Math482Practicetest3solns - UIUC MATH 482 PRACTICE TEST 3...

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Unformatted text preview: UIUC MATH 482 PRACTICE TEST 3 SOLUTIONS (SPRING 2011) Instructions: 0 This test has 9 pages including this cover sheet. 0 Answer 4 of the 5 problems given below. (I will grade all 5 problems and score you on the best 4.) 0 Each problem is scored out of 5 points. 0 This test is out of 20 points. 0 Justify all of your steps to ensure full credit. 0 No calculators or other aids are permitted. 0 Write your name and ID below and on every page. 0 Make your student ID available. NAME: STUDENT ID: 2 1. Perform one iteration of the primal—dual simplex algorithm applied to for the dually feasible vector 7TT = (1/6,—1/2,1/3) to the linear program min 331 +3232 +3x3 +364 2 z subject to 3:51 +4x2 —3:r3 +1174 = 2 3$1 —2$2 +6333 —£U4 = 1 61‘1 +4272 +$4 = 4 $1, £132, £173, SE4 2 0 The dual will be Maximize w = 27r1 + 7r2 + 47r3 subject to 3m + 3n + 67r3 g 1, 47m — 2W2 + 4m 3 3, "37H ‘1' 6W2 S 3, 7T1 — 7T2 + 713 S 1. Plugging in our 7r, we get J : {1, 2,4}. Thus, the tableau for the corresponding Restricted Primal will be :31 9:2 1124 mg 9512‘ mg ——z 0 0 0 0 1 1 1 xi | 2 3 4 1 1 0 0 x3 | 1 3 —2 —1 0 1 0 x; 4 6 4 1 0 0 1 Solving this Restricted Primal, we get the final tableau I 1151 9:2 3:4 as? 9:; :r ——2 —1/2 0 1 O 5/2 3/2 x4 1/2 0 3 1 1/2 —1/2 X, 1/2 1 1/3 0 1/6 1/6 x; 1/2 0 —1 0 —3/2 —1/2 T _ This means that 7r was not optimal, and we need a new vector W = 7r + 67H, where (WT) — (1,1,1) — (5/2,3/2,0) = (—3/2,——1/2,1). To calculate the best 6, we need to check only the third inequality in the Dual, since J = {1, 2,4}. This reduces to the inequality (—3)(1/6 + 9(—3/2)) + 6(—1/2 + 9(—1/2)) g 3, HOOOUJ‘I which gives 19 3 13/3. So, we take 0 = 13/3 and obtain the new NT 2 (~19/3, ~8/3, 14/3). Plugging this 7T into the Dual, we get the new J = {1,3,4}. 1:: 3 2. Determine the value of the following game, together With ROW’S optimal strategy for the game for the following payoff matrix. (Hint: you may want to delete dominated rows/ columns.) 0 5 1 5 —1 1 3 2 1 —2 2 —1 row, the game reduces t ' ' 0 5 ' 0 that With the matrix 1 . Solvmg it (for ) we get that the value is 5/8. The Optimal strategy of the first player is [3/8, 0, 5/8]. The optimal strategy of the second player is [7/8, 1/8, 0, 0]. 4 3. Suppose that $0 is a feasible point of the linear program P min 0T3: subject to and that y satisfies CTy<0, Ay=0, yZO. Prove that P is unbounded. Solution: See solutions to HWG. El 5 4. Solve the following shortest path problem via the graphical pro— cedure derived from the primal—dual method. Start with the feasible, but not optimal solution 71' = (0, 0, . . . ,0) to the dual problem (D) of shortest path problem 711 Brief solution: This is similar to page 112 (Figure 5.4) of the textbook. (Warning, as observed in class, the first step had a typo as @1 = 1.) A shortest s—t path is s —-> 114 —+ 112 —> t of length 8. D 1,1mmf«Mmmewm‘mxm ma mm, 6 (This page is intentionally left blank) 7 5. Explain the max flow solution to the Sports Writer problem dis- cussed in class. Specifically, one wants to know if it is possible for Tampa to obtain first place in the division (i.e., win 83 games and have no one else win 84 games or more) given the standings below. Describe the network and max flow problem that solves this question. Be sure to explain why it works. GAMES WON GAMES LEFT BOS 81 9 NY 77 8 BALT 81 7 TOR 81 7 TAMP 75 8 The games within the division are: | Bos y NY \ BALT \ TOR TAMP—l B08 0 2 3 NY 2 O 1 BALT 3 1 0 TOR 2 2 1 TAMP 2 3 0 8 (This page is intentionally left blank.) \../ k n h 10 R b y H a n .m m m .m S .1 e g a p .E h W ...
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