Consider the following linear programming problem. Min s.t. 4A + 7B 1A + 4B ≤ 21 2A + 1B27 3A +1.5B ≤ 21 -2A + 6B 20 A, B ≥ 0 (a) Find the optimal solution using the graphical solution procedure and the value of the objective function. at (A, B) = (b) Determine the amount of slack or surplus for each constraint. slack for 1A + 4B ≤ 21 surplus for 2A + 1B ≥ 7 slack for 3A + 1.5B ≤ 21 surplus for -2A + 6B ≥ 0 (c) Suppose the objective function is changed to max 8A + 3B. Find the optimal solution and the value of the objective function. at (A, B) =
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Objective Functions and Constraints:
Based on the given details, the objective functions and constraints are
Objective Function:
Constraints:
Subject to
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- Solve Problem 1 with the extra assumption that the investments can be grouped naturally as follows: 14, 58, 912, 1316, and 1720. a. Find the optimal investments when at most one investment from each group can be selected. b. Find the optimal investments when at least one investment from each group must be selected. (If the budget isnt large enough to permit this, increase the budget to a larger value.)This problem is based on Motorolas online method for choosing suppliers. Suppose Motorola solicits bids from five suppliers for eight products. The list price for each product and the quantity of each product that Motorola needs to purchase during the next year are listed in the file P06_93.xlsx. Each supplier has submitted the percentage discount it will offer on each product. These percentages are also listed in the file. For example, supplier 1 offers a 7% discount on product 1 and a 30% discount on product 2. The following considerations also apply: There is an administrative cost of 5000 associated with setting up a suppliers account. For example, if Motorola uses three suppliers, it incurs an administrative cost of 15,000. To ensure reliability, no supplier can supply more than 80% of Motorolas demand for any product. A supplier must supply an integer amount of each product it supplies. Develop a linear integer model to help Motorola minimize the sum of its purchase and administrative costs.Consider the following linear programming model with 4 regular constraints:Maximize 3X + 5Y (a) Draw your graph in the space below:subject to: 4X + 4Y ≤ 48 (constraint #1) 4X + 3Y ≤ 50 (constraint #2) 2X + 1Y ≤ 20 (constraint #3) X ≥ 2 (constraint #4) X, Y ≥ 0 (non-negativity constraints)(a) Which of the constraints is redundant? Constraint #______.Justify by drawing a graph similar to Figure 7.14 on p.263.(b) Is point (9,3) a feasible solution? _____. Explain your answer (by analyzing each of the constraints).Constraint #1: _______________________________________________________________Constraint #2: _______________________________________________________________Constraint #3: _______________________________________________________________Constraint #4: ______________________________________________________________
- Set up the simplex matrix used to solve the linear programming problem. Assume all variables are nonnegative.Maximize f = 5x + 9y subject to 8x + 5y ≤ 200 x + 6y ≤ 250. x y s1 s2 f first constraint second constraint objective functionConsider the following set of constraints: 48Y >= 7296; 0.25 X + 12Y >= 1824, and X + Y <= 152. Pick a suitable statement for this problem: a. Solution to this problem cannot be found without the objective function. b. The feasible region is defined by a single (unique) point. c. It is a non-linear problem - unsuitable for grphical method. d. This problem has two feasible points - one is optimal for miniization problem and other is optimal for maximization problem. e. Feasible region is represented by a line and multiple feasible points are available.Solve these problems using graphical linear programming and answer the questions that follow. Usesimultaneous equations to determine the optimal values of the decision variables.a. Maximize Z = 4x1 + 3x2Subject toMaterial 6x1 + 4x2 ≤ 48 lbLabor 4x1 + 8x2 ≤ 80 hrx1, x2 ≥ 0 b. Maximize Z = 2x1 + 10x2Subject toDurability 10x1 + 4x2 ≥ 40 wkStrength 1x1 + 6x2 ≥ 24 psi Time 1x1 + 2x2 ≤ 14 hrx1, x2 ≥ 0 c. Maximize Z = 6A + 3B (revenue)Subject toMaterial 20A+ 6B ≤ 600 lbMachinery 25A+ 20B ≤ 1,000 hr Labor 20A+ 30B ≤ 1,200 hrA, B ≥ 0 (1) What are the optimal values of the decision variables and Z?
- Find the optimal solution for the following problem. (Round your answers to 3 decimal places.) Minimize C = 14x + 7y + 8z subject to 6x + 12y + 19z ≥ 64 17x + 24y + 9z ≥ 128 and x ≥ 0, y ≥ 0, z ≥ 0. What is the optimal value of x? What is the optimal value of y? What is the optimal value of z? What is the minimum value of the objective function?4. Consider the following linear programming problem: Maximize Z=$15x + $5y, subject to (1) 2x + y ≤ 10 and (2) 4x + 3y ≤ 24 and (3) x, y ≥ 0. Will the optimal solution change if the objective function becomes Maximize Z=$15x + $20y (constraints remain the same)? Select one: a. Can't determine given the information. b. Yes, it will change. c. No, it remains the same.Consider the following LP problem: Min 6X+ 18Y; Subject to : 3 X + 9Y <= 47, and X + Y <= 141. Which one of the following is true?: a. Slack for each constraint is zero. b. Optimal Obj. function value is 94 c. X=70.5, Y=70.5 is the only optimal solution. d. Optimal Obj. function value is 0
- 1. Consider the following linear programming formulation: Min 5x + 2y Subject to (1) 3x + 6y ≥ 18 (2) 5x + 4y ≥ 20 (3) 8x + 2y ≥ 16 (4) 7x + 6y ≤ 42 (5) x, y ≥ 0 a. Solve the problem graphically. Specifically, show each constraint and the feasible region, draw an objective function line and identify an optimal point (the solution). When reporting the optimal solution and the corresponding objective function value, you may estimate the optimal x and y values from the graph. b. What are the optimal values of x and y, using the solver add-in? What is the corresponding value of the objective function? c. How many extreme points does the feasible region have? Enumerate them. Hint: It's from the graph. d. Change the objective function to 15x + 12y.. What is the new optimal solution(s)?Find the optimal solution for the following problem. (Round your answers to 3 decimal places.) Maximize C = 14x + 13y + 5z subject to 9x + 11y + 18z ≤ 61 14x + 15y + 12z ≤ 122 and x ≥ 0, y ≥ 0, z ≥ 0. What is the optimal value of x? What is the optimal value of y? What is the optimal value of z? What is the maximum value of the objective function?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 0