1. MAN 305 OPERATIONS RESEARCH II Week 4 –Sensitivity Analysis with Spreadsheets
DR. KAZIM BARIŞ ATICI

2. Blue Ridge Hot Tubs Company
Howie Jones owns and operates Blue Ridge Hot Tubs, a company that sells two models of hot tubs: the Aqua-Spa and the Hydro-Lux.
Howie purchases prefabricated fiberglass hot tub shells and installs a common water pump and the appropriate amount of tubing into each hot tub.
Every Aqua-Spa requires 9 hours of labor and 12 feet of tubing; every Hydro-Lux requires 6 hours of labor and 16 feet of tubing.
Demand for these products is such that each Aqua-Spa produced can be sold to generate a profit of $350, and each Hydro-Lux produced can be sold to generate a profit of $300.
The company expects to have 200 pumps, 1,566 hours of labor, and 2,880 feet of tubing available during the next production cycle.
The problem is to determine the optimal number of Aqua-Spas and Hydro-Luxes to produce to maximize profits.

3. Blue Ridge Hot Tubs Company
Decision variables:
𝑥 1 : number of Aqua-Spas to produce
𝑥 2 : number of Hydro-Luxes to produce
Objective function (profit):
𝑚𝑎𝑥 350 𝑥 𝑥 2
Constraints:
Pump: 𝑥 1 + 𝑥 2 ≤200
Labor: 9𝑥 1 +6 𝑥 2 ≤1566
Tubing: 12𝑥 𝑥 2 ≤2880
Non-negativity: 𝑥 1,2 ≥0

4. Blue Ridge Hot Tubs Company Mathematical Model
𝑚𝑎𝑥 350 𝑥 𝑥 2
𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜:
𝑥 1 + 𝑥 2 ≤200
9𝑥 1 +6 𝑥 2 ≤1566
12𝑥 𝑥 2 ≤2880
𝑥 1,2 ≥0

5. Blue Ridge Hot Tubs Company Spreadsheet Modeling

6. Sensitivity Analysis
Sensitivity analysis helps answer questions about how sensitive the optimal solution is to changes in various coefficients in an LP model.
Approaches to Sensitivity Analysis:
Change the data and re-solve the model!
Sometimes this is the only practical approach.
Solver also produces sensitivity reports that can answer questions about:
amounts objective function coefficients can change without changing the solution.
the impact on the optimal objective function value of changes in various constrained resources.
the impact on the optimal objective function value of forced changes in certain decision variables.
the impact changes in constraint coefficients will have on the optimal solution.

7. Blue Ridge Hot Tubs Company Sensitivity Report

8. Allowable Increase & Decrease
Values in the “Allowable Increase” and “Allowable Decrease” columns for the Variable (Changing) Cells indicate the amounts by which an objective function coefficient can change without changing the optimal solution, assuming all other coefficients remain constant.
You can verify this by changing the profit coefficient for Aqua-Spas to any value in the range from $300 to $450 and re-solving the model.
Values of zero (0) in the “Allowable Increase” or “Allowable Decrease” columns for the Changing Cells indicate that an alternate optimal solution exists.

9. Reduced Cost
The reduced cost for a nonbasic variable (in a max problem) is the maximum amount by which the variable’s objective function coefficient can be increased before the current basis becomes suboptimal, and it becomes optimal for the nonbasic variable to enter the basis.

10. Shadow Prices
The shadow price of a constraint indicates the amount by which the objective function value changes given a unit increase in the RHS value of the constraint, assuming all other coefficients remain constant.
Shadow prices hold only within RHS changes falling within the values in “Allowable Increase” and “Allowable Decrease” columns.
If the number of available labor hours increased by any amount in the range from 0 to 234 hours, the optimal objective function value changes (increases) by $16.67 for each additional labor hour.
If the number of available labor hours decreased by any amount in the range from 0 to 126 hours, the optimal objective function value changes (decreases) by $16.67 for each lost labor hour.
A similar interpretation holds for the shadow price for the constraint on the number of pumps.
Shadow prices for nonbinding constraints are always zero (In other words, resources in excess supply have a shadow price of zero).

11. A Note about Shadow Prices
Shadow prices only indicate the changes that occur in the objective function value as RHS values change. Changing a RHS value for a binding constraint also changes the feasible region and the optimal solution.
To find the optimal solution after changing a binding RHS value, you must re-solve the problem.
Illustration:
Let us suppose that the RHS value of the labor constraint for our example problem increases by hours (from 1,566 to 1,728) due to the addition of new workers.
Because this increase is within the allowable increase listed for the labor constraint, you might expect that the optimal objective function value would increase by $16.67 × 162=$2,700. That is, the new optimal objective function value would be approximately $68,800 ($66,100 +$16.67 × 162 =$68,800).
When we solve the problem for 1728 labor hours. The new solution involves producing 176 Aqua-Spas and 24 Hydro-Luxes.
The optimal solution to the revised problem is different from the solution to the original problem. This is not surprising, because changing the RHS of a constraint also changes the feasible region for the problem.

12. Adding a new activity
Suppose a new Hot Tub (Typhoon-Lagoon) is being considered. It generates a marginal profit of $320 and requires:
1 pump (shadow price = $200)
8 hours of labor (shadow price = $16.67)
13 feet of tubing (shadow price = $0)
Question: Would it be profitable to produce any?
Answer: $320 - $200*1 - $16.67*8 - $0*13 = -$13.33 = No!

13. Analyzing Changes in Constraint Coefficients
Question: Suppose a Typhoon-Lagoon required only 7 labor hours rather than 8. Is it now profitable to produce any?
Answer: $320 - $200*1 - $16.67*7 - $0*13 = $3.31 = Yes!
Question: What is the maximum amount of labor Typhoon-Lagoons could require and still be profitable?
Answer: We need $320 - $200*1 - $16.67*L3 - $0*13 >=0
The above is true if L3 <= $120/$16.67 = $7.20