MIE376 Lec9 DWD Incomplete.page3

# MIE376 Lec9 DWD Incomplete.page3 - practical algorithm...

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MIE376 Mathematical Programming Lecture Notes Daniel Frances 2011 3 The coordinator would then either decline, partially accept or fully accept the offers, and run some other LP to update the objective function prices to be used by the individual LPs in the next iteration Reformulating the 2-Plant LP Suppose we accept that each of the plants is going to derive their offers based on their own local LP. While the local objective may change based on the signal from the coordinator, the constraint set remains unchanged. Therefore the set of corner point of each of the local constraint set represents the set of possible offers that could be received from each plant: For Plant 1 the set of possible offers is {(P 1 0 , P 2 0 ), (P 1 1 , P 2 1 )…, (P 1 i , P 2 i )…} For Plant 2 the set of possible offers is {(Q 1 0 , Q 2 0 ), (Q 1 1 , Q 2 1 )…, (Q 1 i , Q 2 i )…} Clearly, for any LPs of realistic size it would be computationally infeasible to list all the basic feasible solutions (corner points), and so these sets are only of theoretical importance on the way to deriving a
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Unformatted text preview: practical algorithm. Let’s ASSUME for a moment that we have all the possible (P 1 i ,P 2 i ) and (Q 1 i , Q 2 i ). Then it follows from convexity of each of the constraint sets that • any linear combinations of the (P 1 i ,P 2 i ) offers is also feasible o i.e. for any set of µ i such that Σµ i = 1 an offer Σµ i (P 1 i ,P 2 i ) is also feasible • any linear combinations of the (Q 1 i ,Q 2 i ) offers is also feasible o i.e. for any set of λ i such that Σλ i = 1 an offer Σλ i (Q 1 i ,Q 2 i ) is also feasible Thus any solution (P 1 , P 2 ) can be rewritten as Σµ i (P 1 i ,P 2 i ), or P 1 = Σµ i P 1 i and P 2 = Σµ i P 2 i Similarly we can rewrite Q 1 = Σλ i Q 1 i and Q 2 = Σλ i Q 2 i Substituting in the 2-Plant model we obtain Max 90 Σµ i P 1 i + 80 Σµ i P 2 i + 70 Σλ i Q 1 i + 60 Σλ i Q 2 i subject to 8 Σµ i P 1 i + 6 Σµ i P 2 i + 7 Σλ i Q 1 i + 5 Σλ i Q 2 i ≤ 80; Σµ i =1 Σλ i = 1 µ i , λ i ≥ 0...
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