1. WOOD 492 MODELLING FOR DECISION SUPPORT Lecture 11 Sensitivity Analysis

2. Lab 3 review Sept 28, 2012Wood 492 - Saba Vahid2 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 review

3. 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 492 - Saba Vahid3

4. Example: lumber-chip (Class Example_4) Sept 28, 2012Wood 492 - Saba Vahid4 TB 1 5000 m3 30% Pine 70% Fir $38/m3 TB2 8000 m3 50% Pine 50% Fir $40/m3 Mill Yard Pine Logs (m3) Fir Logs (m3) 0.275 0.235 Pine lumber (MBF) $245/MBF Fir lumber (MBF) $280/MBF Mill Chips (bdu) $43/bdu 0.76 0.78 Change the cutting cost of TS2 to $80/m3 Lumber & Chip LP

5. Example: lumber-chip conversion What should the cutting cost be before we start harvesting from TS2? New cost = old cost – reduced cost New cost = -80 - (-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 492 - Saba Vahid5

6. 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. 0.15 * CB1 + 0.2 * 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 492 - Saba Vahid6

7. 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 492 - Saba Vahid7

8. Next Class Duality Theory 8Wood 492 - Saba VahidSept 28, 2012