The document describes a fuzzy portfolio optimization model using trapezoidal possibility distributions to account for uncertainty in asset returns. The model formulates the portfolio selection problem as a mathematical optimization that maximizes expected return minus risk. Lagrange multipliers and Karush-Kuhn-Tucker conditions are used to derive the optimal solution. Real stock market data is used to provide a numerical example.

1. Fuzzy Portfolio Optimization
Using Carlsson-Fullér-Majlender’s Trapezoidal Possibility
Model
Yuxiang Ou
Abstract
Within the framework of Carlsson-Fullér-Majlender’s Trapezoidal Possibility Model, we
apply Lagrange Multiplier Method and Karush-Kuhn-Tucker Conditions to derive the
optimal solution to fuzzy portfolio selection problem.
Keywords: Porfolio selection; Trapezoidal fuzzy variables; Lagrange method; KKT
conditions
Introduction
The portfolio selection problem concerns how to form an optimal portfolio, that is, how to
decide on the weights of every asset to generate the highest level of investor’s utility. Modern
portfolio analysis was pioneered by [7] Markowitz in 1952. As is known to us, people are
risk-averse return-seekers, which means we all want to maximize the return and minimize the
risk of the investment. There is a balance we need to figure out. The most essential point
Markowitz made in the article is that we can use the expected rate of return (mean) to model
the return and the variance of the rate to represent the risk. Markowitz then derived the
optimal choice with the belief that the investors have complete information, which, in most
instances, is not true. To account for uncertainty, it is better to utilize fuzzy set theory by [9]
Zadeh (1965) and [10] Bellman and Zadeh (1970). Many studies on the portfolio selection
problem using various fuzzy formulations emerge. In terms of membership function of the
fuzzy variable, researchers have proposed linear function, tangent type function, interval
linear function, exponential function, inverse tangent function, etc. to address the problem.

2. 2
This paper is typically based on [1] Carlsson et al (2002), which presumes that the
membership function of fuzzy return is in a trapezoidal form. The model is basically the same
and some expressions might be identical. The novelty of this paper is that we reorganize
the argument and provide some proofs that are omitted in the original paper. Also, we use
real stock market data to give a numerical illustration instead of artificially assigning some
values to the model. That is why we get a different result from the original paper.
Furthermore, we develop some critical thinking of the Carlsson-Fullér-Majlender’s model
and state our concerns.
The rest of the paper goes as follows. In Section 2, we mention some preliminaries on the
related issues. In Section 3, we describe the optimization problem in different ways, apply
Lagrange method to deal with it and employ Karush-Kuhn-Tucker conditions to confirm the
minimizer. We list the generalized algorithm in Section 4 and apply it to realistic situations in
Section 5. The paper will cover some personal suggestions in Section 6 and conclude with
Section 7.
Preliminaries
2.1 Utility theory of portfolio investment
A utility function is viewed as a means of ranking portfolios. Higher utility values are
assigned to portfolios with more attractive risk-return profiles. Based on this rule, we can
design a function as follows:
𝑈 𝑃 = 𝐸 𝑟& − 0.005×𝐴×𝜎.
(𝑟&)
where A is an index of the investor’s risk aversion (𝐴 ≈ 2.46 for an average investor in the
USA), 𝑟& is the rate of return on the portfolio and 𝐸 𝑟& and 𝜎.
(𝑟&) represent its mean value
and variance, respectively. The scaling factor of 0.005 allows us to express the expected
return and variance as percentages rather than decimals.
Note that the sign of 𝐸 𝑟& is positive while that of 𝜎.
(𝑟&) is negative, this utility function is
consistent with reality. Moreover, this utility function prevents us from dealing with
complicated multiobjective optimization problems.

3. 3
2.2 Probability or possibility approach
Investors make decisions on portfolio selection according to their knowledge and anticipation
of capital market, budget constraints and available options. Due to limited or incomplete
information one can gather from the market, there exists uncertainty among the decision-
making process that we need to address.
Probability theory is the standard approach to this issue, with the belief that uncertainty is
equated with randomness. Nevertheless, this is not exactly true. Subjective judgement makes
a huge difference in decision-making but it seems difficult to incorporate it into the
probability theory. The assignment of the probabilities would also be problematic when we
demand a higher precision and more decimal places.
Alternatively, in this paper we will assume that the rates of return on assets are modeled by
possibility distributions. That is, the rate of return on the 𝑖th asset will be represented by a
fuzzy number 𝑟8, and 𝑟8 𝑡 , 𝑡𝜖ℛ will be interpreted as the degree of possibility of the
statement that “𝑡 will be the rate of return on the 𝑖th asset”, which is also named as
membership function. In our method, we will consider only trapezoidal possibility
distributions.
2.3 Trapezoidal fuzzy variable
2.3.1 Membership function
The definition of trapezoidal fuzzy variable is based on the membership function.
Definition. A fuzzy number A is called trapezoidal with tolerance interval [𝑎, 𝑏], left width 𝛼
and right width 𝛽 if its membership function has the following form:
𝐴 𝑡 =
1 −
𝑎 − 𝑡
𝛼
𝑖𝑓 𝑎 − 𝛼 ≤ 𝑡 ≤ 𝑎,
1 𝑖𝑓 𝑎 ≤ 𝑡 ≤ 𝑏,
1 −
𝑡 − 𝑏
𝛽
𝑖𝑓 𝑎 ≤ 𝑡 ≤ 𝑏 + 𝛽,
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
and we denote A by 𝐴 = (𝑎, 𝑏, 𝛼, 𝛽).
This membership function can be visualized as

4. 4
2.3.2 𝜸-level set
A 𝛾-level set of a fuzzy variable is composed of all the possibilities with the grade of
membership higher than 𝛾. Then we can modify Fig. 1 to get a closer look at the issue.
Proposition 1. Let 𝐴 = (𝑎, 𝑏, 𝛼, 𝛽) be a trapezoidal fuzzy variable and [𝐴]O
= [𝑎P 𝛾 , 𝑎.(𝛾)]
be the corresponding 𝛾-level set, then [𝐴]O
= 𝑎P 𝛾 , 𝑎. 𝛾 = 𝑎 − 1 − 𝛾 𝛼, 𝑏 +
1 − 𝛾 𝛽 , ∀𝛾𝜖[0,1].
Proof. It can be easily to check that this proposition holds for 𝛾𝜖{0,1}. Let’s focus on
situations where 0 < γ < 1. From Fig. 2, we observe that 𝛾-level line intersects A’s
membership function at two points, i.e. 𝑎P 𝛾 and 𝑎. 𝛾 . Therefore, we need to derive these
two points here.
For 𝑎P 𝛾 , let 1 −
VWX
Y
= 𝛾. We can get 𝑎P 𝛾 = 𝑡 = 𝑎 − (1 − 𝛾)𝛼;
For 𝑎. 𝛾 , let 1 −
XWZ
[
= 𝛾. We can get 𝑎. 𝛾 = 𝑡 = 𝑏 + (1 − 𝛾)𝛽.
Thus, [𝐴]O
= 𝑎P 𝛾 , 𝑎. 𝛾 = 𝑎 − 1 − 𝛾 𝛼, 𝑏 + 1 − 𝛾 𝛽 .
¢

5. 5
2.3.3 Possibilistic mean
The crisp possibilistic mean value of fuzzy variable A with [𝐴]O
= [𝑎P 𝛾 , 𝑎.(𝛾)] is defined
as
𝐸 𝐴 = 𝛾 𝑎P 𝛾 + 𝑎.(𝛾) 𝑑𝛾
P
]
(1)
Proposition 2. Let A be a trapezoidal fuzzy variable denoted as 𝐴 = (𝑎, 𝑏, 𝛼, 𝛽), then.
𝐸 𝐴 =
𝑎 + 𝑏
2
+
𝛽 − 𝛼
6
(2)
Proof. According to the definition, we can calculate possibilistic mean of trapezoidal fuzzy
variable as follows:
𝐸 𝐴 = 𝛾 𝑎P 𝛾 + 𝑎. 𝛾 𝑑𝛾
P
]
= 𝛾 𝑎 − 1 − 𝛾 𝛼 + 𝑏 + 1 − 𝛾 𝛽 𝑑𝛾
P
]
= 𝛾 𝛼 − 𝛽 𝛾 + 𝑎 + 𝑏 + 𝛽 − 𝛼 𝑑𝛾
P
]
= 𝛼 − 𝛽 𝛾.
𝑑𝛾
P
]
+ 𝑎 + 𝑏 + 𝛽 − 𝛼 𝛾𝑑𝛾
P
]
=
𝛼 − 𝛽
3
+
𝑎 + 𝑏 + 𝛽 − 𝛼
2
=
𝑎 + 𝑏
2
+
𝛽 − 𝛼
2
−
𝛽 − 𝛼
3
=
𝑎 + 𝑏
2
+
𝛽 − 𝛼
6
¢
2.3.4 Possibilistic variance
The crisp possibilistic mean value of fuzzy variable A with [𝐴]O
= [𝑎P 𝛾 , 𝑎.(𝛾)] is defined
as
𝜎.
𝐴 =
1
2
𝛾 𝑎. 𝛾 − 𝑎P(𝛾) .
𝑑𝛾
P
]
(3)
Proposition 3. Let A be a trapezoidal fuzzy variable denoted as 𝐴 = (𝑎, 𝑏, 𝛼, 𝛽), then.

