ESI 6314: Midterm Take Home Exam Answers Question 1 (15 points): True/False and why/why notThe following five questions are true/false. Credit is given onlyif a brief rationale is provided with your answers. (A sentence or two will suffice.) Part a (3 points): Alloptimal solutions to a linear program exist at extreme points. False. Some linear programs have alternative optimal extreme points. Every point on the line between alternative optimal extreme points are also optimal. In this case, there’s an infinite number of optimal solutions, and some are not extreme points. Part b (3 points): In the simplex method for a minimization problem, the objective function strictly decreases at each iteration until an optimal solution is found. (An iteration consists of entering a nonbasic variable and exiting a basic variable.) False. If the problem is degenerate, we could enter a variable with a negative reduced cost, but zero might win the minimum ratio test, and we would not change our solution (or objective) at the next iteration. Part c (3 points): The minimum ratio test is designed to ensure that no variable becomes negative in our solution (except maybe for the z-value itself). True. We wish to make the entering variable as large as possible, because increasing the entering variable improve our objective function. However, we have to stop increasing the entering variable before any of our basic variables become negative. This limit is determined by the minimum ratio test. Part d (3 points): If both a primal problem and its dual have feasible solutions, neither problem can be unbounded. True. A feasible solution to a min problem puts an upper bound on the optimal objective function value, and a feasible solution to a max problem puts a lower bound on the optimal objective function value. Part e (3 points): Suppose that the shadow price for a constraint is equal to 5, and that its range allows an increase of 10 and a decrease of 3 before the basis changes. Then if the right-hand-side for that constraint is increased by 5, the optimal objective function value will increase by 25. True. The shadow price of a constraint tells you the rate of change in the objective function when you change its right-hand-side. If the shadow price is 5, and you increase
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