13
Lecturer Costin-Ciprian POPESCU, PhD The Bucharest Academy of Economic Studies Department of Applied Mathematics E-mail: [email protected] A FUZZY OPTIMIZATION MODEL Abstract. A fuzzy optimization model, mainly aiming the portfolio selection issue, is discussed. Instead of being modeled as piecewise linear fuzzy numbers (triangular, trapezoidal etc.), the input data are presented as quasi S shape fuzzy numbers. In this framework some theoretical and numerical results are given. Key words: fuzzy number, possibilistic mean value, quasi S shape fuzzy data, portfolio optimization. JEL Classification: C020, C440, C610. 1. Introduction The issue of fuzzy mathematical programming is a topic of great interest in literature ([13], [14], [15], [16]). Also the financial market analysis, including portfolio optimization, has numerous approaches ([3], [10], [14], [17], [19]). In fact, the problem of portfolio optimization can be viewed as a problem of decision- making. In this area, the theory of fuzzy sets is a very useful tool [20]. In this paper we deal with both subjects mentioned above. More precisely, we discuss a fuzzy optimization model from the theoretical point of view and we also show its applications to the problem of portfolio selection. It is included in a certain class of models, namely those based on some crisp characteristics of fuzzy data (such as possibilistic mean value or possibilistic variance - these concepts are rigorously defined in [1], [2]). Usually, this type of model has applications in the difficult issue of portfolio selection but it also can be seen as a standalone optimization problem with constraints. In this paper we use a generic model whose objective function is based on possibilistic mean value of a certain fuzzy number. It is used quite a lot in the literature but with various changes in the parameters or constraints (see [4], [5], [9], [14], [18]). However, there is one thing in common: the possibility of transforming the initial problem in some linear optimization problems. Generally, the input data (in the case of portfolio optimization we discuss about the possible return rates) are modeled as triangular or trapezoidal numbers ([4], [5]). Also some generalization in the form of polygonal numbers was given ([9], [18]). By contrast, in this work we propose the use of quasi S shape fuzzy numbers, which have already proven useful in the field of statistical regression (see also [13]). These numbers are partially linearized S shape ones (a

Lecturer Costin-Ciprian POPESCU, PhD The Bucharest …ecocyb.ase.ro/eng/Ciprian Popescu.pdfCostin Ciprian Popescu presentation of the S shape membership functions can be found in [8])

  • Upload
    others

  • View
    16

  • Download
    0

Embed Size (px)

Citation preview

  • Lecturer Costin-Ciprian POPESCU, PhD

    The Bucharest Academy of Economic Studies

    Department of Applied Mathematics

    E-mail: [email protected]

    A FUZZY OPTIMIZATION MODEL

    Abstract. A fuzzy optimization model, mainly aiming the portfolio selection

    issue, is discussed. Instead of being modeled as piecewise linear fuzzy numbers

    (triangular, trapezoidal etc.), the input data are presented as quasi S shape fuzzy

    numbers. In this framework some theoretical and numerical results are given.

    Key words: fuzzy number, possibilistic mean value, quasi S shape fuzzy

    data, portfolio optimization.

    JEL Classification: C020, C440, C610.

    1. Introduction

    The issue of fuzzy mathematical programming is a topic of great interest in

    literature ([13], [14], [15], [16]). Also the financial market analysis, including

    portfolio optimization, has numerous approaches ([3], [10], [14], [17], [19]). In

    fact, the problem of portfolio optimization can be viewed as a problem of decision-

    making. In this area, the theory of fuzzy sets is a very useful tool [20]. In this paper

    we deal with both subjects mentioned above. More precisely, we discuss a fuzzy

    optimization model from the theoretical point of view and we also show its

    applications to the problem of portfolio selection. It is included in a certain class of

    models, namely those based on some crisp characteristics of fuzzy data (such as

    possibilistic mean value or possibilistic variance - these concepts are rigorously

    defined in [1], [2]). Usually, this type of model has applications in the difficult

    issue of portfolio selection but it also can be seen as a standalone optimization

    problem with constraints. In this paper we use a generic model whose objective

    function is based on possibilistic mean value of a certain fuzzy number. It is used

    quite a lot in the literature but with various changes in the parameters or constraints

