Delta Airlines Buys a Refinery: A Mathematical Look at Possible Consequences [138]

FINDING THE OPTIMAL SOLUTION

FOLLING A PATH TO MAXIMIZE PROFIT OR MINIMIZE COST
FOLLING A PATH TO MAXIMIZE PROFIT OR MINIMIZE COST | Source

LOOKING FOR SOMETHING EASIER TO WRITE ABOUT

I THOUGHT I WOULD TURN to something more understandable and simpler than politics; namely mathematics. More speficically, the linear algegra and linear programming I mentioned in my hub, Delta Airlines vs Big Gas No not Big Oil. Now, don't run away screaming yet (although I almost did trying to remember back to my college days in order figure out how it really works), it's not that bad ... really. I know that mathematics, especially with names like linear algebra, is a black hole to many readers, boring to others, scary to even more, or downright mystifying. Because of that, I will do my best to make it amusing, interesting, and meaningful.

What I hope to do is give a little insight into how companies, meaning refineries in the case, make decisions about what to produce and what to charge when they have several choices. In addition, we will run an alternative calculation to see what happens with Delta takes $13 billion off the table for the purchase of jet fuel.and thereby lowers demand.

MY ESOTERIC and FRIENDS REFINERY, INC

SO, WHAT IS ME&FR, INC? It is a ficticious refinery to help demostrate how business decisions are made (in a simplified form, of course) regarding what to produce and how much to charge. ME&FR produces certain quantities of three different products which are distilled (cracked) from oil, gasoline (G), jet fuel (F). and heating oil (H); and sells them at prices 'g', 'f', and 'h', respectively. Guess what, you have just been introduced to ... algebra. How so? Algebra deals with variables and symbols and gasoline, jet fuel, and heating oil as well as their associated prices, are the variables, where G, g, F, f, H, and h are symbols representing those variables.

Knowing this, we can now write our first algebraic equation that will be useful to us, which is T otal Sales (S) = the total barrels of gas produced (G) times its price (g) + the total barrels of jet fuel produced (F) times its price (f) + the total number of barrels of heating oil produced (H) times its price (h) or, in equation form, S = gG + fF+hH, simple, yes? In any case, the point of this paragraph is to introduce you some of the symbols we will be using and what they mean.

Let me now introduce the rest of the variables we will need before we move on the fun stuff.

  • The reason a business operates is to make a Profit (P). It does this by selling things (S) it produces which, in turn, cost a certain amount of money to make; this would be Total Cost (TC)
  • It is also handy to know that Total Cost is made up of two different kinds of costs; one is called Indirect Costs (I), sometimes not quite accurately known as "fixed costs".
  • Indirect costs are those expenses, such as rent, that don't vary depending on how much you produce. For example, if you produce nothing, you will still pay the same rent as if you produced a gazillion things.
  • The other is Variable Costs (V), which, oddly enough, doesn't change either, when expressed as the variable cost per unit produced. What is variable is the total V, because that is dependent on the quatity of the things produced. Now, we can create two more algebraic equations, namely:
  1. Total Cost (TC) = Total Indirect Cost + Total Variable Cost or TC = I + V,
  2. V = gv*G + fv*F + hv*H, where gv, fv, and hv are the unit variable costs (cost per barrel) of G, F, and H, respectively,
  3. gn = g - gv, f - fv, and h - hv are the net prices (unit price - unit variable cost or, said another way, price per barrel - variable cost per barrel)
  4. Profit = Sales - Total Cost or P = S - TC = (gn*G + fn*F + hn*H) - (o*O - I), o and O are the price and quantity of oil purchased, respectively.

(Not too opaque so far, I hope.) Why are these equations important?

Because, as we all know, Profit (P) is what any company, including mine, wants to try to maximize or make as big as possible. From equation (4) you can see that means keeping Sales (S) as large as possible and Total Cost (TC) as small as possible.

It is because of this goal, that many large companies either contract out or create internal departments to employ the theory of linear algegra and the tools of linear programming, tools which were developed to solve these types of problems; one such is an algorithim called the Simplex Method. Using this methodology, it is possible to determine what quantities of G, F, and H to produce, given the market-based prices of g, f, and h in order to maximize P, profit.

From here, we can dig a little deeper.

WHEN BONDAGE IS GOOD?

WHEN IT HELPS US SOLVE A PROBLEM. In this case, we are referring to constraints, which limits the ability of our variables to grow or shrink indefinitely. If even one of variables is without constraint, say the capacity to produce gasoline, then we will never find a practical answer. (Even here, we can get to some answer because, while we can produce as much gasoline as we want, at some point, there won't be enough people to buy it. The answer, of course, would be nonsensical.)

