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WOOD 492 MODELLING FOR DECISION SUPPORT. Lecture 10 Introduction to Sensitivity Analysis. 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
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WOOD 492 MODELLING FOR DECISION SUPPORT Lecture 10 Introduction to Sensitivity Analysis
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? Wood 492 - Saba Vahid
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? Wood 492 - Saba Vahid
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 Wood 492 - Saba Vahid
LP formulation • What is the value of an extra hour of moulding, sanding or labour? • How much more expensive should product 3 be before we start producing it? Z Wood 492 - Saba Vahid
Change to 289 Shadow price (Answering Q1) by Trial & Error Z new = $10,926.25 Z old = $10,920 Difference (shadow price)= $6.25 = value of an additional hour of mouldertime 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.
Increase gradually Reduced Cost (Answering Q2) by Trial & Error Price of product 3 has to increase by $125 before it would be produced. Reduced cost = -125 Reduced Cost: If a variable = 0 in the optimal solution, then its reduced cost is the amount its objective function coefficient (price in this example) needs to change before it will come into the solution (>0). New value of Obj. Coefficient = Old value of Obj. Coeff – Reduced Cost Wood 492 - Saba Vahid
Sensitivity Analysis with Simplex • When LP problems are large with many variables and constraints • Re-solving LPs may require a large computational effort • Simplex algorithm eliminates the need to resolve the LP for every change in parameters • While we won’t get into the details of sensitivity analysis with Simplex method, we can view the results in Excel Solver’s sensitivity report Wood 492 - Saba Vahid
Example of the sensitivity report in Excel • Furniture Manufacturer (refer to Example_3 files on the course website) How sensitive is the solution to this estimated price? How would the solution change if we had more assembly hours? Sensitivity report Wood 492 - Saba Vahid
Lab 3 Preview CB 1 4000 m3 Sawdust (m3) Log deck Head saw Trimmer CB 2 5000 m3 Logs sorted by diameter (m3) Planks and boards (MBF) Lumber products (MBF) Market CB 3 2500 m3 Chips (m3) Lab 3 Preview Wood 492 - Saba Vahid
Lab 3 Preview • Objective: maximize profits ($) • 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) Wood 492 - Saba Vahid
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) Wood 492 - Saba Vahid
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 Wood 492 - Saba Vahid
Next Class • Quiz on Friday, Sept 28 • More sensitivity analysis examples Wood 492 - Saba Vahid