In Exercises 9-20, use the simplex method to solve the given linear programming problem. (In each case the objective function is to be maximized.) 15. Objective function: z = 3x₁ + 4x₂ + x3 + 7x4 Constraints: 8x₁ + 3x₂ + 4x3 + x₁ ≤7 2x₁ + 6x₂ + x3 + 5x₁ = 3 x₁ + 4x₂ + 5x3 + 2x₁ ≤8 X₁ X₂ X3 X4 20
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- You want to take out a 450,000 loan on a 20-year mortgage with end-of-month payments. The annual rate of interest is 3%. Twenty years from now, you will need to make a 50,000 ending balloon payment. Because you expect your income to increase, you want to structure the loan so at the beginning of each year, your monthly payments increase by 2%. a. Determine the amount of each years monthly payment. You should use a lookup table to look up each years monthly payment and to look up the year based on the month (e.g., month 13 is year 2, etc.). b. Suppose payment each month is to be the same, and there is no balloon payment. Show that the monthly payment you can calculate from your spreadsheet matches the value given by the Excel PMT function PMT(0.03/12,240, 450000,0,0).Analyze algebraically what special case in simplex application is present in each of the LP model below. Give an explanation to support your answer. a) Maximize z = 4x1 + 2x2 Subject to: 2x1 - x2 ≤ 2 3x1 - 4x2 ≤ 8 x1, x2 ≥ 0b) Maximize z = 3x1 + 2x2 Subject to: 4x1 - x2 ≤ 8 4x1 + 3x2 ≤ 12 4x1 + x2 ≤ 8 x1, x2 ≥ 04 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 - 3x2
- (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 + 6x21Simplex Method Write in normal form and solve by the simplex method, assuming x, to be nonnegative. 1. The owner of a shop producing automobile trailers wishes to determine the best mix for his three products: flat-bed trailers, economy trailers, and luxury trailers. His shop is limited to working 24 days/month on metal- working and 60 days/month on woodworking for these products. The following table indicates production data for the trailers. Usage per unit of trailer Flat-bed Economy Luxury Available resources 1 2 1 24 Metal work days Wood work days Contribution (Rx100) 60 4 6 2 14 13Use simplex method for linear programming. Show solution for each iteration table. Problem: A small firm that assembles computers is about to start production of two new types of personal computers. Each type will require assembly time, inspection time, and storage space. The amount of each of these resources that can be devoted to the production of the computers is limited. The manager of the firm would like to determine the quantity of each computer to produce in order to maximize the profit generated by sales of these computers. In order to develop a suitable model of the problem, the manager has met with design and manufacturing personnel. As a result of those meetings, the manager has obtained the following information: Type 1 Type 2 Profit per unit $60 $50 Assembly time per unit 4 hours 10 hours Inspection time per…
- . 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 بالـؤالConsider the following linear program. Max 1A + 2B s.t 1A≤ 4 1B≤ 3 2A + 2B = 10 A, B ≥ 0 (a) Show the feasible region. (b) What are the extreme points of the feasible region? (b) What are the extreme points of the feasible region? smaller x-value (A, B) = (______) larger x-value (A,B)= (_____) (c) Find the optimal solution using the graphical procedure. (A, B) = (______)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.
- Indicate 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---Innis 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 products A, B, C, and D, which of the following could be a linear programming objective function? Select one: a. Z = 1A + 2BC + 3D b. Z = 1A + 2AB + 3ABC + 4ABCD c. Z = 1A + 2B + 3C + 4D d. Z = 1A + 2B/C + 3D