WOOD 492 MODELLING FOR DECISION SUPPORT Lecture 11 Sensitivity Analysis

Lab 3 review Sept 28, 2012Wood Saba Vahid2 CB m3 Log deck Logs sorted by diameter (m3) Lumber products (MBF) Planks and boards (MBF) Trimmer Chips (m3) CB m3 CB m3 Head saw Sawdust (m3) Market Lab 3 review

Sensitivity Analysis Results 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. Change in Obj. Fn value = shadow price x change in RHS of constraint 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 Sept 28, 2012Wood Saba Vahid3

Example: lumber-chip (Class Example_4) Sept 28, 2012Wood Saba Vahid4 TB m3 30% Pine 70% Fir $38/m3 TB m3 50% Pine 50% Fir $40/m3 Mill Yard Pine Logs (m3) Fir Logs (m3) Pine lumber (MBF) $245/MBF Fir lumber (MBF) $280/MBF Mill Chips (bdu) $43/bdu Change the cutting cost of TS2 to $80/m3 Lumber & Chip LP

Example: lumber-chip conversion What should the cutting cost be before we start harvesting from TS2? New cost = old cost – reduced cost New cost = (-5)= $ -75 What would be the value of an additional 1000 m3 of logs from TS1? Value of the extra resource = shadow price* units of change value of extra resource = 36 *1000 = $36,000 Sept 28, 2012Wood Saba Vahid5

Material Balance Constraints balance the input-outputs of the model, normally include some positive and one or more negative coefficients Shadow price interpretation is different for material balance constraints –Adding 1 to the RHS actually decreases the variables on the left hand side of the equation * CB * CB2 -1 * SED_10 = 0 (to balance the sed_10 logs) –Changing 0 to 1 → SED_10 variable has to be 1 unit smaller than the sum of 0.15*CB1 and 0.2*CB2 –In this case, shadow price gives the change in value of Obj fn if we have 1 less unit of SED_10 –For economic interpretation, we’ll have to reverse the sign of the shadow price for material balance constraints Sept 28, 2012Wood Saba Vahid6

Economic Interpretations of Sensitivity Analsyis Opportunity cost: a concept related to the value of a lost opportunity. What would have happened to our objective value if we had made a different choice? Can be determined using either shadow prices or reduced costs, depending on what opportunities are being considered: –What is the opportunity cost of not having enough of a resource? (what would happen if we could have access to more of that resource?) shadow price of the constraint related to the resource –What is the opportunity cost of choosing a different variable (e.g. a different cut block for harvest, what would happen if we picked one that is not currently selected?) Reduced cost of the related variable Sept 28, 2012Wood Saba Vahid7

Next Class Duality Theory 8Wood Saba VahidSept 28, 2012