A CHANCE MULTIOBJECTIVE PROBLEM WITH WEIBULL FUZZY DISTRIBUTION

WITH WEIBULL FUZZY DISTRIBUTION

ILIE MARINESCU

Communicated by the former editorial board

We consider that a random variable of the chance constraints, for our selection problem, follows a fuzzy Weibull distribution. Thus, the multiobjective problem will be modeled using fuzzy numbers. We will consider an additional constraint based on the Expers’ judgments that will help us to find a better solution.

AMS 2010 Subject Classification: 60A86, 90C70, 90C26.

Key words: portfolio optimization, concave tranzaction cost, rebalancing, fuzzy probability, Weibull distribution.

1. INTRODUCTION

The selection problem of the rebalancing portfolio, under nonconvex trans- actional costs, was defined in Konno et al. [8, 9], where they considered a branch and bound algorithm for solving a portfolio optimization model under concave transaction costs.

The practitioners are very much concerned about the transaction costs since these have significant effects on the investment strategy. In order to purchase (invest) and/or sell (disinvest) assets, the investor has to pay cer- tain fees. But, unfortunately, the transaction costs are often ignored because the precise treatment of transaction costs leads to a nonconvex minimization problem. A problem of this type hasn’t any efficient method for calculating its exact optimal solution, at least until recently.

In this paper, we model the chance constraints from the multiobjective minimal problem of the cost rebalancing using fuzzy theory approach. And, it is reached to a deterministic equivalent of the selection portfolio problem.

In a stochastic programming problem, the uncertainties in the parameters are represented by probability distributions. This distribution is estimated on the basis of the available observed random data. Here, the parameters are treated as random variables.

MATH. REPORTS16(66),1(2014), 121–132

According to Dash et al. [5] the fuzzy chance constrained programming problem is a chance constrained programming problem in the presence of am- biguous information, where the random variable follows different fuzzy dis- tributions. Thus, fuzzy random variables for chance constrained rebalancing problem is using a linear exponential distribution, which is very efficient from the practical point of view.

The multiobjective minimum cost rebalancing problem with chance con- straints under the mean-absolute deviation (MAD) model can be formulated as it follows :

min







n

X

j=1

c_j1(x_j),

n

X

j=1

c_j2(x_j), ...,

n

X

j=1

c_jk(x_j)





 (1)

subject to P r





n

X

j=1

rj(xj ≤g1



≥q1

γ_j⁰ ≤yj+x⁰_j ≤γj, j = 1, n w₀ ≤

T

X

t=1

p_t

n

X

j=1

(r_jt−r_j)x_j

≤W₀.

where the fees associated with x= (x1, x2, ..., xn) are named transaction cost, x_j represents the amount of investment (or disinvestment) of the assetj (j= 1, ..., n). The transaction cost of the entire investment is Pn

j=1cji(xj) for the i broker, i= 1, ..., k, where cji(xj) is a non-decreasing nonconvex function up to certain pointx_j [8].

It is considered a time horizon composed fromT moments of time, (r1t, r2t, ..., r_nt) is a vector of the return rates of the n assets at the t moment with p_t=P r{(R₁, R₂, ..., R_n) = (r_1t, r_2t, ..., r_nt)}, t= 1, ..., T is known. R_j is a ran- dom variable, which represents the return rate of the j asset. (Pr stands for probability, [6])

The expected rate of the return of the asset j without transaction costs is rj = PT

t=1ptrjt, and the expected rate of the return of the portfolio x = (x₁, x₂, ..., x_n) is given byPn

j=1r_jx_j.

Because the current portfolio may deviate from the present efficient fron- tier, the investors are inclined to ”rebalance” the portfolio due to the change of investment environment.

Let x = (x1, x2, ..., xn) be the new portfolio and x⁰ = (x⁰₁, x⁰₂, ..., x⁰_n) [8, 9] be the portfolio at time 0. We assume that its expected rate of return Pn

j=1r_jx_j is smaller than a constantg₁, because the investor wants to obtain a

maximize expected rate of return, if it is possible, but not assuming any risks.

The risk of the problem is W[r(x)] = p_t

Pn

j=1(r_jt−r_j)x_j

and we con- sider that it is bounded, which means that the investor doesn’t want a very big riskW₀, but neither a small riskw₀, which gives a less expected rate of return.