6. 6
𝜎.
𝐴 = (
𝑏 − 𝑎
2
+
𝛼 + 𝛽
6
).
+
𝛼 + 𝛽 .
72
(4)
Proof. Similarly, we calculate the possibilistic variance:
𝜎.
𝐴 =
1
2
𝛾 𝑎. 𝛾 − 𝑎P 𝛾
.
𝑑𝛾
P
]
=
1
2
𝛾 𝑏 + 1 − 𝛾 𝛽 − 𝑎 + 1 − 𝛾 𝛼 .
𝑑𝛾
P
]
=
1
2
𝛾 𝑏 − 𝑎 + 𝛼 + 𝛽 − 𝛼 + 𝛽 𝛾 .
𝑑𝛾
P
]
=
1
2
𝛾 𝑏 − 𝑎 + 𝛼 + 𝛽 .
− 2 𝑏 − 𝑎 + 𝛼 + 𝛽 𝛼 + 𝛽 𝛾 + 𝛼 + 𝛽 .
𝛾.
𝑑𝛾
P
]
=
1
2
𝑏 − 𝑎 + 𝛼 + 𝛽 .
𝛾𝑑𝛾
P
]
− 𝑏 − 𝑎 + 𝛼 + 𝛽 𝛼 + 𝛽 𝛾.
𝑑𝛾
P
]
+
1
2
𝛼 + 𝛽 .
𝛾`
𝑑𝛾
P
]
=
𝑏 − 𝑎 + 𝛼 + 𝛽 .
4
−
𝑏 − 𝑎 + 𝛼 + 𝛽 𝛼 + 𝛽
3
+
𝛼 + 𝛽 .
8
=
𝑏 − 𝑎 .
4
+
𝑏 − 𝑎 𝛼 + 𝛽
2
+
𝛼 + 𝛽 .
4
−
𝑏 − 𝑎 𝛼 + 𝛽
3
−
𝛼 + 𝛽 .
3
+
𝛼 + 𝛽 .
8
=
𝑏 − 𝑎 .
4
+
𝑏 − 𝑎 𝛼 + 𝛽
6
+
𝛼 + 𝛽 .
24
= (
𝑏 − 𝑎
2
+
𝛼 + 𝛽
6
).
+
𝛼 + 𝛽 .
72
¢
Analysis of portfolio selection problem
3.1 Basic formulation of the optimization problem
Recalling that the utility function of portfolio investment in our model, which is introduced in
Section 2.1, is 𝑈 𝑃 = 𝐸 𝑟& − 0.005×𝐴×𝜎.
(𝑟&).
Assume that
n: the number of available securities;
𝑥8: the proportion invested in security (or asset) 𝑖, 𝑖 = 1,2, … , 𝑛;

7. 7
𝑟8: the rate of return on security 𝑖;
𝑟&: the rate of return on the portfolio.
Then we know that 𝑟& = 𝑟8 𝑥8
e
8fP and 𝑥8
e
8fP = 1. As we do not consider short-selling and
long-buying, we also have 0 ≤ 𝑥8 ≤ 1.
Accordingly, our portfolio selection problem is equivalent to the following mathematical
programming problem:
max
jk
𝑈 𝑃 = 𝐸 𝑟8 𝑥8
e
8fP
− 0.005×𝐴×𝜎.
𝑟8 𝑥8
e
8fP
s. t. { 𝑥8
e
8fP
= 1, 𝑥8 ≥ 0 , 𝑖 = 1,2, … , 𝑛}
(5)
Where 𝑟8 = 𝑎8, 𝑏8, 𝛼8, 𝛽8 , 𝑖 = 1,2, … , 𝑛 are fuzzy variables of trapezoidal form.
3.2 Translations of the optimization problem
Note that in Section 2.3.3 and 2.3.4 we have derived that the possibilistic mean and variance
of a trapezoidal fuzzy variable 𝐴 = (𝑎, 𝑏, 𝛼, 𝛽) are 𝐸 𝐴 =
VnZ
.
+
[WY
o
and 𝜎.
𝐴 =
(
ZWV
.
+
Yn[
o
).
+
Yn[ p
q.
, respectively.
Then for trapezoidal fuzzy number 𝑟8 = 𝑎8, 𝑏8, 𝛼8, 𝛽8 , 𝑖 = 1,2, … , 𝑛, we have
𝐸 𝑟8 =
VknZk
.
+
[kWYk
o
=
P
.
[𝑎8 + 𝑏8 +
P
`
(𝛽8 − 𝛼8)], thus,
𝐸 𝑟8 𝑥8
e
8fP = 𝑥8 𝐸(𝑟8)e
8fP =
P
.
[𝑎8 + 𝑏8 +
P
`
(𝛽8 − 𝛼8)]𝑥8
e
8fP (6)
And since
𝜎.
𝑟8 = (
ZkWVk
.
+
Ykn[k
o
).
+
Ykn[k
p
q.
= (
P
.
𝑏8 − 𝑎8 +
P
`
𝛼8 + 𝛽8 ).
+
Ykn[k
p
q.
, when we
ignore the covariance between rate of returns on different securities, we have
𝜎.
( 𝑟8 𝑥8
e
8fP ) = (
P
.
𝑏8 − 𝑎8 +
P
`
𝛼8 + 𝛽8
e
8fP 𝑥8).
+
P
q.
[ 𝛼8 + 𝛽8 𝑥8
e
8fP ].
(7)
If we introduce the notations as:
𝑢8 =
P
.
[𝑎8 + 𝑏8 +
P
`
(𝛽8 − 𝛼8)],
𝑣8 =
].]]tu
.
𝑏8 − 𝑎8 +
P
`
𝛼8 + 𝛽8 ,
𝑤8 =
].]]tu
q.
(𝛼8 + 𝛽8), then
𝐸 𝑟8 𝑥8
e
8fP =
P
.
[𝑎8 + 𝑏8 +
P
`
(𝛽8 − 𝛼8)]𝑥8
e
8fP = 𝑢8 𝑥8
e
8fP ,

