WOOD 492 MODELLING FOR DECISION SUPPORT

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

WOOD 492 MODELLING FOR DECISION SUPPORT