The multiobjective optimization problem solution is considered to be Pareto-optimal if there are not other solutions that are better for all of the objectives simultaneously. There can be other solutions that are better in sat- isfying one or several objectives, but they must be worse than the Paeto-optimal solution in satisfying the remaining objectives.

Let us introduce the new portfolio at a certain later point x = y+x⁰ made up by all the operations resulted from the rebalancing, a portfolio that has the following meanings:

– ify_j >0, j= 1, ..., n, thenc_ji(y_j) is the associated cost with purchasing y_j units of the assetj,i= 1, ..., k;

– if y_j <0, j = 1, ..., n, thenc_ji(y_j) is the associated cost with selling|y_j| units of the assetj,i= 1, ..., k.

Let zt, z_t⁰, t = 1, . . . , T be nonnegative variables satisfying the following conditions:

zt−z_t⁰ =pt n

X

j=1

(rjt−rj)yj+ (rjt−rj)x⁰_j , z_tz⁰_t= 0, z_t≥0, z_t⁰ ≥0, t= 1, . . . , T.

Then, we have zt+z_t⁰ =

pt n

X

j=1

(rjt−rj)yj+ (rjt−rj)x⁰_j

, t= 1, . . . , T.

Therefore, instead of problem (1) we consider the following problem : (2) min







n

X

j=1

cj1(yj),

n

X

j=1

cj2(yj), . . . ,

n

X

j=1

c_jk(yj)





 subject to

P r





n

X

j=1

r_j(y_j+x⁰_j)≤g₁



≥q₁ γ_j⁰ ≤yj +x⁰_j ≤γj, j = 1, . . . , n w0 ≤

T

X

t=1

zt+z⁰_t

≤W0

zt−z_t⁰ =pt n

X

j=1

[(rjt−rj)yj+ (rjt−rj)x⁰_j] ztz⁰_t= 0, zt≥0, z_t⁰ ≥0, t= 1, . . . , T.

As in [8] we can prove a similar result for problem (2).

Theorem. The complementarity constraint z_tz_t⁰ = 0, z_t≥0, z_t⁰ ≥0, t= 1,2, . . . , T, can be eliminated from problem(P₁). Moreover,PT

t=1 z_t−z_t⁰

= 0.

Thus, the mean-absolute deviation (MAD) model [10] may be represented as a crisp chance constrained programming problem for the minimal cost of the portfolio transaction of the form:

(3) min







n

X

j=1

c_j1(y_j),

n

X

j=1

c_j2(y_j), . . . ,

n

X

j=1

c_jk(y_j)





 subject to

P r





n

X

j=1

rj(yj +x⁰_j)≤g1



≥q1

γ_j⁰ ≤yj+x⁰_j ≤γj, j = 1, . . . , n w₀≤

T

X

t=1

z_t+z_t⁰

≤W₀

w₀≤2

T

X

t=1

z_t≤W₀

pt n

X

j=1

(rjt−rj)yj−zt≤ −p_t

n

X

j=1

(rjt−rj)x⁰_j, t= 1, . . . , T

0≤α≤1,

n

X

j=1

yj = 0,

n

X

j=1

x⁰_j = 1.

where g₁ is uncertain and Pn

j=1r_j(y_j +x⁰_j) represents the expected rate of return for rebalancing portfolio.

We suppose that g1 is a fuzzy random variable and the others real pa- rametersγ_t, γ_t⁰,W₀, w₀ are given. Their significations and definitions are given in [5]. This paper is organized as follows. In Section 2, we review the theo- retical background concerning the triangular fuzzy numbers. In Section 3, we state the minimum cost rebalancing problem under the mean-absolute devia- tion (MAD) model. In Section 4, we solve the deterministic problem subject

to the Expers’judgment. In the last section, we model the programming prob- lem under fuzzy conditional Weibull distribution with different choices of the distribution parameters.

2.SOME PRELIMINARIES

We place a “ e ” over a symbol to denote a fuzzy set. All our fuzzy sets will be fuzzy subsets of the real numbers. So, ae_i,A,e xeall represent fuzzy subsets of the real numbers.

LetFebe the fuzzy number with membership functionµ

Fe:R→[0,1],the set{b|µ_F˜(b)≥α}= ˜F[α] isα-cut ofF ,e ∀0≤α ≤1 . ˜F[0] is separately defined as the closure of the union of all the ˜F[α],0< α≤1.So, a fuzzy numberFe is a fuzzy subset of the real numbers satisfying the following conditions:

– µ_F˜(b) = 1 for some b (normalized);

– ˜F[α] is a closed and bounded interval for 0≤α≤1.

