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Unformatted text preview: ic Programming Extra Economic Information
• Reduced costs: Each variable has an associated reduced cost, or
opportunity cost, which is the amount by which an objective function
coefficient would have to improve (so increase for maximization
problem, decrease for minimization problem) before it would be
possible for its corresponding variable to assume a positive value in
the optimal solution.
• Shadow prices: Each constraint has an associated shadow price
which is the change in the objective value of the optimal solution
obtained by relaxing the constraint by one unit  it is the marginal
utility of modifying the constraint.
• More formally, the shadow price is the value of the Lagrange
multiplier at the optimal solution, which means that it is the
infinitesimal change in the objective function arising from an
infinitesimal change in the constraint.
• The extra economic information arises from the dual model. Refresh your LP knowledge B
Feasible Region 30 E 20 Leather Constraint 10 • Both inequaliFes are saFsﬁed in the shaded area. The extreme points are of the feasible region are B, C, E, and F. Labor Constraint
40 • D 50 max z = 4x1 + 3x2 s.t. x1 + x2 + s1 = 40 2x1 + x2 + s2 = 60 x1, x2, s1, s2 ≥ 0 60 Focus on the relaFonship between extreme points and basic feasible soluFons. max z = 4x1 + 3x2 s.t. x1 + x2 ≤ 40 2x1 + x2 ≤ 60 x1, x2 ≥ 0 The LP (with slack variables) was: X2 C
F 10 20 30 A
40 50 X1 Refresh your LP knowledge The table above shows the correspondence between the basic feasible soluFons to the LP and the extreme points of the feasible region. The basic feasible soluFons to the standard form of the LP correspond in a natural fashion to the LP’s extreme points. Product Mix Example • Why are only 2 products being made? Are products 3, 4, and 5 underpriced? What should
their price be in order to make it worthwhile to manufacture them?
• Why is there extra drilling capacity? Should I pay to increase grinding and manpower
capacity to increase profit? What is the marginal value of these capacities? Product Mix Example
• Shadow prices:
– Grinding Constraint: 6.25
– Drilling Constraint: 0
– Manpower Constraint: 23.75
• The shadow price for Drilling is 0, because it is not binding. Thus
there is no value to increasing this capacity.
• The shadow price for Manpower is much larger than for Grinding, so
it is more advantageous to add 1 unit of Manpower than 1 unit of
Grinding (if they have equal cost), because 1 unit of Manpower
would increase profit by 23.75, whereas Grinding would increase
profit by 6.25.
• The shadow prices are for small changes, performed one at a time.
– How la...
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 Winter '09

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