Lecture 11: Sensitivity Part II: Prices

AGEC 352Spring 2012-February 27

R. Keeney

Constraint Prices

Shadow price = marginal valuation Marginal -> last or next unit Shadow?

an internal price with no actual exchange.

Decision maker is both the supplier and demander.

Use of the Shadow PriceAnswers the question:

If the RHS limitation were expanded by one unit how does the objective variable change? Willingness to pay or accept if

entering the external market Recall the objective variable is

the only measure of success/benefit to the decision maker

Simple Model: RHS +1

Expansion of RHS of x constraint means we can increase our choice of x , which increases Z

The change in Z is our benefit The increased benefit is the maximum we

would pay for the added unit on RHS of x constraint

0,10;11

..max

yxyx

tsyxZ

0,10;10

..max

yxyx

tsyxZ

Simple Model: RHS -1

Reduction of RHS of x constraint means we must reduce our choice of x , which diminishes Z

The change in Z is our loss and is the minimum we should charge to sell x rather than use it

0,10;9

..max

yxyx

tsyxZ

0,10;10

..max

yxyx

tsyxZ

Signs and interpretation

If increasing the RHS increases Z, the shadow price will be positive.

If increasing the RHS decreases Z, the shadow price will be negative.

)()(

xx OldRHSNewRHSOldZNewZpriceShadow

Signs and Interpretation

What if Z doesn’t change?◦Shadow price = 0◦This will be the case for any constraint that does not bind at the optimum…

◦Think about the question: What would we pay for one more unit?

)()(

xx OldRHSNewRHSOldZNewZpriceShadow

Shadow Price SignsObjective Direction

Inequality Direction

<= =>

Maximization Positive

Negative

Minimization Negative

Positive

For a max problem:Increasing the RHS of a <= constraint expands the feasible space, increases the value of Z, generates a positive shadow price.Increasing the RHS of a => constraint contracts the feasible space, reduces the value of Z, generates a negative shadow price.

Shadow Price SignsObjective Direction

Inequality Direction

<= =>

Maximization Positive

Negative

Minimization Negative

Positive

For a min problem:Increasing the RHS of a <= constraint contracts the feasible space, reduces the value of Z, generates a negative shadow price.

Increasing the RHS of a => constraint expands the feasible space, increases the value of Z, generates a positive shadow price.

Mathematical RuleExpanding or reducing the feasible space by adjusting a non-binding constraint has no impact on Z, shadow price = 0.

Let, slack = (Constraint RHS – Constraint LHS)Then we can state the following rule:

(slack)*(shadow price) = 0If shadow price is non-zero, slack must be zero.If shadow price is zero, slack is either a) non-zero or b) zero.

Case b: 0 slack & 0 shadow price

0 1 2 3 4 5 6 7 8 9 10 11 120123456789101112y

x

: 1.0 x + 0.0 y = 10.0

: 0.0 x + 1.0 y = 10.0

: 2.0 x + 1.0 y = 30.0

Payoff: 1.0 x + 1.0 y = 20.0

Optimal Decisions(x,y): (10.0, 10.0): 1.0x + 0.0y <= 10.0: 0.0x + 1.0y <= 10.0: 2.0x + 1.0y <= 30.0 0,

10;10302

..max

yxyx

yxts

yxZ

The first constraint is redundant because it does not add a corner point to the problem.

Plugging x = 10 into 2x + y = 30, gives y=10, which is already a constraint of the problem.

Objective Variable PricesSensitivity of constraints involves

placing an economic value on the resources in the problem◦Look at Excel’s shadow price report

later

Sensitivity of objective coefficients (prices for short) is completely different◦Under what price range does the

optimal plan remain optimal?

Pizza Maker’s ProblemTwo pizza types:

Regular (R) and Deluxe (D) Use available

sauce, dough, sausage, cheese, and mushrooms to make pizzas.

Profit is 2.25 per R pizza, 2.65 per D pizza 0;0:.

1004:500128:27593:10001616:

44088::

65.225.2max

DRnegNonDMushrooms

DRCheeseDRSausageDRDough

DRSaucetosubject

DRP

Feasible Space for Pizza Maker

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R

Payoff: 2.3 R + 2.6 D = 129.7

Optimal Decisions(R,D): (40.0, 15.0) S auce: 8.0R + 8.0D <= 440.0 Dough: 16.0R + 16.0D <= 1000.0 S ausage: 3.0R + 9.0D <= 275.0 Cheese: 8.0R + 12.0D <= 500.0M ushroom: 0.0R + 4.0D <= 1000.0

Regular Pizzas

Deluxe Pizzas

Optimum is R = 40, D = 15

How sensitive is this solution to a change in the price of Deluxe Pizzas?

