Creating system of equations from word problem optimization

[Rifscape]

I have this word problem, and was wondering how I would go about creating a system of equations.

Here is the question:

Problem: You are a small forest landowner, and decide you want to sustainably harvest some of timber on your property. There are costs related to the infrastructure needed to harvest the lumber (permits, machinery, waste disposal, etc.), as well as with the wages you will need to pay the loggers. As a result, you will need to make trade-offs between the two when allocating your budget.

Objectives: You want to maximize the amount of timber you can sustainably harvest. Every logger that you hire is capable of harvesting 120 acres of sustainable woodland, given the minimal necessary equipment. However, for every additional 1000 you spend on infrastructure, a logger can harvest another 1 acre of timber.

Actions: You have the ability to decide how much money you invest in people (often referred to as a Full Time Equivalent or an FTE), and how much you invest in the infrastructure needed to harvest the timber. You do have the ability to hire a logger half time (0.5 of an FTE).

Constraints: • You are required to spend 100,000 to purchase all the permits, the basic equipment needed to harvest the timber, and the minimal number of loggers required by law.

• For every additional logger you hire, you need to invest 50,000 in infrastructure so that they can harvest their 120 acres. If you hire someone part time, then you pay the proportional amount in infrastructure.

• Without hiring additional loggers, you can only handle an additional 50,000 in infrastructure to improve the acreage harvested. In addition, each new logger you hire can only handle up to an additional 100,000 invested in infrastructure. If you hire someone part time, then they can only handle the proportional amount of infrastructure.

• Including insurance and benefits, it costs 100,000 to hire each full time logger. Hint: Each new logger can harvest 120 acres, plus handle between 50,000 to 100,000 worth of additional infrastructure (i.e., 50 to 100 acres of forest).

Problem 1: You have a 250,000 budget. Define a set of equations to define your four constraints and use them to take a graphical approach to determining a feasible set of solutions. What is the optimal way to allocate funds to maximize the timber harvested? How many acres could you harvest?

I know that one equation would be:

250000 >= 100000 + 100000L + 50000I

But what would be the other equations?

I was thinking one would be 50,000I + 50,000 <= 50,000I + 100,000L but I am not too sure.

Thank you for reading

 

[Ray Vickson]

I have this word problem, and was wondering how I would go about creating a system of equations.

Here is the question:

Problem: You are a small forest landowner, and decide you want to sustainably harvest some of timber on your property. There are costs related to the infrastructure needed to harvest the lumber (permits, machinery, waste disposal, etc.), as well as with the wages you will need to pay the loggers. As a result, you will need to make trade-offs between the two when allocating your budget.

Objectives: You want to maximize the amount of timber you can sustainably harvest. Every logger that you hire is capable of harvesting 120 acres of sustainable woodland, given the minimal necessary equipment. However, for every additional 1000 you spend on infrastructure, a logger can harvest another 1 acre of timber.

Actions: You have the ability to decide how much money you invest in people (often referred to as a Full Time Equivalent or an FTE), and how much you invest in the infrastructure needed to harvest the timber. You do have the ability to hire a logger half time (0.5 of an FTE).

Constraints: • You are required to spend 100,000 to purchase all the permits, the basic equipment needed to harvest the timber, and the minimal number of loggers required by law.

• For every additional logger you hire, you need to invest 50,000 in infrastructure so that they can harvest their 120 acres. If you hire someone part time, then you pay the proportional amount in infrastructure.

• Without hiring additional loggers, you can only handle an additional 50,000 in infrastructure to improve the acreage harvested. In addition, each new logger you hire can only handle up to an additional 100,000 invested in infrastructure. If you hire someone part time, then they can only handle the proportional amount of infrastructure.

• Including insurance and benefits, it costs 100,000 to hire each full time logger. Hint: Each new logger can harvest 120 acres, plus handle between 50,000 to 100,000 worth of additional infrastructure (i.e., 50 to 100 acres of forest).

Problem 1: You have a 250,000 budget. Define a set of equations to define your four constraints and use them to take a graphical approach to determining a feasible set of solutions. What is the optimal way to allocate funds to maximize the timber harvested? How many acres could you harvest?

I know that one equation would be:

250000 >= 100000 + 100000L + 50000I

But what would be the other equations?

I was thinking one would be 50,000I + 50,000 <= 50,000I + 100,000L but I am not too sure.

Thank you for reading

(1) You have not written equations---you have written inequalities. There is a huge difference, especially when it comes to solving a problem.
(2) Writing equations and/or inequalitie is useless without definitions of the variables (including their units). What is I? What is L?
(3) I don't know where your inequality 50,000I + 50,000 <= 50,000I + 100,000L comes from, but it can be written more simply as 2*L >= 1. Do you see why?

 

[Rifscape]

My apologies.

The Equation should be 250,000 = 100,000 + 100,000L + 50,000I
Where I = Infrastructure,and L = number of loggers.

I do not think that 3rd equation is correct.

There should be 4 equations; one for each constraint

 

[Ray Vickson]

My apologies.

The Equation should be 250,000 = 100,000 + 100,000L + 50,000I
Where I = Infrastructure,and L = number of loggers.

I do not think that 3rd equation is correct.

There should be 4 equations; one for each constraint

When you say "I = infrastructure", what does that mean? Number of units of infrastructure? Amount of money spend on infrastructure? Something else?

Again: you are probably looking for inequalities, not equations!

