## The Economic Order Quantity – a simple calculus application

The EOQ (Economic Order Quantity) formula is a deceptively simple model. It comes from Zipkin’s “Foundations of Inventory Management” (Irwin/McGraw-Hill, 2000, 0-256-11379-3) and it is the very first model in the book. It was first published 100 years ago, in 1913 – the model, not the book!.

When all is said and done, it’s a simple application of freshman calculus.

Imagine that we sell or use up one product, at a known constant rate $\lambda\$. Periodically, we order more of this product, to replenish our inventory I(t). Further, there is a known constant lead time L – between when we place an order and when we receive it (actually, when we can sell or use it, so this includes unloading and storing). If our inventory will go to zero at t = T, then, at the very latest, we must place an order at T – L:

## Heaviside’s Operational Calculus

I have found a fine introduction to Heaviside’s methods in Spiegel’s “Applied Differential Equations”, 3rd ed, 1981, Prentice Hall. To be specific, pages 204-211. I can’t very well say, “Don’t buy the book just for this” – because that’s exactly what I did!

Let me emphasize that, as far as I know, Heaviside’s methods are now of primarily historical interest. I would not say that Laplace transforms make Heaviside’s methods rigorous – but that Laplace transforms provide a rigorous alternative which, like Heaviside’s, lets us do algebra instead of calculus.

Like Laplace transforms, the quick use of Heaviside’s methods takes advantage of shifting properties, linearity, and tables of known results to speed up calculations. I’m not going to take them that far. With Mathematica® or another symbolic system, I see no need to go beyond the introduction to Heaviside’s methods. What I wanted to see was: just what was Heaviside’s fundamental idea? It turns out that his fundamental idea suffices, given other tools available today.

Of course – as we have seen and as we’ll see below – Mathematica can quickly solve the differential equations to which Heaviside’s methods apply (linear, with constant coefficients). We don’t need Heaviside…

…but it turns out there’s at least one question Heaviside’s methods can answer very, very quickly: find a particular solution (rather than the general solution). I don’t know that I will ever use Heaviside for that, but I know that I could.

One last thing. For my present purposes, it suffices that I will find solutions to a differential equation – and I will confirm that my answers are solutions… but I’m not going to try to prove anything at all. I don’t know what the limitations of this method are – instead of confirming that the differential equation under consideration satisfies some set of conditions, I’ll simply confirm that the answer works.

With that, let me roll up my sleeves and show this to you.
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## Introduction

Down the road, I expect to be using Laplace transforms to set up and solve electric circuits, and for transfer functions in control theory. An obvious starting point is to remind you just what a Laplace transform is.

So I should show you at least one example of solving a differential equation using Laplace transforms.

But if I do that, I really should remind you of the alternative solution, the one you almost certainly learned 1st.

On top of that, I really should show you what Mathematica® can do.

As if all that weren’t enough – though it really won’t take very long – I have seen a nice approach to Heaviside’s operational calculus, and I want to show that to you, too. Ah, by the time I explain it, this will justify a post of its own.

So, I propose to take a typical equation for these methods – linear, with constant coefficients – and I am going to

1. let Mathematica solve it symbolically
3. let Mathematica solve it numerically
4. solve it using Laplace transforms
5. solve the homogeneous equation and then find a particular solution to the inhomogeneous equation

and in a subsequent post I will

• solve it using Heavisisde’s operational calculus

## Calculus – deriving the equations for simple projectile motion

We have worked several examples of simple projectile motion – meaning that the acceleration of gravity was constant and vertical, and there was no acceleration in the horizontal direction. (In particular, there is no air resistance.)

I simply handed us four equations and used them in a firstsecond… and third post. I said I would show how to derive them.

It was junior year in high school that I learned the equations for position and speed as a function of a constant acceleration. I didn’t take calculus until I was a college freshman… and at some point I decided that I knew enough calculus to derive the equations I had been told to memorize two years before.

This is about as elementary as it gets in calculus, but when it was all new to me, it was a thrill to see what it could do for me in physics. I will actually derive them in two slightly different ways.

Here we go.
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## Mechanics: Simple Projectile Motion – 2 (Fort and Ship)

Here is a projectile problem that fascinated me, and I’ve been meaning to show it to you. It comes from Neville de Mestre, “The Mathematics of Projectiles in Sport”, 1990. This is the second post about simple projectile motion, so you might want to look at the first one.

Here we go.

A fort is on top of a cliff h meters directly above the ocean. Approaching the fort is a ship whose guns have the same muzzle velocity vo as the guns at the fort….

Find over what range the ship can be fired on, from the fort, without being able to effectively return the fire.

If gh is small compared with vo^2 show that this distance is approximately double the height of the cliff.

So, we need to find two distances: max from fort to ship, and max from ship to fort.
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## introduction & overview

The purpose of this notebook is to organize the useful techniques of integration which are taught in freshman calculus and then presumed known (ha!) at the beginnings of a course in ordinary differential equations.

Heads up: I’m going to mention hyperbolic trig functions, but until you meet them, they are not relevant and you should ignore them. I’m just trying to be thorough, but I fear that I might be confusing. So I’m going to mention them in the details, but omit them from the summary.

First off, there are three categories of integrals:

1. known
2. special techniques
3. general techniques

Here it is in a nutshell: If an integral is on the “known” list, you’re wasting time trying to use special or general techniques. if an integral can be done using special techniques, you’re wasting time using general techniques.
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## Calculus: where did e come from?

I assembled the following in response to a question from a calculus student. What he asked was literally, “Who first found e?” What he was really asking, I suspect, was more along the lines of: “Where the heck did e come from, and how on earth did anyone find it?”
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