1. WOOD 492 MODELLING FOR DECISION SUPPORT Lecture 10 Introduction to Sensitivity Analysis

2. Why Sensitivity Analysis? LPs are used for finding an optimal plan –Solution of key decision variables that generates the best value for the objective function, based on given parameters 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? Sept 24, 2012Wood 492 - Saba Vahid2

3. Sensitivity Analysis Is extremely useful when making decisions based on the results of an optimization model Helps us answer questions like: –What is the value of additional capacity/resources? –How much would the prices have to change before ….. –How sensitive is the objective value to the estimates of parameter values? –What is the range of parameter values for which the optimal solution stays the same? Sept 24, 2012Wood 492 - Saba Vahid3

4. Example for Sensitivity Anlaysis 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 Vahid4

5. LP formulation 1. What is the value of an extra hour of moulding, sanding or labour? 2. How much more expensive should product 3 be before we start producing it? Sept 24, 2012Wood 492 - Saba Vahid5 Z

6. Shadow price (Answering Q1) by Trial & Error Change to 289 Z new = $10,926.25 Z old = $10,920 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.

7. Reduced Cost (Answering Q2) by Trial & Error Sept 24, 2012Wood 492 - Saba Vahid7 Increase gradually Price of product 3 has to increase by $125 before it would be produced. Reduced cost = -125 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). New value of Obj. Coefficient = Old value of Obj. Coeff – Reduced Cost

8. 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 Vahid8

9. Example of the sensitivity report in Excel Furniture Manufacturer (refer to Example_3 files on the course website) Sept 24, 2012Wood 492 - Saba Vahid9 How sensitive is the solution to this estimated price? How would the solution change if we had more assembly hours? Sensitivity report

10. Lab 3 Preview Sept 24, 2012Wood 492 - Saba Vahid10 CB 1 4000 m3 Log deck Logs sorted by diameter (m3) Lumber products (MBF) Planks and boards (MBF) Trimmer Chips (m3) CB 2 5000 m3 CB 3 2500 m3 Head saw Sawdust (m3) Market Lab 3 Preview

11. Objective: maximize profits ($) Decision variables: –how much to cut from each cut block (m3) –How much logs to process with each sawing pattern(m3) –Which lumber products to produce (mbf) Constraints –Available timber (volume in the cut blocks) –Machine hours available, downtime –Material balance related to conversion factors –No demand constraints –Sawing patterns are not mutually exclusive (we can use a combination of SP1 and SP2) Sept 24, 2012Wood 492 - Saba Vahid11

12. Some unit conversions 1 MBF = volume of a block of 1” by 12” by 12” –1BF= 144 cubic inch –1 BF=0.0254*0.3048*0.3048 cubic meters ≈ 0.002360 m3 –1000 BF=1 MBF= 2.36 m3 Calculating chip conversion factors (m3/m3*, unitless): *Note: In this lab, the chip conversion factors are given in relation to the total log volume, not the lumber volume Total log vol = lumber vol+ sawdust + chips (all units should be in m3) = lumber vol(MBF)*2.360 (m3/MBF)+ 0.1 * total log vol+ chips chips= total log vol - 0.1 total log vol- lumber vol * 2.360 = 0.9 total log volume – (lumber recovery factor* total log vol)*2.360 =total log volume * (0.9 – lumber recovery factor * 2.360) Sept 24, 2012Wood 492 - Saba Vahid12

13. Lab 3 preview Speed of Head saw = 250 ft/minute = 15,000 ft/hr –Accounting for 10% downtime= 15000 *0.9= 13,500 ft/hr –Each log is 8 ft tall and there’s a 4 ft gap between logs so for every 12 ft of conveyor, there is one log passing –13,500/12= 1125 logs/hour (passing through head saw) –Combination of this log/hr rate and log volumes is used to calculate processing time (hr/m3) for the head saw, as shown in the Excel file Speed of Trimmer= 95 lugs/minute = 5,700 lugs/hr –Accounting for 90% coverage= 5,700 *0.9=5,130 boards/hr –Accounting for 10% downtime= 5,130 *0.9= 4,617 boards/hr –Combination of this board/hr rate and lumber volumes is used to calculate processing time (hr/MBF) for the trimmer, as shown in the Excel file Sept 24, 2012Wood 492 - Saba Vahid13

14. Next Class Quiz on Friday, Sept 28 More sensitivity analysis examples 14Wood 492 - Saba VahidSept 24, 2012