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


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