Consider the following LP problem: Maximize 51 + 32 + 43 Subject to: 21 + 2 + 3

Question

Consider the following LP problem: Maximize 51 + 32 + 43

Subject to: 21 + 2 + 3 20 31 + 2 + 23 30 (1, 2,3 ) 0

a. Use slack/surplus variables to rewrite the problem so that all the constraints are equal-to constraints. (2 points)

b. Identify the different sets of basic variables that might be used to obtain a solution to the problem. Of the possible sets of basic variables, which lead to feasible solutions, and what are the values of the variables at each of these solutions? What is the value of the objective function at each of the basic feasible solutions? Summarize your results in a table. (10 points)

c. What is the optimal solution to the LP problem? (4 points)

d. Which constraints are binding at the optimal solution? (3 points)

Explanation / Answer

By Using Lagrange function

L= 5X1 + 3X2 + 4X3 - ( 2X1 + X2 + 2X3 - 20 )

change in L/ change in X1 = 5 - 2= 0

Therefore, = 5 / 2

change in L/ change in X2 = 3 - = 0

Therefore, = 3

change in L/ change in X3 = 4 - 2 =0

= 2

change in L/ change in = -2 X1 - X2 - 2X3 = 0

B ) L= 5X1 + 3X2 + 4X3 - ( 3X1 + X2 + 2X3 - 30 )

change in L/ change in X1 = 5 - 3= 0

Therefore, 5/3 =

change in L/ change in X2 = 3 - = 0

Therefore, 3 =

change in L/ change in X3 = 4 - 2 =0

Therefore, 2 =

change in L/ change in = -3 X1 -X2 - 2X3 = 0

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