A triangular fuzzy number Fe is a triplet (y¹, y², y³) ∈ R³. µ

Fe(y) rep- resents a number in [0,1] , which is the membership function of Fe evaluated iny.

µFe(y) =



























0, y ∈(−∞, y¹)

x−y¹

y²−y¹, y∈[y¹, y²]

x−y³

y²−y³, y∈(y², y³] 0, y ∈(y³,∞)

For any fuzzy numberFe, theα-cut ofFeis a closed and bounded interval for 0≤ α ≤1, i.e. F([α]) = [qe ∗(α), q^∗(α)]. We define a partial order relation between two fuzzy numbers Fe1 and Fe2 using α-cuts Fe1[α] and Fe2[α]. Let Fe₁[α] =

q_∗¹(α), q^1∗(α)

andFe₂[α] =

q_∗²(α), q^2∗(α)

. Then, Fe₁ Fe₂ iffq_∗¹(α)≥ q^2∗(α) , for each α∈[0,1].

3. THE MUTIOBJECTIVE PROBLEM UNDER FUZZY WEIBULL DISTRIBUTION

Letg₁ be a continuous random variable with probability density function Weibull

f(g₁;λ, µ;s) =







e^−λsg1µ λ^sµg1µ−1

if g1>0, s≥1, µ≥1;

0 otherwise;

where λ and µ are uncertain parameters, describing the probability density function.

3.1. THE PARAMETERλIS A FUZZY NUMBER

Let eg1 be fuzzy random variable with fuzzy density Weibull distribution, eλa fuzzy parameter andµis a parameter real nonnegative given. So the fuzzy chance constrained multiobjective problem for the minimal cost of the portfolio transaction is of the form:

(4) min







n

X

j=1

c_j1(y_j),

n

X

j=1

c_j2(y_j), ...,

n

X

j=1

c_jk(y_j)





 subject to

P re





n

X

j=1

rj(yj +x⁰_j)≤eg1



eq1

γ_j⁰ ≤y_j+x⁰_j ≤γ_j, j = 1, ..., n w₀≤2

T

X

t=1

z_t≤W₀ p_t

n

X

j=1

(r_jt−r_j)y_j−z_t≤ −p_t

n

X

j=1

(r_jt−r_j)x⁰_j, t= 1, . . . , T.

Let Pn

j=1r_j(y_j+x⁰_j)≤eg₁ be a fuzzy event. Then the probability of this eventP re

Pn

j=1rj(yj+x⁰_j)≤eg1

is a fuzzy number. Itsα-cut is the set

P re





n

X

j=1

rj(yj +x⁰_j)≤eg1



[α]

= Z ∞

Pn

j=1rj(yj+x⁰_j)

e^−λsg1µ λ^sµg1µ−1 dg1

λ∈eλ[α]

=

q_∗¹(α), q^1∗(α) , for 0≤α≤1.Denoteqe1[α] = [q1∗(α), q₁^∗(α)] theα-cut of the fuzzy numberqe1. Lemma 1. If eg₁ is a fuzzy random variable Weibull distributed with µ real nonnegativ parameter given, and eλ1 is a fuzzy number, then the minimum α-cut of the fuzzy probability is

q_∗¹(α) =e^{− λ∗1(α) s n}

X

j=1

r_j(y_j+x⁰_j)µ

for α∈[0,1], where eλ₁[α] = [λ1∗(α), λ^∗₁(α)].

Proof.

P re





n

X

j=1

r_j(y_j+x⁰_j)≤eg₁



[α]

= Z ∞

Pn

j=1rj(yj+x⁰_j)

λ₁^sµg₁^µ−1e^−λ1sg1µdg₁

λ₁ ∈λe₁[α]

=

e^−λ1s

Pn

j=1rj(yj+x⁰_j)µ

λ1∗(α)≤λ₁ ≤λ^∗₁(α)

=

q¹_∗(α), q^1∗(α) . Because the function

f1(

n

X

j=1

rj(yj +x⁰_j);λ1) =e^{− λ1(α) s n}

X

j=1

rj(yj+x⁰_j)µ

is decreasing at infinite, such as the minimum is attained inλ^∗₁(α). So it results that the minimum α-cut of the fuzzy probability is

e^{− λ∗1(α)} s

u^µ.

