ch.5 Sensitivity Analysis Notes

1 ECON3300 Sensitivity Analysis Learning Objectives: We will Interpret Excel’s Sensitivity Report to Assess the Impact of Changes to a Standard LP Modeling Questions: • The purpose of Sensitivity Analysis. • Changes of objective function coefficients. Examples. • Concept of shadow price. • Concept of binding and nonbinding constraints. • Changes of RHS values of constraints. Examples. • Simultaneous Changes. Example A Maximization Example : Beaver Creek Pottery Company produces authentic clay bowls and mugs. The two primary resources used by the company are special pottery clay and skilled labor. Given these limited resources, the company desires to know how many bowls and mugs to produce each day in order to maximize profit. The Pottery faces resource constraints: • The hours of labor time used cannot exceed 40 hours per day • Clay in lbs used per day cannot exceed 120 pounds per day The objective is to Maximize Profit under the recourses constraints. Step 1 : Clearly define the decision variables Decision Variables: x 1 = number of bowls to produce per day x 2 = number of mugs to produce per day Step 2 : Construct the objective function Objective Function: Maximize Z = $40x 1 + $50x 2 Where Z = profit per day Step 3 : Formulate the constraints Resource Availability: 40 hrs of labor per day 120 lbs of clay Resource Constraints: 1x 1 + 2x 2  40 hours of labor 4x 1 + 3x 2  120 pounds of clay Non-Negativity Constraints: x 1  0; x 2  0 Full Linear Programming Model: Maximize Z = $40x 1 + $50x 2 subject to: 1x 1 + 2x 2  40 4x 2 + 3x 2  120 x 1 , x 2  0 Excel “Solver” Results Screen

2 Solver results screen Answer Report Sensitivity Report

4 Let’s find sensitivity range for the first OF coefficient (profit of one bowl). Upper limit 40+26.67=66.67 Low limit 40-15=25. Conclusion: profit of one bowl can vary between $25 and $66.67 and the combination x 1 = 24 and x 2 =8 (24 bowls and 8 mugs produced) will still be optimal. Let’s find sensitivity range for the second Objective Function coefficient (profit of one mug). Upper limit 50+30=80 Low limit 50-20=30. For the second coefficient of objective function c 2 (profit for 1mug) the allowable increase of 30 results in an upper limit of 80, whereas the allowable decrease of 20 results in a lower limit of 30. For coefficient c 2 sensitivity range is [30,80]. Conclusion: profit of one mug can vary between $30 and $80 and the combination x 1 = 24 and x 2 =8 (24 bowls and 8 mugs produced) will still be optimal.

5 Sensitivity Analysis of Right-Hand Side Values Question 2. RHS constraint Changes In sensitivity analysis of right-hand sides of constraints we are interested in the following questions: • Keeping all other factors the same, how much would the optimal value of the objective function (for example, the profit) change if the right-hand side of a constraint changed by one or more units? • For how many additional or fewer units will this per unit change be valid? Shadow Price Assuming there are no other changes to the input parameters, the change (increase or decrease) to the objective function value per unit increase to a right hand side of a constraint is called the “ Shadow Price ”. Further Interpretation of Shadow Price • The shadow price is the value of an extra unit of the resource • That is, the unit contribution to the objective function. • The shadow price can be used as a maximum unit price to pay for extra resource. • Other words, shadow price is defined as the marginal value of one additional unit of resource. Sensitivity Range for a Constraint Quantity The sensitivity range for a constraint quantity value is also the range over which the shadow price is valid . Constraint Quantity Value Ranges by Computer Excel Sensitivity Range for Constraints Beaver Creek Pottery Example The second table provides details about the constraints. Allowable Increase and Allowable Decrease have to be added to and subtracted from the Constraint Right Hand Side to obtain the limits where Shadow prices are valid.

7 Example 3. If 40 extra hours of labor can be obtained, what is its total value of profit? Example 3. If 40 extra hours of labor can be obtained, what is its total value of profit? Solution: The shadow price (or marginal value) for labor is $16 per hour. ($16/hr)(40 hr.) = $640. In other words, profit is increased by $640 if 40 extra hours of labor can be obtained. The shadow price (or marginal value) for labor is $16 per hour. ($16/hr.)(40 hr.) = $640. In other words, profit is increased by $640 if 40 extra hours of labor can be obtained. New Optimal Value of Objective Function= =Current Value of OF+Shadow Price*40=$1360+640=$2000

8 Constraint Quantity Value Ranges. Summary. Very important piece of information that is provided by the sensitivity ranges for the constraint quantity values is the range over which the shadow price remains valid. When RHS value of constraint increases past the upper limit of the sensitivity range or decreases below the lower limit, the shadow price will change. Excel Sensitivity Report for Beaver Creek Pottery Solution Screen Constraint Quantity Value Range Example 4. Find Constraint Quantity Value Sensitivity Range for labor constraint of Beaver Creek Pottery Example. Upper limit is 40+ 40 =80. Low limit is 40- 10 =30. Conclusion: The number of hours of labor in the right-hand side of labor constraint can vary from 30 till 80 and shadow price $16 per unit recourse will remain the same.

