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6.1 Forestry club fund raising. The school forestry club is planning the year's merrymaking, and three traditional fund raising activities are again considered.

1. Catering soda and bean feeds to fraternities and sororities at a net profit of $150 per party.

2. Selling delivered cords of oak firewood from the school forest to university faculty at a net profit of $100 per cord.

3. Selling Christmas trees grown on the school forest at a net profit of $5 per tree.

The club wants to make as much net income for the year as possible. An inventory of its resources shows some limits to what they can do. Revenues are delayed, so the operating cost of all activities has to be paid from the club's $1500 treasury balance left from last year. A willing-worker survey of club members showed 20 hours of skilled bean cooks, 100 hours of skilled labor, and 500 hours of general purpose grunt labor to be available. The Forestry Department equipment manager said the club could use 500 miles of yellow cattle truck for free, but additional miles must be rented from the department at $0.50 per mile. Selling Sam, the club vice-president, got enthused at these ideas and arranged contracts for two soda and bean feeds and 150 Christmas trees before club members locked him up in an equipment locker for fear that their resources might be inadequate and the club might be sued for breach of contract. The forest economics class analyzed the production functions and found the following inputs were required per unit output:

1.Soda and bean feeds: $75 capital, 5 hours cooks' labor

2.Firewood: $30 capital, 5 hours skilled labor, 10 hours grunt labor, 50 miles of truck use

3.Christmas trees: $2 capital, 0.2 hour skilled labor, 1 hour grunt labor, 2 miles of truck use

Your task is to help the forestry club by

(a) Defining the decision variables.

(b) Presenting the problem as either a set of equations or a detached coefficient matrix, labeling each row or equation. (Hint: The truck rental activity takes the most thought).

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6.2 Land use allocation. A 100-acre management unit of public wildland spans a small valley and is distributed into three types as follows:

1. North slopes, 700 acres

2. South slopes, 210 acres

3. Creek bottom, 90 acres

Acres in each of these tracts can be devoted to timber production, forage, or set aside for recreational purposes. The forest ranger wants to decide how to best allocate his land between the three uses. The decision variables are defined as follows:

Prices for products and services produced from this tract are as follows:

1. Timber, $200 per MBF.

2. Forage, $5 per animal unit month (AUM).

3. Recreation, $1 per recreation visitor day (RVD) on the slopes; $3 per RVD in the creek bottom.

The total available budget is $5000 per year. The average annual yields and costs per acre for each land use and land type possibility are as follows:

Formulate and write linear equations to reflect the following:

(a) A contract has been given to the local sawmill to supply a minimum of 50,000 bd.ft. of logs per year.

(b) Sam Grazemore has a historical permit to use the two slope tracts for a total of 100 AUM per year.

(c) The budget constraint.

(d) The objective function is to maximize annual net revenue.

(e) For legally mandated aesthetic and ecological balance reasons, the government requires at least 20 percent of the total land area to be in each land use.

(f) For every acre allocated to recreation, at least 10 acres should be allocated to either forest or range to meet the government's new economic goals to produce more dollar outputs from public land.

(g) Because this is cattle country, a county ordinance states that for every 1000 bd.ft. sold, at least 20 AUM must be offered for sale.

(h) Is this problem feasible if constraints (a) to (g) are all applicable? Explain.

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6.3 Habitat suitability index problem. A forest in the pine flatwoods of Florida consists of four timber stands and the owners wanted to plan their harvest over the next three 10-year period using linear programming. The 12 decision variables were defined as acres cut by stand by period:

	Period thinned
Stand	1	2	3	Total acres
A	X11	X12	X13	190
B	X21	X22	X23	200
C	X31	X32	X33	185
D	X41	X42	X43	210

The owners wanted to maximize the total habitat suitability of the forest for the gopher tortoise (Gopherus polyphemus); in addition, they are interested in the timber volume goals yet have no specific goal level. They will only use thinnings, and hope to thin each stand once over the next thirty years. After doing some computations, you have determined that the average habitat suitability for each stand, using the appropriate thinning timing and thinning prescription is

The potential thinning volumes (cords/acre) are

You also decide that there should be at least some harvest volume in each time period, and that each stand, when cut, must be cut entirely within a single planning period.

(a) What is the optimum management schedule for the owners?

(b) What is the average habitat suitability for the management plan?

(c) How much wood will they harvest over the three periods?

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6.4 Elk habitat. Reformulate the elk habitat problem (pages 278-281) as follows: the wildlife manager is unsatisfied with the distribution of cover and forage that resulted from the previous formulation, especially in terms of ensuring that the best cover (oldest stands) is located close to the best forage (recently cutover stands). She decides to drop the cost minimization objective function and substitute one that maximizes the age differences between management units located next to (adjacent to) one another. Also, she wishes to extend the planning horizon from two to three periods to enable a further look into the future. Reformulate the elk habitat problem to incorporate this new objective function and the longer planning horizon.