A Production Problem: Shutting down a Solvents Factory

An Interactive Applet powered by Sage and MathJax.

(By Prof. Gregory V. Bard.)


You are a high-level manager in a chemical company, and your firm has recently bought out a major rival. Most of the factories of the rival will be kept open, but there's a small plant that is out-of-date and redundant, and moreover has a poor environmental track record. That factory, in Smithville, will be closed.

The Smithville factory makes four different paint-thinner products, each of which has some generic, benign, standard ingredients (a chemist would say reactants or reagents) but each of which uses four highly toxic solvents. What is worse, is that the plant has a large stockpile of these toxic solvents---because they were not using lean manufacturing.

Your goal is to schedule one last (huge) production run, that will use up as much of the highly toxic solvents as possible, making useful products for sale. There are companies that professionally do toxic waste disposal, but the fees are enormous and should be minimized.


There are three things that you need to know in order to solve this problem. You need to know what supplies are on hand (the solvents). You need to know how much (of each Solvent) that each Product requires. Then you need to know how the disposal fees are going to be calculated.

First, there's the question of what supplies are already on hand:

The disposal company that your conglomerate has contracted with (for cleaning up Smithville) is going to charge 1000 dollars per liter of these highly toxic solvents.


First, try to move the sliders around and find a plan that is feasible (in other words, one that does not use more supplies than actually exist) but that minimizes the fines. See how low you can go, based on intuition alone.

Second, try to see if you can find a way, using algebra, to make the fine zero dollars. If you get stuck, click the "Show the Math" check box.

Notes from a Mathematician's Perspective

This is a beautiful example of a simple problem where intuition and common sense will get you a very sub-optimal solution. When I move the sliders around, I get around 5 million dollars in fines. That is very different from zero dollars.

A minor "hello" must be offered to a branch of pure mathematics, which is rarely studied today but which was once very widely studied. I refer to Number Theory, which is all about problems dealing with integers, divisibility, prime numbers, and so forth. It turns out that linear systems of equations (like the above) where the coefficients are prime numbers, are very hard to solve by guesswork or mental arithmetic. This is true even for modestly sized prime numbers, as we saw above.

What is Lean Manufacturing?

The old-style of management for factories would involve huge stockpiles. Every possible supply would be stored and warehoused in enormous quantities, which had to be paid for, warehoused, and moved around. In the worst case, the stockpiles could be disorganized and clutter up the factory to the point of impeding operations. Basically, managers were paranoid about avoiding shortages which would stop production. They were biased in favor of having extra supplies (for a small cost) and biased against having too few supplies (which would result in not meeting a production quota).

"Lean manufacturing" is the exact opposite. You only keep in stock exactly what you need, no more and no less. This is actually far harder than it may seem. It requires one to know exactly how much of each supply you need for each stage of the process. Most steps in the process have other steps before them, and those have steps even before them! Modern manufacturing is a web of inter-connected stages. Also, the timing has to be right, because the different stages will occur at different times and deliveries of supplies can come late on occasion.

While the computational and organizational burden is disturbingly complex, the payoff in terms of efficiency and cost-effectiveness can be phenomenal. Because I am only writing a few paragraphs here, about a topic which normally would be an entire book, I am definitely leaving out many details. For example, eliminating excessive supplies and stockpiles is only one of several forms of waste targeted by Lean Manufacturing. Another would be errors in the production system, which render parts (manufactured during intermediate stages) to be rejected by quality control. Often these errors can be prevented or rarified with simple low-cost techniques.

You can read more about "Lean Manufacturing" in the Wikipedia article of the same name.

Last modified on December 11th, 2013.