app3 - APPENDIX 3 Answers to Selected Problems Chapter 2...

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APPENDIX 3 Answers to Selected Problems Chapter 2 SECTION 2.1 1a ± A ² ±² b 3 A ² ± ² c A ³ 2 B is undefned. d A T ² ± ² e B T ² f AB ² ± ² g BA is undefned. 2 ² ± ² 2.2 1 ² or ±³ ² 2.3 1 No solution. 2 Infnite number oF solutions oF the Form x 1 ² 2 ± 2 k , x 2 ² 2 ³ k , x 3 ² k . 3 x 1 ² 2, x 2 ²± 1. 4 6 8 ± 1 ± 1 ± 3 1 2 1 4 6 8 x 1 x 2 ± 1 1 3 1 2 1 x 1 x 2 x 3 0.10 0.30 0.60 0 0.70 0.30 0.50 0.30 0.20 y 1 y 2 y 3 6 15 24 4 10 16 1 2 ± 0 ± 1 1 2 7 8 9 4 5 6 1 2 3 9 18 27 6 15 24 3 12 21 ± 3 ± 6 ± 9 ± 2 ± 5 ± 8 ± 1 ± 4 ± 7 2.4 1 Linearly dependent. 2 Linearly independent. 2.5 2 A ± 1 ² 3 A ± 1 does not exist. 8a ´ 1 1 00 ´ B ± 1 . 2.6 2 30. REVIEW PROBLEMS 1 Infnite number oF solutions oF the Form x 1 ² k ± 1, x 2 ² 3 ± k , x 3 ² k . 3 ² ± ² 4 x 1 ² 0, x 2 ² 1. 13 Linearly independent. 14 Linearly dependent. 15 a Only iF a , b , c , and d are all nonzero will rank A ² 4. Thus, A ± 1 exists iF and only iF all oF a , b , c , and d are nonzero. b Applying the Gauss-Jordan method, we fnd iF a , b , c , and d are all nonzero, A ± 1 ² 18 ± 4. 0 0 0 ´ 1 d ´ 0 0 ´ 1 c ´ 0 0 ´ 1 b ´ 0 0 ´ 1 a ´ 0 0 0 U t T t 0 0.90 0.75 0.20 U t ³ 1 T t ³ 1 ± ´ 1 2 ´ ± 3 ± ´ 1 2 ´ ± ´ 1 2 ´ ± 2 ± ´ 1 2 ´ ± ´ 1 2 ´ ± 1 ± ´ 1 2 ´
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APPENDIX 3 Answers to Selected Problems 1371 Chapter 3 SECTION 3.1 1 max z ± 30 x 1 ² 100 x 2 1 s.t. x 1 ² x 2 ³ 7 (Land constraint) 4 x 1 ² 10 x 2 ³ 40 (Labor constraint) 1 s.t. 10 x 1 + 10 x 2 ´ 30 (Government constraint) x 1 , x 2 ´ 0 2 No, the government constraint is not satisfied. b No, the labor constraint is not satisfied. c No, x 2 ´ 0 is not satisfied. 3.2 1 z ± $370, x 1 ± 3, x 2 ± 2.8. 3 z ± $14, x 1 ± 3, x 2 ± 2. 4a We want to make x 1 larger and x 2 smaller, so we move down and to the right. b We want to make x 1 smaller and x 2 larger, so we move up and to the left. b We want to make both x 1 and x 2 smaller, so we move down and to the left. 3.3 1 No feasible solution. 2 Alternative optimal solutions. 3 Unbounded LP. 3.4 1 For i ± 1, 2, 3, let x i ± tons of processed factory i waste. Then the appropriate LP is min z ± 15 x 1 ² 10 x 2 ² 20 x 3 s.t. 0.10 x 1 ² 0.20 x 2 ² 0.40 x 3 ´ 30 (Pollutant 1) s.t. 0.45 x 1 ² 0.25 x 2 ² 0.30 x 3 ´ 40 (Pollutant 2) x 1 , x 2 , x 3 ´ 0 It is doubtful that the processing cost is proportional to the amount of waste processed. For example, processing 10 tons of waste is probably not 10 times as costly as processing 1 ton of waste. The Divisibility and Certainty Assumptions seem reasonable. 3.5 1 Let x 1 ± number of full-time employees (FTE) who start work on Sunday, x 2 ± number of FTE who start work on Monday ,..., x 7 ± number of FTE who start work on Saturday; x 8 ± number of part-time employees (PTE) who start work on Sunday x 14 ± number of PTE who start work on Saturday. Then the appropriate LP is min z ± 15(8)(5)( x 1 ² x 2 ²µµµ² x 7 ) min z ± ² 10(4)(5)( x 8 ² x 9 x 14 ) s.t. 8( x 1 ² x 4 ² x 5 ² x 6 ² x 7 ) ² 4( x 8 ² x 11 ² x 12 ² x 13 ² x 14 ) ´ 88 (Sunday) s.t. 8( x 1 ² x 2 ² x 5 ² x 6 ² x 7 ) ² 4( x 8 ² x 9 ² x 12 ² x 13 ² x 14 ) ´ 136 (Monday) s.t. 8( x 1 ² x 2 ² x 3 ² x 6 ² x 7 ) ² 4( x 8 ² x 9 ² x 10 ² x 13 ² x 14 ) ´ 104
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This note was uploaded on 11/02/2010 for the course ORIE 1101 taught by Professor Trotter during the Fall '10 term at Cornell.

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app3 - APPENDIX 3 Answers to Selected Problems Chapter 2...

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