Jacksonville Trust Co. LTD invests in various types of securities. They have $9

Question

Jacksonville Trust Co. LTD invests in various types of securities. They have $9 million for immediate investment and wish to maximize the interest earned over the next year. Risk is not a factor. Preferred investment alternatives include Bonds (B), Stocks (S), Gold (G), Land (L), BitCoin* (C), and famous Art Masterpieces (M). The expected return for Bonds is 5%, Stocks 12%, Gold 11%, Land 4%, BitCoin 2%, and Art Masterpiece 3%. The maximum amount to be invested for each alternative is $2 million for Bonds, $1 million for Stocks, $2 million for Gold, $1 million for Land, $2 million for BitCoin, and $3 million for Art Masterpiece. To structure the portfolio, the Senior VP has specified that at least 20% of the investment should be in corporate Bonds, common Stocks, and BitCoin; and no more than 35% of the investment is to be in Gold and Land. In addition, the total investments on Bond, Stocks, and Gold should be greater than the total amount of money invested on Land, BitCoin, and Art Masterpiece. The objective is to maximize the return. Formulate a linear model.

*BitCoin: an innovative consensus payment network that enables a new payment system and a completely digital money.

(a) What are the Decision Variables?

(b) What is the Objective Function?

(c) What are Constraint Equations including non-negativity constraints?

Explanation / Answer

a) What are the Decision Variables?

Answer - Decision variable are constraints.

(b) What is the Objective Function?

Answer – The objective function can be written as

Let X, Y, Z, P, Q, R be the notation for each of the investment modes as below

X - Bonds (B), - 5% - 2 millions

Y - Stocks (S), - 12% - 1 million

Z - Gold (G), -11% - 2 millions

P - Land (L), - 4% - 1 million

Q - BitCoin* (C), - 2% - 2 million

R - famous Art Masterpieces (M) – 3%. – 3 millions

Then the objective function is contribution of each investment returns from each mode as per their mentioned percentages

Maximize objective Function Z1 =    5% X + 12% Y+11% Z+4% P+ 2% Q+ 3% R

Maximize objective Function Z1 = 0.05 X + 0.12 Y + 0.11 Z + 0.04 P + 0.02 Q + 0.03 R

This is the objective function, as we need to maximize the return from each investment. The return from each investment is in the form of percentage as given in the question. So the total return on investment will be the sum of the return on each investment mode.

Then we need to maximize the investment from this investments hence the

Maximize investment mode as below

Maximize objective Function Z1 = 0.05 X + 0.12 Y + 0.11 Z + 0.04 P + 0.02 Q + 0.03 R

(c) What are Constraint Equations including non-negativity constraints?

Constraints which limits the maximization of this objective function are as below

Constraint 1

The Max investment can be done for each mode is given as below

X - Bonds (B), = Max 2 millions

Y - Stocks (S), =Max 1 million

Z - Gold (G), =Max 2 millions

P - Land (L), = Max 1 million

Q - BitCoin* (C), = Max 2 millions

R - famous Art Masterpieces (M) = Max 3 millions

0 < X <,= 2 ,

0 < Y <,= 1 ,

0 <Z <,= 2 ,

0 < P <,= 1 ,

0 <Q <,= 2 ,

0 <R <,= 3

The investment value for each of the mode should be less thanor equal to above limits, as it calls for maximum investment for these modes.

Constraint 2

At least 20% of the investment should be in corporate Bonds, common Stocks, and BitCoin;

X + Y + Q >,= 20% of 9 million

X + Y + Q >,= 1.8

Constraints 3

No more than 35% of the investment is to be in Gold and Land.

Z + P <, = 35% of 9 millions

Z + P <, = 3.15

Constraint 4

The total investments on Bond, Stocks, and Gold should be greater than the total amount of money invested on Land, BitCoin, and Art Masterpiece.

X + Y + Z > P + Q + R

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