A linear programming problem is given as below. What is the optimal solution (X1, X2)? Maximize: Z = 5X1 + 3X2 subject to: X1+ X2≤ 12 X1+2X2 ≤ 20 X2 ≥ 3 X1, X2 ≥ 0
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7. A linear programming problem is given as below. What is the optimal solution (X1, X2)?
Maximize: Z = 5X1 + 3X2
subject to:
X1+ X2≤ 12X1+2X2 ≤ 20
X2 ≥ 3
X1, X2 ≥ 0
(9, 3)
(20, 0)
(0, 12)
(0, 10)
(4, 8)
(2, 10)
7.
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- Let A = {0, 1, 2, 3} and define a relation R on A as follows: R = {(0, 0), (0, 2), (0, 3), (2, 2), (2, 0), (1, 1), (2, 1), (3, 3)}. (a) Draw the directed graph of R. (b) Is R reflexive? Explain. (c) Is R symmetric? Explain. (d) Is R transitive? ExplainSuppose an assignment problem, where there are 3 employees who must perform 3 different tasks. The times (in hours) that employees take to perform each of the tasks is shown below. Each employee must perform only one task, and a task can only be performed by one employee. All employees are paid exactly the same wage per hour of work. Minimize the total time to make all tasks. Task 1 Task 2 Task 3 Employee 1 154 29 40 Employee 2 18 137 12 Employee 3 94 13 31 How many variables does the problem have? How many constraints does the problem have? The objective function corresponds to:22. Suppose that the feasible region of a maximization LP problem has corners of (0,0), (10,0), (5,5), and (0,7). If profit is given to be $X + $3Y what is the maximum profit the company can earn? Part 2 A. $10 B. $0 C. $14 D. $21 E. $15
- Problem 3: Let L(x, y) be the statement “x loves y”, where the domain for both x and y consists of all people in the world. Use quantifiers to express each of the following statements.1. Everybody loves Jerry.2. Everybody loves somebody.3. There is somebody whom everybody loves.4. Nobody loves everybody.5. There is somebody whom Lydia does not love.6. There is somebody whom no one loves.7. There is exactly one person whom everybody loves.8. There are exactly two people whom Lynn loves.9. Everybody loves himself or herself.10. There is someone who loves no one besides himself or herself.Example 1. The Big M Method solve LP with mixed constraints Minimize: Z = 4x1 + 2x2 + x3 Subject to: 2x1 + 3x2 + 4x3 ≤ 14 3x1 + x2 + 5x3 ≥ 4 x1 + 4x2 + 3x3 ≥ 6 x1, x2, x3≥ 0b) Maximize Z = −40X1 −100X2s.t 10X1 + 5X2 ≤ 2502X1 + 5X2 ≤ 1002X1 + 3X2 ≤ 90X1, X2 ≥ 0Solve by simplex method, what are the solutions? Show that this problem hasmultiple solutions and find the solutions?
- (5) 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: _______________________________________________________________(c) Which of the following points yields the best solution? Underline the best solution: (7,5), (9,2), (6,6).Justify your answer (using the data from the above LP…Determine whether the problem has multiple solutions, unbounded solutions, or no feasible solutions.Maximize z = 10x1 + 11x2 + 7x3, subject to 2x1 + 3x2 − 9x3 ≤ 72 2x1 + 5x2 − 10x3 ≥ 100 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0 The problem has multiple solutions. The problem has unbounded solutions. T he problem has no feasible solutions.3. A company produces three products: A,B and C. Each product requires two raw materials; steel and aluminum. The following LP model describes the company’s product mix problem x1, x2, and x3 represent the number of units of product A, B and C respectivelyMax Z = 30x1 + 10x2 + 50x3Subject to6 x1+ 3 x2 + 5 x3 ≤ 450 (steel)3 x1 + 4 x2+ 5 x3 ≤ 300 (alumimium), X1, x2, x3 ≥ 0 Identify the two production methods that optimize the company’s profit (hint: Use simplex method)