So, what is a constraint in mathematics, anyway? It is an equation which limits a variable to a certain set of values. I introduced the idea of one in the last paragraph, a constraint on how much of a product, say gasoline, a given refinery is capable of producing in a period of time. Another will be the total Indirect Costs (I). Now, let me present a few of the constraints we will need:

  • I = $1,000,000 (indirect costs will be $1 million)
  • G, F, H >= 0 (which says you can't produce negative amounts)
  • g, f, h, we will assume, are initially set by the market at $155.40 per barrel ($3.70/gal).
  • gv, fv, hv are assumed to be $14.70, $13.86, and $13.60 per barrel of G, F, and H respectively.
  • That means gn, fn, and hn are $140.70, $141.54, and $142.80,
  • For simplicity, we will further assume that ME&FR can produce no more than 46,000 bbls per year or G, F, H <= 46,000 bbls
  • W stands for the waste by-products of the cracking process and is equal to 22% of the oil (O) purchased or W = .22O
  • Further, we will assume that ME&FR will buy a million barrels of oil annually, that is O = 1,000,000 bbls.
  • Finally, we need to know how much G, F, and H can be produced from one barrel of oil, which is 44% - 48%, 20% - 24%, and 8% - 12%, respectively for G, F, and H.

Given all of the above, we can now write the equations that are used to calculate the optimal mix of products needed to maximize ME&FR's profit. It is:

Maximize P = 140.7G + 137.54F + 142.8H + 0W - 100O - I subject to
O = 1000000 bbld
I = $1,000,000


G <= 480000 bbls
F <= 480000 bbls
H <= 480000 bbls (sets the maximum amount of each product that can be produced.)
.22O = W (this says that waste = 22% of the amount of oil used.)

G - .44O >= 0
G - .48O <= 0
F- .24O <= 0
F - .2O >= 0
H - .12O <= 0
H - .08O >= 0 (all of these "inequalities" establish this limits of how much of each product can be made from a barrel of oil.)

O - G - F - H - W >= 0 (which simply says that the sum of barrels of the distillates must be less than or equal to the number of barrels of oil used to produce them.)

Those are fairly easy constraints to understand because they make common sense and what you have above is common sense in equation form. In words, what all that mess above says is that subject to the following constraints:

  • you bought a million barrels of oil
  • your indirect costs are $1,000,000
  • you can only produce so much of each product
  • 22% of the oil purchased is wasted (the percentages I used are close to reality, btw)
  • you can only produce so much of each product out of a barrel of oil
  • and, you can't produce more by-products of oil than the oil used (not really quite true, but ignoring the exact truth doesn't hurt anything here.)

It is as simple as that. Now, on to how this is all used.

SOLVING THE PROBLEM

OK, now that we did all of those mental gymnastics, let's put or effort to work. To solve the equations listed in the last section, I turned to a Simplex tool I found on the Internet.

The first thing to do, of course, is to solve the equations as I have just set them up. When you do that, you will find you maximize ME&FR's

  • profit at $9,182,800 if you produce
  • 440,000, 220,000, and 120,000 barrels of gasoline, aviation fuel, and heating oil respectively.

This the case when the price on the market is the same for each product is the same, and you have all of the other listed constraints. Now, what happens when Delta Airlines buys a refinery that wasn't producing anything to start with?

  • The first thing that happens is the demand for aviation fuel goes down as well
  • The next thing that happens is the price of aviation fuel declines because there is less demand

So let's let the price of aviation fuel fall to $3.65/gal or $139.44/bbl from $3.70/gal; what happens?

  • Profit at ME&FR falls to $8,748,000
  • The amount of aviation fuel produce drops to 200,000 bbls, the amount of gas produced is increased to 460,000 bbl, and
  • The amount of heating fuel produced remains the same.

OK, let's suppose most of the refineries around the world do the same, what happens now? The supply of gasoline goes up and what occurs when that happens? Well, the price of gasoline falls, of course. Let's let that fall, then, to say $3.68/gal. That would bring the net price of gas down to $139.86/bbl. When you plug this into our equations, finally, nothing changes.

Looking back, then, to my original hub on Delta and their purchase of an oil refinery, I made the unsupported observation that Delta's action could ultimated bring down the price of gas at the pump. Here we have evidence of how that may happen; the removal of demand of aviation fuel brought down the price of this commodity from $3.70 to $3.65gal, which in turn brought down the price of gas, in this example, to $3.68 from $3.70/gal.

Neat, isn't it?

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