# Quantitative Methods in Business – Linear Programming

1- Statement of the Problem:

Middle East for investment offers a bundle of investment options in many types of securities. Mr. Brown, an investor, would like to invest $ 5 million in various securities. He wishes to maximize his yearly profit over the next year. The investment company offered him a portfolio including Bonds, Stocks, Gold and Land. The expected return is 6% for Bonds, 14% for Stocks, 10% for Gold and 5% for Land. For diversification purposes, the maximum amount to be invested is $ 3 million in Bonds, $ 3 million in Stocks, $ 2 million in Gold and $ 1 million in Land. In addition to that, the investment company specifies that at least 40% of the total investment should be in Bonds and Stocks, and no more than 20% of the total investment should be in land.

2- Formulation:

The decision variables are:

B= Total Dollar amount to be invested in Bonds.

S= Total Dollar amount to be invested in Stocks.

G= Total Dollar Amount to be invested in Gold.

L= Total Dollar Amount to be invested in Land.

The objective function that maximizes the return per year is: Z= 0.06B+0.14S+0.1G+0.05L

Subject to:

1)B+S+G+L≤5,000,000

2)B≤3,000,000

3)S≤3,000,000

4)G≤2,000,000

5)L≤1,000,000

6)B+S≥0.40(B+S+G+L)

→ B+S≥0.4B+0.4S+0.4G+0.4L

→ 0.6B+0.6S-0.4G-0.4L≥0

7)L ≤0.2(B+S+G+L)

→L ≤0.2B+0.2S+0.2G+0.2L

→ -0.2B-0.2S-0.2G+0.8L ≤0

Given that B; S; G and L are positive values

3- POM-QM Output:

4- Analysis:

That is, Mr. Brown needs to invest $ 3 million in stocks and $ 2 million in Gold. The total return would be $ 620,000 and the total return on investment (ROI) would be: Total return/Total amount invested= 620,000/5,000,000= 12.4% The reduced cost for the amount invested in Bonds (B), which is at zero level, is 0.04 (4%). This means that the expected return on Bonds should increase by 4% (from 6% to 10%) in order for B to become non-zero.

The reduced...

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