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Lecture 11.5: Sensitivity Part III: Ranges AGEC 352 Fall 2012—October 22 R. Keeney
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 Optimum is R = 40, D = 15 Deluxe Pizzas How sensitive is this solution to a change in the price of Deluxe Pizzas? Regular Pizzas
How does changing the price of the deluxe pizza affect this problem? Objective Equation: 2.25R + 2.65D = 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 2.65. If the objective line gets flat enough, the optimal point will switch to the next corner point immediately leftward.
Deluxe Price Increase Price increase makes this line flatter. If it changes enough we will have a new optimal combination of R and D pizzas. Deluxe Pizzas Regular 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 This corner point is where the cheese and sauce constraints cross. Deluxe Pizzas Cheese constraint Sauce constraint Regular Pizzas
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 $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?