Question

A client of an investment firm has $10000 available for investment. He has instructed that his money be invested in three stocks, so that no more than $5000 is invested in any one stock but at least $1000 be invested in each stock. He has further instructed the firm to use its current data and invest in the manner that maximizes his overall gain during a one-year period. The stocks, the current price per share and the firm’s predicted stock price a year from now are summarized below:

Stock Current  Price Projected Price 1 year
James $25 $35
QM $50 $60
Del Candy $100 $125

Required:
Formulate the problem as a linear programming model including decision variables, objective function and the constraints. Use the first letter of each variable to represent the decision variable.

Answer 1

Decision variables:

J = Number of James stocks

Q= Number of QM stocks

D = Number od Del Candy stocks

Objective Function:

G=10J+10Q+25D

Constrains:

`25J+50Q+100D \leq 10,000`

`J \leq 200`

`Q \leq 100`

`D \leq 50`

`J \geq 40`

`Q \geq 20`

`D \geq 10`

Step-by-step explanation:

In order to define the decision variables we take the first letter of each Stock, as the problem indicates. We have three Stocks: James, QM and Del Candy, so:

J = Number of James stocks

Q= Number of QM stocks

D = Number od Del Candy stocks

Now, to get the objective function, we need to know how much each stock is going to earn. For the James stocks, we know that the original value is $25 and the future value is $35, therefore, each stock will gain: $35-$25=$10.

That's where the 10J came from.

For the QM stocks, we know that the original value is $50 and the future value is $60, therefore, each stock will gain: $60-$50=$10.

That's where the 10Q came from.

And finally. For the Del Candy stocks, we know that the original value is $100 and the future value is $125, therefore, each stock will gain: $125-$100=$25.

That's where the 25D came from.

So we put them all together to get our objective function, which will represent the overall gain during the one year period:

G=10J+10Q+25D

For the constrains, we know that the client wishes to invest $10,000 and that James stock's price is $25, QM's price is $50 and Del Candy's price is $100 so the first constrain will be:

`25J+50Q+100D \leq 10,000`

It would be less than or equal because they have a top of $10,000 to invest. They could invest less though if that maximizes the profit.

Next, the client said that no more than $5,000 should be invested in any one stock, so we take the price of each stock and the number of shares to be bought for each stock and build our inequalities:

`25J \leq 5,000`

`50Q \leq 5,000`

`100D \leq 5,000`

and solve for each variable so we get:

`J \leq 200`

`Q \leq 100`

`D \leq 50`

the client also said that at least $1,000 should be invested in each stock, so we get the following inequalities:

`25J \geq 1,000`

`50Q \geq 1,000`

`100D \geq 1,000`

and then we solve each inequality for the given variable>

`J \geq 40`

`Q \geq 20`

`D \geq 10`