Chapter 11 Questions and Answers
______________________________________________________________________________________________
11.1 Sarah Shope Problem. Sarah Shope, the owner of a 120-acre tract in Georgia, contacted a consulting forester and asked him to prepare a harvest schedule that would maximize the total volume of harvested timber from her property over the next 4 decades subject to certain constraints. A survey of the forest provided the data on the next page. (All land is the same site quality.) For analysis purposes, there are four planning periods of 10 years each. All harvests and cultural treatments are at the midpoint of each decade.
A close discussion revealed that the following constraints were important to Sarah:
1. Harvest volume control: the harvest in a decade cannot deviate by more than 20 percent from the harvest in the preceding decade.
2. The inventory of the merchantable standing timber after the fourth-decade harvest must be at least 1600 MBF.
Stand |
Condition |
Area (acres) |
Age (years) |
Volume per acre (MBF) |
|---|---|---|---|---|
| A | Poorly stocked, diseased loblolly | 40 | 35 | 8 |
| B | Fully stocked, vigorous loblolly | 80 | 55 | 32 |
3. After existing stands have been regeneration harvested, they are immediately planted and established. All regenerated stands are given the same prescription of plant, a thinning harvest at age 20, and a final harvest at age 30.
4. A 20-acre park in stand B is to be set aside uncut as a possible future homesite.
The consultant then evaluated the mid-decade yield of existing and future stands in MBF:
| Existing stands | Future stands | ||||||
|---|---|---|---|---|---|---|---|
| Decade cut |
Stand A |
Stand B |
Decades of growth |
Age when cut |
Inventory before cut |
Inventory after cut |
Volume cut |
| 1 | 10 | 40 | 0 | 0 | 0 | 0 | 0 |
| 2 | 13 | 50 | 1 | 10 | 0 | 0 | 0 |
| 3 | 15 | 58 | 2 | 20 | 12 | 7 | 5 |
| 4 | 16 | 65 | 3 | 30 | 30 | 0 | 30 |
With this information the consultant formulated the problem in a model I structure (see page 640).
Questions for consultant's formulation and analysis:
(a) The consultant provided the linear programming solution printout (see pages 641-642). Explain to her what each of the six numbered items means.
(b) Answer the following five questions using the solution as your reference.
1. Why does the dual price equal zero for the park constraint? Doesn't it seem reasonable that this should be a binding constraint?
2. If the ending inventory was increased from 1600 to 2000 MBF, what is the estimated opportunity cost in terms of total timber harvest?
3. How can the constraints on H 4 (rows 11 and 12) be satisfied when the decision variables A 4 and B 4 have a zero value?
4. Is this a model I or a model II formulation, and exactly how do you know for sure which it is?
5. Suppose new wildlife and visual concerns required that each stand be represented in the total harvest for the first period in proportion to how much of the total area is occupied by the stand with a maximum deviation of +/-10 percent. Write constraint equations for the problem to ensure that this condition is satisfied.
(c) Recast the problem in a model II structure. Define decision variables and present a complete detached coefficient matrix.
______________________________________________________________________________________________
11.2 Idaho Stud Problem. The Idaho Stud Company of Coeur D'Alene recently purchased 10,000 acres of fir and larch stands which it planned to manage as a unit. The tracts purchased had three stand types, each with different site quality.
Stand |
Condition |
Area (acres) |
Site |
Current volume per acre (MBF) |
|---|---|---|---|---|
| A | Young plantation | 2000 | I | 4 |
| B | Poorly stocked second growth | 5000 | II | 12 |
| C | Poorly stocked second growth with old growth residuals | 3000 | III | 31 |
Corporate management wants a harvest plan for 5 decades that tells how many acres should be harvested in each stand and the total volume harvested in each decade. For planning purposes, all harvests are assumed to occur at the mid-decade times of 5, 15, 25, 35, and 45 years from now.
The forestry staff reviewed their stands, species, and the economic situation and selected three possible prescriptions for existing stands and two prescriptions for regenerated stands.
