Proceedings

selection had the strongest impact over modern financial mathematics. Hence,
Markowitz’s theory over portfolio management analyses individual agents of the
financial markets. This theory combines concepts of probability theory and
optimizations in order to model the agents’ behaviour during various times of
evolution in economy. The theory assumes that agents are searching for a balance
between maximizing income and minimizing investment risks. Income is calculated
with medium efficiency, and the risk is a variance of the assets that form the portfolio.
Such mathematical representations of income and risk have allowed the
implementation of optimization techniques in studies concerning portfolio
management. The two objectives of the investors – maximizing profits and
minimizing risks – are oriented so that they maximize the income’s expected value
and minimize the variance of the portfolio’s value. The solution will hence be a
function of the risk level to the resulted income. Although recent models present new
and varied perspectives of the mathematical definitions of risk and economic agents’
income (investors, in this case), the relationship between income and risk has always
been one of the problems that the financial theories have tried to resolve.
Although the fact that the analysis of the problems described by the portfolio
selection, related to the model efficiency mean-variance can be resolved in a
polynomial time interval, historically speaking, many efforts have been made in order
to reduce the effective computation time. Sharpe’s model (1963) represents one of the
main successes in this field. The model reduces the estimation of the number of the
variance-covariance coefficients to a much smaller total value. Considering the
hypothesis according to which investors’ decisions are influenced only by the income
risk (which is smaller than the income mean), the Mean-Semivariance (M-S) method
was suggested, in order to develop the model known today as Markowitz-Mao-
Swalm. In this case, the definition of semivariance is the expected value of the square
of the deviation from the mean or, more generally speaking, the value selected as
critical by the decision taker.
In order to solve the problem of large-size portfolio optimization models, Konno and
Yamazaki (Konno and Yamazaki, 1991) have analysed the efficiency mean-absolute
deviation – type approach for investment risk measurement. Using the data supplied
by the Tokyo stock market, Konno and Yamazaki have compared the performance
resulted from the efficiency mean-variance model, with respect to the efficiency
mean-absolute deviation model, and discovered that the efficiency levels of the two
were extremely close. Feinstein and Thapa (Feinstein and Thapa, 1993) have
rephrased the above-mentioned model, equivalent to the one developed by Konno
(Konno et al., 1993) and that simultaneously halves the restrictions on the number of
non-zero assets of the optimal portfolio. Also, while Konno demonstrated that the
efficiency mean-absolute deviation does not need a covariance matrix, Simaan
(Simaan, 1997) discovers that this should lead to a much higher estimated risk and
exceed the benefits.
1. THE BI-OBJECTIVE MODEL FOR MIXED ASSETS PORTFOLIO
SELECTION
In order to illustrate our method, we assume that an investor assigns his wealth in
projects and traditional assets. Therefore, regarding the mixed assets portfolio
selection, the available assets for investments are divided in two types. The first class
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