8. 8
𝜎.
𝑟8 𝑥8
e
8fP
=
1
2
𝑏8 − 𝑎8 +
1
3
𝛼8 + 𝛽8
e
8fP
𝑥8
.
+
1
72
𝛼8 + 𝛽8 𝑥8
e
8fP
.
=
1
0.005𝐴
𝑣8
e
8fP
𝑥8
.
+
1
72
72
0.005𝐴
𝑤8 𝑥8
e
8fP
.
=
1
0.005𝐴
𝑣8
e
8fP
𝑥8
.
+
1
0.005𝐴
𝑤8
e
8fP
𝑥8
.
thus,
𝑈 𝑃 = 𝐸 𝑟8 𝑥8
e
8fP
− 0.005×𝐴×𝜎.
𝑟8 𝑥8
e
8fP
= 𝑢8 𝑥8
e
8fP
− 0.005𝐴×
1
0.005𝐴
𝑣8
e
8fP
𝑥8
.
+
1
0.005𝐴
𝑤8
e
8fP
𝑥8
.
= 𝑢8 𝑥8
e
8fP
− 𝑣8
e
8fP
𝑥8
.
− 𝑤8
e
8fP
𝑥8
.
and the optimization problem becomes
max
jk
𝑈 𝑃 = 𝑢8 𝑥8
e
8fP
− 𝑣8
e
8fP
𝑥8
.
− 𝑤8
e
8fP
𝑥8
.
s. t. { 𝑥8
e
8fP
= 1, 𝑥8 ≥ 0 , 𝑖 = 1,2, … , 𝑛}
(8)
Here the 𝑖th asset is represented by a triplet (𝑣8, 𝑤8, 𝑢8), where 𝑢8 denotes its possibilistic
expected value, and 𝑣8
.
+ 𝑤8
.
denotes its possibilistic variance multiplied by the constant
0.005×𝐴.
The convex hull of { 𝑣8, 𝑤8, 𝑢8 : 𝑖 = 1,2, … , 𝑛}, denoted by 𝑇, and defined by
𝑇 = 𝑐𝑜𝑛𝑣 𝑣8, 𝑤8, 𝑢8 : 𝑖 = 1,2, … , 𝑛
= { 𝑣8
e
8fP
𝑥8, 𝑤8
e
8fP
𝑥8, 𝑢8 𝑥8
e
8fP
: 𝑥8
e
8fP
= 1, 𝑥8 ≥ 0 , 𝑖 = 1,2, … , 𝑛}
is a convex polyhedron in ℛ`
. We can move to any point in the polytope by varying the value
of 𝑥8. In other words, let 𝑣] = 𝑣8
e
8fP 𝑥8, 𝑤] = 𝑤8
e
8fP 𝑥8, and 𝑢] = 𝑢8
e
8fP 𝑥8, we need to
find the point within the polytope generating the highest value of 𝑢] − 𝑣]
.
− 𝑤]
.
. Then
problem (8) turns into the following three-dimensional non-linear programming problem:
max
yz,{z,|z
𝑈 𝑃 = 𝑢] − 𝑣]
.
− 𝑤]
.
s. t. 𝑣], 𝑤], 𝑢] 𝜖 𝑇
(9)
Or, equivalently,
min
yz,{z,|z
𝑈 𝑃 = 𝑣]
.
+ 𝑤]
.
− 𝑢]
s. t. 𝑣], 𝑤], 𝑢] 𝜖 𝑇
(10)

9. 9
Note that 𝑇 is a compact and convex subset of ℛ`
, and the implicit function
𝑔€ 𝑣], 𝑤] = 𝑣]
.
+ 𝑤]
.
− 𝑐
is strictly convex for any 𝑐 𝜖 ℛ. This means that any optimal solution to (10) must be on the
boundary of 𝑇. As 𝑇 is a polyhedron of ℛ`
and the optimal solution must be on the boundary
of 𝑇, then any optimal solution can be obtained as a convex combination of at most 3 extreme
points of 𝑇. [1] Carlsson, Fullér and Majlender (2002) presented an algorithm for finding
such an optimal solution. In the algorithm, one should calculate: (i) the (exact) solutions to all
conceivable 3-asset problems with non-collinear assets, (ii) the (exact) solutions to all
conceivable 2-asset problems with distinguishable assets, and (iii) the utility value of each
asset. Then one can compare the utility values of all feasible solutions and portfolios with the
highest utility value will be chosen as optimal solutions to the portfolio selection problem.
3.3 Optimal solutions
3.3.1 3-asset problems
Consider three noncollinear assets 𝑣8, 𝑤8, 𝑢8 , i = 1,2,3,
Proposition 4. For any noncollinear assets 𝑣8, 𝑤8, 𝑢8 , i = 1,2,3, ∄ 𝛼P, 𝛼. 𝜖ℛ.
, 𝛼P, 𝛼. ≠ 0,
such that
𝛼P
𝑣P
𝑤P
𝑢P
+ 𝛼.
𝑣.
𝑤.
𝑢.
− 𝛼P + 𝛼.
𝑣`
𝑤`
𝑢`
= 0.
Proof. Suppose there exists 𝛼P, 𝛼. 𝜖ℛ.
, 𝛼P, 𝛼. ≠ 0, such that
𝛼P
𝑣P
𝑤P
𝑢P
+ 𝛼.
𝑣.
𝑤.
𝑢.
− 𝛼P + 𝛼.
𝑣`
𝑤`
𝑢`
= 0, then
𝛼P
𝑣P − 𝑣`
𝑤P − 𝑤`
𝑢P − 𝑢`
+ 𝛼.
𝑣. − 𝑣`
𝑤. − 𝑤`
𝑢. − 𝑢`
= 0, that is, we have
𝑣P − 𝑣`
𝑤P − 𝑤`
𝑢P − 𝑢`
= −
Yp
Yƒ
𝑣. − 𝑣`
𝑤. − 𝑤`
𝑢. − 𝑢`
if 𝛼P ≠ 0;
or
𝑣. − 𝑣`
𝑤. − 𝑤`
𝑢. − 𝑢`
= −
Yƒ
Yp
𝑣P − 𝑣`
𝑤P − 𝑤`
𝑢P − 𝑢`
if 𝛼. ≠ 0.
We find collinearity in both cases, which contradicts our noncollinear assumptions.
¢

10. 10
Then the 3-asset optimal portfolio selection problem with not-necessarily non-negative
weights is
min
jƒ,jp,j„
𝑈 𝑃 = (𝑣P 𝑥P + 𝑣. 𝑥. + 𝑣` 𝑥`).
+ (𝑤P 𝑥P + 𝑤. 𝑥. + 𝑤` 𝑥`).
− (𝑢P 𝑥P + 𝑢. 𝑥. + 𝑢` 𝑥`)
s. t. 𝑥P + 𝑥. + 𝑥` = 1
(11)
Let
𝐿 𝑥, 𝜆 = 𝑣P 𝑥P + 𝑣. 𝑥. + 𝑣` 𝑥`
.
+ 𝑤P 𝑥P + 𝑤. 𝑥. + 𝑤` 𝑥`
.
− 𝑢P 𝑥P + 𝑢. 𝑥. + 𝑢` 𝑥` + 𝜆(𝑥P + 𝑥. + 𝑥` − 1)
(12)
be the Lagrange function of the constrained optimization problem (11). Then the Kuhn-
Tucker necessity conditions are
2𝑣P 𝑣P 𝑥P + 𝑣. 𝑥. + 𝑣` 𝑥` + 2𝑤P 𝑤P 𝑥P + 𝑤. 𝑥. + 𝑤` 𝑥` − 𝑢P + 𝜆 = 0
2𝑣. 𝑣P 𝑥P + 𝑣. 𝑥. + 𝑣` 𝑥` + 2𝑤. 𝑤P 𝑥P + 𝑤. 𝑥. + 𝑤` 𝑥` − 𝑢. + 𝜆 = 0
2𝑣` 𝑣P 𝑥P + 𝑣. 𝑥. + 𝑣` 𝑥` + 2𝑤` 𝑤P 𝑥P + 𝑤. 𝑥. + 𝑤` 𝑥` − 𝑢` + 𝜆 = 0
𝑥P + 𝑥. + 𝑥` − 1 = 0
(13)
Proposition 5. The Karush-Kuhn-Tucker necessity conditions listed as (13) can be
transformed into the following linear equality system:
𝑞P
.
+ 𝑟P
.
𝑞P 𝑞. + 𝑟P 𝑟.
𝑞P 𝑞. + 𝑟P 𝑟. 𝑞.
.
+ 𝑟.
.
𝑥P
𝑥.
=
1
2
𝑢P − 𝑢` − 𝑞P 𝑣` − 𝑟P 𝑤`
1
2
𝑢. − 𝑢` − 𝑞. 𝑣` − 𝑟. 𝑤`
𝑤ℎ𝑒𝑟𝑒 𝑞P = 𝑣P − 𝑣`, 𝑞. = 𝑣. − 𝑣`, 𝑟P = 𝑤P − 𝑤` 𝑎𝑛𝑑 𝑟. = 𝑤. − 𝑤`
(14)
Proof. From the third equation in (13), we have
𝜆 = 𝑢` − 2𝑣` 𝑣P 𝑥P + 𝑣. 𝑥. + 𝑣` 𝑥` − 2𝑤` 𝑤P 𝑥P + 𝑤. 𝑥. + 𝑤` 𝑥` ,
from the fourth equation, we have 𝑥` = 1 − 𝑥P − 𝑥..
Substituting 𝜆 and 𝑥` with these two expressions,
the first equation in (13) becomes
2𝑣P 𝑣P 𝑥P + 𝑣. 𝑥. + 𝑣` 1 − 𝑥P − 𝑥. + 2𝑤P 𝑤P 𝑥P + 𝑤. 𝑥. + 𝑤` 1 − 𝑥P − 𝑥. − 𝑢P + 𝑢`
− 2𝑣` 𝑣P 𝑥P + 𝑣. 𝑥. + 𝑣` 1 − 𝑥P − 𝑥.
− 2𝑤` 𝑤P 𝑥P + 𝑤. 𝑥. + 𝑤` 1 − 𝑥P − 𝑥. = 0
that is,
2 𝑣P − 𝑣` (𝑣P − 𝑣`)𝑥P + (𝑣.−𝑣`)𝑥. + 𝑣` + 2(𝑤P
− 𝑤`) (𝑤P − 𝑤`)𝑥P + (𝑤.−𝑤`)𝑥. + 𝑤` − 𝑢P + 𝑢` = 0
recalling that 𝑞P = 𝑣P − 𝑣`, 𝑞. = 𝑣. − 𝑣`, 𝑟P = 𝑤P − 𝑤` 𝑎𝑛𝑑 𝑟. = 𝑤. − 𝑤`, we have
2𝑞P(𝑞P 𝑥P + 𝑞. 𝑥. + 𝑣`) + 2𝑟P(𝑟P 𝑥P + 𝑟. 𝑥. + 𝑤`) − 𝑢P + 𝑢` = 0

