Lecture 12: Sensitivity Examples (Shadow Price Interpreted) AGEC 352 Spring 2012 – February 29 R. Keeney

Shadow Price signs  Signs on shadow prices differ whether the inequality constraint is ≤ or ≥.  They also differ for maximization and minimization problems. MaximizationMinimization ≤PositiveNegative ≥ Positive

Less than (<=) case A boundary that is <= (upper bound) We use +1 definition of shadow price ◦ The +1 will always ‘relax’ the upper bound A 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 bound A 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 bound Minimization ◦ 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 bound Maximization ◦ 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 bound Minimization ◦ 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 Problem 4 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 LHSRHSShadow Price Nitrogen201.3>= Potash300.0>= Phosphate80.0<= Total Weight1000=

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 calcium Nutritional Requirement LHSRHSShadow Price Thiamine1>= Niacin5>= Calories722.2>=

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 however Nutritional Requirement RHS1 % increase Shadow Price SP * 1% increase Thiamine Niacin Calories

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…

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

Planting Problem Shadow price for land is 2X labor ◦ 1 unit of land is usually worth more than a unit of labor Compare them as 1% increase in our resource base (labor > land > allot)