# Linear Programming

Linear Programming

Project description
The question:

You are an Independent Financial Advisor providing expert advice to individuals on request. A client has 150,000 which he wishes to invest in income-generating securities and government bonds so as to maximise his annual return. He has selected five possible investments, all of which he considers to have reasonably high yields and stability:

Oil company A stock, paying 9 % annual dividend.
Oil company B stock, paying 7.7% annual dividend.
Electric utility A stock, paying 5% annual dividend.
Electric utility B stock, paying 6.2% annual dividend.
Government bonds, paying 5.5 % annual interest.

Since he has no plans to sell his stock in the near future, he is not concerned with their selling price. Based on the various risk levels involved, he has made the following decisions:

The total investment in oil stocks may not exceed 32,000.
The total investment in electric utilities may not exceed 60,000.
The investment in oil company A may not exceed 28,000.
The total investment in electric utility A may not exceed 35,000.
The total investment in oil stocks may not exceed the total investment in electric utilities.
The investment in oil company A and electric utility A combined may not exceed the investment in Government bonds.

As the clients financial adviser you should advise him of the amount of money he should invest in the various investments in order to maximise his annual return.

1. Formulate the problem clearly defining the variables, the objective function and the constraints.
2.Solve the problem by linear programming using an Excel spreadsheet model, remembering that the Client’s objective is to maximise total annual income. State the optimal investments plan clearly, giving the values of all the problem variables. Include a copy of your spreadsheet, making sure that the layout of the spreadsheet is easy to follow and is carefully annotated. The layout should be your own design.
3. Produce an Excel Sensitivity Report

( I have done questions 1-3 which I shall attach as a word document and excel document, I would just need for the person to check to confirm that I am right and/or make any changes).
What I need help with is the last bit of the work:

4. Give an interpretation of the values in the Variables and Constraints tables of the report. Give three examples to test the sensitivity ranges given in Excel report by changing the original problem values. Each example should consider changing only one factor (i.e., variable or constraint). State why you have chosen the particular factors that you used and include copies of any appropriate Excel output and explain your answers and conclusions.

Microsoft Excel 14.4 Sensitivity Report
Worksheet: [Workbook3]Sheet1
Report Created: 19/12/2014 21:44:38

Variable Cells
Final    Reduced    Objective    Allowable    Allowable
Cell    Name    Value    Cost    Coefficient    Increase    Decrease
\$B\$14    CONSTRAINTS X1    28000    0    0.09    1E+30    0.013
\$C\$14    CONSTRAINTS X2    4000    0    0.077    0.013    0.022
\$D\$14    CONSTRAINTS Y1    0    -0.012    0.05    0.012    1E+30
\$E\$14    CONSTRAINTS Y2    60000    0    0.062    1E+30    0.007
\$F\$14    CONSTRAINTS Z1    58000    0    0.055    0.007    0.055

Constraints
Cell    Name    Value    Price    R.H. Side    Increase    Decrease
\$G\$15    X1 + X2 + Y1 + Y2 + Z1 <= 150000 USED    150000    0.055    0    1E+30    30000
\$G\$16    X1 + X2 <= 32000 USED    32000    0.022    0    28000    4000
\$G\$17    Y1 + Y2 <= 60000 USED    60000    0.007    0    30000    28000
\$G\$18    X1 <= 28000 USED    28000    0.013    0    4000    28000
\$G\$19    Y1 <= 35000 USED    0    0    0    1E+30    35000
\$G\$20    X1 + X2 <= Y1 + Y2 USED    32000    0    0    1E+30    28000
\$G\$21    X1 + Y1 <= Z1  USED    28000    0    0    1E+30    30000

MA5051Project Management Group ZInterim Report Part 2

Linear Programming Formulation

Problem Variables:
Let X1 be the amount invested in the oil company A stock
Let X2 be the amount invested in the oil company B stock
Let Y1 be the amount invested in the electric utility A stock
Let Y2 be the amount invested in the electric utility B stock
Let Z1 be the amount invested in Government Bonds

Constraints:
X1 + X2 + Y1 + Y2 + Z1 = £150000
X1 + X2 = £32000           maximum invested in oil stock
Y1 + Y2 = £60000           maximum invested in electric utilities
X1 = £28000                   maximum invested in oil company stock A
Y1 = £35000                   maximum invested in electric utility stock A
X1 + X2 = Y1 + Y2    investment in oil stock may not exceed investment in electric utilities
X1 + Y1 = Z1  investment in oil stock A and electric utility stock A combined may not exceed investment in government bonds
X1, X2, Y1, Y2, Z1 = 0

Objective Function:
Maximise total profit: P ? Max P = 0.09X1 + 0.077X2 + 0.05Y1 + 0.062Y2 + 0.055Z1

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