We suppose the triangular fuzzy numbers eλ1 = (λ⁰₁, λ1, λ⁰⁰₁) and eq1 = (q⁰₁, q₁, q₁⁰⁰).The deterministic equivalent of the problem (4) using of Lemma 1 is given such as it follows :

(5) min







n

X

j=1

c_j1(y_j),

n

X

j=1

c_j2(y_j), ...,

n

X

j=1

c_jk(y_j)





 subject to

e

−

λ⁰⁰₁−α(^λ001−λ₁)s

Pn

j=1rj(yj+x⁰_j)

µ

−q⁰⁰₁ +α q₁⁰⁰−q₁

≥0 γ_j⁰ ≤yj+x⁰_j ≤γj, j = 1, ..., n

w0≤2

T

X

t=1

zt≤W0

pt n

X

j=1

(rjt−rj)yj−zt≤ −p_t

n

X

j=1

(rjt−rj)x⁰_j, t= 1, . . . , T

0≤α≤1,

n

X

j=1

yj = 0,

n

X

j=1

x⁰_j = 1.

Thus the problem (5) is a nonlinear multiobjective programming problem.

4.THE TRANSACTION COST MODEL CONSTRAINED

Knowing the medium of the assets profitableness at a certain moment, we will concentrate our assets with the rates of return bigger than the medium.

We make the following notation:

NT ={j|r_j ≥rT, j = 1, . . . , n}

wherer_T =Pn j=1

rj

n.So that we impose the constraint:

M2 ≤ X

j∈N_T

(yj+x⁰_j)≤M1

where M₁ and M₂ are the parameters given by the decision maker. It results that the problem (5) has the following equivalent form:

(6) min







n

X

j=1

c_j1(y_j),

n

X

j=1

c_j2(y_j), ...,

n

X

j=1

c_jk(y_j)





 subject to

e

−

λ⁰⁰₁−α(^λ001−λ₁)s

Pn

j=1rj(yj+x⁰_j)

^µ

−q⁰⁰₁ +α q₁⁰⁰−q₁

≥0 M₂≤ X

j∈N_T

(y_j+x⁰_j)≤M₁ γ_j⁰ ≤y_j+x⁰_j ≤γ_j, j = 1, ..., n w₀≤2

T

X

t=1

z_t≤W₀ p_t

n

X

j=1

(r_jt−r_j)y_j−z_t≤ −p_t

n

X

j=1

(r_jt−r_j)x⁰_j, t= 1, . . . , T

0≤α≤1,

n

X

j=1

y_j = 0,

n

X

j=1

x⁰_j = 1.

5.FUZZY CONDITIONAL DISTRIBUTION WEIBULL

We consider the fuzzy chance constrained programming problem, where we impose that the minimum gain should be equal with a certain fuzzy value.

So that the problem has the following form :

(7) min







n

X

j=1

c_j1(y_j),

n

X

j=1

c_j2(y_j), ...,

n

X

j=1

c_jk(y_j)







subject to P re

n

X

j=1

rj(yj+x⁰_j)≤eg1

n

X

j=1

r_j⁰(yj+x⁰_j) =eg2

qe2

M₂≤ X

j∈NT

(y_j+x⁰_j)≤M₁ γ_j⁰ ≤yj+x⁰_j ≤γj, j = 1, ..., n w₀≤2

T

X

t=1

z_t≤W₀ p_t

n

X

j=1

(r_jt−r_j)y_j−z_t≤ −p_t

n

X

j=1

(r_jt−r_j)x⁰_j, t= 1, . . . , T

0≤α≤1,

n

X

j=1

y_j = 0,

n

X

j=1

x⁰_j = 1.

where (g₁, g₂) is a uncertain random variable with probability density function

f(g₁, g₂;λ, µ;s) =











e^−λsg1µλ

s(¹µ+1)

Γ 1

µ+1

µg1µ−1 if g1 > g2>0, λ >0, s≥1, µ≥1;

0 otherwise;

and

fg2(g2;λ, µ;s) =











e^{−λsg2µ λ}

sµ

Γ

1

µ+1 if g₂ >0, λ >0, s≥1, µ≥1;

0 otherwise;

is the marginal density function of g₂, and Pn

j=1r_j⁰(y_j +x⁰_j) represents the expected rate of return (minim accepted) for rebalancing portfolio. We sup- pose that (eg₁,eg₂) is a fuzzy random variable and the others real parameters γ_j, γ_j⁰, W₀, w₀, j = 1, . . . , n are given. Their significations and definitions are given in [8, 9, 10].

