A.
For each warehouse i, add a warehouse i’ that corresponds to the amount shipped
with the surcharge.
Let x
ij
be the amount shipped along route (i,j) at the original
cost, and x
ij
Objective function
’ be the amount shipped along(i’,j) with the surcharge.
Supply constraints:
Demand constraint:
Nonnegativity:
Because in this example, it is cheaper to ship from the i warehouse before the
corresponding i’ warehouse, at optimality, the supply for the i warehouses will be
completely filled before the corresponding i’ warehouses will ship, which will give,
as an optimal solution, an answer that adheres to our restriction that the first n
i
units must all be shipped before the surcharge is in effect.
B.
No.
This formulation gave correct solutions only because it was cheaper to ship the
first n
i
units first.
If p
i
is negative, it now becomes cheaper to ship from the i’
warehouses, which would mean the supply for the i’ warehouses will fill before the
units from the i warehouses are shipped.
This leads to a solution where units with
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 Fall '09
 TROTTER
 Optimization, Shipping, vij, feasible solution changes

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