Existing Stands
1. Clearcut and plant in periods 1, 2, or 3.
2. Commercially thin in period 1. Clearcut and plant in period 3.
3. In period 1, permanently assign the land to uneven-aged management.
(Note that prescription 2 cannot be applied to stand A because of insufficient volume.)
Regenerated Stands
1. Regeneration harvest at age 20.
2. Regeneration harvest at age 30.
These prescription options along with detailed plot data describing the average composition of the three stand types were evaluated in a yield simulator to produce yield forecasts for the three stands.
Yields (MBF)
| Years from now | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| Existing stands | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 |
| Even-aged prescriptions | |||||||||
| Unthinned stand A | 6 | 10 | 14 | 22 | 29 | 37 | 44 | 48 | |
| Thinned stand B (thin 4 MBF at year 5) | 11 | 15 | 20 | 24 | 28 | 33 | 36 | 38 | |
| Unthinned stand B | 15 | 19 | 24 | 27 | 29 | 33 | 36 | 38 | |
| Thinned stand C (thin 10 MBF at year 5) | 24 | 27 | 30 | 34 | 36 | 38 | 39 | 40 | |
| Unthinned stand C | 34 | 36 | 38 | 39 | 40 | 40 | 41 | 41 | |
| Uneven-aged prescriptions | |||||||||
| Stand A - Harvest | 0 | 4 | 8 | 12 | 12 | ||||
| Stand A - RGS | 6 | 10 | 12 | 12 | 12 | ||||
| Stand B - Harvest | 5 | 8 | 9 | 10 | 10 | ||||
| Stand B - RGS | 10 | 13 | 13 | 14 | 14 | ||||
| Stand C - Harvest | 10 | 8 | 6 | 6 | 6 | ||||
| Stand C - RGS | 24 | 22 | 20 | 18 | 18 | ||||
| Years from birth | |||||||||
| Regenerated stands | 0 | 10 | 20 | 30 | 40 | ||||
| Site I (stand A) | 0 | 5 | 10 | 27 | 40 | ||||
| Site II (stand B) | 0 | 2 | 7 | 19 | 28 | ||||
| Site III (stand C) | 0 | 0 | 3 | 9 | 18 | ||||
Management constraints. At the end of 10 years, management will reconsider the plan. Until then, the following goals and constraints are to be used to guide the harvest plan over the next 5 decades.
1. The driving goal is to maximize harvest in the first period.
2. At least 20,000 MBF must be harvested in each period.
3. A minimum of 2000 acres must be in uneven-aged management for stream buffers and protection of visually sensitive zones.
4. A maximum of 5000 acres is permitted in uneven-aged management because company foresters do not believe it to be as productive as even-aged management.
5. Stand C is currently a good habitat for the rare and endangered Idaho bearcat, and by law some acreage must be left uncut for at least two periods while the researchers figure out what to do. Therefore a maximum of 1000 acres can be clearcut harvested in each of the first two periods.
6. As an extra guarantee against overcutting, at least 10,000 MBF of residual inventory after the fifth period harvest in 45 years is required.
Questions
(a) Make a list defining a set of model I decision variables for this problem.
(b) Write a set of equations to present the objective function and constraints of this problem in a model I formulation.
(c) If possible, solve the problem and evaluate the solution and harvest schedule for corporate management.
Excel spreadsheet associated with Problem 11.2
______________________________________________________________________________________________
11.3 Financial analysis of fertilization. Using the data of table 11.16, do a financial analysis on a per acre basis to determine the rate of return from the $200 investment in fertilizing stand types B and C. Is this result consistent with the results in table 11.17 when we analyze the whole forest under different policies, given the assumption that all type B and C land is fertilized to achieve type A productivity?
Answers to Problem 11.3This page is kindly hosted by the Warnell School of Forest Resources, University of Georgia
The content and opinions expressed on this Web page do not necessarily reflect the views of nor are
they endorsed by the University of Georgia or the University System of Georgia.