There is a large chunk of information necessary as a preface to my question, so bare with me for a paragraph or two. I work for a pond treatment company. We have a set number of ponds we treat during a month, some are contracted to be treated once a month, some are treated twice. The question is simple "How do I maximize profit during a given month of treatment?" Seems simple on the surface, but I am having a lot of trouble with it.(adsbygoogle = window.adsbygoogle || []).push({});

As of now, we have two hourly workers in two separate trucks treating 5 days a week. My boss has three "treatment routes" setup for us to use. Each route consists of a specific set of locations across the city. The first route's members are all those ponds that are under contract to be treated once a month; we can call this set A. The second route's members are all those ponds that are under contract to be treated twice a month, excluding a specific set C. We can call the set associated with the second route B. The last route is precisely the specific set C mentioned before. The routes have been generated using a program that finds the most optimal route between any collection of addresses, such that one of the addresses is used as both the origin and the termination of the route (this address is the address of the company office/shop. This is where we start our treatments everyday). The problem here is that we never finish routes A or B in a single day, and only occasionally do we finish route C in a single day. As a result, we have to make a trip back to the shop (we can call this the origin I suppose) at the end of the day, and then the next day make the trip back to the point on the route at which we stopped the day before. Intuitively, this seems inefficient.

The bottom line is to optimize the amount of money made in a month considering only cost of driving, cost of workers, and income from pond treatments.

All of the locations on a given route are actually sets themselves. Each of these subsets represents a property. Each property is billed for the treatment of their pond/ponds monthly. The amount billed to each property remains constant through out the season. Each property can be billed for treatment only after all members of this subset have been treated as contracted. These subsets will contain at most 8 members (our largest property has 8 ponds). A day of treatment will never be ended in the middle of one of these subsets. Daily routes will only ever be terminated at a given member (pond) if all other members of that subset (property) have been visited.

So, all members of A must be visited once in a given month. All members of B must be visited twice a month. The reason that there exists a set C is because C is technically a single property that cannot be billed till all the members of C have been visited twice. However, there are over 30 ponds in set C, so my boss wants it separate because the completion of treatments for all members of set C brings in a large sum. This only occurs once every member of C has been visited twice for a given month (since C is contracted for treatments twice a month). We see no money from C until this is completed, regardless of weather we have treated some of its subsets already.

A couple values that I am sure will come into play (some will be constants, some will be variable functions):

Wage of worker A

Wage of worker B

Average miles per gallon of truck A

Average miles per gallon of truck B

Derivative of ponds treated with respect to time

Distance between endpoint of a given day to the origin for both workers

Time worked in a given day

Price of gas

My real question here is "How do I even start this problem?" I have a vague notion of how all these pieces fit together but I can't seem to get beyond that. Take the price of gas for example. That changes daily, and not in a well defined manner either. Same goes for ponds treated per day. Some days we move quickly, some days we get stuck spending the better half of that day at one pond. Mileage of the vehicles is variant as well (although I couldn't tell you exactly how it varies). Need I use highway mpg during parts of the route involving the highway and city mpg for the rest? Or will an average mpg suffice? Although there is a huge amount of variation possible in the setup I have given so far, many of the variables will have conditions imposed upon them (for example, time worked in a given day can't possibly exceed 24 hours, but will likely never exceed 12, and most frequently be [itex](8\pm1.5)[/itex] hours. How do I account for or model these types of 'probabilistic' variables?)

I've always been interested in mathematics, so if concepts beyond calculus arise as a result of what I have explained above, do not be afraid to incorporate them in your response. I know this is a complicated question, but I also know that there must be some way to encapsulate all of these numbers and relations into a mathematical model. Any direction will help. I don't need you to solve this, I just need to know how to solve this. Thanks in advance.

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