5.1. THE PARAMETERλIS A FUZZY NUMBER

Let (eg₁,eg₂) be fuzzy random variable with fuzzy density function Weibull ,eλa fuzzy parameter and µis a parameter nonnegative given.

Thus,α-cut of fuzzy event (Pn

j=1r_j(y_j+x⁰_j)≤eg₁

Pn

j=1r⁰_j(y_j+x⁰_j) =eg₂) is the set

P re





n

X

j=1

rj(yj+x⁰_j)≤eg1

n

X

j=1

r⁰_j(yj+x⁰_j) =eg2



[α]

= Z ∞

Pn

j=1rj(yj+x⁰_j)

f(g1, g2;λ, µ;s) fg2(g2;λ, µ;s) dg1

λ ∈eλ[α], Z ∞

0

f(g1, g2;λ, µ;s)

f_g2(g₂;λ, µ;s) dg₁= 1

=

q_∗²(α), q^2∗(α)

Lemma 2. If (eg₁,eg₂) is a fuzzy random variable with probability density function f(g1, g2;eλ2, µ;s) and fg2

g2;eλ2, µ;s

is the fuzzy marginal density function of g2 , where µ is a real parameter given, and eλ2 is a fuzzy num- ber. Then we have the following relation for the minimum α-cut of the fuzzy probability:

e^{− λ∗2(α)} s n

X

j=1

r_j(y_j+x⁰_j)µ

−

n

X

j=1

r_j⁰(y_j+x⁰_j)µ for α∈[0,1], where eλ₂[α] = [λ2∗(α), λ^∗₂(α)].

Proof.

P re





n

X

j=1

rj(yj +x⁰_j)≤eg1

n

X

j=1

r_j⁰(yj +x⁰_j) =eg2



[α]

=

Z ∞ Pn

j=1rj(yj+x⁰_j)

λ₂^sµg₁^µ−1e^{−λ2s g1µ−}

Pn

j=1r⁰_j(yj+x⁰_j)^µ dg₁

λ₂∈eλ₂[α]

=

e^{−λ2s g1µ−Pnj=1r0j(yj+x0j)µ}

λ2∗(α)≤λ₂ ≤λ^∗₂(α)

=

q_∗²(α), q^2∗(α) . Because the function e^{−λ2s g1µ−}

Pn

j=1r⁰_j(yj+x⁰_j)^µ

is decreasing at infinite in λ2, such as the minimum is attained inλ^∗₂(α) . So it results that the minimum α-cut of the fuzzy probability is

q_∗²(α) =e^{− λ∗2(α)} s

Pn

j=1rj(yj+x⁰_j)µ

− Pn

j=1r_j⁰(yj+x⁰_j)µ

.

We suppose the triangular fuzzy numbers eλ₂ = (λ⁰₂, λ₂, λ⁰⁰₂) and eq₂ = (q⁰₂, q₂, q₂⁰⁰).and such that using Lemma 2 the deterministic equivalent of (7) is

the following:

(8) min







n

X

j=1

c_j1(y_j),

n

X

j=1

c_j2(y_j), ...,

n

X

j=1

c_jk(y_j)





 subject to

e

−

λ⁰⁰₂−α(^λ002−λ2)s

Pn

j=1rj(yj+x⁰_j)

^µ

−

Pn

j=1r⁰_j(yj+x⁰_j)

^µ

−q⁰⁰₂+α q⁰⁰₂−q₂

≥0 M₂≤ X

j∈N_T

(y_j+x⁰_j)≤M₁ γ_j⁰ ≤y_j+x⁰_j ≤γ_j, j = 1, ..., n w₀≤2

T

X

t=1

z_t≤W₀ p_t

n

X

j=1

(r_jt−r_j)y_j−z_t≤ −p_t

n

X

j=1

(r_jt−r_j)x⁰_j, t= 1, . . . , T

0≤α≤1,

n

X

j=1

y_j = 0,

n

X

j=1

x⁰_j = 1.

From [4] and [11], it can be given a solution for nonlinear multiobjective programming problems (5) and (8).

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Received 30 November 2011 University of Bucharest, Faculty of Mathematics and Computer Science,

Str. Academiei 14, 010014 Bucharest,

Romania