A case study of the J.D Williams Investment Strategy Problem. This is an assignment for IIUM students who took Decision Science (MGT 3050) in Semester 2, 2014/2015.

1. DECISION SCIENCE
(MGT 3050)
GROUP ASSIGNMENT
CASE STUDY:
INVESTMENT STRATEGY
NAME MATRIC NO.
NUR HASEENA BT HANAFE 1322762
AFIFAH NABILAH BT MOHAMAD SAFEI 1321976
KHAN AKMAM BINTE ARIF 1124392
SITI HAJAR BINTI MUSTAFFA 1328680
Lecturer’s Name : Dr. Rafikul Islam
Section : 7
Session : Semester 2, 2014/2015

2. TABLE OF CONTENTS
CONTENTS PAGES
TABLE OF CONTENTS 2
SUMMARY OF THE CASE PROBLEM 3
ANSWER
CONCLUSION 12
APPENDIX 13 - 21
REFERENCES 22

3. SUMMARY
The case problem is about J.D. Williams which specializes in investment advisory. It
basically suggests its clients on how much of their capital to invest in each of the three
different type of funds which are the growth fund, money fund and the income fund. It also
opts for a diversified portfolio for each client by implementing certain requirements for each
type of fund. For instance growth fund must be more than 20% but less than 40% and at least
30% should be allocated to the money fund.
The case revolves around one client with a fund of $800000. Apart from the
requirements of the portion of each type of fund on the client’s portfolio, another constraint of
risk index has been placed. The clients overall risk index should not be over 0.05. Also,
individual risk index for each type of fund has been given. The company predicted the annual
yield of growth, income and money funds to be 18%, 12.5% and 7.5%.
We are required to develop a model where the yield from the client’s portfolio is
maximized given the constraints.
Let growth fund be X1, income fund X2 and money fund X3.
The formulation of the above problem is:
Objective Function:
Maximize Z= 0.18 X1 + 0.125 X2 + 0.075 X3 (maximizing yield)
Subject to:
1 X1 + 1 X2 + 1 X3 ≤ 800000 (available fund)
0.80 X1 -0.20 X2 -0.20 X3 ≥ 0 (requirement of X1 to be more than 20%)
0.60 X1 -0.40 X2 -0.40 X3 ≤ 0 (requirement of X1 to be less than 40%)
-0.20 X1 + 0.80 X2 -0.20 X3 ≥ 0 (requirement of X2 to be greater than 20%)
-0.50 X1 + 0.50 X2 -0.50 X3 ≤ 0 (requirement of X2 to be less than 50%)
-0.30 X1 -0.30 X2 + 0.70 X3 ≥ 0 (requirement of X3 to be at least 30%)
0.05 X1 + 0.02 X2 -0.04 X3 ≤ 0 (requirement of overall risk index to be not more than 0.05)
X1, X2, X3 ≥ 0

4. The first question of the case deals with solving the stated problem. The second
question asks for the suggestion when the overall risk index is increased by 0.005. The third
question asks whether there will be any change in the previous optimal solution if the yield
for the growth fund is decreased by 2% first and then a further 2%. The fourth question deals
with a constraint where the growth fund cannot be more than the income fund. The last
question asks to what extent we feel this asset allocation model is useful.
Apart from the last question, we will be solving each of them by using
PHPSimplexfrom www.phpsimplex.com/en/. The solution and comments of each question
will be discussed in the subsequent sections.

5. QUESTION 1
Model formulation:
Maximize Z = 0.18 X1 + 0.125 X2 + 0.075 X3
subject to 1X1 + 1 X2 + 1 X3 ≤ 800000
0.8 X1 - 0.2 X2 - 0.2 X3 ≥ 0
0.6 X1 - 0.4 X2 - 0.4 X3 ≤ 0
-0.2 X1 + 0.8 X2 - 0.2 X3 ≥ 0
-0.5 X1 + 0.5 X2 - 0.5 X3 ≤ 0
-0.3 X1 - 0.3 X2 + 0.7 X3 ≥ 0
0.05 X1 + 0.02 X2 - 0.04 X3 ≤ 0
X1, X2, X3 ≥ 0
The problem is converted to canonical form by adding slack, surplus and artificial variables as
appropriate.
 As the constraint 1 is of type '≤' we should add the slack variable X4.
 As the constraint 2 is of type '≥' we should add the surplus variable X5 and the
artificial variable X11.
 As the constraint 3 is of type '≤' we should add the slack variable X6.
 As the constraint 4 is of type '≥' we should add the surplus variable X7 and the
artificial variable X12.
 As the constraint 5 is of type '≤' we should add the slack variable X8.
 As the constraint 6 is of type '≥' we should add the surplus variable X9 and the
artificial variable X13.
 As the constraint 7 is of type '≤' we should add the slack variable X10.
Maximize: 0.18 X1 + 0.125 X2 + 0.075 X3 + 0 X4 + 0 X5 + 0 X6 + 0 X7 + 0 X8 + 0 X9 + 0
X10 + 0 X11 + 0 X12 + 0 X13
1 X1 + 1 X2 + 1 X3 + 1 X4 = 800000
0.8 X1 - 0.2 X2 - 0.2 X3 -1 X5 + 1 X11 = 0
0.6 X1 - 0.4 X2 - 0.4 X3 + 1 X6 = 0