How does changing the price of the deluxe pizza affect this problem?

Objective Equation: 2.25R + 2.65D = PRewrite this as:

D = P/2.65 – R*(2.25/2.65)

The slope of the objective line will flatten if we increase the price of deluxe pizzas above 2.65.

If the objective line gets flat enough, the optimal point will switch to the next corner point immediately leftward.

Deluxe Price Increase

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R

Payoff: 2.3 R + 2.6 D = 129.7

Optimal Decisions(R,D): (40.0, 15.0) S auce: 8.0R + 8.0D <= 440.0 Dough: 16.0R + 16.0D <= 1000.0 S ausage: 3.0R + 9.0D <= 275.0 Cheese: 8.0R + 12.0D <= 500.0M ushroom: 0.0R + 4.0D <= 1000.0

Regular Pizzas

Deluxe Pizzas

Price increase makes this line flatter. If it changes enough we will have a new optimal combination of R and D pizzas.

Called Allowable increase in Excel’s Sensitivity Report

The size of the price increase determines whether the slope of the objective line gets flat enough to shift to the leftward corner point.

This is what the allowable increase on objective coefficients is measuring.

The allowable decrease does the same in the opposite direction.

Sensitivity Report on Pizza Prices:Prices increase->Profit/pizza goes up

The allowable increase says that if the profit/deluxe pizza goes up by more than 72.5 cents we should shift to a new combination of R and D pizzas (more D, less R).

If profit/deluxe pizza goes down by more than 40 cents make more R and less D.

Important point: Any change in the profits/pizza will change the objective value, but if in the allowable range, the best choices do not adjust.

Constraint Sensitivity

Cheese and Sauce are binding constraints with positive shadow prices

We would pay to have more cheese or sauce available to make pizzas with because we could increase profits

Binding constraints

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R

Payoff: 2.3 R + 2.6 D = 129.7

Optimal Decisions(R,D): (40.0, 15.0) Sauce: 8.0R + 8.0D <= 440.0 Dough: 16.0R + 16.0D <= 1000.0 Sausage: 3.0R + 9.0D <= 275.0 Cheese: 8.0R + 12.0D <= 500.0M ushroom: 0.0R + 4.0D <= 1000.0

Regular Pizzas

Deluxe Pizzas

This corner point is where the cheese and sauce constraints cross.

Sauce constraint

Cheese constraint

Constraint Ranges

Excel’s constraint sensitivity report also reports allowable increase and decrease

These values indicate the magnitude of changes allowed to the RHS quantity without changing the marginal valuation (shadow price)

Expanding the sauce constraint

Adding 1 to the RHS of the sauce constraint expands the feasible space

Moves the corner point rightward allowing for a higher objective variable value

The shadow price says every time we expand this constraint by one unit, we gain about $0.18 of profits

Allowable increase tells us how long we can keep making these 1 unit moves in the constraint

Expanding sauce capacity

If we kept moving the Sauce constraint to the right what would happen?

Eventually, sauce would not be limiting.

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R

S auce: 8.0 R + 8.0 D = 440.0

Sausage: 3.0 R + 9.0 D = 275.0

Cheese: 8.0 R + 12.0 D = 500.0

Payoff: 2.3 R + 2.6 D = 0.0Optimal Decisions(R,D): ( 0.0, 0.0) S auce: 8.0R + 8.0D <= 440.0

S ausage: 3.0R + 9.0D <= 275.0 Cheese: 8.0R + 12.0D <= 500.0

Deluxe

Regular

Sauce is no longer limiting

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R

Sauce: 8.0 R + 8.0 D = 510.0 Sausage: 3.0 R + 9.0 D = 275.0

Cheese: 8.0 R + 12.0 D = 500.0

Payoff: 2.3 R + 2.6 D = 0.0Optimal Decisions(R,D): ( 0.0, 0.0) S auce: 8.0R + 8.0D <= 510.0

S ausage: 3.0R + 9.0D <= 275.0 Cheese: 8.0R + 12.0D <= 500.0

D

R

With a RHS value of 510, the sauce constraint is no longer on the boundary of the feasible space. This is the information provided by the allowable increase of the constraint.

Typical QuestionsPizza maker wants to sell her

excess dough. What is the minimum amount she can charge?

Pizza maker can buy 200 units of sauce for $15.00. Should she do it?

Pizza maker has a sale on deluxe pizzas reducing profit per unit by 15%. Should she change the production plan for this week?