Question

In a portfolio problem, X1, X2, and X3 represent the number of shares purchased of stocks 1, 2, and 3 which have selling prices of $15, $45, and $100, respectively. The expected returns on investment of the three stocks are 10%, 8%, and 13%. The investor has up to $40,000 to invest. The stockbroker suggests limiting the investments so that no more than $10,000 is invested in stock 2 or the total number of shares of stocks 2 and 3 does not exceed 350. The investor stimulates that stock 1 must not account for more than 35% of the number of shares purchased.

1. An appropriate objective function is

1. Max 15x1 + 45x2 + 100x3
2. 15x1 + 45x2 + 100x3 <= 40,000
3. x1 +x2 +x3 <= 40,000
4. 0.08 *15x1 + 0.1 * 45x2 + 0.13 * 100x3 <= 40,000

2. An appropriate constraint would be:

1. 0.65 x1 – 0.35 x2 – 0.35 x3 <= 0.35
2. 0.65 x1 – 0.35 x2 – 0.35 x3 <= 0
3. 0.65 x1 – 0.35 x2 – 0.35 x3 >= 0
4. X1 <= (0.35) (x1+x2+x3)

3. Two appropriate constraint would be:

1. 45 X2 <= 1000 and X2 + X3 <= 350
2. X2 >= 1000 and (0.1) x2 + (0.13) x3 >= 350
3. 45 x2 >= 1000 and x2 +x3 >= 350
4. 45 x2 >= 1000 and 45x2 + 100x3 >= 350

are these correct ?

Answer 1

Objective function: (max)Z = 0.1*15*X1 + 0.08*45*X2 + 0.13*100*X3 (In terms of return)

Constraint:   15*X1 + 45*X2 + 100*X3 40000

  45*X2   10000

X2 + X3 350

X1   0.35*(X1 + X2 + X3)

1. (max)Z = 0.1*15*X1 + 0.08*45*X2 + 0.13*100*X3

2. X1   0.35*(X1 + X2 + X3)

3. 45*X2   10000 and X2 + X3 350