11. 11
i.e. 2[ 𝑞P
.
+ 𝑟P
.
𝑥P + 𝑞P 𝑞. + 𝑟P 𝑟. 𝑥. + (𝑞P 𝑣` + 𝑟P 𝑤`)] − 𝑢P + 𝑢` = 0
simplifying and rearranging,
𝑞P
.
+ 𝑟P
.
𝑥P + 𝑞P 𝑞. + 𝑟P 𝑟. 𝑥. =
1
2
𝑢P − 𝑢` − 𝑞P 𝑣` − 𝑟P 𝑤`
Similarly, for the second equation, we substitute 𝜆 and 𝑥` and then simplify and rearrange it
to
𝑞P 𝑞. + 𝑟P 𝑟. 𝑥P + 𝑞.
.
+ 𝑟.
.
𝑥. =
1
2
𝑢. − 𝑢` − 𝑞. 𝑣` − 𝑟. 𝑤`
therefore, the equation system (13) is equivalent to
𝑞P
.
+ 𝑟P
.
𝑥P + 𝑞P 𝑞. + 𝑟P 𝑟. 𝑥. =
1
2
𝑢P − 𝑢` − 𝑞P 𝑣` − 𝑟P 𝑤`
𝑞P 𝑞. + 𝑟P 𝑟. 𝑥P + 𝑞.
.
+ 𝑟.
.
𝑥. =
1
2
𝑢. − 𝑢` − 𝑞. 𝑣` − 𝑟. 𝑤`
which can be expressed in matrix form as
𝑞P
.
+ 𝑟P
.
𝑞P 𝑞. + 𝑟P 𝑟.
𝑞P 𝑞. + 𝑟P 𝑟. 𝑞.
.
+ 𝑟.
.
𝑥P
𝑥.
=
1
2
𝑢P − 𝑢` − 𝑞P 𝑣` − 𝑟P 𝑤`
1
2
𝑢. − 𝑢` − 𝑞. 𝑣` − 𝑟. 𝑤`
¢
Now we try to figure out the solution to equation (14). Before that, it would be helpful to
consider the uniqueness of the solution.
Proposition 6. If 𝑣8, 𝑤8, 𝑢8 , i = 1,2,3 are not collinear, then equation (14) has a unique
solution.
Proof. Suppose that the solution to equation (14) is not unique, i.e.
𝑑𝑒𝑡
𝑞P
.
+ 𝑟P
.
𝑞P 𝑞. + 𝑟P 𝑟.
𝑞P 𝑞. + 𝑟P 𝑟. 𝑞.
.
+ 𝑟.
. = 0
that is,
𝑑𝑒𝑡
𝑞P
.
+ 𝑟P
.
𝑞P 𝑞. + 𝑟P 𝑟.
𝑞P 𝑞. + 𝑟P 𝑟. 𝑞.
.
+ 𝑟.
. = 𝑞P
.
+ 𝑟P
.
𝑞.
.
+ 𝑟.
.
− 𝑞P 𝑞. + 𝑟P 𝑟.
.
= 𝑞P
.
𝑞.
.
+ 𝑞P
.
𝑟.
.
+ 𝑟P
.
𝑞.
.
+ 𝑟P
.
𝑟.
.
− 𝑞P
.
𝑞.
.
+ 2𝑞P 𝑞. 𝑟P 𝑟. + 𝑟P
.
𝑟.
.
= 𝑞P
.
𝑟.
.
+ 𝑟P
.
𝑞.
.
− 2𝑞P 𝑞. 𝑟P 𝑟.
= 𝑞P 𝑟. − 𝑟P 𝑞.
.
= 𝑑𝑒𝑡
𝑞P 𝑟P
𝑞. 𝑟.
.
= 0

12. 12
i.e. 𝑑𝑒𝑡
𝑞P 𝑟P
𝑞. 𝑟.
= 0.
Thus, the row of
𝑞P 𝑟P
𝑞. 𝑟.
are not linearly independent: ∃ (𝛼P, 𝛼.) ≠ 0 such that
𝛼P 𝑞P, 𝑟P + 𝛼. 𝑞., 𝑟. = 0 ⇔
𝛼P 𝑣P − 𝑣`, 𝑤P − 𝑤` + 𝛼. 𝑣. − 𝑣`, 𝑤. − 𝑤` = 0
(15)
Suppose 𝛼P ≠ 0, then 𝑞P = −
Yp
Yƒ
𝑞. and 𝑟P = −
Yp
Yƒ
𝑟., (14) turns into
(−
𝛼.
𝛼P
𝑞.).
+ (−
𝛼.
𝛼P
𝑟.).
−
𝛼.
𝛼P
𝑞.
.
−
𝛼.
𝛼P
𝑟.
.
−
𝛼.
𝛼P
𝑞.
.
−
𝛼.
𝛼P
𝑟.
.
𝑞.
.
+ 𝑟.
.
𝑥P
𝑥.
=
1
2
𝑢P − 𝑢` +
𝛼.
𝛼P
𝑞. 𝑣` +
𝛼.
𝛼P
𝑟. 𝑤`
1
2
𝑢. − 𝑢` − 𝑞. 𝑣` − 𝑟. 𝑤`
that is,
(𝑞.
.
+ 𝑟.
.
)
𝛼.
.
𝛼P
. −
𝛼.
𝛼P
−
𝛼.
𝛼P
1
𝑥P
𝑥.
=
1
2
𝑢P − 𝑢` +
𝛼.
𝛼P
𝑞. 𝑣` +
𝛼.
𝛼P
𝑟. 𝑤`
1
2
𝑢. − 𝑢` − 𝑞. 𝑣` − 𝑟. 𝑤`
i.e.
(𝑞.
.
+ 𝑟.
.
)
𝛼.
.
−𝛼P 𝛼.
−𝛼P 𝛼. 𝛼P
.
𝑥P
𝑥.
= 𝛼P
1
2
𝛼P 𝑢P − 𝑢` + 𝛼.(𝑞. 𝑣` + 𝑟. 𝑤`)
1
2
𝛼P 𝑢. − 𝑢` − 𝛼P(𝑞. 𝑣` + 𝑟. 𝑤`)
Multiplying both sides by [𝛼P, 𝛼.] we get
0 = 𝛼P
𝛼P 𝛼.
1
2
𝛼P 𝑢P − 𝑢` + 𝛼. 𝑞. 𝑣` + 𝑟. 𝑤`
1
2
𝛼P 𝑢. − 𝑢` − 𝛼P 𝑞. 𝑣` + 𝑟. 𝑤`
= 𝛼P
.
1
2
𝛼P 𝑢P − 𝑢` + 𝛼. 𝑞. 𝑣` + 𝑟. 𝑤` + 𝛼P 𝛼.
1
2
𝛼P 𝑢. − 𝑢` − 𝛼P 𝑞. 𝑣` + 𝑟. 𝑤`
=
1
2
𝛼P
.
[𝛼P 𝑢P − 𝑢` + 𝛼. 𝑢. − 𝑢` ]
Note that we suppose 𝛼P ≠ 0, then we get 𝛼P 𝑢P − 𝑢` + 𝛼. 𝑢. − 𝑢` = 0. Combine this
with equation (15) we have
𝛼P
𝑣P
𝑤P
𝑢P
+ 𝛼.
𝑣.
𝑤.
𝑢.
− 𝛼P + 𝛼.
𝑣`
𝑤`
𝑢`
= 0
i.e. 𝑣8, 𝑤8, 𝑢8 , i = 1,2,3, which contradicts our noncollinearity assumption.
If 𝛼P = 0 then 𝛼. ≠ 0, and from equation (15) we know that 𝑞. = 𝑟. = 0. Then equation
(14) becomes

