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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.

lonza leggiera
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joshua clark
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  • Why do you think that M has to be either 0 or 80,000? Does the statement of the problem allow for the 80,000 acres to have 20,000 acreas of pine, 20,000 acres of aspen and 40,000 acres of mixed? – Brian Borchers May 21 '19 at 04:06
  • The way I read the problem It has to be all pine, all aspen, or all of the mixed category. Since there is an LEV value for Pine, Aspen, and Mixed and the pine and aspen are listed as having to be the full 80,000 or nothing Im assuming that the Mixed category would be the same. – joshua clark May 21 '19 at 04:15
  • Linear programming can't express that set of constraints, but you can achieve the desired effect using integer linear programming. – Brian Borchers May 21 '19 at 04:18
  • Thank you for your input. I May be viewing the problem in the wrong way. I will move forward and see what I can do. – joshua clark May 21 '19 at 04:20

1 Answers1

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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})

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

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).

George
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