Let's consider imaginary problem called up by the famous novel “Dune” (by Frank Herbert). A valuable substance often referred to as "the spice" which allows instantaneous travel to anywhere in the universe is found only on the desert planet Arrakis. We have the Emperor’s permission to start the spice output, moreover we have a monopoly; however there are 3 bad things. Firstly, the surface of the planet is of little use for construction and we have found only 1000 flat places for mining plants. Secondly, the planet is inhabited by the giant sandworms which easily destroy plants and the faster we’re extracting the spice the more aggressive they become. Our biologists found a formula for the probability of plant’s destruction during one day:
where ω is an extraction speed (kg/day), a changeable parameter, and Ω = 200 kg/day – operation speed of a plant (a fixed parameter for this kind of plants).
The average exploitation period of a plant (denote it T) is 10 days and the probability of breakdown grows with exploitation time. The formula was found by our engineers:
where t is the average age of a plant (days).
Broken and destroyed plants will be restored every night. We want to get the maximal profit and so we have to find the optimum extraction speed which helps us to extract the spice as much as possible and on the other hand to prevent destructions. For this calculation we will use Almagrid. First of all we should make out a term tree.
The price of the spice is 10$ per kg. It’s clear that every day we get revenue equal to number of plants multiplied by extraction speed (kg/day), multiplied by the price of the spice.
Our daily profit is the difference between revenue and expenses. The cost of a plant construction is 800$ so expenses are equal to the number of broken and destroyed plants multiplied by this cost.
Now let’s find the optimum speed. Firstly set ω = 150 kg/day. This is the iteration grid:
We can see that the system stabilizes after the 13th day (of course expenses also stabilize and become equal to 530603,1$ per day ), and we will have gotten 20365439,3$ by the 22 day.
If we try to set the speed ω = 250 kg/day, we will receive error report (because all plants will be destroyed). The closest to operation speed which doesn’t give us error is ω = 179 kg/day. Look at the iteration grid in this case:
We can see that after stabilization the daily expenses are 1,5 more than in the previous case (they are 790554,8 $ per day), but in spite of this we get bigger profit: we will have gotten 20988352,3 $ by the 22 day.
This table shows us how profit by the 22 day depends on the extraction speed.
|ω, kg/day ||Profit by 22 day, $
This is a graph of this dependence.
We can see that the optimum speed is ω = 174,9 kg/day. Now if we decide to start the extraction we will get the maximal profit.
The imaginary problem described earlier includes a big amount of real problems which exist in production: how to find the optimum regime for equipment to slow down its tear and destruction by aggressive environment or slow down pollution of environment, and simultaneously to get the maximal profit.
In what follows we are going to change the problem to include nonlinear effects in it (and this will be very important for applications). We hope that our solutions will attract people who want to optimize their production.