## 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:

## introduction

It can be difficult to find a clear statement of what the Capital Asset Pricing Model (henceforth CAPM) is. I’m not trying to do much more than provide that. In particular, I did not find the wiki article to be useful, even after acquiring a couple of recent books on the subject.

I own six references:

• Sharpe, Wiliam F.; “Investments”, Prentice Hall, 1978; 0-13-504605-X.
• Reilly, Frank K.; “Investments”, CBS College Publishing (The Dryden Press), 1980; 0-03-056712-2.
• Gringold, Richard C and Kahn, Ronald N.; Active Portfolio Management, McGraw-Hil, 2000; 0-07-024882-6.
• Roman, Steven; Introduction to the Mathematics of Finance, Springer, 2004; 0-387-21364-3.
• Benninga, Simon; Financial Modeling, 3rd ed. MIT, 2008; 0-262-02628-7.
• Ruppert, David; Statistics and Data Analysis for Financial Engineering; Springer 2011; 978-1-4419-7786-1.

There is more than one version of the CAPM… Roman (p. 62) tells me that “The major factor that turns Markowitz portfolio theory into capital market theory is the inclusion of a riskfree asset in the model…. generally regarded as the contribution of William Sharpe, for which he won the Nobel Prize…. the theory is sometimes referred to as the Sharpe-Lintner-Mossin (SLM) capital asset pricing model.”

Then Benninga (p. 265) told me about “Black’s zero-beta CAPM… in which the role of the risk-free asset is played by a portfolio with a zero beta with respect to the particular envelope portfolio y.” (We’ll come back to this, briefly.)
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## A Minimization problem

Let me work a minimization problem rather than a maximization. Like the previous problem, this one comes from Loomis & Turban, “Applied programming for management”, 003-078240-6 (pp. 97-99).

A customer asks a butcher to grind up several cuts of beef to form a blend of not less than 17.6% protein and 14.8% fat.

What he has available is

so that table gives us the A matrix and the c vector; the protein and fat requirements give us the b vector. Let the variables be x1, x2, x3.
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## Linear programming – Getting the final tableau given the answer

(That may sound strange. Bear with me.)

## Introduction

Every once in a while, I pick up and play with linear programming. This post will show you a couple of elementary ways to set up and solve a small linear programming problem….

But that is not the main purpose of this post.

For the record, Mathematica® has some special–purpose commands for solving large linear programming problems. I believe it handles them as data sets in a standardized format. I also believe its smallest example has more than 30 variables. So, if you need to do linear programming professionally, you should look at Mathematica’s linear programming command.

But if, like me, you pick up linear programming as a student who wants to work his way through a textbook, then you need something else.

Working through a textbook will almost certainly involve tabular displays of the initial problem, a sequence of tabular displays of intermediate non-optimal solutions, and a display of the final optimal solution. Each of these tables is typically called by the French “tableau”, plural tableaux.

You need to know how to get the final tableau, given Mathematica’s solution and the initial tableau (i.e. the initial data).
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