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Lecture 12: Sensitivity Examples (Shadow Price Interpreted)
AGEC 352Spring 2012 – February 29
R. Keeney
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Shadow Price signsSigns on shadow prices differ whether
the inequality constraint is ≤ or ≥.They also differ for maximization and
minimization problems.
Maximization
Minimization
≤ Positive Negative
≥ Negative Positive
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Less than (<=) case A boundary that is <= (upper
bound)We use +1 definition of shadow
price◦The +1 will always ‘relax’ the upper
boundA decision maker facing a less
restrictive choice set◦Can be better off (binding constraint)◦Can be unaffected (slack constraint)
Better off depends on max vs. min
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Great than (>=) case A boundary that is >= (lower
bound)We use +1 definition of shadow
price◦The +1 will always ‘tighten’ a lower
boundA decision maker facing a more
restrictive choice set◦Can be worse off (binding constraint)◦Can be unaffected (slack constraint)
Better off depends on max vs. min
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Example (Upper/Max)Upper bound
◦Maximization◦Land available to plant
Shadow price = the change in returns generated by a +1 to the land constraint
Shadow price = Maximum rent that can be paid Use extra profits from additional resources to
acquire the resource
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Example (Upper/Min)Upper boundMinimization
◦Fertilizer mix phosphate limit◦Shadow price = the change in costs
from a 1 unit increase in the phos limit
◦Shadow price = discount the mixer could offer to the buyer to expand the phos limit Pass some of cost savings to buyer
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Example (Lower/Max)Lower boundMaximization
◦Every 10 acres of corn planted requires 1 acre left fallow (set aside) Shadow price = change in profits from
increasing set-aside by 1 Shadow price = payment farmer must
receive to participate
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Example (Lower/Min)Lower boundMinimization
◦Calcium requirement in a daily diet Shadow price = change in cost of
requiring an extra unit of calcium Shadow price = maximum price that can
be paid per unit of non-food calcium supplement
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Lab Assignment Problem4 Fertilizers (see lab 5 for
fertilizer info)◦Different compositions of nitrogen, potash, and phosphate
◦Meet an order (at minimum cost) by mixing the four fertilizers that has: Exactly 1000 units of fertilizer At least 20% (by weight) nitrogen At least 30% (by weight) potash At most 8% (by weight) phosphate
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Shadow Prices in Fert. Problem
Fertilizer Component
LHS RHS Shadow Price
Nitrogen 201.3 >= 200 0.00
Potash 300.0 >= 300 10.00
Phosphate
80.0 <= 80 -14.00
Total Weight
1000 =1000 11.70
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Interpretation of Potash Potash constraint Required to have a minimum amount of
potash in the fertilizer mix Increasing the RHS of the potash
constraint makes the problem more restrictive, higher percentage of potash required
Shadow price is positive because costs will increase with the increase of RHS
Interpret this as the amount we would be willing to pay to avoid having the RHS increase
Also, the discount we could offer for a mix that had 0.1% less potash content
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Interpretation Phosphate
Phosphate constraint Upper limit on the phosphate content Increasing the RHS of the phosphate
constraint makes the problem less restrictive, higher percentage of phosphate allowed
Shadow price is negative because costs will decrease with the increase of RHS
Interpret this as the amount we would be willing to pay to relax the RHS by one unit
Also, the markup we should charge if someone required 0.1% less phosphate in their fertilizer mix
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Interpretation in general
Always should be in context of the problem◦Signs are actually trivial if you understand the problem (better off/worse off)
◦Does an increase in the RHS improve or worsen the objective? If it improves, then we know the
willingness to pay for increasing the RHS
If it worsens, then we know the willingness to pay to avoid having the RHS increase
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Advanced Analysis: Which constraint is the most costly?
Recall the cereal problem from lecture◦Two cereals mixed to meet minimum requirements on thiamine, niacin, and calciumNutritional Requirement
LHS RHS Shadow Price
Thiamine 1 >= 1 14.44
Niacin 5 >= 5 2.36
Calories 722.2 >= 500 0.00
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Rather than comparing units, we want to compare % of RHS
1 mg of thiamine and 1 mg of niacin are not directly comparable
% increases in the RHS of constraints are howeverNutritional
Requirement
RHS 1 %increas
e
Shadow
Price
SP * 1%
increase
Thiamine 1 0.01 14.44 0.14
Niacin 5 0.05 2.36 0.12
Calories 500 5 0 0.00
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Ranking the constraints
Thiamine was the most costly constraint to meet We would have judged this the same
just comparing shadow prices, but that could be misleading
Similar to elasticity interpretations Elasticity of demand for food versus
cars Requires that you understand the problem and interpretation to make the comparisons
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Fertilizer Problem
Consider◦Is total comparable to others?◦How to deal with positive vs negative
shadow prices? Compare relaxations of constraints…
Fert Component RHS Shadow Price 1 pct Value of 1Pct IncreaseN 200 0 2 0K 300 10 3 30P 80 -14 0.8 -11.2
total 1000 11.7 10 117
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Common percentage and direction (of objective variable)
Cost saving, 1% change in K◦Total cost reduces $30.00
Cost saving, 1% change in P◦Total cost reduces by $11.20
Fert Component RHS Shadow Price 1 pct Value of 1Pct IncreaseN 200 0 2 0K 300 10 3 30P 80 -14 0.8 -11.2
total 1000 11.7 10 117
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Planting Problem
Shadow price for land is 2X labor◦1 unit of land is usually worth more than
a unit of laborCompare them as 1% increase in our
resource base (labor > land > allot)
1% S Price SP * 1%Land LHS 500 5 13.75 68.75Rowcrop Land LHS 400 4 0 0Wht Allot LHS 120 1.2 12 14.4Jan-Apr Labor LHS 1600 16 6.25 100May-Aug Labor LHS 2000 20 0 0Sep-Dec Labor LHS 1600 16 0 0