1. WOOD 492 MODELLING FOR DECISION SUPPORT Lecture 9 Intro to Sensitivity Analysis

2. Example: Seasonal lumber inventory Lumber Inventory system Can store lumber and sell in future periods Maximum capacity of the warehouse Maximum sales in each period assumption1: handling and storage costs applied together in each period assumption2: Lumber purchased and sold at the same period does not need storing Sept 24, 2012Wood 492 - Saba Vahid2 Lumber Inventory

3. Quiz 2 problem Day-care food planning to minimize costs How much of each food item to give to children Nutritional requirements: –must receive between 400 and 600 total calories –No more than 30% of calories should come from fat –must receive a minimum of 60 mg of Vitamin C and 12 gr of Protein –need at least 2 slices of bread (to make a sandwich) Sept 24, 2012Wood 492 - Saba Vahid3

4. Sept 24, 2012Wood 492 - Saba Vahid4 Variable Name Bread (slice) P.B. (tbsp) Jam (tbsp) Milk (cup) Juice (cup) total calories (cal) Sign RHS Answer Objective 54 71535 tot. cal. balance 7010050150100 =0 tot. cal. min 1 >=400 tot. cal. max 1 <= 600 tot. cal. From fat 10750700 -30% <= 0 Vitamin C 0032120 >= 60 Protein 34081 >=12 Bread 1 >=2

5. Use of LPs Finding an optimal plan –Solution of key decision variables that generates the best value for the objective function Infeasible Solutions –cannot find a feasible solution –Mostly due to a mistake Unbounded Problems –LP can increase/decrease without bound Solvable Models –Gives a set of values to decision variables, based on the parameters used in the formulation Sept 24, 2012Wood 492 - Saba Vahid5

6. Why Sensitivity Analysis? LP solutions are based on the “certainty” assumption –What if we are not 100% sure about the parameter values? –How can we determine the impact of parameter changes on the optimal solution? –Which parameters are the most important one to estimate correctly based on the sensitivity of the objective function? Answering “what-if” questions –What is the value of additional capacity/resources? –How much would the prices have to change before ….. –How sensitive is the total profits to the recovery factor estimates? Sept 24, 2012Wood 492 - Saba Vahid6

7. Example of sensitivity analysis Producing 5 products using 3 moulders, 2 sander, and labour Factory works 2 shifts, 8 hours per shift, 6 days / week Each product takes 20 hrs of labour 8 workers working 48 hours / week Sept 24, 2012Wood 492 - Saba Vahid7

8. LP formulation 1. What is the value of an extra hour of moulding, sanding or labour? 2. How much more expensive should products 3, 4 & 5 be in order before we would start producing them? Sept 24, 2012Wood 492 - Saba Vahid8 Z

9. Shadow price (Answering Q1) by Trial & Error Change to 289 Z new = $10,926.25 Z old = $10,620 Difference (shadow price)= $6.25 = value of an additional hour of moulder time Shadow price: marginal value of a resource/constraint. Can be calculated by adding 1 to the RHS of a constraint and calculating the difference in the objective function.

10. Reduced Cost (Answering Q2) by Trial & Error Sept 24, 2012Wood 492 - Saba Vahid10 Increase gradually Price of product 3 has to increase by $125 before it would be produced. Reduced Cost: If a variable = 0 in the optimal solution, then its reduced cost is the amount its objective function coefficient (price in this example) needs to change before it will come into the solution (>0).

11. Sensitivity Analysis with Simplex When LP problems are large with many variables and constraints –Re-solving LPs may require a large computational effort –Simplex algorithm eliminates the need to resolve the LP for every change in parameters –While we won’t get into the details of sensitivity analysis with Simplex method, we can view the results in Excel Solver’s sensitivity report Sept 24, 2012Wood 492 - Saba Vahid11

12. Next Class Sensitivity Analysis in Excel Solver 12Wood 492 - Saba VahidSept 24, 2012