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SEFYDLIAD MATHEMATEG A FFISEG
INSTITUTE OF MATHEMATICS AND PHYSICS
SEMESTER 1 EXAMINATIONS, JANUARY 2010
MA33110 โ Linear Programming
Time allowed โ 2 hours
- All questions may be attempted.
- Marks gained from questions in section B will be given greater consideration in assessing a first class performance.
- Calculators are permitted, provided they are silent, self-powered, without communications facilities, and incapable of holding text or other material that could be used to give a candidate an unfair advantage. They must be made available on request for inspection by invigilators, who are authorised to remove any suspect calculators.
MA33110: Linear Programming January 2010 Page 2 of 7
Questions 1-4 are based on the following diagram. The diagram is also printed on a separate sheet appended to this examination paper. You may use that sheet for your answers, in which case you should detach it and insert it in your answer book. Remember to write your University number in the appropriate space.
Related linear system โX 1 + X 2 X 3 = 3 X 1 +4X 2 X 4 = 28 8X 1 โ3X 2 X 5 = 35 3X 1 + X 2 X 6 = 11 โ2X 1 +5X 2 X 7 = 4
MA33110: Linear Programming January 2010 Page 4 of 7
(^5) Use the Simplex method to show that the linear programming problem Maximise 10X 1 +20X 2 +15X 3 subject to 3X 1 + 2X 2 โค 60
- X 1 + X 2 + 4X 3 โค 10 2X 1 โ 2X 2 + 5X 3 โค 50 X 1 , X 2 , X 3 โฅ 0 has its solution at (X 1 , X 2 , X 3 ) = (8,18,0). What is the smallest value that the objective coefficient of X 3 would have to be increased to before X 3 is non-zero in the solution?
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6 When and why are^ artificial variables^ used? The non-negative variables X 1 , X 2 , X 3 are constrained as follows: โX 1 +2X 2 +X 3 โค 1 โX 1 +2X 3 โฅ 4 3X 1 โX 2 +2X 3 = 4 It is required to minimise X 1 +2X 2 +3X 3. Give the initial objective function if the problem is to be solved using (i) the two-phase method; (ii) the Big-M method.
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7 The following is a payoff matrix in a two person zero sum game.
2 6 1 2 2 5 4 3 2 0 3 0
Explain what this means and, by referring to this matrix explain also the meanings of the terms dominance and saddle point. What is the long term outcome of this game? [10]
8 Each of four decorators has tendered for four jobs; the following table shows the prices quoted. Alf Ben Cat Dai Job 1 6 6 7 3 Job 2 2 4 2 3 Job 3 6 9 8 7 Job 4 3 5 4 2 If it is required to allocate one of the jobs to each of the decorators, use the Hungarian method to find the cheapest allocation. [10]
MA33110: Linear Programming January 2010 Page 5 of 7
Section B
(^9) A manufacturer of aluminium cans must produce monthly at least 2400 cases of a
Standard can and a minimum of 2800 cases of Toughened cans. The company has three manufacturing processes it can use; the first uses a special pure grade aluminium, whilst the other two allow for some use of recycled aluminium. To comply with the conditions of government funding already received, the company must use at least 600kg of recycled aluminium in its monthly production. The characteristics of each process are shown in the following table: Input (per run) Output (per run) Recycled aluminium used (kg)
Standard cans (cases)
Toughened cans (cases)
Cost (per run)
Process 1 0 6 8 65 Process 2 2 12 12 150 Process 3 3 10 15 200 The company manager must decide how many runs of each process to initiate per month to run the plant at minimum cost. With Xi denoting the number of runs of the i-th process per month, for i=1,2,3 the following table shows the initial and final tableaux when the problem was solved using the dual simplex method: X 1 X 2 X 3 X 4 X 5 X 6 RHS Initial tableau
0 โ2 โ3 1 0 0 โ โ6 โ12 โ10 0 1 0 โ โ8 โ12 โ15 0 0 1 โ โ65 โ150 โ200 0 0 0 โQ Final tableau
โ3/4 0 1 โ3/4 1/8 0 150 9/8 1 0 5/8 โ15/80 0 75 โ23/4 0 0 โ15/4 โ3/8 1 350 โ185/4 0 0 โ225/4 โ25/8 0 โQ+ (a) Explain how the initial tableau was derived and give the solution. (b) How would the solution change if the legal requirement of recycled aluminium usage was reduced to 560 kg per month? (c) If the cost of using Process 3 changes to ยฃ(200+d), for what range of values of d does the solution remain valid? (d) How would you interpret the value โโ25/8โ in the objective row of the final tableau?
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MA33110: Linear Programming January 2010 Page 7 of 7
13 A supermarket chain has available 25 pallets of a particular product available at each of their three warehouses. They needs to transport 15 pallets to the outlet at Alpha Avenue and 20 pallets to each of the shops at Beta Bay, Gamma Grove and Delta Drive. The following table displays the transportation costs per pallet from each warehouse to each outlet: Outlet Alpha Avenue Beta Bay Gamma Grove Delta Drive
Warehouse
W1 2 8 22 17
W2 5 10 25 18
W3 1 4 19 20
(a) Use the Northwest Corner rule to determine a basic feasible transportation schedule. Verify that the cost of this is ยฃ1080. (b) A colleague comes up with the alternative schedule: From W1: 15 to Alpha Avenue, 10 to Gamma Grove From W2: 5 to Beta Bay, 20 to Delta Drive From W3: 15 to Beta Bay, 10 to Gamma Grove which costs just ยฃ910. Apply the Method of Fictitious Costs to show that this schedule is in fact optimal.
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University Number: _________
- MA33110: Linear Programming Examination
- Diagram for Questions 1- - โX 1 + X 2 X 3 = Related linear system - X 1 +4X 2 X 4 = - 8X 1 โ3X 2 X 5 = - 3X 1 + X 2 X 6 =