    (see [4], [5], [9], [14], [18]). However, there is one thing in common: the

    possibility of transforming the initial problem in some linear optimization

    problems. Generally, the input data (in the case of portfolio optimization we

    discuss about the possible return rates) are modeled as triangular or trapezoidal

    numbers ([4], [5]). Also some generalization in the form of polygonal numbers was

    given ([9], [18]). By contrast, in this work we propose the use of quasi S shape

    fuzzy numbers, which have already proven useful in the field of statistical

    regression (see also [13]). These numbers are partially linearized S shape ones (a

  • Costin Ciprian Popescu

    presentation of the S shape membership functions can be found in [8]). The

    linearization of some parts of the membership function is necessary for the model

    proposed here. In this way the data fulfill the necessary conditions that allow us to

    apply for our case some general results about binary operations (as sum, scalar

    multiplication or even some comparisons between fuzzy numbers - see [6], [7],

    [11], [12], [15], [16]). In section 2 these theoretical aspects are emphasized in a

    more detailed presentation. We will also show the steps that must be taken in order

    to find a solution to the model described here. In section 3 the applicability of the

    proposed algorithm is verified for a set of numerical data.

    2. Theoretical results

    At first we give some results regarding quasi S shape fuzzy numbers [13],

    which are derived from S shape numbers [8]. Because the membership function of

    a S shape number has an asymptotic behavior regarding the lines 0y , 1y , it

    appears to be difficult to make certain operations with them. However, their partial

    linearization allows us to use the general results regarding sum or scalar (that is a

    real number) multiplication ([6], [11], [16]). So consider the numbers R, ,

    2

    1,0, 2221 ll , 1,

    2

    1, 1211 ll and

    11

    1111

    1ln

    l

    l,

    21

    2121

    1ln

    l

    l,

    12

    1212

    1ln

    l

    l,

    22

    2222

    1ln

    l

    l.

    Also consider two bijective functions:

    1,01

    ,1

    1: 11

    11

    21

    21 llf , 1,0

    1

    1,

    1: 22

    22

    12

    12 llg

    given by

    11

    11

    1111

    11

    1121111

    1121

    2121

    21

    21

    21

    2122121

    1, ,

    1ln

    , ,1

    1

    ,1

    1 ,

    1ln

    lxl

    l

    lxll

    xe

    lxl

    l

    lxll

    xfx

    and

  • A Fuzzy Optimization Model

    . 1

    1, ,

    1ln

    , ,1

    1

    ,1

    ,1

    ln

    22

    22

    2222

    22

    2222222

    2212

    1212

    12

    12

    12

    1221212

    lxl

    l

    lxll

    xe

    lxl

    l

    lxll

    xgx

    A quasi S shape fuzzy number, u~ , can be completely defined by its membership

    function. Let 1,0:Ru be this function, as follows:

    . 1

    ln1

    ,1

    ln1

    ,1

    1ln

    1

    1,

    1ln

    1 ,

    1ln

    1,

    1ln

    1

    1 ,

    ,1

    ln1

    11ln

    1

    1, ,0

    12

    12

    1211

    11

    11

    22

    22

    2212

    12

    12

    11

    11

    1121

    21

    21

    22

    22

    2221

    21

    21

    l

    l

    ll

    l

    lx

    l

    l

    ll

    l

    lxxg

    l

    l

    ll

    l

    lxxf

    l

    l

    ll

    l

    lx

    xu

    Figure 1. A quasi S shape fuzzy number

  • Costin Ciprian Popescu

    The condition for the existence of such a number is [13]:

    12

    12

    1211

    11

    11

    1ln

    11ln

    1

    l

    l

    ll

    l

    l

    or

    1211

    1211

    1211

    1211 11lnll

    ll

    ll

    ll.

    In the case of equality we obtain a unique point, 0x , where the membership

    function is equal to 1. This is equivalent to

    1211

    1211

    1211

    1211 11lnll

    ll

    ll

    ll.