13. 13
𝑞P
.
+ 𝑟P
.
0
0 0
𝑥P
𝑥.
=
1
2
𝑢P − 𝑢` − 𝑞P 𝑣` − 𝑟P 𝑤`
1
2
𝑢. − 𝑢`
Multiplying both sides by [0,1], we obtain
P
.
𝑢. − 𝑢` = 0, i.e. 𝑢. − 𝑢` = 0.
Note that 𝑞. = 𝑟. = 0 which implies 𝑣. − 𝑣` = 𝑤. − 𝑤` = 0, thus
𝑣. − 𝑣` = 𝑤. − 𝑤` = 𝑢. − 𝑢` = 0. This means that 𝑣8, 𝑤8, 𝑢8 , i = 1,2,3 are collinear,
which contradicts our noncollinearity assumption.
¢
Now we turn to the search for this unique solution.
Using the general matrix inverse formula:
𝑡P 𝑡.
𝑡` 𝑡Š
WP
=
1
𝑡P 𝑡Š − 𝑡. 𝑡`
𝑡Š −𝑡.
−𝑡` 𝑡P
we find the optimal solution to (14) is
𝑥P
∗
𝑥.
∗
=
1
𝑞P
.
+ 𝑟P
.
𝑞.
.
+ 𝑟.
.
− 𝑞P 𝑞. + 𝑟P 𝑟.
.
𝑞.
.
+ 𝑟.
.
− 𝑞P 𝑞. + 𝑟P 𝑟.
−(𝑞P 𝑞. + 𝑟P 𝑟.) 𝑞P
.
+ 𝑟P
.
×
1
2
𝑢P − 𝑢` − 𝑞P 𝑣` − 𝑟P 𝑤`
1
2
𝑢. − 𝑢` − 𝑞. 𝑣` − 𝑟. 𝑤`
=
1
𝑞P 𝑟. − 𝑟P 𝑞.
.
𝑞.
.
+ 𝑟.
.
− 𝑞P 𝑞. + 𝑟P 𝑟.
−(𝑞P 𝑞. + 𝑟P 𝑟.) 𝑞P
.
+ 𝑟P
.
×
1
2
𝑢P − 𝑢` − 𝑞P 𝑣` − 𝑟P 𝑤`
1
2
𝑢. − 𝑢` − 𝑞. 𝑣` − 𝑟. 𝑤`
(16)
thus, we know that 𝑥∗
= 𝑥P
∗
, 𝑥.
∗
, 𝑥`
∗
= (𝑥P
∗
, 𝑥.
∗
, 1 − 𝑥P
∗
− 𝑥.
∗
) is a candidate for a constrained
minimizer. To ensure that our portfolio selection function given by equation (11) minimizes
at 𝑥 = 𝑥∗
, it is necessary to check the Karush-Kuhn-Tucker sufficiency condition.
Proposition 7. 𝑥∗
= 𝑥P
∗
, 𝑥.
∗
, 𝑥`
∗
= (𝑥P
∗
, 𝑥.
∗
, 1 − 𝑥P
∗
− 𝑥.
∗
) satisfies the Kuhn-Tucker
sufficiency condition and constitutes a minimal solution to problem (11) if 𝑥P
∗
≥ 0, 𝑥.
∗
≥ 0
and 𝑥`
∗
≥ 0.
Proof. Recalling that

14. 14
𝐿 𝑥, 𝜆 = 𝑣P 𝑥P + 𝑣. 𝑥. + 𝑣` 𝑥`
.
+ 𝑤P 𝑥P + 𝑤. 𝑥. + 𝑤` 𝑥`
.
− 𝑢P 𝑥P + 𝑢. 𝑥. + 𝑢` 𝑥` + 𝜆(𝑥P + 𝑥. + 𝑥` − 1)
(12)
We need to show that 𝐿′′(𝑥, 𝜆) is a positive definite matrix at 𝑥 = 𝑥∗
in the subset defined by
{𝑦 = 𝑦P, 𝑦., 𝑦` 𝜖ℛ`
: 𝑦P + 𝑦. + 𝑦` = 0}.
Since
∇j 𝐿 𝑥, 𝜆 =
2𝑣P 𝑣P 𝑥P + 𝑣. 𝑥. + 𝑣` 𝑥` + 2𝑤P 𝑤P 𝑥P + 𝑤. 𝑥. + 𝑤` 𝑥` − 𝑢P + 𝜆
2𝑣. 𝑣P 𝑥P + 𝑣. 𝑥. + 𝑣` 𝑥` + 2𝑤. 𝑤P 𝑥P + 𝑤. 𝑥. + 𝑤` 𝑥` − 𝑢P + 𝜆
2𝑣` 𝑣P 𝑥P + 𝑣. 𝑥. + 𝑣` 𝑥` + 2𝑤` 𝑤P 𝑥P + 𝑤. 𝑥. + 𝑤` 𝑥` − 𝑢P + 𝜆
then
∇j
.
𝐿 𝑥, 𝜆 = 2
𝑣P
.
+ 𝑤P
.
𝑣P 𝑣. + 𝑤P 𝑤. 𝑣P 𝑣` + 𝑤P 𝑤`
𝑣P 𝑣. + 𝑤P 𝑤. 𝑣.
.
+ 𝑤.
.
𝑣. 𝑣` + 𝑤. 𝑤`
𝑣P 𝑣` + 𝑤P 𝑤` 𝑣. 𝑣` + 𝑤. 𝑤` 𝑣`
.
+ 𝑤`
.
= 2(
𝑣P
𝑣.
𝑣`
𝑣P
𝑣.
𝑣`
•
+
𝑤P
𝑤.
𝑤`
𝑤P
𝑤.
𝑤`
•
)
hence, the following inequality
𝑦•
∇j
.
𝐿 𝑥, 𝜆 𝑦 =
𝑦P
𝑦.
𝑦`
•
×2
𝑣P
𝑣.
𝑣`
𝑣P
𝑣.
𝑣`
•
+
𝑤P
𝑤.
𝑤`
𝑤P
𝑤.
𝑤`
•
×
𝑦P
𝑦.
𝑦`
= 2
×
𝑦P
𝑦.
𝑦`
•
×
𝑣P
𝑣.
𝑣`
×
𝑣P
𝑣.
𝑣`
•
×
𝑦P
𝑦.
𝑦`
+
𝑦P
𝑦.
𝑦`
•
×
𝑤P
𝑤.
𝑤`
×
𝑤P
𝑤.
𝑤`
•
×
𝑦P
𝑦.
𝑦`
= 2× 𝑣P 𝑦P + 𝑣. 𝑦. + 𝑣` 𝑦`
.
+ 𝑤P 𝑦P + 𝑤. 𝑦. + 𝑤` 𝑦`
.
≥ 0
(17)
holds for any 𝑦 = 𝑦P, 𝑦., 𝑦` 𝜖ℛ`
. That is, ∇j
.
𝐿 𝑥, 𝜆 is a positive semidefinite matrix.
If 𝑦•
∇j
.
𝐿 𝑥, 𝜆 𝑦 = 0, then from (17) we have
𝑣P 𝑦P + 𝑣. 𝑦. + 𝑣` 𝑦` = 0, 𝑤P 𝑦P + 𝑤. 𝑦. + 𝑤` 𝑦` = 0;
Suppose for some 𝑦 = 𝑦P, 𝑦., 𝑦` ≠ 0, 𝑦P + 𝑦. + 𝑦` = 0, these two equalities satisfy,
that is,
𝑣P 𝑣. 𝑣`
𝑤P 𝑤. 𝑤`
1 1 1
𝑦P
𝑦.
𝑦`
=
0
0
0
has nonzero solutions, which implies
𝑑𝑒𝑡
𝑣P 𝑣. 𝑣`
𝑤P 𝑤. 𝑤`
1 1 1
= 𝑑𝑒𝑡
𝑣P − 𝑣` 𝑣. − 𝑣` 𝑣`
𝑤P − 𝑤` 𝑤. − 𝑤` 𝑤`
0 0 1
= 𝑑𝑒𝑡
𝑣P − 𝑣` 𝑣. − 𝑣`
𝑤P − 𝑤` 𝑤. − 𝑤`
= 𝑑𝑒𝑡
𝑞P 𝑞.
𝑟P 𝑟.

