Optimizing Investment Portfolio: Mathematical Programming Techniques

This assignment focuses on applying mathematical programming techniques to optimize investment portfolios. It involves formulating linear programming models to maximize returns and ratings under various constraints, such as investment limits, rating requirements, and specific fund allocations. T...Read More

Mathematical Programming Techniques Assignment (Part A)
.
Institution Name:
Student Name:
Arrivals into the workshop

a. i. Maximize the return from your portfolio of investments
Defining Variables:
Let; X1= the amount of return from fund A
X2= the amount of return from fund B
X3= the amount of return from fund C
X4= the amount of return from fund P
X5= the total amount of return from R
X6= the amount of return from fund S
X7= the amount of return from fund T
X8= the amount of return from fund X
X9= the amount of return from fund Z
a.ii. Maximum return r= 20x1+ 30x2 + 20 x3 + 60 x4+ 40x5 + 35 x6 + 40 x7
+ 5x8 + 8x9 . This is the objective function for obtaining the total returns from
each fund.
Where; 20x1 = Total Returns from fund A
30X2= the amount of return from fund B
20X3= the amount of return from fund C
60X4= the amount of return from fund P
40x5=the amount of return from fund Q
35X6= the amount of return from fund R
40X7= the amount of return from fund X
5X8= the amount of return from fund Y
8X9= the amount of return from fund Z
The Constraints are;
X1, x2, x3, x4, x5, x6, x7, x8 and x9 ≥0. This means that the individual returns from each
fund must be greater than or equal to 0.
X1 + X2 +X3+ X4+ X4 + X5 + X6 + X7 + X8 + X9≤ 10000. This means that the sum
returns from each of the individual funds must not be less than or equal to 10000 (i.e. the
amount must not be more than 10000).
i. Maximize the total rating for your portfolio of investments
Defining Variables:
Let; X1= the rating of fund A
X2= the rating of fund B
X3= the rating of fund C
X4= the rating of fund P
X5= the rating of fund R
X6= the rating of fund S
X7= the rating of fund T

X8= the rating of fund X
X9= the rating of fund Z
` Maximum rating R= 5x1+ 3x2 + 3x3 + 4x4+ 5x5 + 1x6 + 2x7 + 3x8 + 3x9
. This is the objective function for obtaining the total rating of each fund.
Where; 5X11 = the rating of fund A
3X2= the rating of fund B
3X3= the rating of fund C
5X4= the rating of fund P
4X5=the rating of fund Q
5X6= the rating of fund R
1X7= the rating of fund X
2X8= the rating of fund Y
3X9= the rating of fund Z
The Constraints are;
X1, x2, x3, x4, x5, x6, x7, x8 and x9 ≥0. This means that the individual of each
fund must be greater than or equal to 0.
X1 + X2 +X3+ X4+ X4 + X5 + X6 + X7 + X8 + X9≤ 45 This mean that the sum
ratings of each of the individual funds must not be less than or equal to 5(i.e. the
amount must not be more than 5).
a.iii. Maximize the return given that the portfolio should have an average rating of
at least 3.7.
Defining Variables:
Let; X1= the rating of fund A
X2= the rating of fund B
X3= the rating of fund C
X4= the rating of fund P
X5= the rating of fund R
X6= the rating of fund S
X7= the rating of fund T
X8= the rating of fund X
X9= the rating of fund Z
Maximum rating R= 5x1+ 3x2 + 3x3 + 4x4+ 5x5 + 1x6 + 2x7 + 3x8 + 3x9 . This
is the objective function for obtaining the total rating of each fund.
Where; 5X11 = the rating of fund A
3X2= the rating of fund B
3X3= the rating of fund C
5X4= the rating of fund P
4X5=the rating of fund Q
5X6= the rating of fund R

1X7= the rating of fund X
2X8= the rating of fund Y
3X9= the rating of fund Z
The Constraints are;
X1, x2, x3, x4, x5, x6, x7, x8 and x9 ≥0. This means that the individual of each
fund must be greater than or equal to 0.
X1 + X2 +X3+ X4+ X4 + X5 + X6 + X7 + X8 + X9≤ 5This mean that the sum
ratings of each of the individual funds must not be less than or equal to 3.7 (i.e.
the amount must not be more than 3.7).
b. Investigate how each of these portfolios would change if
The funds invested with C&R must be greater than those invested with the other
fund providers.
The change of portfolio is determined using the excel solver and the set conditions
outlined above. The table below shows how the conditions will change. The funds will
have a composition of similar to the one in the column used. It is clearly demonstrated
that P, Q, and R will have larger shares .The calculations are available in the excel
appendices.
B.ii. The portfolio must include investments in funds B and Z.
For the portfolio to include B and Z, the table below shows the optimal conditions. B
must be 60 while Z must be 16. This will give a maximum return and rating.The
calculations are available in the excel appendices.
c. Extend your investigation to include the restriction that the total investment
with funds
Carrying a “5 rating” must be greater than the total investment in the funds
carrying a
Rating 1 or 2 or 3.
The table below shows the changes in the variables.
Maximizing Returns Number to make
1. Objective Function 2 5
2. Input Variables
3. Decision Variables
4. Constraints
Required used 1 3 3
A 20 0 40 <= 0
B 30 0 60 <= 0
C 20 0 40 <= 0