6. -0.2 X1 + 0.8 X2 - 0.2 X3 -1 X7 + 1 X12 = 0
-0.5 X1 + 0.5 X2 - 0.5 X3 + 1 X8 = 0
-0.3 X1 - 0.3 X2 + 0.7 X3 - 1 X9 + 1 X13 = 0
0.05 X1 + 0.02 X2 - 0.04 X3 + 1 X10 = 0
X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13 ≥ 0
There is infinitely many values of X1, X2, X3 for the optimal value Z = 0, which are contained
in the part of the plane 0 X1 + 0 X2 + 0 X3 = 0 that satisfies the constraints of the problem.
One is:
X1 = 248888.88888889
X2 = 160000
X3 = 391111.11111111
Thus, the allocation of the funds is as follows:
Growth Fund, X1 = $ 248,889
Income Fund, X2 = $160,000
Money Market Fund, X3 = $391,111
The allocation of fund will provide maximum annual yield at $94,133
QUESTION 2
If the client’s risk index is increased by one half of a percentage point, from 0.05 to
0.005, then the annual yield of the investment would also be increased by $4,667. This means
that the total annual yield will increases from $94,133 to a new projection of $98,800.
The investment recommendation would change as the following:

7. Fund Allocation Projected Annual Yield
Growth fund, X1 = $293,333 $293,333 x 0.18 = $52,800
Income Fund, X2 = $160,000 $160,000 x 0.125 = $20,000
Money Market Fund, X3 = $346,667 $346,667 x 0.075 = $26,000
Total = $800,000 Total = $ 98,800
Thus, the new total annual yield will be $98,800 compared to before which was $94,133
QUESTION 3
When annual yield of the growth fund would decrease from 18% to 16%, keeping everything
else constant, the only change in our formulation would be the coefficient of X1 in the
objective function. The new formulation would be:
Maximize: 0.16 X1 + 0.125 X2 + 0.075 X3
Subject to:
1 X1 + 1 X2 + 1 X3 ≤ 800000
0.80 X1 -0.20 X2 -0.20 X3 ≥ 0
0.60 X1 -0.40 X2 -0.40 X3 ≤ 0
-0.20 X1 + 0.80 X2 -0.20 X3 ≥ 0
-0.50 X1 + 0.50 X2 -0.50 X3 ≤ 0
-0.30 X1 -0.30 X2 + 0.70 X3 ≥ 0
0.05 X1 + 0.02 X2 -0.04 X3 ≤ 0
X1, X2, X3 ≥ 0
Tableau Form:
Maximize: 0.16 X1 + 0.125 X2 + 0.075 X3 + 0 X4 + 0 X5 + 0 X6 + 0 X7 + 0 X8 + 0 X9 + 0 X10
+ 0 X11 + 0 X12 + 0 X13
Subject to:
1 X1 + 1 X2 + 1 X3 + 1 X4 = 800000

8. 0.8 X1 -0.2 X2 -0.2 X3 -1 X5 + 1 X11 = 0
0.6 X1 -0.4 X2 -0.4 X3 + 1 X6 = 0
-0.2 X1 + 0.8 X2 -0.2 X3 -1 X7 + 1 X12 = 0
-0.5 X1 + 0.5 X2 -0.5 X3 + 1 X8 = 0
-0.3 X1 -0.3 X2 + 0.7 X3 -1 X9 + 1 X13 = 0
0.05 X1 + 0.02 X2 -0.04 X3 + 1 X10 = 0
X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13 ≥ 0
Solving this by the simplex method gives the optimal solution:
Z = 89155.555555556
X1 (growth fund) = 248888.88888889
X2 (income fund) = 160000
X3 (money fund) = 391111.11111111
As we can see, the optimal solution did not change from the original recommendation
even though the annual yield decreased to 16%. However, if it is reduced to 14% the
recommendation will change to optimize Z. Here the coefficient of X1 in the objective
function will change again giving us:
Maximize: 0.14 X1 + 0.125 X2 + 0.075 X3
Subject to:
1 X1 + 1 X2 + 1 X3 ≤ 800000
0.80 X1 -0.20 X2 -0.20 X3 ≥ 0
0.60 X1 -0.40 X2 -0.40 X3 ≤ 0
-0.20 X1 + 0.80 X2 -0.20 X3 ≥ 0
-0.50 X1 + 0.50 X2 -0.50 X3 ≤ 0
-0.30 X1 -0.30 X2 + 0.70 X3 ≥ 0
0.05 X1 + 0.02 X2 -0.04 X3 ≤ 0
X1, X2, X3 ≥ 0

