Lecture 11.5: Sensitivity Part III: Ranges

AGEC 352Fall 2012—October 22

R. Keeney

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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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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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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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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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?