For each of the following programs: (1) Sketch the feasible region of the program and the direction of the objective function. (2) Use you sketch to find an optimal solution to the program. State the optimal solution give the objective value for this solution. If an optimal solution does not exist, state why. (a) (b) maximize subject to maximize subject to - 2x1 + x2 2₁-2₂ ≤ 1, 2x₁-₂ ≥ 1, 2x1 + 2x₂ ≥ 4, I1, I₂ 20 ₁ + 2x₂ -₁ + 2x₂ ≤ 6, x+3x₂ ≤ 12, I1, I2 20
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- Consider the following LP problem: Min 6X+ 27Y Subject to : 2 X + 9Y => 25, and X + Y <= 75. Pick a suitable statement for this problem: a. X=37.5, Y=37.5 is the only optimal solution. b. Optimal Obj. function value is 75 c. X = 0, Y = 0 is the only optimal solution. d. Optimal Obj. function value is 0Innis Investments manages funds for a number of companies and wealthy clients. The investment strategy is tailored to each client's needs. For a new client, Innis has been authorized to invest up to $1.2 million in two investment funds: a stock fund and a money market fund. Each unit of the stock fund costs $50 and provides an annual rate of return of 10%; each unit of the money market fund costs $100 and provides an annual rate of return of 4%. The client wants to minimize risk subject to the requirement that the annual income from the investment be at least $60,000. According to Innis' risk measurement system, each unit invested in the stock fund has a risk index of 8, and each unit invested in the money market fund has a risk index of 3. The higher risk index associated with the stock fund simply indicates that it is the riskier investment. Innis's client also specified that at least $300,000 be invested in the money market fund. Refer to the computer solution shown below. Optimal…1. Write a linear program for the problem. 2. What is the optimal production mix among different coffee varieties and the profit associated with it? 3. What should be the per bag profit of Tarrazu which will make it beneficial to produce this variety? 4. One Cup is considering increasing the weekly bags to 150. If it costs One Cup $3.00 per bag to procure beans of any variety. Conduct sensitivity analysis and determine if One Cup should consider increasing the per week production. 5. Identify the range over which per bag profit of Sumatra and Kona may vary and still retain the current production plan as optimal for One Cup. 6. Once roasted the coffee beans must be preserved properly. To ensure that, One Cup Coffee is considering investing in a vacuum packaging machine for 24 hours/ week. It takes 10, 15 and 10 minutes to pack a bag of Sumatra, Kona and Tarazzu respectively. Identify if the current production plan will suit the installation of the new packaging scheme.
- 4 For each of the following, determine the direction in which the objective function increases: a z = 4x, - x2 b z = -x, + 2x2 C z = -x - 3x2Indicate which of the following is an all-integer linear program and which is a mixed-integer linear program. Write the LP Relaxation for the problem but do not attempt to solve. (a) 70x1 + 65x2 (b) Is this linear program an all-integer linear program or a mixed-integer linear program? This is an all-integer linear program. This is a mixed-integer linear program. Max Write the LP Relaxation for the problem. s.t. (1) (2) (3) Max s.t. ---Select--- (1) 9x₁ + 4.5x₂ ≤ 400 (2) 4.5X₁ + 2x₂ ≤250 (3) x₁ + x₂ ≤ 150 X₁, X₂20 and X₂ integer Min s.t. (1) Is this linear program an all-integer linear program or a mixed-integer linear program? This is an all-integer linear program. This is a mixed-integer linear program. (2) Min s.t. Write the LP Relaxation for the problem. 7x1 + 8x2 (1) 6X₁ + 8X₂ ≥ 8 (2) 6X₁ + 10x₂ ≥ 12 X₁, X₂20 and integer ---Select---Elaborate/ Explain Integer Linear Optimization and put examples
- (b) Indicate whether the following linear program is an all-integer linear program or a mixed-integer linear program. Max 3x1 + 4x2 s.t. 2x1 + 4x2 2 8> 2x1 + 6x2 2 12 x1, x2 2 0 and integer This is a Max s.t. Write the LP Relaxation for the problem but do not attempt to solve. If required, round your answers to one decimal place. Its LP Relaxation is 2x1 + x1, x2 linear program. xz x1 + 6x21a. What is the orrect graph that shows the feasible region for the problem. b. What are the extreme points of the feasible region? c. What is the correct graph that shows the optimal solution for the problemFind the optimal solution for the following problem. (Round your answers to 3 decimal places.) Minimize C = 14x + 7y + 8z subject to 6x + 12y + 19z 2 64 17x + 24y + 9z 2 128 x 2 0, y 2 0, z 2 0. and a. What is the optimal value of x? b. What is the optimal value of y?
- A survey was conducted to 12 first time voters on their preferred candidate. The results are: BBM, BBM, LR, IM, PL, PL, IM, IM, BBM, BBM, LR, LR. Which statement is true? The Borda score of PL is two points. BBM wins by plurality method. The Condorcet winner is IM. The modes are LR and IM Which of the following is a property of all linear programming problems? alternate courses of action to choose from minimization of some objectives a computer program usage of graphs in the solutionScenario You are going to plant a rectangular flower bed consisting of tulips in the middle surrounded by daisies on the outside. You have the same amount of each flower and will need an equal area for each. You want the border of daisies to be uniform around the tulips in the middle, as shown in the diagram below:3. (Note: This is a variation of problem 6 of chapter 16 in your textbook.) Kenya and Dionne live on adjacent plots of land. Each has two potential uses for their land, the present values of each of which depend on the use adopted by the other, as summarized in the table. All the values in the table are known to both parties. Dionne Rental housing Bee keeping Kenya Apple growing A: $200 B: $700 A: $400 B: $650 Pig farming A: $450 B: $400 A: $450 B: $500 a. What is the efficient outcome? b. If there are negotiation costs of $150, what activities will the two pursue on their land? c. If there are no negotiation costs and the two negotiate, what activities will the two pursue on their land? How might a benevolent planner help reduce the costs of negotiating to encourage the optimal combination of land uses?