Prove or disprove each of the following, using complementary slackness.
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- Consider the following linear program : z = 6x1 + 6x2 + 10x3 → min s.t. 4x1 + 3x2 + 2x3 ≥ 16 x1 + 2x2 + 5x3 ≥ 16 x1, x2, x3 ≥ 0 a) a) Build the dual problem corresponding to the given problem! b) Solve the dual problem graphically! Determine the exact coordinates by computing the intersection point of two appropriate lines! c) Conclude for the solution (objective and variables) of the primal problem by using the complementary slackness conditions.Consider the LP model and its corresponding graph below. Maximize z = 5x + 4ySubject to:3x + 2y <= 12x + 2y >= 8x >= 0y >= 0 From the given graph, what are the corner points of the feasible region? What are the values of x and y for the optimal solution? What is the maximum value of z?Are you able to use complementary slackness and your answer to part (a) to find solution(s) to the primal LPP? Why or why not?answer to part (a): attached here No, it does not have an optimal solution.
- Find r and h that minimize S subject to the constraint V = 54n.Consider the following maximization problem and select the correct number of slack variables required to solve the problem using the simplex method. Maximize: P=x+4y−2zP=x+4y−2z Subject to: x+2y−3z≤4x+2y−3z≤4 5x+6y+7z≤85x+6y+7z≤8 9x+10y+11z≤129x+10y+11z≤12 12+14y+15z≤1612+14y+15z≤16 x≥0, y≥0, z≥0Justify answer using complementary slackness and give optimal solution also to the dual aswell.
- Consider the following linear program: Max Z = -4x1 + 2x2 Subject To: -2x1 + 2x2 ≤ 7 x1 ≥ 2 x1 - 4x2 ≤ 0 2x1 + 2x2 ≥ 10 x1, x2 ≥ 0 Part A: Write the LP in standard equality form. Part B: Solve the original LP graphically (to scale). Clearly identify the feasible region and, if one or more exist, the optimal solution(s) (provide exact values for x1, x2, and Z).Consider the following constrained maximization problem:maximize y = x1 + 5 ln x2subject to k - x1 - x2 = 0,where k is a constant that can be assigned any specific value.a. Show that if k = 10, this problem can be solved as one involving only equality constraints.b. Show that solving this problem for k = 4 requires that x1 = 1.c. If the x’s in this problem must be nonnegative, what is the optimal solution when k = 4?d. What is the solution for this problem when k = 20? What do you conclude by comparing thissolution to the solution for part (a)?Note: This problem involves what is called a “quasi-linear function.” Such functions provide importantexamples of some types of behavior in consumer theory—as we shall see.