** 3. ** | Cells : B10:B12; Constraints: B10:B12 0, G6 F6, G7 F7; Profit: = B10 ^{ * } C4+B11 ^{ * } D4+B12 ^{ * } E4; Target cell = B13. |

(a and b) max. Z = 12 x _{ 1 } + 16 x _{ 2 } + 0 s _{ 1 } + 0 s _{ 2 } ; s.t. 3 x _{ 1 } + 2 x _{ 2 } + s _{ 1 } = 500, 4 x _{ 1 } + 5 x _{ 2 } + s _{ 2 } = 800, x _{ i } 0, s _{ i } |

(a and b) c _{ 1 } , 12.8, 15 c _{ 2 } , 320 q _{ 1 } , 0 q _{ 2 } 1,250; (c) $0, $3.20 |

(a) x _{ 1 } = 4, x _{ 2 } = 3, Z = 57, A: s _{ 1 } = 40, s _{ 2 } = 0, B: s _{ 1 } = 0, s _{ 2 } = 0, C: s _{ 1 } = 0, s _{ 2 } = 20; (b) x _{ 1 } = 2, x _{ 2 } = 4; (c) solution point same for profit = $15, new solution x _{ 1 } = 0, x _{ 2 } = 5, Z = 100 for profit = $20 |

** 11. ** | (a and b) max. Z = 2.25 x _{ 1 } + 3.10 x _{ 2 } + 0 s _{ 1 } + 0 s _{ 2 } + 0 s _{ 3 } ; s.t. 5.0 x _{ 1 } + 7.5 x _{ 2 } + s _{ 1 } = 6,500, 3.0 x _{ 1 } + 3.2 x _{ 2 } + s _{ 2 } = 3,000, x _{ 2 } + s _{ 3 } = 510, x _{ i } 0, s _{ j } |

(a) Additional processing time, $0.75/hr.; (b) 0 c _{ 1 } 2.906, 2.4 c _{ 2 } , 6,015 q _{ 1 } , 1,632 q _{ 2 } 3,237, 0 q _{ 3 } 692.308 |

(a) x _{ 1 } = 4, x _{ 2 } = 0, s _{ 1 } = 12, s _{ 2 } = 0, s _{ 3 } = 11, Z = 24,000; (b) x _{ 1 } = 1, x _{ 2 } = 3, Z = 28,500; (c) C still optimal |

** 17. ** | (a and b) max. Z = 300 x _{ 1 } + 520 x _{ 2 } ; s.t. x _{ 1 } + x _{ 2 } = 410, 105 x _{ 1 } + 210 x _{ 2 } = 52,500, x _{ 2 } = 100, x _{ i } |

(a) no, max. price = $80; (b) $2.095 |

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** 21. ** | (a) x _{ 1 } = 300, x _{ 2 } = 100, s _{ 1 } = 0 lb., s _{ 2 } = 15 lb., s _{ 4 } = 0.6 hr., Z = 230; (b) c _{ 1 } = $0.60, x _{ 1 } = 257, x _{ 2 } = 143, Z = 240; (c) x _{ 1 } = 300, x _{ 2 } = 125, Z = 242.50 |

** 23. ** | (a and b) min. Z = 50 x _{ 1 } + 70 x _{ 2 } ; s.t. 80 x _{ 1 } + 40 x _{ 2 } = 3,000, 80 x _{ 1 } = 1,000, 40 x _{ 2 } = 800, x _{ i } |

** 25. ** | (a) personal interviews, $0.625/interview; (b) 25 c _{ 2 } , 1,800 q _{ 1 } |

(a) x _{ 1 } = 333.3, x _{ 2 } = 166.7, s _{ 1 } = 100 gal., s _{ 2 } = 133.3 gal., s _{ 3 } = 83.3 gal., s _{ 4 } = 100 gal., Z = 1,666; (b) any values of c |

** 29. ** | max. Z = 1.20 x _{ 1 } + 1.30 x _{ 2 } ; s.t. x _{ 1 } + x _{ 2 } 95,000, .18 x _{ 1 } + .30 x _{ 2 } 20,000, x _{ i } |

(a) 5%, $16,111.11; (b) x _{ 1 } = 0, x _{ 2 } = 66,666.7, Z = 86,666.67 |

** 33. ** | (a and b) min. Z = 400 x _{ 1 } + 180 x _{ 2 } + 90 x _{ 3 } s.t. x _{ 1 } 200, x _{ 2 } 300, x _{ 3 } 100, 4 x _{ 3 } x _{ 1 } x _{ 2 } 0, x _{ 1 } + x _{ 2 } + x _{ 3 } = 1,000, x _{ i } 0; (c) x _{ 1 } = 200, x _{ 2 } = 600, x _{ 3 } = 200, Z = 206,000 |

** 35. ** | max. Z = 0.50 x _{ 1 } + 0.75 x _{ 2 } ; s.t. 0.17 x _{ 1 } + 0.25 x _{ 2 } 4,000, x _{ 1 } + x _{ 2 } 18,000, x _{ 1 } 8,000, x _{ 2 } 8,000, x _{ 1 } , x _{ 2 } 0; (b) max. Z = 0.50 x _{ 1 } + 0.75 x _{ 2 } ; 0.17 x _{ 1 } + 0.25 x _{ 2 } = 4,000, x _{ 1 } + x _{ 2 } = 18,000, x _{ 1 } = 8,000, x _{ 2 } = 8,000, x _{ i } 0. |

x _{ 1 } = 8,000, x _{ 2 } = 10,000, Z = 11,500; (a) $375; increase, x _{ 1 } = 8,000, x _{ 2 } = 10,500, Z = $11,875; x _{ 1 } = 8,000, x _{ 2 } = 10,560, Z = $11,920 |

** 39. ** | (a) purchase land; (b) not purchase land |

** 41. ** | (a) $0.78, 360 cartons; (b) $0; (c) x _{ 1 } = 108, x _{ 2 } = 54, x _{ 3 } = 162, Z = 249.48, no discount |

** 43. ** | x _{ 1 } = 3, x _{ 3 } = 6, Z = 3,600; (a) more assembly hr.; (b) additional profit = $600; (c) no effect |

** 45. ** | x _{ 1 } = 1,000, x _{ 2 } = 800, x _{ 3 } = 200, Z = 760; (a) increase by 100, $38 in additional profit; (b) x _{ 1 } = 1,000, x _{ 2 } = 1,000, Z = 770; (c) Z = 810; x _{ 1 } = 1,600, x _{ 2 } = 200, x _{ 3 } = 200 |

** 47. ** | x _{ 1 } = 20, x _{ 2 } = 33.33, x _{ 3 } = 26.67, Z = $703,333.4 |

** 49. ** | x _{ 13 } = 350, x _{ 21 } = 158.33, x _{ 22 } = 296.67, x _{ 23 } = 75, x _{ 31 } = 610, x _{ 42 } = 240, Z = $77,910 |

** 51. ** | max Z = 12 x _{ 1 } + 8 x _{ 2 } + 10 x _{ 3 } + 6 x _{ 4 } ; s.t. 2 x _{ 1 } + 9 x _{ 2 } + 1.3 x _{ 3 } + 2.5 x _{ 4 } 74; 2 x _{ 1 } + .25 x _{ 2 } + x _{ 3 } + x _{ 4 } 24; 2 x _{ 1 } + 5 x _{ 3 } + 2 x _{ 4 } 36; 45 x _{ 1 } + 35 x _{ 2 } + 50 x _{ 3 } + 16 x _{ 4 } 480, x _{ i } 0. |