Background:

You are the supervisor of a National Forest district in the Pacific Northwest. Your task is to develop a long$-$term management plan for this district. Your general objective is to maintain a diverse forest landscape in order to protect the forest against epidemics and fires, to provide habitat for varied wildlife, and for aesthetic reasons. However, you must report on the economic implications of any management plan you suggest.

The property consists of a mosaic of even$-$aged stands of Douglas$-$fir, of which a large part is old growth. The stands are classified in age classes of $20$ years each. When part of a stand is cut, reforestation follows immediately by planting whatever area was cut with one$-$year$-$old seedlings. The price of timber is $$100/$Mbf$,$ the cost of reforestation is $$200$ per acre, and the interest rate is $3$ percent per year.

The yield per acre of a stand, Yi$,$ depends only on its age$-$class, i$,$ as shown in column B$.$ The initial forest state is shown column C$.$ At the beginning of period $1,$ there are $500$ acres in age class $1($trees aged $1$ to $20$ years$),500$ acres in age class $2,$ and so on$,$ up to $4000$ acres in age class $8($trees $141$ years old and older; there is no upper age limit for trees in age class $8).$ You will design a management plan over the next $100$ years $($p $=5$ periods of $20$ years each$).$ This plan is defined by the area cut in each age class in each period $($Xij$).$ To begin, you plan on cutting $25\%$ of the total forest area in each $20-$year period, preferentially cutting from the oldest age classes. You will then explore other management plans that depend on the objectives you set.

Maximize Landscape Diversity

Task $4$

Develop a plan that maximizes landscape diversity in the sense that it maximizes the smallest area in any age class and period $($Amin$).$ You are no longer required to cut $25\%$ of the forest each period. The plan should also end with a steady$-$state $($sustainable$)$ forest.

$$ Create a new worksheet by copying the current worksheet. Name it $$Landscape Diversity$.$

$$ In cell A$16,$ write $$Minimum acres per age class:$$ Your objective is to use the Solver to find the plan that maximizes Amin, subject to constraints imposed by the real$-$world context of the problem, as indicated below.

$$ Indicate the decision variables in the solver. Remember you will be asking the solver to change both the cut $($Xij$)$ as well as the minimum acres per age class.

$$ What constraints do you need for this? Enter them into the Solver. Questions to think about:

o What is the range of feasible values for the area you can cut?

o What is the minimum value you can stock, given your objective $$ how does the stock table relate to your objective?

$$ To ensure that the forest at the end of the plan is sustainable, specify that the state at the beginning of period $5$ must be the same as that at the beginning of period $6.$ Add this steady$-$state constraint to the Solver.

All told, you will have four classes of constraints.

Note: The solver may not find a solution with the standard level of precision. If that is the case, decrease the Solver precision to $0\mathrm{.}01.$

Copy the resulting cut and stock tables into your lab report and justify the objective, variable cells, and constraints that you set in the Solver. What is the minimum acreage value that you achieved? Should everyone in class get the same solution to this problem? Why or why not?

Maximize Economic Forest Value

Task $5$

Develop a plan that maximizes economic forest value. You can cut as much as you like from any age class as long as it still leads to a $$sustainable$($steady$-$state cutting plan$)$ forest.

$$ Copy and move the current spreadsheet. Call it $$Max Economic Value.$$

$$ Change the objective function to maximize forest value.

What is your forest worth in this framing? How does this plan compare with the one that maximized landscape diversity? What is the opportunity cost of maximizing landscape diversity? Put your answers in your lab report.

Compromise Plan

Task $6$

Observing that maximizing landscape diversity has a high opportunity cost, you decide to develop a compromise plan. This will still seek high landscape diversity by maximizing the smallest area in any age class and period. But the forest value must be at least half that of the maximum forest value obtained above.

$$ Copy and move the current spreadsheet. Name it 'Compromise'.

$$ Change the objective function and constraints as needed to reflect the new objective.

How does this plan compare with the one that maximized diversity? With the one that maximized forest value? Are you satisfied that this is an acceptable compromise? Put your answer in your lab report.