    Remark that the parameters and control the left spreading and the right

    spreading, respectively. The parameters 21l , 11l determine the length of the

    linearization on the left side of the membership function. Similarly, the parameters

    22l , 12l determine the length of the linearization on the right side of the

    membership function. We denote the quasi S shape fuzzy number u~ by

    22211211 ,,,,,~ llllu . It can also be defined as a triplet rruru ,, (like any

    fuzzy number with certain properties, as it is shown in [6], [7], [11], [16]), where

    R1,0:, ruru and [13]:

    , 1, ,1

    ln1

    1

    1

    1

    , ,1

    ln

    ,0 ,1

    ln1

    1

    1

    1

    11

    11

    11

    111111

    1121

    21

    21

    21

    212121

    lrl

    l

    lr

    ll

    llrr

    r

    lrl

    l

    lr

    ll

    ru

    . 1, ,1

    ln1

    1

    1

    1

    , ,1

    ln

    ,0 ,1

    ln1

    1

    1

    1

    12

    12

    12

    121212

    1222

    22

    22

    22

    222222

    lrl

    l

    lr

    ll

    llrr

    r

    lrl

    l

    lr

    ll

    ru

  • A Fuzzy Optimization Model

    Some crisp characteristics of a general fuzzy number rraraa ,,~ are

    introduced in [1]: the lower possibilistic mean value (

    1

    0

    2~ drraraM ), the

    upper possibilistic mean value (

    1

    0

    2~ drraraM ) and the possibilistic mean

    value (2

    ~~~ aMaMaM ). By applying these definitions to

    22211211 ,,,,,~ llllu , we obtain the following results:

    21

    112111

    11

    21111

    21

    221

    1

    01

    ln3

    2

    132~

    l

    lll

    l

    ll

    l

    ldrruruM ,

    12

    221222

    12

    21212

    22

    222

    1

    0

    1ln

    3

    2

    132~

    l

    lll

    l

    ll

    l

    ldrruruM

    and (see also [13])

    21

    112111

    11

    21111

    21

    221

    1ln

    2

    1

    26

    2

    162

    ~

    l

    lll

    l

    ll

    l

    luM

    12

    221222

    12

    21212

    22

    222 1ln

    2

    1

    26

    2

    162 l

    lll

    l

    ll

    l

    l

    12

    21212

    22

    222

    11

    21111

    21

    221 2

    1

    2

    16

    1

    2 l

    ll

    l

    l

    l

    ll

    l

    l

    2112

    221112222111

    1

    1ln

    2

    1

    2 ll

    llllll.

    For example, if we analyze the number 3.0,2.0,7.0,9.0,10,3~u (Figure 1) we

    obtain: 0915.2~uM , 2571.9~uM and 5828.3~uM . We also remark that

    if 1211 ll and 2221 ll (i.e. same linearization levels on both sides) we obtain

    2

    ~uM . Taking into account some previous results from literature ([4],

    [14], [18]), we consider the fuzzy optimization model S , as below. Notice that in

    our problem the input data are modeled as quasi S shape numbers instead of

    polygonal ones. So the initial problem is formulated as:

  • Costin Ciprian Popescu

    nip

    p

    tup

    upM

    iii

    n

    i

    i

    n

    i

    ii

    n

    i

    iin

    ,1 ,

    1

    ~Pos

    s.t.

    ~max

    S

    1

    1

    1Rp

    where npp ,...,1p is a vector for which 11

    n

    i

    ip , 1,0 , Rt and iu~ are

    quasi S shape fuzzy numbers. When such a model is used for improving a

    portfolio selection, the parameters gain some concrete meanings: for example p

    show how an investor must share this capital between n (possibly risky) assets. In

    order to solve S , in the next steps we change the form of this optimization model.

    In [1] it is proved that n

    i

    i

    n

    i

    i aMaM11

    ~~ and abMabM ~~ (where Rb

    and 1~a ,…, na

    ~ , a~ are fuzzy numbers). Applying these results for our case, we

    conclude that the objective function can be transformed as follows: n

    i

    iii

    n

    i

    ii pupM11

    2

    1~

    n

    i

    ipl

    ll

    l

    l

    l

    ll

    l

    l

    112

    21212

    22

    222

    11

    21111

    21

    221 2

    1

    2

    16

    1

    n

    i

    i

    n

    i

    i pll

    llpllll

    12112

    2211

    1

    122221111

    1ln

    2

    1

    2

    1.