10 Geometrically, the boundary of a binding constraint will pass through the optimal solution, and the boundary of a non-binding constraint will not pass through the optimal solution. A non-binding constraint will always have a shadow price of 0 and changing the RHS of a non- binding constraint by any amount within its allowable increase or decrease will have no impact on the optimal solution and no impact on the optimal value of the objective function. A binding constraint can have a positive, negative, or 0 shadow price. Simultaneous Changes Some analysis of simultaneous changes can be possible using the 100 percent rule. Par Inc. is a small manufacturer of golf equipment. It produces two types of golf bags: Standard and Deluxe. Each bag type requires the following operations (and production times) to produce one unit:   Production Time (hours) Time Available Operation Standard Deluxe (hours/month) Cutting and Dyeing 0.7 1 630 Sewing 0.5 5/6 600 Finishing 1 2/3 708 Inspection 0.1 0.25 135 Time available refers to the production capacity for each of the above operations. For example, 630 total hours a month are available for cutting and dyeing, which will be distributed for the production of the two types of bags. Every standard bag makes a profit of $10 , and every deluxe bag makes a profit of $9 . The problem is to determine the optimal number of standard bags and deluxe bags to produce every month to maximize the profit contribution. The Sensitivity Report for the Par Inc. problem is shown below.

11 X=540, Y=252; Z=10X+9Y. Suppose that in the Par, Inc. problem the correct profit contributions are $11.50 and $8.25 for standard bags and deluxe bags, respectively. Use the 100 percent rule to determine whether these new values will affect the optimal solution. Interpreting Excel’s Sensitivity Report to Assess the Impact of Changes to a Standard LP model Note : Below, the “current optimal solution” refers to the optimal solution provided by Excel’s SOLVER tool. The upper table of the Sensitivity Report can be used to identify changes to one or more objective function coefficients for which the current optimal solution will remain optimal. The table provides for each objective function coefficient an allowable increase and an allowable decrease.  If, making no other changes , one objective function coefficient is increased by any positive amount up to its allowable increase or decreased by any positive amount up to its allowable decrease, then:| • the current optimal solution will remain optimal; and • the resulting optimal value of the objective function can be calculated using the “new” objective function coefficient  If, making no other changes , multiple objective function coefficients are changed, each either increased by some percentage of its allowable increase or decreased by some percentage of its allowable decrease, and if the sum of those percentages is < 100%: • the current optimal solution will remain optimal; and • the resulting optimal value of the objective function can be calculated using the “new” set of coefficients The 100 percent rule states that if the sum of the percentage changes does not exceed 100%, the optimal solution will not change. If the sum exceeds 100%, we cannot conclude that the solution has changed; we need to re-run SOLVER. Solution: The allowable increase for Standard Bag Profit Contribution is 3.5. The increase of 1.5 (from 10 to 11.50) represents 1.5/3.5 = 42.86% of the allowed increase. The allowable decrease for Deluxe Bag Profit Contribution is 2.333. The decrease of 0.75 (from 9 to 8.25) represents 0.75/2.333 = 32.14%. The sum of 42.86% + 32.14% is 75%, which is less than 100%. Therefore, the optimal solution X=540, Y=252 will not change. Note: the value of the total profit will change since the profit contributions have changed. A similar version of the 100 percent rule also applies to simultaneous changes in the constraint right-hand sides. The lower table of the Sensitivity Report can be used to assess the impact of changing (within limits) the RHS of one or more constraints. The table provides for each constraint a shadow price (rate of change in the optimal value of the objective function per unit increase in the constraint’s RHS), an allowable increase, and an allowable decrease.

13 and the objective function will improve by (20)( 4.37 ) + (100)( 6.94 ) = $781.40. Step 4. Compute the new value the objective function. Current objective function value + the value of improvement $7,668+ $ 781.40 = $8,449.4 Approaches to Sensitivity: Summary The Excel solver also produces sensitivity reports that can answer questions about: • by how much the objective function coefficient can change without affecting the optimal solution? • how much the optimal objective function value will change as the right-hand side of a constraint is varied? • how much is worth to add one unit of the constrained resource (i.e. constraint’s RHS)?