Answer to Question #133137 in Operations Research for xxxx

Question

A business executive has the option to invest money in two plans: Plan A guarantees that each peso invested will earn PhP0.70 a year later, and plan B guarantees that each peso invested will earn PhP2 after two years. In plan A, investments can be made annually, and in plan B, investments are allowed for periods that are multiples of two years only.

Formulate the LP model in order for the executive to determine how to invest his PhP1,000,000 to maximize the earnings at the end of four years. (CLUE: How much should you invest in plan A and plan B at the start of years 1, 2, 3, and 4?)
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The challenge in this problem is that the return (interest) of plan A and plan B occur in different time periods: plan A’s interest is recognized annually (every year) while plan B’s interest is recognized only every two years.

Answer 1

Assume x₁, x₂, x₃ and x₄ are the investments in plan A in year 1 to 4. Plan B allows investments for periods that are multiples of 2. Hence, assume y₁ as the investment for 4 years and y₂ as the investment for 2 years.

Total investment constraint : x₁+x₂+x₃+x₄+y₁+y₂ = 1,000,000

Total earning at the end of 4^th year from

plan A : E_A = (x₁*1.7⁴ - x₁) + (x₂*1.7³ -x₂) + (x₃*1.7² -x₃) + (x₄*1.7 - x₄)

plan B : E_B = (y₁*3² -y₁) + (y₂*3 -y₂)

Objective is to Max (E_A+ E_B) subject to the investment constraint and

x₁ , x_2, x_3, x_{4 ,} y_{1 ,} y₂ >= 0

Since the total investment is 1,000,000. We can simplify / looks it differently. Total earning will be

= return - investment = x₁*1.7⁴ + x₂*1.7³ + x₃*1.7² + x₄*1.7 + y₁*3² + y₂*3 - 1,000,000