15. 15
= 𝑑𝑒𝑡
𝑞P 𝑟P
𝑞. 𝑟.
= 0
As our proof of Proposition 6, this contradicts our noncollinearity assumption of
𝑣8, 𝑤8, 𝑢8 , i = 1,2,3. So 𝑦•
∇j
.
𝐿 𝑥, 𝜆 𝑦 ≠ 0.
Then from inequality (17), we know that 𝑦•
∇j
.
𝐿 𝑥, 𝜆 𝑦 > 0, i.e. 𝐿′′(𝑥, 𝜆) is a positive
definite matrix at 𝑥 = 𝑥∗
, thus 𝑥 = 𝑥∗
is a minimizer of the utility function in problem (11).
¢
3.3.2 2-asset problems
Now consider a 2-asset problem with two assets, denoted as 𝑣P, 𝑤P, 𝑢P and 𝑣., 𝑤., 𝑢. ,
such that 𝑣P, 𝑤P, 𝑢P ≠ 𝑣., 𝑤., 𝑢. . The optimization problem turns into
min
jƒ,jp,
𝑈 𝑃 = (𝑣P 𝑥P + 𝑣. 𝑥.).
+ (𝑤P 𝑥P + 𝑤. 𝑥.).
− (𝑢P 𝑥P + 𝑢. 𝑥.)
s. t. 𝑥P + 𝑥. = 1
(18)
The Lagrange function of this constrained problem is
𝐿 𝑥, 𝜆 = 𝑣P 𝑥P + 𝑣. 𝑥.
.
+ 𝑤P 𝑥P + 𝑤. 𝑥.
.
− 𝑢P 𝑥P + 𝑢. 𝑥. + 𝜆(𝑥P + 𝑥. − 1) (19)
The Karush-Kuhn-Tucker necessity conditions are
2𝑣P 𝑣P 𝑥P + 𝑣. 𝑥. + 2𝑤P 𝑤P 𝑥P + 𝑤. 𝑥. − 𝑢P + 𝜆 = 0
2𝑣. 𝑣P 𝑥P + 𝑣. 𝑥. + 2𝑤. 𝑤P 𝑥P + 𝑤. 𝑥. − 𝑢. + 𝜆 = 0
𝑥P + 𝑥. − 1 = 0
(20)
Subtract the second equation from the first one in (20), we get
2 𝑣P − 𝑣. 𝑣P 𝑥P + 𝑣. 𝑥. + 2 𝑤P − 𝑤. 𝑤P 𝑥P + 𝑤. 𝑥. − (𝑢P − 𝑢.) = 0
and we substitute 𝑥. using the third equation:
2 𝑣P − 𝑣. [𝑣P 𝑥P + 𝑣.(1 − 𝑥P)] + 2 𝑤P − 𝑤. [𝑤P 𝑥P + 𝑤.(1 − 𝑥P)] − (𝑢P − 𝑢.) = 0
2 𝑣P − 𝑣.
.
𝑥P + 𝑤P − 𝑤.
.
𝑥P + 2 𝑣P − 𝑣. 𝑣. + 2 𝑤P − 𝑤. 𝑤. = (𝑢P − 𝑢.)
i.e.
𝑣P − 𝑣.
.
+ 𝑤P − 𝑤.
.
𝑥P =
1
2
𝑢P − 𝑢. − 𝑣P − 𝑣. 𝑣. − 𝑤P − 𝑤. 𝑤. (21)
If 𝑣P − 𝑣.
.
+ 𝑤P − 𝑤.
.
≠ 0 then we find the solution 𝑥∗
= 𝑥P
∗
, 𝑥.
∗
= (𝑥P
∗
, 1 − 𝑥P
∗
)
where
𝑥P
∗
=
1
𝑣P − 𝑣.
. + 𝑤P − 𝑤.
.
×[
1
2
𝑢P − 𝑢. − 𝑣P − 𝑣. 𝑣. − 𝑤P − 𝑤. 𝑤.] (22)
Otherwise, if 𝑣P = 𝑣. and 𝑤P = 𝑤. then from equation (21) we find 𝑢P = 𝑢., which
contradicts the assumption that the two assets are not identical. Therefore, we can always

16. 16
have a candidate solution to the constrained minimizer problem. The only question is whether
this candidate solution minimizes our selection function or not.
Similarly, we take a look at 𝐿′′(𝑥, 𝜆).
Since
𝐿 𝑥, 𝜆 = 𝑣P 𝑥P + 𝑣. 𝑥.
.
+ 𝑤P 𝑥P + 𝑤. 𝑥.
.
− 𝑢P 𝑥P + 𝑢. 𝑥. + 𝜆(𝑥P + 𝑥. − 1) (19)
then
∇j 𝐿 𝑥, 𝜆 =
2𝑣P 𝑣P 𝑥P + 𝑣. 𝑥. + 2𝑤P 𝑤P 𝑥P + 𝑤. 𝑥. − 𝑢P + 𝜆
2𝑣. 𝑣P 𝑥P + 𝑣. 𝑥. + 2𝑤. 𝑤P 𝑥P + 𝑤. 𝑥. − 𝑢. + 𝜆
so
∇j
.
𝐿 𝑥, 𝜆 = 2
𝑣P
.
+ 𝑤P
.
𝑣P 𝑣. + 𝑤P 𝑤.
𝑣P 𝑣. + 𝑤P 𝑤. 𝑣.
.
+ 𝑤.
. = 2(
𝑣P
𝑣.
𝑣P
𝑣.
•
+
𝑤P
𝑤.
𝑤P
𝑤.
•
)
hence,
𝑦•
∇j
.
𝐿 𝑥, 𝜆 𝑦 =
𝑦P
𝑦.
•
×2
𝑣P
𝑣.
𝑣P
𝑣.
•
+
𝑤P
𝑤.
𝑤P
𝑤.
•
×
𝑦P
𝑦.
= 2× 𝑣P 𝑦P + 𝑣. 𝑦.
.
+ 𝑤P 𝑦P + 𝑤. 𝑦.
.
≥ 0
holds for any 𝑦 = 𝑦P, 𝑦. 𝜖ℛ.
.
If 𝑦•
∇j
.
𝐿 𝑥, 𝜆 𝑦 = 0 then 𝑣P 𝑦P + 𝑣. 𝑦. = 0 and 𝑤P 𝑦P + 𝑤. 𝑦. = 0.
For any 𝑦 = 𝑦P, 𝑦. 𝜖ℛ.
such that 𝑦P, 𝑦. ≠ 0 and 𝑦P + 𝑦. = 0, then 𝑦. = −𝑦P ≠ 0.
From 𝑣P 𝑦P + 𝑣. 𝑦. = 0 we have 𝑣P 𝑦P − 𝑣. 𝑦P = 𝑣P − 𝑣. 𝑦P = 0.
Note that 𝑦P ≠ 0, thus 𝑣P − 𝑣. = 0. Also, we can derive that 𝑤P − 𝑤. = 0.
As our proof of proposition 6, we find that the two assets are identical, which contradicts the
assumption. So 𝑦•
∇j
.
𝐿 𝑥, 𝜆 𝑦 > 0, i.e. 𝐿′′(𝑥, 𝜆) is a positive definite matrix at 𝑥 = 𝑥∗
, and
𝑥 = 𝑥∗
is a minimizer of the utility function in problem (18).
Generalized algorithm for n-asset problem
For n-asset selection problem, we can break it down into 3-asset or 2-asset problems as what
we have discussed and provide a generalized algorithm for it. This algorithm will terminate in
𝑜(𝑛`
) steps.
Step 1: Let 𝑐 ≔ +∞ and 𝑥€ ≔ [0, … ,0].
Step 2: Choose three points from the bag { 𝑣8, 𝑤8, 𝑢8 , i = 1, … , 𝑛} which have not been
considered yet. If there are no such points then go to Step 9, otherwise denote these three
points by 𝑣“, 𝑤“, 𝑢“ , 𝑣”, 𝑤”, 𝑢” and 𝑣•, 𝑤•, 𝑢• . Let 𝑣P, 𝑤P, 𝑢P ≔ 𝑣“, 𝑤“, 𝑢“ ,
𝑣., 𝑤., 𝑢. ≔ 𝑣”, 𝑤”, 𝑢” and 𝑣`, 𝑤`, 𝑢` ≔ 𝑣•, 𝑤•, 𝑢• .

