Name: THE SIMPLEX METHOD: STANDARD MAXIMIZATION PROBLEMS

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1 Name: 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. STANDARD MAXIMIZATION PROBLEMS meet the following conditions:. The objective function is maximized 2. All variables in the problem are non-negative.. 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:. Set up the initial simplex tableau: (a) Create slack variables. (b) Rewrite the objective function so that the coefficient of P is. 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. (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 ) pivot element: The element in both the pivot column and pivot row. (b) Convert pivot element to 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 2.. 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. (b) Variables heading columns not in unit form are assigned the value of. c Melanie Yosko This page may not be scanned, copied, or redistributed without permission. Page

2 Math 2 Section 6. Continued. Solve the linear programming problem using the Simplex Method. Maximize P = x + y + 2z Objective Function Subject to: x + y z 2 ] 5x y z Constraints (these get slack variables) x + y + z x ] y These say these variables are non-negative z (a) Is this a standard maximization problem? i. Is the objective function maximized? ii. Are all variables in the problem non-negative? Translation: are these inequalities present: x, y, z for every variable in the problem? iii. Can each constraint be written so that the expression containing the variables is less than or equal to a non-negative constant? Translation: Do all inequalities look like: (sum/difference of variable terms) or (sum/difference of variable terms) positive number iv. We see these three criteria are satisfied so, yes! We have a standard maximization problem. (b) Now to adjust the constraints so they can also go in the tableau: Now we create slack variables. We introduce a new variable to "take up the slack" so we can change our inequalities to equations. i. Let s just talk through how this works. Looking at the first constraint, x + y z 2, we see we do have an inequality. To demonstrate how to change to =, let s pull some numbers out of the air for demonstration purposes. ii. Suppose x =, y =, and z = 5. If we plug in those values for each variable, we get () + () (5) We see this is a true statement. iii. We will replace with +s =. So what number do we need to add to the left-hand side so that we have a true equation? 2 In this case, we need to add : + = 2 iv. Let s generalize this. We could have chosen many different values for x, y, z, and each would have given a different number needed to pick up the slack. As we change the values for x, y, z, the number needed changes too. We can also say it varies. We can call it the slack variable (able to vary) and represent it in general just like we did variables x, y, z. v. Thus, x + y z 2 becomes x + y z +s = 2 vi. Constraint becomes: x + y z +s = 2 vii. Constraint 2 becomes: 5x y z +s 2 = viii. Constraint becomes: x + y + z +s = (c) Putting everything in the initial simplex tableau: Equations with slack variables that came from the constraints go first. The objective function goes last. i. Rearrange the objective function so that the coefficient of P is all variables and P are on the same side of the equal sign. Translation: Move all terms on the right-hand side to the left-hand side. c Melanie Yosko This page may not be scanned, copied, or redistributed without permission. Page 2

3 ii. P = x + y + 2z becomes x y 2z + P = Maximize P = x + y + 2z Put this at the bottom Subject to: x + y z 2 ] x + y z +s = 2 5x y z These become 5x y z +s 2 = x + y + z x + y + z +s = x ] x y 2z + P = y Not in simplex tableau z iii. Now take these new equations and put them in a matrix, which we call our (d) Now let s find our pivot element! i. Look at the bottom row of the initial simplex tableau, to the right of the vertical line. Put an arrow under the most negative entry. Hint: Look at all numbers with a negative sign. If they were all positive, which is the largest? ii. Did you put an arrow under the negative 2 in the z column? If so, you are correct! That means the z column is the pivot column PC We will look at the z column and the constant column. We will divide like this: find the smallest non-negative ratio. constant entry PC entry to c Melanie Yosko This page may not be scanned, copied, or redistributed without permission. Page

4 Ratio 2 2 = # X 5 = # X PR (smallest non-negative ratio) 2 PC 2 5 PR 2 PC Now we pivot on the circled element. Remember what pivot means? We will perform row operations, making the circled entry and all other entries in that column. Scratch 2 5 ( )R R 2 c Melanie Yosko This page may not be scanned, copied, or redistributed without permission. Page

5 . Now we make a unit column by changing the, entry to a : R +R R Scratch: () a b = ( ( ) a ) top top b = bottom bottom R : +R : 2 new R :???????? Write R as fractions with common denominators 2 R : +R : new R : 2 9 Does the in the, 8 entry look familiar? It should! It was our smallest pivot ratio! Change the 2, entry to a : R +R 2 R R : +R 2 : 5 new R 2 :???????? Write R 2 as fractions with common denominators 2 R : 55 +R 2 : new R 2 : 2 9 Change the, entry to a : R +R R R : 2 +R : 2 new R :???????? Write R 2 as fractions with common denominators 2R : +R : new R : Successful pivot on, entry (made unit column). c Melanie Yosko This page may not be scanned, copied, or redistributed without permission. Page 5

6 . Has an optimal solution been reached? No. There are still negatives in the bottom row, to the left of the vertical line. Thus, we find the next pivot element. Do this in the tableau given to the right. 5 (a) PIVOT COLUMN: Put an arrow under the most negative entry in the bottom row, to the left of the vertical line. (b) PIVOT ROW: Put an arrow next to ( the smallest ) non-negative ratio constant entry. PC entry (c) Circle PIVOT ELEMENT (d) Now check your answer below. Ratio PR 6 2 X PC 5. Pivot on the circled element: R R Fill in the missing entries (see the blanks below): c Melanie Yosko This page may not be scanned, copied, or redistributed without permission. Page 6

7 R +R 2 R 2 Does the in the, 8 entry look familiar? It should! It was our smallest pivot ratio! Scratch: 2 R : 6 +R 2 : new R 2 :???????? 96 2 Write R 2 as fractions with common denominators 2 R 26 : +R 2 : new R 2 : The process will continue using row operations. Do you trust us? Can we just give you the final tableau?? Here is the final simplex tableau: Draw a line through the columns that are not unit columns. Those variables are assigned a value of.. Thus, we know: x = s = s = 8. Now look at the columns remaining in the table. Read off the answers like we did with matrices: y = 9 8 z = 2 s 2 = 6 P = 5 8 Here are all the variables: x = y = 9 8 z = 2 s = s 2 = 6 s = P = 5 8 c Melanie Yosko This page may not be scanned, copied, or redistributed without permission. Page

8 Now that you ve gotten the hang of it, let s try some more!. Put the linear programming problem into an initial simplex tableau: Maximize P = x + y Subject to: x + y 2x + y 5 x y 2. Put the linear programming problem into an initial simplex tableau: Maximize P = 8x + y + z Subject to: 2x + y + z 2x y + 5z 5 x + 9y + z 8 x y z c Melanie Yosko This page may not be scanned, copied, or redistributed without permission. Page 8

9 . Given the following simplex tableau, select the next pivot element and pivot once (and only once). x y z s s 2 P constant Given the final simplex tableau, read off the solutions: x = y = z = s = s 2 = s = P = 5. Given the final simplex tableau, read off the solutions: x = y = z = s = s 2 = s = P = c Melanie Yosko This page may not be scanned, copied, or redistributed without permission. Page 9