enter, type the multiplier, and hit Enter. On paper we record this row operation as R.. PIVOT To get a zero above or below a one in a pivot position, we add a multiple of one row to another. This is called pivoting. Pivoting clears a column of all nonzero non-pivot entries. For example, if there is a one in the pivot position R C, in R C, and in R C, we would clear the first column by doing the following, assuming we have gotten a one in the pivot position first: i. To get a zero at R C, we would multiply the pivot row (R ) by and add the result to the row we are getting a zero in (R ). On paper we would record this operation as R + R. ii. To get a zero at R C, we would multiply the pivot row (R ) by and add the result to the row we are getting a zero in (R ). On paper we would record this operation as R + R

3 To do this in the Rowops program, we would select To Pivot, hit Enter, type the row number where the pivot is located, hit Enter, type the column number where the pivot is located, hit Enter. This operation will clear the entire column (replace every non-pivot number in the column with zero). When pivoting in Rowops, we do not enter the manual row operations, just the location of the pivot. On paper, however, we must record the appropriate manual row operations to get every zero in the column. Hint: If you have a one in the pivot position and A in the position you want to clear, the pivoting operation is: A R p + R c If you have B in the pivot position and A in the position you want to clear, the pivoting operation is: A R p + B R c [R p = pivot row R c = row you are getting a zero in] Note: You are required to record the manual row operation and the resulting matrix at each step of the simplex method. Examples of manual row operations include R and R + R. It is not o.k. to record a pivoting operation as pivot on R C.. Stop When you have reached the final tableau, select To Stop to exit the program and then turn your calculator off to completely disengage the program. III. Solving the Problem 6. To identify the pivot column, find the most negative indicator (number in the bottom row to the left of the vertical bar). If there is a tie, choose the one farther to the left. For this matrix, the most negative indicator is 0, so the pivot column is column. 7. To identify the pivot, form test quotients by taking each number in the pivot column and dividing it into the corresponding constant. You are looking for the number which produces the smallest non-negative quotient. Disregard quotients with zero or a negative number in the denominator. If all quotients must be disregarded, no optimum solution exists. If there is a tie between two test quotients, choose the pivot closest to the top of the matrix. Circle the pivot in every matrix Test Quotients: = 6 = < 6 Conclusion: The pivot is the located in row column. : Initial Simplex Tableau (Matrix ) with the first pivot circled 8. To get a one in the pivot position, multiply the pivot row by the reciprocal of the number in the pivot position. Record this row operation in front of the resultant matrix. Since there is a in the pivot position, we must multiply the pivot row (R ) by

4 a. Select To Multiply. Hit ENTER b. Enter the row number Hit ENTER c. Enter the multiplier Hit ENTER Record the row operation and the resultant matrix on your paper. R Initial Matrix (Matrix ) row operation Resultant Matrix (Matrix ) 9. To get a zero above or below the pivot position, multiply the pivot row by the opposite of the number you want to become zero and add the result to the row you are getting the zero in, A R p + R c Be sure you record the appropriate manual row operation to get every zero in the column To get a zero in R C manually, we would multiply the pivot row (row ) by and add that to row one. On paper we record R + R. To get a zero in R C manually, we would multiply the pivot row (row ) by 0 and add that to row three. On paper we record 0R + R. a. Hit ENTER to advance the program. b. Select To Pivot. Hit ENTER c. Enter the row number the pivot is in. d. Enter the column number the pivot is in.

5 Record the row operations and the resultant matrix. x x s s R + R 0-0R + R Matrix row operations Matrix z Repeat steps 6 9 until all of the indicators are either positive or zero. This is called the final tableau. Read the solution from the final tableau. We still have a negative indicator, so we have not reached the final tableau. Repeating Step 6: The most negative indicator is the - in R C, so the pivot column is column. Repeating Step 7: Test Quotients: 60 = = = 60 = 7 < 7 Conclusion: The pivot is the located in R C. Repeating Step 8: Since there is a in the pivot position, we must multiply the pivot row (row ) by. a. Hit ENTER to advance the program. b. Select To Multiply. Hit ENTER c. Enter the row number Hit ENTER d. Enter the multiplier Hit ENTER Record the row operation and the resultant matrix on your paper.

6 R Matrix row operation Matrix Repeating Step 9: To get a zero in R C manually, we would multiply the pivot row (row ) by and add that to row two. On paper we record R + R. To get a zero in R C manually, we would multiply the pivot row (row ) by and add that to row three. On paper we record R + R. a. Hit ENTER to advance the program. b Select To Pivot. Hit ENTER c. Enter the row number the pivot is in. d. Enter the column number the pivot is in. Record the row operations and the resultant matrix on your paper. R R x x s s z Matrix row operations Final Tableau + R + R 0 8 Since the indicators of this matrix are all either positive or zero, we have reached the final matrix. We can now read the solution from the matrix. Basic variables: x = x = 0 Nonbasic variables: s = 0 s = 0 Conclusion: A maximum z-value of 8 occurs when x = and x = 0 6