Linear Programing For Forest Management

Question

The question I am working on is as follows,

You are in charge of managing a $200,000$-acre forest. Of these $200,000$ acres, $80,000$ can only grow pine and $40,000$ can only grow aspen. The remaining $80,000$ acres can grow either pine or aspen, or a mixture of both. The Land Expectation Value (LEV) for pine stands in the forest is $90$ Dollars per ac, for aspen stands the LEV is $50$ Dollars per ac, and for mixed stands it is $70$ Dollars per ac. You have three critical species in the forest, American Marten, Broadwing Hawk, and Three-toed Woodpecker. Your habitat conservation plan requires you to maintain habitat for $500$ American Marten breeding pairs, $600$ Broadwing Hawk breeding pair, and $450$ Three-toed Woodpecker breeding pairs. American Marten is found only on the pine and mixed forest types at an average density of $5$ breeding pairs per $1,000$ acres in pine and $2$ breeding pairs per $1,000$ acres in mixed forest. Broadwing Hawk is found at average densities of $1$ breeding pair per $1,000$ acres of pine, $3$ breeding pairs per $1,000$ acres in mixed forest, and $6$ breeding pairs per $1,000$ acres in aspen. Three-toed Woodpeckers are found at average densities of $2$ breeding pairs per $1,000$ acres of pine, $4$ breeding pairs per $1,000$ acres in mixed forest, and $1$ breeding pair per $1,000$ acres in aspen. In addition to meeting these three habitat goals, you would like to maximize the LEV of the forest.

I am having trouble with a certain part of the problem. It states that you can have a $200,000$ acre forest that has $80,000$ dedicated to pine and $40,000$ dedicated to aspen. There is the remaining $80,000$ that can be mixed or either pine or Aspen. I have written the $z$ function like this,

Link to all the data from problem organized for ease

$$Max(Z) = P(\text{acres pine}) * 90 + A(\text{acres aspen}) * 50 + M(\text{acres mixed}) * 70$$

with constraints that I have written so far,

$$0.005*P + 0.002*A + 0*M = 500$$ (marten)

$$0.001*P + 0.003*A + 0.006*M = 600$$ (Hawk)

$$0.002*P + 0.003*A + .001*M = 450$$ (WoodPecker)

$$P >= 80,000$$

$$A >= 40,000$$

$$P >= 0$$

$$A >= 0$$

$$M >= 0$$

Where I'm getting confused is with the remaining $80,000$ acres that can be mixed, pine, or aspen. I don't know how to write it out so $M$ is either $0$ or $80,000$. I may be approaching this problem incorrectly, I'm just learning how to do linear programming. Any help would be greatly appreciated.

Answer 1

That looks like a good attempt. I would suggest re-defining your variables to account for the mixed area (the area that could hold a mixture of both types of tree):

Let there be three areas:

Area 1 (Only Pine), total of 80k Acres Area 2 (Only Aspen), total of 40k Acres Area 3 (Pine/Aspen/Mixed), total of 80k Acres

Then I would define the variables to be:

American Marten can only grow in Pine & Mixed (i.e. areas 1 & 3)

Let Xi be the number in thousands of acres of American Marten (Where i is in { 1 = Area 1, 3 = Area 3})

Broadwing Hawk can grow in either area (i.e. areas 1, 2 & 3)

Let Yi be the number in thousands of acres of Broadwing Hawk (Where i is in { 1 = Area 1, 2 = Area 2, 3 = Area 3})

Three-toed Woodpeckers can grow in either area (i.e. areas 1, 2 & 3)

Let Zi be the number in thousands of acres of Three-toed Woodpeckers (Where i is in { 1 = Area 1, 2 = Area 2, 3 = Area 3})

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Once the variables are defined, the constraints become:

Total Acres (In Thausands)

Constrain the total number of Acres (in thousands) of Pine

X1 + Y1 + Z1 <= 80

Constrain the total number of Acres (in thousands) of Apen

Y2 + Z2 <= 40

Constrain the total number of Acres (in thousands) of Mixed

X3 + Y3 + Z3 <= 80

Breeding Pairs

Constrain the number of American Marten breeding pairs

5*X1 + 2*X3 >= 500

Constrain the number of Broadwing Hawk breeding pairs

Y1 + 6*Y2 + 3*Y3 >= 600

Constrain the number of Three-toed Woodpecker breeding pairs

2*Z1 + Z2 + 4*Z3 >= 450

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Finally the Objective function becomes:

MAX (X1 + Y1 + Z1)*90000 + (Y2 + Z2)*50000 + (X3 + Y3 + Z3)*70000

Note how the $/Acre is multiplied by 1000, since the variables are defined per 1000 Acres (and the pricing is per Acre).