Answer to Question #265939 in Operations Research for mathman

Question

A business executive has the option of investing money in two plans. Plan A guarantees that each dollar invested will earn 70 cents a year hence, and plan B guarantees that each dollar invested will earn $2 two years hence. Plan A allows yearly

investments, while in plan B, only investments for periods that are multiples of two

years are allowed. How should the executive invest $100,000 to maximize the earnings at the end of three years? Formulate this problem as a linear programming

problem

Answer 1

Let X_IA=Amount invested in year i using plan A,i=1,2,3

Let X_IB=Amount invested in year i using plan B,i=1,2,3

Let us first formulate the constraint as follows: 

X_1A+X _1B$=$ 100,000

The earning at the end of the first year is X_1A+0.7X_1A=1.7X_1A

Investment at the beginning of the 2nd year is X_2A+X_2B ,which must be equal to the earning at the end of the first year, that is;

X_2A+X_2B=1.7X_1A

The earning at the end of the 2nd year is 3X_{1B +}1.7X_2A

Investment at the beginning of the 3rd year is X_{3A ,}which must be equal to the earning at the end of the 2nd year, that is, 3X_{1B +}1.7X_2A=X_3A

The total earning at the end of the 3rd year is 3_x2B+1.7_x3A

Therefore, LP model of the problem is ;

Maximize Z=3_x2B+1.7_x3A

Subject to;

X_1A+X _1B$=$ 100,000

X_2A+X_2B=1.7X_1A

3X_{1B +}1.7X_2A=X_3A

X_1A , X_1B, X_2A,X_2B, X_3A$\ge$ 0