Production equation
Each week a company produces X units and sells p per cent of its
stock. How should the company plan its warehouse space?
Age
16 to 18
Challenge level
Problem
Each week a company produces $X$ units and sells $p$ per cent of its stock. How should the company plan its warehouse space? Will the stock fluctuate, or increase or decrease over time, or tend to a limit?
Initially the company has no stock. Show that, over a long period of time, the amount of stock tends to the limit ${100X\over p}$.
Getting Started
Taking $X_n$ as the amount of stock at the end of week $n$, you
need to solve the difference equation (recurrence relation)
$$X_{n+1} = X + (1 - {p\over 100})X_n$$ Put $X_n = Y_n - C$ then
choose $C$ such that $$Y_{n+1} = (1 - {p\over 100})Y_n$$ and
consider the values of this expression for $Y_n, Y_{n-1}, Y_{n-2},
... Y_1, Y_0$.
Student Solutions
Simon from Elizabeth College, Guensey and Andrei from Tudor Vianu National College, Romania both solved this problem using geometric series. Here is Simon's solution:
The equation to show stock levels in week $n$ is: $$U_n = X + \left(1 - {p\over 100}\right)U_{n-1}.$$ Substituting for $U_{n-1}$ and simplifying:
\begin{eqnarray} U_n &=& X + \left(1 -
{p\over 100}\right) \left(X + \left(1 - {p\over 100}\right)
U_{n-2}\right) \\ &=& X + \left(1 - {p\over 100}\right)X +
\left(1 - {p\over 100}\right)^2U_{n-2}. \end{eqnarray}
Originally there was no stock so the initial condition is $U_0 =
0$. To make the expressions easier to read we write $s= \left(1 -
{p\over 100}\right)$.Writing $U_n$ in terms of $U_0$: $$U_n = X(1 + s + s^2 + ... + s^{n-1})+ s^nU_0 = \sum_{r=0}^{n-1}Xs^r.$$ Summing this geometric series $$U_n = {X(1-s^n)\over 1 - s} = {100X\over p}\left(1 - \left(1 - {p\over 100 }\right)^n\right).$$ To find the limit of $U_n$ over a very long period of time, as $(1 - p/100)< 1$, we have $$\lim_{n\to \infty}\left(1- {p\over 100}\right)^n \to 0$$ and so $$\lim_{n\to \infty}U_n = {100X \over p}.$$ The company should ensure that they have enough warehouse space for ${100X\over p}$ items. However, the chances are that sales and production will vary during the year, and therefore they should ensure that they have a buffer zone so that they can have more stock if necessary. Also, they are unlikely to be in business for an infinite length of time and therefore should cater said limit to their needs based on the financial situation.
Teachers' Resources
The general method of solving problems where there is a formula for
the transition from one phase to the next is to solve either a
difference equation (such as this), or a differential equation,
according to whether the model is discrete or continuous.