17. 17
Step 3: If 𝑑𝑒𝑡
𝑞P 𝑟P
𝑞. 𝑟.
= 𝑑𝑒𝑡
𝑣P − 𝑣` 𝑤P − 𝑤`
𝑣. − 𝑣` 𝑤. − 𝑤`
= 0
then go to Step 2, otherwise go to Step 4.
Step 4: Compute the first two components, [𝑥P
∗
, 𝑥.
∗
], of the optimal solution to (11) using
equation (16).
Step 5: If [𝑥P
∗
, 𝑥.
∗
, 1 − 𝑥P
∗
− 𝑥.
∗
] > 0 then go to Step 6, otherwise go to Step 2.
Step 6: If 𝑈 𝑥P
∗
, 𝑥.
∗
, 1 − 𝑥P
∗
− 𝑥.
∗
< 𝑐 then go to Step 7, otherwise go to Step 2.
Step 7: Let 𝑐 = 𝑈 𝑥P
∗
, 𝑥.
∗
, 1 − 𝑥P
∗
− 𝑥.
∗
, and let
𝑥€ = [0, … ,0, 𝑥P
∗
“–—
, 0, … ,0, 𝑥.
∗
”–—
, 0, … ,0, 𝑥`
∗
•–—
, 0, … ,0]
Step 8: Go to Step 2.
Step 9: Choose two points from the bag { 𝑣8, 𝑤8, 𝑢8 , i = 1, … , 𝑛} which have not been
considered yet. If there are no such points then go to Step 16, otherwise denote these two
points by 𝑣“, 𝑤“, 𝑢“ and 𝑣”, 𝑤”, 𝑢” . Let 𝑣P, 𝑤P, 𝑢P ≔ 𝑣“, 𝑤“, 𝑢“ and 𝑣., 𝑤., 𝑢. ≔
𝑣”, 𝑤”, 𝑢” .
Step 10: If 𝑣P − 𝑣.
.
+ 𝑤P − 𝑤.
.
= 0 then go to Step 9, otherwise go to Step 11.
Step 11: Compute the first component, 𝑥P
∗
, of the optimal solution to (18) using equation (22).
Step 12: If 𝑥P
∗
, 𝑥.
∗
= 𝑥P
∗
, 1 − 𝑥P
∗
> 0 then go to Step 13, otherwise go to Step 9.
Step 13: If 𝑈 𝑥P
∗
, 1 − 𝑥P
∗
< 𝑐 then go to Step 14, otherwise go to Step 9.
Step 14: Let 𝑐 = 𝑈 𝑥P
∗
, 1 − 𝑥P
∗
, and let
𝑥€ = [0, … ,0, 𝑥P
∗
“–—
, 0, … ,0, 𝑥.
∗
”–—
, 0, … ,0]
Step 15: Go to Step 9.
Step 16: Choose a point from the bag { 𝑣8, 𝑤8, 𝑢8 , i = 1, … , 𝑛} which has not been considered
yet. If there is no such point then go to Step 20, otherwise denote this point by 𝑣“, 𝑤“, 𝑢“ .
Step 17: If 𝑈 𝑣“, 𝑤“, 𝑢“ = 𝑣“
.
+ 𝑤“
.
− 𝑢“ < 𝑐 then go to Step 18, otherwise go to Step 16.
Step 18: Let 𝑐 = 𝑈 𝑣“, 𝑤“, 𝑢“ = 𝑣“
.
+ 𝑤“
.
− 𝑢“, and let
𝑥€ = [0, … ,0, 1
“–—
, 0, … ,0]
Step 19: Go to Step 16.
Step 20: 𝑥€ is an optimal solution and – 𝑐 is the optimal value of the original portfolio
selection problem (8).

18. 18
Numerical illustration
We now use real-life data to demonstrate the proposed algorithm.
For simplicity, we consider a 3-asset problem. In order to alleviate the impact of correlation
between distinct assets, we look for companies from uncorrelated or less correlated industrial
sectors. Hence, we choose Facebook Inc. (FB), Exxon Mobil Corporation (XOM), and The
Coca-Cola Company (KO). Since Facebook held its initial public offering (IPO) on May 18,
2012, we pick monthly quotes of these three stocks from May, 2012 to April, 2016. All the
data are collected from http://finance.yahoo.com.
We first compute monthly rate of returns using the stock quotes by the following equation:
𝑟8X% = 100×
𝑃8XnP − 𝑃8X
𝑃8X
%
where 𝑟8X is the percentage of return on asset 𝑖. Note that in the utility function (see Section
2.1) we add up a scaling factor of 0.005 to avoid decimals, we now need to use percentages
rather than decimals of returns on the asset. There are 48 monthly stock quotes and thus we
can obtain 47 monthly percentage returns on each asset.
As 𝑟8 are assumed to be trapezoidal fuzzy variables with possibilistic distributions, we need
to figure out the exact trapezoidal forms. Normally, the researcher can use the Delphi Method
[4] to decide the trapezoidal form. In our illustration, we use the frequency statistic method
(see [3] Gupta et al, 2008) to estimate the trapezoidal fuzzy return rates.

19. 19
The percentage returns on Facebook Inc. (FB) can be graphed as:
From Fig. 3 we observe that most of the historical data fall into the intervals [−12.0, −4.0],
[−4.0, 4.0], [4.0, 12.0] and [12.0, 20.0]. We take the mid-points of the intervals
[−12.0, −4.0] and [12.0, 20.0] as the left and the right end points of the tolerance interval,
respectively. Thus, the tolerance interval of the fuzzy percentage returns is [−8.0, 16.0]. By
going through all the historical data, we find the minimum possible value -30.2 and the
maximum possible value 47.9 and view them as the limits of uncertain percentage returns in
the future, respectively. Therefore, the left spread is 22.2 and the right spread is 31.9, and the
trapezoidal percentage returns on FB is 𝑟P = [−8.0, 16.0, 22.2, 31.9].
Likewise, we can obtain the trapezoidal returns on XOM, which is 𝑟. = [−4.6, 3.8, 4.3, 7.5],
and KO, which is 𝑟` = [−4.5, 4.5, 3.9, 3.9].
Assume that 𝐴 = 2.46, we can calculate
𝑣P, 𝑤P, 𝑢P = (2.331, 0.707, 5.617),
𝑣., 𝑤., 𝑢. = (0.684, 0.154, 0.133),
𝑣`, 𝑤`, 𝑢` = (0.643, 0.102, 0.000).
First consider the 3-asset problem with 𝑣P, 𝑤P, 𝑢P , 𝑣., 𝑤., 𝑢. and 𝑣`, 𝑤`, 𝑢` .
Since 𝑑𝑒𝑡
𝑞P 𝑟P
𝑞. 𝑟.
= 𝑑𝑒𝑡
𝑣P − 𝑣` 𝑤P − 𝑤`
𝑣. − 𝑣` 𝑤. − 𝑤`
= 𝑑𝑒𝑡
1.688 0.605
0.041 0.052
= 0.063 ≠ 0,

