Lecture 11: Sensitivity Part II: Prices AGEC 352 Spring 2011 – March 2 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 Price  Answers 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

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

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.

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?

Shadow Price Signs Objective Direction Inequality Direction <==> MaximizationPositiveNegative MinimizationNegativePositive 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 Signs Objective Direction Inequality Direction <==> MaximizationPositiveNegative MinimizationNegativePositive 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 Rule Expanding 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) = 0 If 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 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 Prices Sensitivity 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 Problem  Two 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

Feasible Space for Pizza Maker 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 D = P Rewrite 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 If the objective line gets flat enough, the optimal point will switch to the next corner point immediately leftward.

Deluxe Price Increase 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 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. Deluxe Regular

Sauce is no longer limiting 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 Questions Pizza maker wants to sell her excess dough. What is the minimum amount she can charge? Pizza maker can buy 200 units of sauce for $ 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?