9. Tableau Form:
Maximize: 0.14 X1 + 0.125 X2 + 0.075 X3 + 0 X4 + 0 X5 + 0 X6 + 0 X7 + 0 X8 + 0 X9 + 0 X10
+ 0 X11 + 0 X12 + 0 X13
Subject to:
1 X1 + 1 X2 + 1 X3 + 1 X4 = 800000
0.8 X1 -0.2 X2 -0.2 X3 -1 X5 + 1 X11= 0
0.6 X1 -0.4 X2 -0.4 X3 + 1 X6= 0
-0.2 X1 + 0.8 X2 -0.2 X3 -1 X7 + 1 X12 = 0
-0.5 X1 + 0.5 X2 -0.5 X3 + 1 X8 = 0
-0.3 X1 -0.3 X2 + 0.7 X3 -1 X9 + 1 X13 = 0
0.05 X1 + 0.02 X2 -0.04 X3 + 1 X10 = 0
X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13 ≥ 0
Solving this gives us the optimal solution:
Z = 85066.666666667
X1 (growth fund) = 160000
X2 (income fund) = 293333.33333333
X3 (money fund) = 346666.66666667
As we can see, the optimal solution is quite different from the previous cases. If
annual yield of growth fund decreases to14%, investment in growth and money fund will
decrease, first drastically and the latter slightly, and investment in income fund will increase
by a significant amount in order to maximise the client’s return.
QUESTION 4
The current optimal solution shows the amount of money invested in growth fund is
more than in the income fund which are $248889 and $160000 respectively. However, in
question 4, the amount invested in growth fund is not allowed to exceed the amount invested
in income fund. Therefore, to add this new requirement, we will add a new constraint to the
equation:

10. Maximize: 0.18 X1 + 0.125 X2 + 0.075 X3
Subject to constraints:
1 X1 + 1 X2 + 1 X3 ≤ 800000
0.8 X1 - 0.2 X2 - 0.2 X3 ≥ 0
0.6 X1 - 0.4 X2 - 0.4 X3 ≤ 0
-0.2 X1 + 0.8 X2 - 0.2 X3 ≥ 0
-0.5 X1 + 0.5 X2 - 0.5 X3 ≤ 0
-0.3 X1 - 0.3 X2 + 0.7 X3 ≥ 0
0.05 X1 + 0.02 X2 - 0.04 X3 ≤ 0
1 X1 - 1 X2 + 0 X3 ≤ 0 New Constraint
X1, X2, X3 ≥ 0
Tableau Form
Maximize: 0.18 X1 + 0.125 X2 + 0.075 X3 + 0 X4 + 0 X5 + 0 X6 + 0 X7 + 0 X8 + 0 X9 + 0 X10
+ 0 X11 + 0 X12 + 0 X13 + 0 X14
Subject to:
1 X1 + 1 X2 + 1 X3 + 1 X4 = 800000
0.8 X1 - 0.2 X2 - 0.2 X3 - 1 X5 + 1 X12 = 0
0.6 X1 - 0.4 X2 - 0.4 X3 + 1 X6 = 0
-0.2 X1 + 0.8 X2 - 0.2 X3 - 1 X7 + 1 X13 = 0
-0.5 X1 + 0.5 X2 - 0.5 X3 + 1 X8 = 0
-0.3 X1 - 0.3 X2 + 0.7 X3 - 1 X9 + 1 X14 = 0
0.05 X1 + 0.02 X2 - 0.04 X3 + 1 X10 = 0
1 X1 - 1 X2 + 1 X11 = 0 New Constraint
X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14 ≥ 0
Solving this using the Simplex Method in Tableau format, we got:
The optimal solution value is Z = 93066.666666667
X1 (Growth Fund) = 213333.33333333
X2 (Income Fund) = 213333.33333333
X3 (Money Fund) = 373333.33333333

11. When the new constraint is added to the equation, we found out that the amount to be
invested in both growth fund and income fund will be the same at the risk of 0.18 and 0.125
respectively. And both amounts are decreased from $248889 and $160000 to
$213333.33333333 respectively. The projected annual yield when this constraint is applied is
also much lower than the original of $94,133, which is $93067. This is only a decrease of
0.2% in yield . Since the yield decrease is so small, we may prefer this portfolio.
QUESTION 5
We would recommend the use of this model only when the potential new clients meet
the requirement and outlined criteria of similar objectives and constraints. The company’s
mission, however is to provide the professional, financial advice that best meets the individual
investors’ needs. We would therefore, not recommend the use of this asset allocation model as
a general guide to financial investment.

12. REFERENCES
 Anderson, Sweeney, Williams, Martin; An Introduction to Management Science:
Quantitative Approaches to Decision Making (International Edition); 13th Edition;
South-Western Cengage Learning.
 PHP Simplex; http://www.phpsimplex.com/simplex/simplex.htm?l=en