20. 20
we get
𝑥P
∗
𝑥.
∗
=
1
𝑞P 𝑟. − 𝑟P 𝑞.
.
𝑞.
.
+ 𝑟.
.
− 𝑞P 𝑞. + 𝑟P 𝑟.
−(𝑞P 𝑞. + 𝑟P 𝑟.) 𝑞P
.
+ 𝑟P
. ×
1
2
𝑢P − 𝑢` − 𝑞P 𝑣` − 𝑟P 𝑤`
1
2
𝑢. − 𝑢` − 𝑞. 𝑣` − 𝑟. 𝑤`
=
1
0.063.
0.004 −0.100
−0.100 3.214
1.661
0.035
=
0.792
−13.5072
.
Notice that 𝑥.
∗
< 0, which is not feasible, then we found no qualified 3-asset candidate for an
optimal solution to (10).
Now we turn to all conceivable 2-asset problems:
○1 For the combination of FB and XOM,
since 𝑣P − 𝑣.
.
+ 𝑤P − 𝑤.
.
= 3.018 ≠ 0, we get
𝑥P
∗
=
1
𝑣P − 𝑣.
. + 𝑤P − 𝑤.
.
×
1
2
𝑢P − 𝑢. − 𝑣P − 𝑣. 𝑣. − 𝑤P − 𝑤. 𝑤.
=
1
3.018
×1.530 = 0.507
Thus, [0.507, 0.493, 0] is a qualified candidate for an optimal solution to (10), where
𝑈 0.507, 0.493, 0 = −0.417.
○2 For the combination of FB and KO,
since 𝑣P − 𝑣`
.
+ 𝑤P − 𝑤`
.
= 3.214 ≠ 0, we get
𝑥P
∗
=
1
𝑣P − 𝑣`
. + 𝑤P − 𝑤`
.
×
1
2
𝑢P − 𝑢` − 𝑣P − 𝑣` 𝑣` − 𝑤P − 𝑤` 𝑤`
=
1
3.214
×1.661 = 0.517
Thus, [0.517, 0, 0.483] is a qualified candidate for an optimal solution to (10), where
𝑈 0.517, 0, 0.483 = −0.435.
○3 For the combination of XOM and KO,
since 𝑣. − 𝑣`
.
+ 𝑤. − 𝑤`
.
= 0.004 ≠ 0, we get
𝑥P
∗
=
1
𝑣. − 𝑣`
. + 𝑤. − 𝑤`
.
×
1
2
𝑢. − 𝑢` − 𝑣. − 𝑣` 𝑣` − 𝑤. − 𝑤` 𝑤`

21. 21
=
1
0.004
×0.035 = 8.75 > 1
Thus, this cannot be a qualified candidate for an optimal solution to (10).
Finally, we compute the utility values of all the 1-asset options:
𝑈 1, 0, 0 = 𝑣P
.
+ 𝑤P
.
− 𝑢P = 0.316;
𝑈 0, 1, 0 = 𝑣.
.
+ 𝑤.
.
− 𝑢. = 0.359;
𝑈 0, 0, 1 = 𝑣`
.
+ 𝑤`
.
− 𝑢` = 0.424.
Comparing the function values of all feasible solutions we find that the optimal portfolio
would be 𝑥∗
= [0.517, 0, 0.483], i.e. the combination of Facebook (51.7%) and Coca-Cola
(48.3%).
Remarks on Carlsson-Fullér-Majlender’s model
6.1 Assumption of covariance
To calculate possibilistic variance of the linear combination of fuzzy variables, we shall use
the following theorem ([6] Sánta, 2012):
𝑉𝑎𝑟 𝜆] + 𝜆8 𝐴8
e
8fP
= 𝜆8
.
𝑉𝑎𝑟 𝐴8
e
8fP
+ 2 𝜆8 𝜆“ 𝐶𝑜𝑣 𝐴8, 𝐴“
e
8fP
𝑤ℎ𝑒𝑟𝑒 𝐴8 𝑎𝑟𝑒 𝑓𝑢𝑧𝑧𝑦 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠 𝑎𝑛𝑑 𝜆8 𝑎𝑟𝑒 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠, 𝑖 = 1, … , 𝑛.
(23)
A simple two-variable theorem has also been presented in [5] (Carlsson, 2001).
From these definitions, we learn that we need to take the covariance terms into account when
calculating the variance of fuzzy number combinations.
Note that in Carlsson-Fullér-Majlender’s model when we derive possibilistic variance of the
whole portfolio, we actually ignore the intercorrelation between different assets. That is, we
assume the covariance to be zero. The subsequent discussion is based on this hypothesis. In
that case, Carlsson-Fullér-Majlender’s model only applies to cases where the optional assets
are uncorrelated or significantly less correlated. In fact, however, it is unreal to find assets
that are totally uncorrelated, so this model is not as applicable and effective as we might
expect. We need to pick out the assets from differentiating industrial sectors carefully to
comply with the zero-covariance assumption.

22. 22
6.2 Possibilistic variance of portfolio
Even if we rule out the covariance terms, there might still be some confusion in equation (7).
Note that 𝑥8 is a real number and 𝑟8 is a fuzzy number. Using the formula in (23), we can
derive the portfolio variance as
𝜎.
𝑟8 𝑥8
e
8fP
= 𝑥8
.
𝜎.
𝑟8
e
8fP
= 𝑥8
.
e
8fP
𝑏8 − 𝑎8
2
+
𝛼8 + 𝛽8
6
.
+
𝛼8 + 𝛽8
.
72
from equation 4
= 𝑥8
.
e
8fP
1
2
𝑏8 − 𝑎8 +
1
3
𝛼8 + 𝛽8
.
+
𝛼8 + 𝛽8
.
72
= 𝑥8
.
1
2
𝑏8 − 𝑎8 +
1
3
𝛼8 + 𝛽8
.e
8fP
+
𝛼8 + 𝛽8
.
𝑥8
.
72
e
8fP
=
1
2
𝑏8 − 𝑎8 +
1
3
𝛼8 + 𝛽8 𝑥8
.e
8fP
+
1
72
𝛼8 + 𝛽8 𝑥8
.
e
8fP
≠
1
2
𝑏8 − 𝑎8 +
1
3
𝛼8 + 𝛽8
e
8fP
𝑥8
.
+
1
72
𝛼8 + 𝛽8 𝑥8
e
8fP
.
which is given by equation (7).
If equation (7) is not true, neither is the rest of the discussion. The whole Carlsson-Fullér-
Majlender’s model, therefore, does not seem convincing to me. Considering that this is a
well-known model in fuzzy optimization, I am not sure if I have taken this the wrong way or
not.
If equation (7) is corrected to
𝜎.
𝑟8 𝑥8
e
8fP
=
1
2
𝑏8 − 𝑎8 +
1
3
𝛼8 + 𝛽8 𝑥8
.e
8fP
+
1
72
𝛼8 + 𝛽8 𝑥8
.
e
8fP
(24)
then the optimization problem turns into
max
jk
𝑈 𝑃 = 𝑢8 𝑥8
e
8fP
− 𝑣8 𝑥8
.
e
8fP
− 𝑤8 𝑥8
.
e
8fP
s. t. { 𝑥8
e
8fP
= 1, 𝑥8 ≥ 0 , 𝑖 = 1,2, … , 𝑛}
(25)
rather than problem (8), which is
max
jk
𝑈 𝑃 = 𝑢8 𝑥8
e
8fP
− 𝑣8
e
8fP
𝑥8
.
− 𝑤8
e
8fP
𝑥8
.
s. t. { 𝑥8
e
8fP
= 1, 𝑥8 ≥ 0 , 𝑖 = 1,2, … , 𝑛}
(8)
and thus the solutions will change correspondingly.

23. 23
6.3 Feasibility of the solution
As we disregard short-selling and long-buying, the feasible set of the solution should be
{𝑥8 𝜖ℛ: 𝑥8 𝜖 0,1 , 𝑖 = 1, … , 𝑛}. However, we do not include this condition into the constraints
of our optimization problem. The Carlsson-Fullér-Majlender’s model, in fact, computes the
not-necessarily feasible weights, so we need to check the feasibility every time we obtain a
candidate of the solution. This may cause some incovenience.
Conclusions
In this paper, we introduce the Carlsson-Fullér-Majlender’s trapezoidal possibility model to
address fuzzy portfolio selection problem. We devise a utility function based on portfolio
selection theory formulated by [7] (Markowitz, 1952). Using some properties of trapezoidal
fuzzy variable as well as optimization theory, we translate the optimization problem into a
non-linear prgramming problem, in which we can employ the Lagrange Multiplier Method
and Karush-Kuhn-Tucker (KKT) Conditions to calculate the optimal solutions. We provide a
generalized algorithm for the problem and then use some real data for illustration. We end the
paper with some personal thinking of the model, including its limitations or even some faults.

24. 24
References
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[4] Linstone, Harold A., and Murray Turoff, eds. The Delphi method: Techniques and
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