lecture13 - Review Modeling PPP Review Modeling PPP MILP...

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Review Modeling PPP IE426: Optimization Models and Applications: Lecture 13 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University October 17, 2006 Jeff Linderoth IE426:Lecture 13 Review Modeling PPP MILP Relaxations Branch and Bound PPP – Integer An engineering plant can produce five types of products: p 1 , p 2 , . . . p 5 by using two production processes: grinding and drilling. Each product requires the following number of hours of each process, and contributes the following amount (in hundreds of dollars) to the net total profit. p 1 p 2 p 3 p 4 p 5 Grinding 12 20 0 25 15 Drilling 10 8 16 0 0 Profit 55 60 35 40 20 Jeff Linderoth IE426:Lecture 13 Review Modeling PPP MILP Relaxations Branch and Bound PPP – More Info Each unit of each product take 20 manhours for final assembly. The factory has three grinding machines and two drilling machines. The factory works a six day week with two shifts of 8 hours/day. Eight workers are employed in assembly, each working one shift per day. Jeff Linderoth IE426:Lecture 13 Review Modeling PPP MILP Relaxations Branch and Bound PPP maximize 55 x 1 + 60 x 2 + 35 x 3 + 40 x 4 + 20 x 5 (Profit/week) subject to 12 x 1 + 20 x 2 + 0 x 3 + 25 x 4 + 15 x 5 288 (Grinding) 10 x 1 + 8 x 2 + 16 x 3 + 0 x 4 + 0 x 5 192 (Drilling) 20 x 1 + 20 x 2 + 20 x 3 + 20 x 4 + 20 x 5 384 Final Assembly x i 0 i = 1 , 2 , . . . 5 Jeff Linderoth IE426:Lecture 13
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Review Modeling PPP MILP Relaxations Branch and Bound Restrictions on PPP Now suppose we wish to add the constraint that we wish to make at most two products. At most two of the five x j can be positive. z j = 1 Make product j 0 Otherwise ( x j > 0 z j = 1) !! Add constraints 1 5 j =1 z j 2 . 2 x j M j z j j = 1 , 2 , . . . 5 . Jeff Linderoth IE426:Lecture 13 Review Modeling PPP MILP Relaxations Branch and Bound PPP – Make no more than two model maximize 55 x 1 + 60 x 2 + 35 x 3 + 40 x 4 + 20 x 5 (Profit/week) subject to 12 x 1 + 20 x 2 + 0 x 3 + 25 x 4 + 15 x 5 288 10 x 1 + 8 x 2 + 16 x 3 + 0 x 4 + 0 x 5 192 20 x 1 + 20 x 2 + 20 x 3 + 20 x 4 + 20 x 5 384 z 1 + z 2 + z 3 + z 4 + z 5 2 x i M i z i i = 1 , 2 , . . . 5 x i 0 i = 1 , 2 , . . . 5 z i { 0 , 1 }∀ i = 1 , 2 , . . . 5 Jeff Linderoth IE426:Lecture 13 Review Modeling PPP MILP Relaxations Branch and Bound What about the M’s? M i = 10 4 i = 1 , 2 , . . . 5 ? Can we make M i smaller? Smaller M i are good! It gives a tighter relaxation. Jeff Linderoth IE426:Lecture 13 Review Modeling PPP MILP Relaxations Branch and Bound Small M’s Good. Big M’s Baaaaaaaaaaaaaad! Let’s look at the geometry P = { x R + , z ∈ { 0 , 1 } : x Mz, x u } LP ( P ) = { x R + , z [0 , 1] : x Mz, x u } conv ( P ) = { x R + , z ∈ { 0 , 1 } : x uz } Jeff Linderoth IE426:Lecture 13
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Review Modeling PPP MILP Relaxations Branch and Bound P z M u x 0 1 P = { x R + , z ∈ { 0 , 1 } : x Mz, x u } Jeff Linderoth IE426:Lecture 13 Review Modeling PPP MILP Relaxations
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