When I used to teach this stuff (in a graduate-level Operations Research program) I would emphasize a number of points:

(1) Give sensible, easily-interpreted names to variables---not just $x_1, x_2$ of $x, y$.

So, if the problem involves the number of chairs and desks to produce, a sensible terminology such as C and D can help a lot; it is even permissible to spell them out, so one could use Chair and Desk as names of variables.

Your use of "I" for infrastructure and "L" for loggers is good, so you did well on that score.

(2) Define the variables, including their units of measurements. In your case, is "I" a number of units or a monetary amount?

If your I is a monetary amount It might be better to express it in units of $100,000 and let I = expenses (in $000,000). That makes all the numbers smaller, and can help with accuracy when seeking a numerical solution. (Believe it or not, a formulation having mixtures of many different orders of magnitude coefficients and right-hand-sides can seriously degrade the ability to obtain an accurate solution. Some sophisticated optimization software will even have "scaling" options that allow the program itself to re-write the input before submitting it to a solver.)

(3) You speak of the option of hiring 1/2-time loggers, and that raises the following issues: (i) do we just allow L to have fractional values, so that L = 4.63 is allowed? or (ii) must L be either an integer or a half-integer, so that L = 4 or L = 4.5 are OK, but not L = 4.63? In case (ii) it may be better to have two types of "logger" variables: Lf = number of full-time loggers to hire, and Lh = number of half-time loggers to hire. Now your problem may become a so-called "mixed-integer" programming, where we require that Lf and Lh be non-negative integers.

This method can raise some difficulties: is it OK to have Lh=2 and Lf=0 instead of Lf=1 and Lh=0? If so, you might as well just eliminate the variable Lf and work with Lh instead. However, if that is not "allowed", you need to try to ensure that the maximum allowed full-time loggers is used before you start using any additional half-time loggers. If half-time loggers are more expensive (so Lh=2 costs more than Lf=1) that will come out automatically in the solution. However, if their costs are really the same, you can proceed in one of two ways: (i) artificially inflate the cost of part-time loggers, at least by a little bit, then take off the artificial premium when actually evaluating the profit figure; or (ii) introduce a constraint that says you are allowed to have Lh > 0 only when Lf is at its maximum allowed value. (That can be handled by introducing additional binary variables, but it makes the problem harder to solve. Nevertheless, such techniques are done in real-world industrial modelling situations.)

 

[Rifscape]

When you say "I = infrastructure", what does that mean? Number of units of infrastructure? Amount of money spend on infrastructure? Something else?

Again: you are probably looking for inequalities, not equations!

When I used to teach this stuff (in a graduate-level Operations Research program) I would emphasize a number of points:

(1) Give sensible, easily-interpreted names to variables---not just $x_1, x_2$ of $x, y$.

So, if the problem involves the number of chairs and desks to produce, a sensible terminology such as C and D can help a lot; it is even permissible to spell them out, so one could use Chair and Desk as names of variables.

Your use of "I" for infrastructure and "L" for loggers is good, so you did well on that score.

(2) Define the variables, including their units of measurements. In your case, is "I" a number of units or a monetary amount?

If your I is a monetary amount It might be better to express it in units of $100,000 and let I = expenses (in $000,000). That makes all the numbers smaller, and can help with accuracy when seeking a numerical solution. (Believe it or not, a formulation having mixtures of many different orders of magnitude coefficients and right-hand-sides can seriously degrade the ability to obtain an accurate solution. Some sophisticated optimization software will even have "scaling" options that allow the program itself to re-write the input before submitting it to a solver.)

(3) You speak of the option of hiring 1/2-time loggers, and that raises the following issues: (i) do we just allow L to have fractional values, so that L = 4.63 is allowed? or (ii) must L be either an integer or a half-integer, so that L = 4 or L = 4.5 are OK, but not L = 4.63? In case (ii) it may be better to have two types of "logger" variables: Lf = number of full-time loggers to hire, and Lh = number of half-time loggers to hire. Now your problem may become a so-called "mixed-integer" programming, where we require that Lf and Lh be non-negative integers.

This method can raise some difficulties: is it OK to have Lh=2 and Lf=0 instead of Lf=1 and Lh=0? If so, you might as well just eliminate the variable Lf and work with Lh instead. However, if that is not "allowed", you need to try to ensure that the maximum allowed full-time loggers is used before you start using any additional half-time loggers. If half-time loggers are more expensive (so Lh=2 costs more than Lf=1) that will come out automatically in the solution. However, if their costs are really the same, you can proceed in one of two ways: (i) artificially inflate the cost of part-time loggers, at least by a little bit, then take off the artificial premium when actually evaluating the profit figure; or (ii) introduce a constraint that says you are allowed to have Lh > 0 only when Lf is at its maximum allowed value. (That can be handled by introducing additional binary variables, but it makes the problem harder to solve. Nevertheless, such techniques are done in real-world industrial modelling situations.)

You are right that this question is very vague.

After some more deliberation, I was able to discover that there should be four equations; one for each constraint.

Based on what my professor said, I think the first equation is:
y = costs
1. y = 100,000 since this is a fixed value
2. y = 50,000*FTE - for each full time employee, there can only be either 1 or 0.5 employees. Not any other decimals
3. ? I am unsure on how to create an equation for this
4. y = 100,000*FTE I am not sure this is correct.

Your point on standardizing the scale is very helpful. I still haven't found the answer yet but I think I am close.

If you have any other advice on how to figure out the other equations please let me know.

Thank you for your help so far