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formulations_lecture - WORK SCHEDULING PROBLEM I A post...

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WORK SCHEDULING PROBLEM I A post office requires FULL TIME workers during a week as follows: M Tu W Th F Sat Sun 17 13 15 19 14 16 11 Each employee works five consecutive days, and two days off. Formulate a LP model to minimize full time employees. Decision Variables: x i = full time employees starting work on day i , i = 1 , · · · , 7. Workers available Monday: x 1 + x 4 + x 5 + x 6 + x 7 . IOE 310 Fall 2007
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THE LP MODEL I Minimize z , x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 = z x 1 + x 4 + x 5 + x 6 + x 7 17 x 1 + x 2 + x 5 + x 6 + x 7 13 x 1 + x 2 + x 3 + + x 6 + x 7 15 x 1 + x 2 + x 3 + x 4 + x 7 19 x 1 + x 2 + x 3 + x 4 + x 5 14 x 2 + x 3 + x 4 + x 5 + x 6 16 x 3 + x 4 + x 5 + x 6 + x 7 11 x 1 0 x 2 0 x 3 0 x 4 0 x 5 0 x 6 0 x 7 0 The solution of this model is: z x 1 x 2 x 3 x 4 x 5 x 6 x 7 LP Sol 67 3 4 3 10 3 2 22 3 0 10 3 5 Rounded Sol 25 2 4 2 8 0 4 5 IP Sol 23 4 4 2 6 0 4 3 IOE 310 Fall 2007
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CAPITAL BUDGETING PROBLEM I Concept: Net Present Value (NPV). In 1 In 2 In 3 In 4 In 5 Time 0 cash outlay ($) 11 53 5 5 29 Time 1 cash outlay ($) 3 6 5 1 34 NPV ($) 13 16 16 14 39 $40 million available now. $20 million available a year from now. Whole or any fraction of each investment can be purchased. IOE 310 Fall 2007
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LP MODEL I DECISION VARIABLES: x i Fraction of investment i purchased. Maximize z , 13 x 1 + 16 x 2 + 16 x 3 + 11 x 4 + 39 x 5 = z 11 x 1 + 53 x 2 + 5 x 3 + 5 x 4 + 29 x 5 40 3 x 1 + 6 x 2 + 5 x 3 + x 4 + 34 x 5 20 0 x 1 1 0 x 2 1 0 x 3 1 0 x 4 1 0 x 5 1 SOLUTION: x 1 = x 3 = x 4 = 1, x 2 = 0 . 201, x 5 = 0 . 288 and z = 57 . 449.
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