Hello there! In this tutorial, I’ll be formulating a
linear programming model for this investment problem.
XYZ Trust has to invest $8 million in Petroleum, Stocks, Gold, and Real Estate,
given these projected rates of return, these risk factors,
and subject to these guidelines. The question is “how much should XYZ invest
in each of these alternatives in order to maximize dollar return on investment.
Let’s begin by defining the decision variables. Since return on investment depends on how
much is invested in each alternative, we can define the decision variables as follows:
Let P equal the dollar amount to invest in Petroleum,
S, the amount to invest in stocks, G, the amount in Gold,
And R the amount in Real Estate. Note here that it will be insufficient to
simply say “Let P = Petroleum”. What about Petroleum does P actually represent?
Does it represent the number of barrels of petroleum,
or the cost per barrel of petroleum, the risk associated with petroleum,
or the return on investment in Petroleum. We need to know the units.
Next we state the objective function: Since Return on investment is clearly the
product of the rate of return and the amount invested, we can state the objective as follows:
Maximize 0.14P + 0.165S … and so on. For the constraints, let’s first write one
for the total investment. Since P, S, G, and R represent amount to invest
in these alternatives, we can write P + S + G + R = 8000000
We use equality here because we assume the trust is going to invest the entire 8 million.
If the 8 million is a budget that may not be fully exhausted, we could use a less than
or equal sign instead. For part a) of the guidelines,
“No single investment alternative should account for more than 40% of budget”.
So for Petroleum we can write P ≤ 0.4(P + S + G + R), since this is total
investment. If total investment does not have to equal
8m, we might have to leave this as is. But since P + S + G + R = 8000000, we can
simply replace this by 0.4(8000000). which we can rewrite this as P ≤ 3200000.
We do the same for Stocks, Gold, and Real Estate.
Note that I did not just write one line for these 4 constraints as depicted here.
You might understand what it means, but most software for solving LPs usually don’t.
And because these are 4 distinct constraints, we usually state them individually.
For b) The amount invested in Gold and Real Estate
combined must be at least twice that invested in Petroleum.
And we write G + R ≥ 2P,
with G+R representing “Gold and Real Estate combined”,
≥ representing “at least”, and 2P representing twice the amount in Petroleum.
This is rewritten as G + R - 2P ≥ 0
The requirement in c) states that “The risk associated with Stocks should
not exceed twice the risk associated with Real Estate.”
And we write 7S ≤ 2(3R)
because the risk associated with stocks is 7 and that for real estate is 3.
And that can be rewritten as 7S – 6R ≤ 0
For d), Investment in real estate should exceed investment in Stocks by no more than $1M.
So we write R ≤ S + 1 000 000. which basically means that the difference
between R and S has to be less than or equal to a million.
For part e), the average risk factor should be no more than 5.
Since the risk factors are 5, 7, 3 and 2, we can begin by writing
5P + 7S + 2G + 3R A common error here is the tendency to want
to divide this sum by 4 (because an average is required). But that will be incorrect.
What we need here is actually a weighted average, with the weights being the investment amounts.
So instead, we divide by total investment and state that the average should be less
or equal to 5. We can then cross multiply the denominator,
and change the right side to a numerical value. To finish up, we just add the non-negativity
constraints. And that concludes the LP formulation.
Thanks for watching.