    Moreover, by using the substitutions

    12

    21212

    22

    222

    11

    21111

    21

    221

    1

    2

    1

    2

    16

    1

    l

    ll

    l

    l

    l

    ll

    l

    lL ,

    1222211122

    1llllL ,

    2112

    22113

    1

    1ln

    2

    1

    ll

    llL

    and

    2

    iii (for ni ,1 )

  • A Fuzzy Optimization Model

    we can write:

    n

    i

    ii

    j

    j

    n

    i

    ii pLupM1

    3

    11

    ~ .

    Taking into account the definition of the possibility of ba~ (where a~ is a fuzzy

    number and Rb ) [15], we can find some characteristics of a quasi S shape

    fuzzy number u~ . So, for Rt , we have:

    1) 0~Pos tu

    when

    21

    21

    21

    1ln

    1

    1

    l

    l

    lt ;

    2) 21

    2121212121212121

    1ln111~Pos

    l

    lllllltlltu

    when

    21

    21

    21

    21

    21

    1ln

    1ln

    1

    1

    l

    lt

    l

    l

    l;

    3) te

    tu1

    1~Pos

    when

    11

    11

    21

    21 1ln1

    lnl

    lt

    l

    l;

    4) 11

    1111111111111111

    1ln111~Pos

    l

    lllllltlltu

    when

    11

    11

    1111

    11 1ln11

    lnl

    l

    lt

    l

    l;

    5) 1~Pos tu

    when

    11

    11

    11

    1ln

    1

    l

    l

    lt .

    Remark that the subset of quasi S shape fuzzy numbers, with the usual two binary

    operations (addition and scalar multiplications), is not necessarily closed. For a

    convenient conversion of the first constraint, we need to discuss the inequality

    tupn

    i

    ii

    1

    ~Pos . We see that there are several possibilities, as follows:

  • Costin Ciprian Popescu

    1) If n

    i

    ii pl

    l

    lt

    1 21

    21

    21

    1ln

    1

    1

    then

    0~Pos1

    tupn

    i

    ii .

    2) If n

    i

    ii

    n

    i

    ii pl

    ltp

    l

    l

    l1 21

    21

    1 21

    21

    21

    1ln

    1ln

    1

    1

    then

    n

    i

    i

    n

    i

    iin

    i

    ii

    p

    pl

    l

    ltll

    tup

    1

    1 21

    21

    21

    22121

    1

    1ln

    1

    1

    ~Pos .

    Thus

    tupn

    i

    ii

    1

    ~Pos

    is equivalent to

    tpl

    l

    lll

    n

    i

    ii

    1 21

    21

    2212121

    1ln

    1

    1.

    3) If n

    i

    ii

    n

    i

    ii pl

    ltp

    l

    l

    1 11

    11

    1 21

    21 1ln1

    ln

    then

    n

    i

    i

    n

    i

    ii

    p

    pt

    n

    i

    ii

    e

    tup

    1

    1

    1

    1~Pos1

    .

    Thus

    tupn

    i

    ii

    1

    ~Pos

    is equivalent to

    tpn

    i

    ii

    1

    1ln .

  • A Fuzzy Optimization Model

    4) Assume that n

    i

    ii

    n

    i

    ii pl

    l

    ltp

    l

    l

    1 11

    11

    111 11

    11 1ln11

    ln .

    It follows that

    tupn

    i

    ii

    1

    ~Pos

    is equivalent to

    tpl

    l

    lll

    n

    i

    ii

    1 11

    11

    2111111

    1ln

    1

    1.

    5) If n

    i

    ii pl

    l

    lt

    1 11

    11

    11

    1ln

    1

    then

    1~Pos1

    tupn

    i

    ii .

    Taking into account the previous lines, we can conclude that the solution for the

    initial model can be obtained by solving three linear models (that will be denoted

    by (S1), (S2), (S3)), which have the same objective function. The solution for (S) is

    chosen such that the common objective function reaches its maximum. In other

    words, if 1Sp , 2Sp and 3Sp are the solution vectors for (S1), (S2), (S3) then the

    solution for the initial problem is one of these three but so that the objective

    function takes the greatest possible value. Therefore 1Sp , 2Sp and 3Sp can be

    viewed only as preliminary results for the initial model, only one being the optimal

    solution. On the other hand, there is the possibility that one of these problems has

    no admissible solutions, as we shall see in the next section. In this case we

    compare only the existing solutions. We have:

  • Costin Ciprian Popescu

    , ,1 ,

    1

    1ln

    1ln

    1

    1

    1ln

    1

    1

    s.t.

