3. Use the Simplex method to find the optimal solution to the problem below (15

Question

3. Use the Simplex method to find the optimal solution to the problem below (15 pts). Investment: An investor has at most $45,000 to invest in government bonds, mutual funds and money market funds. The average yield for government bonds, mutual funds and money market funds, are 4%, 7% and 9% respectively. The investor's policy requires that the total amount invested in mutual and money market funds must not exceed the amount invested in government bonds. How much should be invested in each type of investment in order to maximize the return? What is the maximum return? You may use your calculator if you like. Please show me at least the following work: - all decision variables, xi, 2..etc. and indicate what each variable represents - the objective function equation - the initial simplex matrix the final solution matrix - the answers to the two questions asked for this problem

Explanation / Answer

Let x1 denotes the amount invested in government bonds

x2 denotes the amount invested in mutual funds

x3 denotes the amount invested in money market Funds

Obejctive Function Maximize Z = 0.04x1 + 0.07x2 + 0.09x3

Constraints:

x1 >= (x2+x3)

x1 + x2 + x3 = 45000

The initial simplex matrix will be

Tableau #1
x1 x2 x3 s1 s2 s3 s4 s5 s6 p
1 1 1 1 0 0 0 0 0 0 45000  
1 -1 -1 0 -1 0 0 0 0 0 0   
1 0 0 0 0 -1 0 0 0 0 0   
0 1 0 0 0 0 -1 0 0 0 0   
0 0 1 0 0 0 0 -1 0 0 0   
1 1 1 0 0 0 0 0 -1 0 45000  
-0.04 -0.07 -0.09 0 0 0 0 0 0 1 0   

The final simplex matrix will be with solution steps

Tableau #1
x1 x2 x3 s1 s2 s3 s4 s5 s6 p
1 1 1 1 0 0 0 0 0 0 45000  
1 -1 -1 0 -1 0 0 0 0 0 0   
1 0 0 0 0 -1 0 0 0 0 0   
0 1 0 0 0 0 -1 0 0 0 0   
0 0 1 0 0 0 0 -1 0 0 0   
1 1 1 0 0 0 0 0 -1 0 45000  
-0.04 -0.07 -0.09 0 0 0 0 0 0 1 0   

Tableau #2
x1 x2 x3 s1 s2 s3 s4 s5 s6 p
1 1 1 1 0 0 0 0 0 0 45000  
-1 1 1 0 1 0 0 0 0 0 0   
1 0 0 0 0 -1 0 0 0 0 0   
0 1 0 0 0 0 -1 0 0 0 0   
0 0 1 0 0 0 0 -1 0 0 0   
1 1 1 0 0 0 0 0 -1 0 45000  
-0.04 -0.07 -0.09 0 0 0 0 0 0 1 0   

Tableau #3
x1 x2 x3 s1 s2 s3 s4 s5 s6 p
1 1 1 1 0 0 0 0 0 0 45000  
-1 1 1 0 1 0 0 0 0 0 0   
-1 0 0 0 0 1 0 0 0 0 0   
0 1 0 0 0 0 -1 0 0 0 0   
0 0 1 0 0 0 0 -1 0 0 0   
1 1 1 0 0 0 0 0 -1 0 45000  
-0.04 -0.07 -0.09 0 0 0 0 0 0 1 0   

Tableau #4
x1 x2 x3 s1 s2 s3 s4 s5 s6 p
1 1 1 1 0 0 0 0 0 0 45000  
-1 1 1 0 1 0 0 0 0 0 0   
-1 0 0 0 0 1 0 0 0 0 0   
0 -1 0 0 0 0 1 0 0 0 0   
0 0 1 0 0 0 0 -1 0 0 0   
1 1 1 0 0 0 0 0 -1 0 45000  
-0.04 -0.07 -0.09 0 0 0 0 0 0 1 0   

Tableau #5
x1 x2 x3 s1 s2 s3 s4 s5 s6 p
1 1 1 1 0 0 0 0 0 0 45000  
-1 1 1 0 1 0 0 0 0 0 0   
-1 0 0 0 0 1 0 0 0 0 0   
0 -1 0 0 0 0 1 0 0 0 0   
0 0 -1 0 0 0 0 1 0 0 0   
1 1 1 0 0 0 0 0 -1 0 45000  
-0.04 -0.07 -0.09 0 0 0 0 0 0 1 0   

Tableau #6
x1 x2 x3 s1 s2 s3 s4 s5 s6 p
0 0 0 1 0 0 0 0 1 0 0   
0 2 2 0 1 0 0 0 -1 0 45000  
0 1 1 0 0 1 0 0 -1 0 45000  
0 -1 0 0 0 0 1 0 0 0 0   
0 0 -1 0 0 0 0 1 0 0 0   
1 1 1 0 0 0 0 0 -1 0 45000  
0 -0.03 -0.05 0 0 0 0 0 -0.04 1 1800

Tableau #7
x1 x2 x3 s1 s2 s3 s4 s5 s6 p
0 0 0 1 0 0 0 0 1 0 0   
0 1 1 0 0.5 0 0 0 -0.5 0 22500  
0 0 0 0 -0.5 1 0 0 -0.5 0 22500  
0 -1 0 0 0 0 1 0 0 0 0   
0 1 0 0 0.5 0 0 1 -0.5 0 22500  
1 0 0 0 -0.5 0 0 0 -0.5 0 22500  
0 0.02 0 0 0.025 0 0 0 -0.065 1 2925

Tableau #8
x1 x2 x3 s1 s2 s3 s4 s5 s6 p
0 0 0 1 0 0 0 0 1 0 0   
0 1 1 0.5 0.5 0 0 0 0 0 22500  
0 0 0 0.5 -0.5 1 0 0 0 0 22500  
0 -1 0 0 0 0 1 0 0 0 0   
0 1 0 0.5 0.5 0 0 1 0 0 22500  
1 0 0 0.5 -0.5 0 0 0 0 0 22500  
0 0.02 0 0.065 0.025 0 0 0 0 1 2925

Investment must be:

Government bonds = $22500

Mutual Funds = $0

Money Market funds = $22500

Maximum Profit = 0.04(22500) + 0.07(0) + 0.09(22500) = $2925

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