The optimal solution of this linear programming problem is at the intersection of constraints 1 and 2. Max 3x₁ + x₂ s.t. 4x₁ + x₂ ≤ 400 4x + 3x₂ ≤ 600 X₁ + 2x₂ ≤ 300 X1, X₂20 (a) Over what range can the coefficient of x₁ vary before the current solution is no longer optimal? (Round your answers to two decimal places.) to (b) Over what range can the coefficient of x₂ vary before the current solution is no longer optimal? (Round your answers to two decimal places.) to (c) Compute the dual value for the first constraint. Compute the dual value for the second constraint. Compute the dual value for the third constraint.
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- . Consider the following linear programming problem: Maximize 12X + 10Y Subject to: 4X + 3Y = 480 2X + 3Y 360 all variables 20 Which of the following points (X,Y) is not feasible? a (70,70) b. (20,90) c. (100,10) d. (0,100) كلا أجرب الأرقام بالـؤالInnis 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…For the remaining questions, consider the following problem description: An oil company is considering exploring new well sites S₁, S2, ..., S10 with respective costs C1, C2, C10. And in particular they want to find the least-cost selection of 5 out of the 10 possible sites. The binary decision variables x₁,x2,..., X10 denote the decision to explore the corresponding site.
- Solve the following Linear Programming model using the graphical method (USING EXCEL){Write the steps of construction} Q1)MaximizeH = x + 3y Objective functionsubject tox + y ≤ 502x + y ≤ 60 x ≥ 0, y ≥ 0Combined-cycle power plants use two combustion turbines to produce electricity. Heat from the first turbine’s exhaust is captured to heat waterand produce steam sent to a second steam turbine that generates additional electricity. A 968-megawatt combined-cycle gas fired plant can be purchased for $450 million, has no salvage value, and produces a net cash flow(revenues less expenses) of $50 million per year over its expected 30-year life. Solve, a. If the hurdle rate (MARR) is 12% per year, how profitable an investment is this power plant? b. What is the simple payback period for the plant? Is this investment acceptable?What combination of x and y will yield the optimum for this problem? Maximize Z = $3x + $15y Subject to: Multiple Choice x= 0, y=4 x= 0, y=3 x= 0, y=0 x= 2y=0 O x=1,y=25 2x + 4y ≤ 12 5x + 2y ≤ 10
- Consider the following Pareto maximization problem with decision variables x and y: vmax (x2 + x, - 2y) s.t. 0< x< 10 0< y< 5. What is the unique efficient point for this problem? (0,0) (10,5) (10,0) (110,0)1. If constraint has a shadow price of $6, Right-Hand-Side (RHS) is 12, allowable increase is 2, allowable decrease is 4. How would objective function change if the RHS of this constrains changes from 12 to 9? Answer___________Dickie Hustler has $2 and is going to toss an unfair coin(probability .4 of heads) three times. Before each toss, hecan bet any amount of money (up to what he now has). Ifheads comes up, Dickie wins the number of dollars he bets;if tails comes up, he loses the number of dollars he bets.Use dynamic programming to determine a strategy thatmaximizes Dickie’s probability of having at least $5 afterthe third coin toss.
- 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?For a "Dual minimization problem" with optimal point equal to (x1 = 5, x2 = 1) and best Z = 56, its respective "Primal maximization problem " would have Please choose the option that would best fit the empty space above. Its respective best Z also equal to 56. Shadow prices equal to 1 (x1) and 5 (x2), respectively. Shadow prices exactly the same as the shadow prices obtained in the Primal problem. Exactly the same optimal point. None of the aboveb) The Scott Tractor Company ships tractor parts from Omaha to St. Louis by railroad. However, a contract limits the number of railroad cars the company can secure on each branch/arc during a week. Given these limiting conditions, the company wants to know the maximum number of railroad cars containing tractor parts that can be shipped from Omaha to St. Louis during a week. Phoenix Des Moines 3 5 6. 6. 1. Dallas 2 Omaha St. Louis 7. 3 Denver 6. ActivateW O 31°C