Lecture 12: Sensitivity Examples (Shadow Price Interpreted)

AGEC 352Spring 2012 – February 29

R. Keeney

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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