Use Simplex method to Maximize 1 2 Ζ = 4x +10x Subject to 1 2 2x + x ≤ 50 1 2 2x + 5x ≤100 1 2 2x + 3x ≤ 90 1 2 x , x ≥ 0 QUESTION TWO
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A Manager has the problem of assigning four new machines to three production facilities.
The respective profits derived are as shown. If only one machine is assigned to a production
facility, determine the optimal assignment.
Profits ($’000) production facility
Machine A B C
A 10 10 14
B 10 11 13
C 12 10 10
D 13 12 11
(c) Use Simplex method to
Maximize 1 2 Ζ = 4x +10x
Subject to 1 2 2x + x ≤ 50
1 2 2x + 5x ≤100
1 2 2x + 3x ≤ 90
1 2 x , x ≥ 0
QUESTION TWO
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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.)A decision problem has the following three constraints: 70X + 6Y <= 420; 24X + 3Y= 72; and 11X - Y <= 14 . The objective function is Min 17X + 38Y . The objective function value is : a. 338 b. 676 c. unbounded d. infeasible e. 0A Manager has the problem of assigning four new machines to three production facilities. The respective profits derived are as shown. If only one machine is assigned to a production facility, determine the optimal assignment. Profits (K’000) production facility Machine A B C A 10 10 14 B 10 11 13 C 12 10 10 D 13 12 11
- Compute the objective function value for the following problem: Min 30X + 150Y subject to : 2X>=0 ;2X + 10Y = 20; X+Y>=0 a. 0 b. 54 c. infeasible d. 300 e. unboundedA company has three plants at location A,B and C which produce the same product. It has to supply this to buyers located at P, Q and R. The weekly plant capacities for A, B and C are 250, 800 and 350 units respectively, while the buyer requirements are 700, 200 and 500 for P, Q and R respectively. The unit shipping cost (in Ksh) are given as Plant Buyer P Buyer Q Buyer R A 8 4 10 B 9 7 9 C 6 5 8 Determine the distribution for the company so as to minimize the cost of transportation using least cost method.Consider a small Oil production firm with 5 competing oil production projects, A - E. The table below shows the estimated long-term profit (Net Present Value) for each project as well as the amount of investment capital required to start the project. You have been contacted to help select the best combination of projects to maximize the Net Present Value subject to the capital investment limit of $32 million. Production Project A B C D E Estimated Profit (millions) 25 20 19 28 21 Capital Required (millions) 11 8 14 19 13 Formulate a Binary Integer Programming (BIP) model on a spreadsheet. Solver the model using Solver.
- Four proposals are being considered with investment and NPW shown below. C can only beconsidered if B is implemented. B and D and E are mutually exclusive. The budget is $80.A B C D EInvestment $10 $20 $30 $40 $50NPW $34 $28 $20 $16 $32a) Formulate this as a 0-1 integer program (objective function and constraints).b) Use trial and error to find all feasible solutions, and indicate which is optimal.c) Set up this problem in Excel, and use Solver to find the same optimal solutionA firm operates two types of machines A and B as shown in the table. Type A Type B Maximum Space Available Floor Space Available 3 2 18 Number of Workers 5 3 30 The ratio of machines of type A to those of type B should be greater than one. Tasks Define the term linear programming Write down four inequalities to represent the information above. If the profit from using type A machine is Shs 6000 per hour and that of B is Shs 9000 per hour, write down the expression for the profit in terms of and Find the number of machines of each type that should be in operation to give the maximum profit. Hence find the maximum profit What advice do you give to this company’s managerWhat combination of x and y will yield the optimum for this problem? Maximize $10x + $4y, subject to (1) 5x + 3y ≤ 15 and (2) 3x + 6y ≤ 18 and (3) x, y ≥ 0.
- Mr. Smith is a rentier, he lives on renting apartments. However, these days he has to sell one of his apartments to get a big money. He hesitates which of three apartments of his to sell. Help Mr. Smith to make the right decision if you know the following: The present prices per m2 in the three locations equal: A) 4000, B) 5000, C) 6000 goldies, respectively. However, the long-run prices of those real estates are: A) 5000, B) 6000, and C) 7000 goldies, whereas the respective standard deviations are A) 400, B) 500, and C) 250 goldies.Ann and Ben are getting divorced and want todetermine how to divide their joint property: retirementaccount, home, summer cottage, investments, and miscellaneous assets. To begin, Ann and Ben are told to allocate100 total points to the assets. Their allocation is as shownin Table 69.Assuming that all assets are divisible (that is, a fractionof each asset may be given to each person), how should theassets be allocated? Two criteria should govern the asset allocation:Criteria 1 Each person should end up with the same number of points. This prevents Ann from envying Ben and Benfrom envying Ann.Criteria 2 The total number of points received by Ann andBen should be maximized.If assets could not be split between people, what problemarises? TAB LE 69PointsItem Ann’s Ben’sRetirement account 50 40Home 20 30Summer cottage 15 10Investments 10 10Miscellaneous 5 10Find the minimum value of the function z=2x+2y subject to the following constraints. x≤17 y≤16 5x+2y≥42 3x+11y≥84