    max

    S1

    1

    1 21

    21

    1 21

    21

    21

    1 21

    21

    2212121

    1

    3

    1

    nip

    p

    tpl

    l

    tpl

    l

    l

    tpl

    l

    lll

    pL

    iii

    n

    i

    i

    n

    i

    ii

    n

    i

    ii

    n

    i

    ii

    n

    i

    ii

    j

    jnRp

    nip

    p

    tpl

    l

    tpl

    l

    tp

    pL

    iii

    n

    i

    i

    n

    i

    ii

    n

    i

    ii

    n

    i

    ii

    n

    i

    ii

    j

    jn

    ,1 ,

    1

    1ln

    1ln

    1ln

    s.t.

    max

    S2

    1

    1 11

    11

    1 21

    21

    1

    1

    3

    1Rp

    and

    . ,1 ,

    1

    1ln

    1

    1ln

    1ln

    1

    1

    s.t.

    max

    S3

    1

    1 11

    11

    11

    1 11

    11

    1 11

    112111111

    1

    3

    1

    nip

    p

    tpl

    l

    l

    tpl

    l

    tpl

    l

    lll

    pL

    iii

    n

    i

    i

    n

    i

    ii

    n

    i

    ii

    n

    i

    ii

    n

    i

    ii

    j

    jnRp

    3. Numerical test

    We will study the case of a virtual portfolio with five assets. Assume that

    the return rates (including the possibility of some losses) are modeled as five quasi

    S shape fuzzy data (see Fig. 2), as follows: 3.0,2.0,7.0,9.0,10,3~1u ,

    3.0,2.0,7.0,9.0,9,1~2u , 3.0,2.0,7.0,9.0,6,2~

    3u , 3.0,2.0,7.0,9.0,7,5.0~

    4u ,

    3.0,2.0,7.0,9.0,5.6,5.1~5u . For each number iu~ ( 5,1i ) we consider the pair

  • A Fuzzy Optimization Model

    RRii , , which gives the left point and the right point where the

    membership function intersects the horizontal axis. For our numbers, we obtain:

    276.12,6363.5 , 276.11,6363.3 , 2759.8,6363.4 , 2759.9,1363.3 and

    7759.8,1363.4 , respectively. The level Rt is taken equal to ii 5,1max i.e.

    1363.3t . The other parameters can be chosen depending on the level of risk

    acceptable to the investor. So we consider 5.0 and 1.0i , 5.0i , for all

    5,1i . Taking into account those stated in section 2, we proceed directly to the

    formulation of problems (S1), (S2) and (S3). After calculations, we obtain:

    5,1 ,5.01.0

    1

    1363.38863.28863.13863.33863.23863.4

    1363.31363.41363.36363.46363.36363.5

    1363.30113.101129.05113.151129.05113.2

    s.t.

    5828.23328.30828.20828.45828.3max

    S1

    54321

    54321

    54321

    54321

    543215

    ip

    ppppp

    ppppp

    ppppp

    ppppp

    ppppp

    i

    Rp

    5,1 ,5.01.0

    1

    1363.36972.06972.11972.01972.18028.0

    1363.38863.28863.13863.33863.23863.4

    1363.35.15.023

    s.t.

    5828.23328.30828.20828.45828.3max

    S2

    54321

    54321

    54321

    54321

    543215

    ip

    ppppp

    ppppp

    ppppp

    ppppp

    ppppp

    i

    Rp

    5,1 ,5.01.0

    1

    1363.38083.18083.23083.13083.23083.0

    1363.36972.06972.11972.01972.18028.0

    1363.37472.37472.22472.42472.32472.5

    s.t.

    5828.23328.30828.20828.45828.3max

    S3

    54321

    54321

    54321

    54321

    543215

    ip

    ppppp

    ppppp

    ppppp

    ppppp

    ppppp

    i

    Rp

    The solution for (S1) is 1.0,1.0,1.0,5.0,2.01Sp , for which the maximum of the

    objective function is 5578.3 . The solution for the problem (S2) is

    1.0,1.0,1.0,375.0,325.02Sp , for which the maximum of the common objective

  • Costin Ciprian Popescu

    function is 4953.3 . The problem (S3) has no admissible solutions. Due to the fact

    that 5578.34953.3 , the final result for the initial model (S) is 1Sp .

