WOOD 492 MODELLING FOR DECISION SUPPORT

1. WOOD 492 MODELLING FOR DECISION SUPPORT Lecture 13 Duality Theory

2. The Dual Problem • Another way of looking at a linear program • Has the same optimal value for the objective function as the primal model • The shadow prices of the dual model, are the optimal values of the decision variables in the primal model and vice versa • Extremely helpful when we can solve the dual model but not the primal one (e.g. large primal model which are computationally intensive) Wood 492 - Saba Vahid

3. Re-visiting the last example Wood 492 - Saba Vahid

4. The constraints • What does this constraint mean? 12 Y1 + 10 Y2 + 20 Y3 =>550 • Y1 is the marginal value of resource 1 (moulding) when the optimum profits are being produced • Y2 and Y3 are defined similarly • (12 Y1 + 10 Y2 + 20 Y3) = total marginal value of producing 1 unit of product 1 • $550 = marginal profit of 1 unit of product 1 • If the total value of resources used and the total profits are the same, it is worth producing product 1, other wise it’s not worth it Wood 492 - Saba Vahid

5. Y1 Y2 Y3 16 Y2 + 20 Y3 is valued at $475 (with the current valuation as shown in the answer row) Product 3 produces $350 profits, so it doesn’t make sense to use these resources for making product 3 (for example, you are better off selling these resources in a “market” at their current value) Wood 492 - Saba Vahid

6. Shadow prices and reduced costs • The results of the dual model are the shadow prices for the primal model and vice versa • Shadow prices show a form of internal pricing/valuation for the resources at the optimal level: • How much is a resource worth for producing the optimal result • E.g. an hour of labor is worth $23.75 to the firm • Reduced cost is calculated from the dual model as: Reduced cost = RHS (in the dual problem) – actual constraint value e.g. reduced cost of product 3 = 350 – 475 = -125 • So the actual value of the constraint (e.g. 475) shows the price that will make it feasible to produce product 3 Wood 492 - Saba Vahid

7. Dual-primal relationship Primal (Maximize) Dual (Minimize) ith constraint <= ith variable => 0 ith constraint => ith variable <= 0 ith constraint = ith variable unrestricted j th variable =>0 j th constraint => j th variable <= 0 j th constraint <= j th variable unrestricted j th constraint = ith constraint is binding ith variable is non-zero jth variable is non-zero jth constraint is binding Wood 492 - Saba Vahid

8. Lab 4 preview • Harvest scheduling • 3 stands with different ages and species • Different harvest regimes (based on the period of first cut, first cut can happen in any period*) * Note this is different from what we discussed in the class. There is now no restrictions on when the first cut can happen • Considering the total forest inventory (standing tree volume) • Even-flow restrictions • Beginning and ending inventory limits • Growth and yield volumes (how much trees grow in each period) • Our objective is to maximize the total harvest (over 8 decades) • Our decision variable is the total area (ha) in each stand to manage with each harvest regime • e.g. how many ha of stand 1 should we harvest in the first period (S1_P1) • e.g. how many ha of stand 3 should we leave uncut (S3_uncut) Wood 492 - Saba Vahid

9. Data tables • The first table is related to existing (natural) stands • Shows the available timber volume in the stands in each decade (decade 1 shows the current age of the stands) • The second table is related to regenerated stands (natural stands after being harvested once) • Shows the available timber volume in the stands after different periods of growth Wood 492 - Saba Vahid

10. Data tables • Yield tables show the harvest volumes per ha in each decade, depending on the period in which the first cut happens (the column headings correspond to our decision variables) • e.g. for stand 1, if we choose to enter in the first period (first column), we will harvest 490 m3/ha (value extracted from the natural stands table), and then we will have to wait three decades before being able to harvest again, in which case we will get 364 m3/ha (value extracted from the regenerated stands table) Fill the rest of the data table Wood 492 - Saba Vahid

11. Data tables • Ending inventory tables, show the remaining volume of timber in the stands (m3/ha) at the end of each period, depending on when the first cut happens • E.g. when we enter stand 1 in the first period, the remaining volume at the end of decade 1 is “0” because we have just harvested it. At the end of decade2 and 3, there is a growing inventory of standing timber (42 and 224 m3/ha, values extracted from the regenerated stands table), and since we harvest again in decade 4, the inventory drops to “0” again. Fill the rest of the data table Lab 4 matrix Wood 492 - Saba Vahid

12. Next Class • Economic interpretations of duality Wood 492 - Saba Vahid