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– What if more than one change is made? Shadow Prices • If we added 1 hour of grinding time, we would increase profit by
6.25. We could add as much as 96 (=384288) hours, and gain 96 x
6.25 = 600.
• If we added 1 hour of manpower time, we would increase profit by
23.75. We could add as much as 22.1(=406.1384) hours, and gain
22.1 x 23.75=524.875
• In either situation, the amount produced will change. Change in Manpower Adjacent Solutions
• • • Notice the extreme solutions, at
most two products are made. Is it
ever optimal to make three different
products? More than three?
– It is possible for three products
to be in the optimal solution.
– However, there will never be
more than three nonzero
products because there are
only three constraints.
In a linear model with m
constraints, there will never be
more than m nonzero variables in
an optimal basic (extreme point)
solution.
In a nonlinear model, the optimal
solution is not always an extreme
point solution (may be interior). Reduced Costs and Obj Ranging SensiFvity Analysis: Types of LP Changes Six types of changes in an LP’s parameters change the
optimal solution:
1. Changing the objective function coefficient of a
nonbasic variable.
2. Changing the objective function coefficient of a basic
variable.
3. Changing the righthand side of a constraint.
4. Changing the column of a nonbasic variable.
5. Adding a new variable or activity.
6. Adding a new constraint. Multiple Changes: Parametric Programming Multiple Changes: Parametric Programming Multiple Changes: The 100% Rule 100% Rule for Changing Objective Function Coefficients
Depending on whether the objective function coefficient of any variable
with a zero reduced cost in the optimal tableau is changed, there are
two cases to consider:
Case 1 – All variables whose objective function coefficients are changed
have nonzero reduced costs in the optimal row 0.
• the current basis remains optimal if and only if the objective function
coefficient for each variable remains within the allowable range.
• If the current basis remains optimal, both the values of the decision
variables and objective function remain unchanged.
• If the objective coefficient for any variable is outside the allowable
range, the current basis is no longer optimal.
Case 2 – at least one variable whose objective function coefficient is
changed has a reduced cost of zero.
• Apple the 100% rule. Diet Problem Define:
BR = # of Brownies
IC = # of Scoops of Ice Cream
COLA = # of Cans of Cola
PC = # of Pieces of Pineapple
MIN 50 BR + 20 IC + 30 COLA + 80 PC
SUBJECT TO
400 BR + 200 IC + 150 COLA + 500 PC >= 500 (Calorie constraint)
3 BR + 2 IC
>= 6 (Chocolate
constraint)
2 BR + 2 IC + 4 COLA + 4 PC
>= 10 (Sugar constraint)
2 BR + 4 IC + COLA + 5 PC
>= 8 (Fat constraint)
END OBJECTIVE FUNCTION VALUE 1) 90.00000 VARIABLE VALUE REDUCED COST BR 0.000000 27.500000...
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 Winter '09

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