P 60 0 120 <= 0
Q 40 0 80 <= 0
R 35 0 70 <= 0
X 40 0 80 <= 0
Y 5 0 10 <= 0
Z 8 0 16 <= 0
258 0 Return 516
d. Investigate how the models could be modified so that the solution maximizes
both
Return and rating.
To maximize both return and rates, the table below shows the output. The values of A, B,
C, P, Q, R, X, Y, and Z are shown below. When both the return and the ratings are
maximized, the return becomes 258. This is the maximum return that can be obtained.
Maximizing Returns Number to make
1. Objective Function 1 1
2. Input Variables
3. Decision Variables
4. Constraints
Required used 1 3 3
A 20 0 20 <= 0
B 30 0 30 <= 0
C 20 0 20 <= 0
P 60 0 60 <= 0
Q 40 0 40 <= 0
R 35 0 35 <= 0
X 40 0 40 <= 0
Y 5 0 5 <= 0
Z 8 0 8 <= 0
258 0 Return 258

Part b. Simulation Modelling Assignment
i. Consider the effect on the firm of increasing (or decreasing demand)
recommending any changes to the departmental staffing.
Time Queues
Q1 Q2 Q3
1 17 16 15
2 18 17 17
3 15 15 15
4 17 17 16
5 20 20 17
6 21 24 16
7 17 21 17
8 17 17 17
9 21 21 13
10 17 25 18
11 13 20 14
12 20 19 16
13 24 19 16
14 25 25 14
15 16 27 17
1 13 18 17
2 25 16 15
3 20 9 9
4 29 14 14
5 35 8 8
6 40 14 14
7 39 5 5
8 47 5 5
9 55 8 8
10 64 8 8
11 73 14 14
12 72 5 5
13 77 5 5
14 92 14 14
15 95 14 14
Table1: Before Any Change is made

0 2 4 6 8 10 12 14 16
0
5
10
15
20
25
30
35
Queueing Plots
Q1 Q2 Q3
Time Queues
Q1 Q2 Q3
1 5 5 5
2 17 17 14
3 20 23 13
4 20 30 12
5 5 23 16
6 13 20 13
7 5 12 12
8 20 13 13
9 24 23 14
10 21 22 13
11 13 22 16
12 13 19 12
13 17 24 19
14 13 18 16
15 20 22 18
1 13 12 12
2 22 14 13
3 25 9 9
4 22 14 13
5 21 6 6

6 33 14 14
7 24 14 14
8 27 14 14
9 18 5 5
10 30 15 15
11 35 15 15
12 33 8 8
13 45 14 14
14 44 8 8
15 41 8 8
Table 2: Demand Reduced
0 2 4 6 8 10 12 14 16
0
5
10
15
20
25
30
35
Queueing Plots
Q1 Q2 Q3
Time Queues
Q1 Q2 Q3
1 13 13 13
2 17 17 16
3 17 17 17
4 21 21 16
5 20 25 17

6 13 21 18
7 13 16 18
8 13 13 13
9 13 13 13
10 13 13 13
11 5 5 5
12 5 5 5
13 17 17 16
14 17 18 17
15 17 18 16
1 5 7 7
2 13 13 13
3 5 5 5
4 13 13 13
5 20 8 8
6 17 15 15
7 7 7 7
8 17 8 8
9 29 8 8
10 34 15 15
11 32 15 15
12 30 8 8
13 27 15 15
14 25 14 14
15 24 8 8
Table 3: Demand Increased
0 2 4 6 8 10 12 14 16
0
5
10
15
20
25
30
35
Queueing Plots
Q1 Q2 Q3

Summary:
 Increasing demand increases queueing time.
 Decreasing demand decreases queuing time.
 Decrease/ increase in demand has no effects on arrival time.
ii. Investigate the effect of changes to the production capacities per time unit
of the machines at some of the stages
0 2 4 6 8 10 12 14 16
0
5
10
15
20
25
30
35
Queueing Plots
Q1 Q2 Q3
Fig. 1: No change in Production Capacity
0 2 4 6 8 10 12 14 16
0
5
10
15
20
25
30
35
Queueing Plots
Q1 Q2 Q3
Fig. 2: Decrease in Production Capacity

0 2 4 6 8 10 12 14 16
0
5
10
15
20
25
30
35
Queueing Plots
Q1 Q2 Q3
Fig. 3: Increase in production Capacity
Summary:
o Increasing production decreases the queuing time.
o Increasing production increases the queuing time.
Appendices
Appendix 1: Investigating Change of variables.
Maximizing Returns Number to make
1. Objective Function 3 6
2. Input Variables
3. Decision Variables
4. Constraints
Required used Min used Available
A 20 0 60 <= 0
B 30 0 90 <= 0
C 20 0 60 <= 0
P 60 0 180 <= 0
Q 40 0 120 <= 0
R 35 0 105 <= 0
X 40 0 120 <= 0

Y 5 0 15 <= 0
Z 8 0 24 <= 0
258 0 Return 774
Appendix 2: Portfolio with funds B and Z
Maximizing Returns Number to make
1. Objective Function 2 5
2. Input Variables
3. Decision Variables
4. Constraints
Required used Min Used Available
A 20 0 40 <= 0
B 30 0 60 <= 0
C 20 0 40 <= 0
P 60 0 120 <= 0
Q 40 0 80 <= 0
R 35 0 70 <= 0
X 40 0 80 <= 0
Y 5 0 10 <= 0
Z 8 0 16 <= 0
258 0 Return 516