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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 we are solving a maximization problem andthe variable xr is about to leave the basis.a What is the coefficient of xr in the current row 0?b Show that after the current pivot is performed, thecoefficient of xr in row 0 cannot be less than zero.c Explain why a variable that has left the basis on agiven pivot cannot re-enter the basis on the next pivot.Consider the problem of calculating the median m of the following numbers: 4,5,9. (a) Write down the linear program that you could solve, to find the median m. Then put this LP in the form: min c^T x s.t. Ax ≥ b, x ≥ 0, where x is your list of variables. (b) Now write down the dual of the problem above. Call the list of dual variables y. (Hint: there are 6 components of y.)
- Consider the following linear programming model: maximize Z = 3x1 + 2x2 subject to : x1 +x2 ≤ 1 x1 + x2 ≥ 2 x1,x2 ≥ 0 a) Write this model in a standard (augmented) form. (i.e. Introduce slack/surplus, artificial etc.)b) Constract the initial simplex tableau and carry on your calculations to solve this model using the simplex method. Interpret your result.1. A city is reviewing the location of its fire stations. The city is made up of a number of neighborhoods, as illustrated in the figure below. A fire station can be placed in any neighborhood. It is able to handle the fires for both its neighborhood and any adjacent neighborhood (any neighborhood with a non-zero border with its home neighborhood). The objective is to minimize the number of fire stations used. Solve this problem. Which neighborhoods will be hosting the firestations?Show the complete Linear Programming Model. Show solutions (i.e. decision variables, objective function, subject to constraints, etc.) Q. A gold processor has two sources of gold ore, source A and source B. In order to keep his plant running, at least three tons of ore must be processed each day. Ore from source A costs $1,000 per ton to process, and ore from source B costs $500 per ton to process. Costs must be kept to less than $4,000 per day. Moreover, Government Regulations require that the amount of ore from source B cannot exceed twice the amount of ore from source A. If ore from source A yields 2 oz. of gold per ton, and ore from source B yields 3 oz. of gold per ton, how many tons of ore from both sources must be processed each day to maximize the amount of gold extracted subject to the above constraints? Formulate the LP model.
- Find the minimum value of the function z=2x+2y subject to the following constraints. x≤17 y≤16 5x+2y≥42 3x+11y≥8421. A linear programming problem has two constraints 2X + 4Y ≤ 100 and 1X + 8Y ≤ 100, plus nonnegativity constraints on X and Y. Which of the following statements about its feasible region is TRUE? Part 2 A. The graphical origin (0, 0) is not in the feasible region. B. The feasible region includes all points that satisfy one constraint, the other, or both. C. The feasible region cannot be determined without knowing whether the problem is to be minimized or maximized. D. The two corner points are (0, 0) and (50, 12.5). E. There are four corner points including (50, 0) and (0, 12.5)27 In order to manufacture 3,000 pairs of shoes in a week, a firm can use 3,000 workers and 100 machines or 200 machines and 4,000 workers Which method is considered more technically efficient? a 4,000 workers and 200 machines b Both are equally efficient c 3,000 workers and 100 machines d Neither could be considered efficient 33 A manufacturing business can use 100 workers and 20 machines, 140 workers and 18 machines, or 150 workers and 18 machines to produce 80 chairs If each worker costs $40 and each machine is rented for $1000, the economically efficient input combination is: a 100 workers and 20 machines b 150 workers and 18 machines c none of these input combinations d 140 workers and 18 machines
- Suppose a company must service customers lying inan area of A sq mi with n warehouses. Kolesar and Blumhave shown that the average distance between a warehouseand a customer is An Assume that it costs the company $60,000 per year tomaintain a warehouse and $400,000 to build a warehouse.(Assume that a $400,000 cost is equivalent to foreverincurring a cost of $40,000 per year.) The company fills160,000 orders per year, and the shipping cost per order is$1 per mile. If the company serves an area of 100 sq mi,then how many warehouses should it have?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.2. Provident Capital Corp. specializes in investment portfolios designed to meet the specific risk tolerances of its clients. A client contacted Provident with P2,000,000 available to invest. Provident’s investment advisor recommends a portfolio consisting of two investment funds: the Dynamic fund and the Diversified fund. The Dynamic fund has a projected annual return of 10%, and the Diversified fund has a projected annual return of 8%. The investment advisor requires that at most P1,400,000 of the client’s funds should be invested in the Dynamic fund. Provident’s services include a risk rating for each investment alternative. The Dynamic fund, which is the more risky of the two investment alternatives, has a risk rating of 6 per P40,000 invested. The Diversified fund has a risk rating of 4 per P40,000 invested. For example, if P400,000 is invested in each of the two investment funds, Provident’s risk rating for the portfolio would be 6(10) + 4(10) = 100. Finally, Provident developed…