    Figure 2. Five data of quasi S shape type

    4. Conclusion

    From the general set of fuzzy numbers, the subset of quasi S shape

    elements was considered. Some theoretical features (such as possibilistic mean

    value, some binary operations) of these elements were emphasized. An

    optimization model in which the data are presented in the form of quasi S shape

    fuzzy numbers was discussed. A numerical application, in order to test the

    applicability of the model, was given.

    REFERENCES

    [1] Carlsson, C., Fullér, R. (2001), On Possibilistic Mean Value and Variance of Fuzzy Numbers; Fuzzy Sets and Systems, 122, 315-326;

    [2] Carlsson, C., Fullér, R. (2003), A Fuzzy Approach to Real Option Valuation; Fuzzy Sets and Systems 139, 297-312;

    [3] Carlsson, C., Fullér, R., Heikkilä, M., Majlender, P. (2007), A Fuzzy Approach to R&D Project Portfolio Selection; International Journal of

    Approximate Reasoning, 44, 93-105;

    [4] Chen, G., Chen, S., Fang, Y., Wang, S. (2006), A Possibilistic Mean VaR Model for Portfolio Selection; Advanced Modeling and Optimization, 1,

    99-107;

  • A Fuzzy Optimization Model

    [5] Deng, X., Li R. (2014), Gradually Tolerant Constraint Method for Fuzzy Portfolio Based on Possibility Theory; Information Sciences 259, 16-24;

    [6] Dubois, D., Prade, H. (1978), Operations on Fuzzy Numbers; International Journal of Systems Science, 9, 612-626;

    [7] Dubois, D., Prade, H. (1980), Fuzzy Sets and Systems: Theory and Applications; Mathematics in Science and Engineering, 144, Academic

    Press;

    [8] Fang, Y., Lai, K.K., Wang, S.Y. (2006), Portfolio Rebalancing Model with Transactions Costs Based on Fuzzy Decision Theory; European Journal of

    Operational Research, 175, 879-893;

    [9] Fulga, C., Dedu, S., Popescu, C.C. (2011), Managementul riscului. Aplicaţii în optimizarea portofoliilor, ASE Bucharest Publishing House;

    [10] Geambaşu, C., Şova, R., Jianu, I., Geambaşu, L. (2013), Risk Measurement in Post-modern Portfolio Theory: Differences from Modern

    Portfolio Theory, Economic Computation and Economic Cybernetics

    Studies and Research, 1, 113-132; ASE Publishing, Bucharest;

    [11] Goetschel, R., Voxman, W. (1986), Elementary Fuzzy Calculus; Fuzzy Sets and Systems, 18, 31-43;

    [12] Hanns, M. (2005), Applied Fuzzy Arithmetic. An Introduction with Engineering Applications, Springer;

    [13] Iacob, A.I., Popescu, C.C. (2012), An Optimization Model with Quasi S Shape Fuzzy Data; AWER Procedia Information Technology and Computer

    Science, 2, 132-136;

    [14] Inuiguchi, M., Ramik, J. (2000), Possibilistic Linear Programming: A Brief Review of Fuzzy Mathematical Programming and a Comparison with

    Stochastic Programming in Portfolio Selection Problem; Fuzzy Sets and

    Systems, 111, 3-28;

    [15] Liu, B. (1999), Uncertain Programming; Wiley; [16] Ming, M., Friedman, M., Kandel, A. (1997), General Fuzzy Least

    Squares; Fuzzy Sets and Systems, 88, 107-118;

    [17] Novotná, M. (2013), A multivariate Analysis of Financial and Market-based Variables for Bond Rating Prediction; Economic Computation and

    Economic Cybernetics Studies and Research, 2, 67-84, ASE Publishing;

    [18] Popescu, C.C., Fulga, C. (2011), Possibilistic Optimization with Applications to Portfolio Selection; Proceedings of the Romanian Academy

    Series A: Mathematics, Physics, Technical Sciences, Information Science, 2,

    88-94;

    [19] Zeng, S., Baležentis, A., Su, W. (2013), The Multi-criteria Hesitant Fuzzy Group Decision Making with MULTIMOORA Method; Economic

    Computation and Economic Cybernetics Studies and Research, 3, 171-184;

    ASE Publishing, Bucharest.