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Consider the following linear program: Max 2X + 3Y s.t. 5X +5Y ≤ 400 -1X+ 1Y ≥ 10 1X + 3Y ≥ 90 X, Y ≥ 0 a. Use the graphical solution procedure to find the optimal solution. b. Conduct a sensitivity analysis to determine the range of optimality for the objective function coefficients X & Y. c. What are the binding constraints? d. If the right-hand-side of the binding constraints are marginally increased, what will be the Dual Value?
Let (IP) be the following integer program: max (0, -1). s.t. 1-1 -107 -10 10 -11</-0.5 1-10 / 0 z integer Let (LP) be the LP relaxation of (IP). We draw the feasible region of (LP) here. 2. . . . . . . to (LP), and prove that it is optimal by providing a (a) Determine an optimal solution certificate of optimality.
Question 1. (30 points) You are provided with the following integer program: max := 3x + y 8.t. x + 1.6y S8 - 5 + 6x = 15 35.5 x,y20 and integer (a) On the graph provided on the following page, use the graphical solution method to identify the feasible points on your graph. (Use the scale 1 by 1 for each small square so that you can visually detect the feasible integer solutions.) (b) Enumerate the feasible extreme points...
Consider the following linear program Max 3xl +2x2 S.t 1x1 + 1x2 〈 10 3x1 1x2 〈 24 1xl t 2x2< 16 And xl, x2> 0. a) Use Excel Solver to find the optimal solution to this problem. State the optimal values of xl, x2, and Z. b) Assume that the objective function coefficient for xl changes from 3 to 5. Does the optimal solution change? c) Assume that the objective function coefficient for x1 remains 3, but the objective...
Question 3 Solve the following linear program: Max 3x+2y s.t. 2x+2y <8 A 3x+2y < 12 B 1x+0.5y < 3C x,y> 0 w How much slack is in constraint B? 2 units of slack O 10 units of slack O 2 units of surplus 10 units of surplus
Consider the following constraints and the corresponding graph below: Constraint 1 Constraint 2:x+2ys8 Constraint 3 x-3y 2-2 2x-v21 2r-y-1 4 4 6 b. (3 points) The objective function is Minimize 2x-3y. Mark the optimal solution(s) n the above graph. Do not calculate the x and y coordinates at optimal solution(s). Draw the optimal objective function line through the optimal solution(s)
M 4. Consider the utility maximization problem max U(x,y) = x +y s.t. x + 4y = 100. (a) Using the Lagrange method, find the quantities demanded of the two goods. (b) Suppose income increases from 100 to 101. What is the exact increase in the optimal value of U(x, y)? Compare with the value found in (a) for the Lagrange multiplier. (C) Suppose we change the budget constraint to px + y = m, but keep the same utility...
Consider the following constraints and the corresponding graph below Constraint 1: 2x-y21 Constraint 2:x+2y S8 Constraint 3: x-3y 2-2 2x-y-1 4 x +2y 8 4 7 a. (3 points) Shade the feasible region in the graph provided above. b. (3 points) The objective function is Minimize 2x-3y. Mark the optimal solution(s) in the above graph Do not calculate the x and y coordinates at optimal solution(s). Draw the optimal objective function line through the optimal solution(s).
9. Consider the utility maximization problem max x + y s.t. px + y =m, where the constants p, 9, and m are positive, and the constant a € (0,1). (a) Find the demand functions, x* (p, m) and y* (p, m). (b) Find the partial derivatives of the demand functions w.r.t. p and m, and check their signs. (c) How does the optimal expenditure on the x good vary with p?8 (d) Put a = 1/2. What are the...
Consider the following linear program: Max 2A + 10B s.t. 3A ≤ 15 B ≤ 6 4A + 4B = 28 A, B ≥ 0 a. draw graph that shows the feasible region for the problem b. What are the extreme points of the feasible region c. Draw graph that shows the optimal solution for the problem