In the spirit of probability - based multi - objective optimization, each objective can be analogically represented as a single event in a system (Zheng et al, 2022a) and the whole event of multi - objective simultaneous optimization corresponds to the product of all single objectives (events). All performance utility indexes of a candidate are preliminarily classified into two types: i.e., beneficial type and unbeneficial type, in accordance with the role and preference of a candidate in the optimization, respectively. Specifically, the assessment of the preferable probability P_ij of both beneficial indicators and unbeneficial indicators can be carried out according to the evaluation procedure in Fig. 1 (Zheng et al, 2022a).

The meanings of the quantities and the factors in Fig. 1 are as follows:

P_ij indicates the partial preferable probability of the j-th performance utility indicator of the i-th alternative scenario, X_ij; n expresses the total number of the alternative scenario; m reflects the total number of the performance (objective); represents the arithmetic value of the j-th performance utility indicator; X_jmax and X_jmin show the maximum and minimum values of the j-th performance utility indicator, respectively; α _j and β _j express the normalized factors of the j-th performance utility indicator X_ij in the beneficial status and in the unbeneficial status, individually; the beneficial status or the unbeneficial status of the j-th performance utility indicator X_ij is determined according to its specific role or preference in the instant problem; and P_i represents the total (overall) preferable probability of the i-th alternative scenario (Zheng et al, 2022a).

Here, as to the portfolio investment problem, a simultaneous maximization of the rate of return and minimization of the variance of the rate of return is a typical bi-objective problem which contains a beneficial indicator and an unbeneficial indicator, respectively.

Figure 1

Рис. 1 – Процедура оценки метода многокритериальной оптимизации, основанной на вероятности

Слика 1 – Поступак евалуације метода вишекритеријумске оптимизације засноване на вероватноћи

In this bi - objective portfolio investment problem, the total preferable probability is the decisive objective function which needs to be maximized in a high-dimensional space, so the complex data treatment might be involved; thus uniform design for experiments with mixtures (UDEM) can be employed to simplify data processing rationally.

Uniform design for experiments with mixtures (UDEM), based on the good lattice point (GLP), was proposed by Fang et al (2018). The method of UDEM can be used to create a set of effective sampling points for experimental design with the restraint of x_l + x₂ + x₃．．．+ x_s =1 for the proportion x_i with the total number of s (Fang et al, 2018); therefore, it can be used as an efficient sampling method for the portfolio investment problem here to conduct the simplification of data processing with discretization.

In addition, Fang especially developed uniform design tables and their usage tables for proper application (Fang et al, 2018).

According to Fang et al (2018), the concrete steps of uniform design for experiments with mixtures (UDEM) are generally as follows:

According to Markowitz's study, the return rate function f1 and the risk function f2 could be introduced, and their expressions are, respectively,

(3)

(4)

In Eq. (3), is the rate of return of the i-th security and s is the total number of the securities. In Eq. (4), is the correlation coefficient between the i-th security and the j-th security; is the risk of the i-th security.

According to the objective evaluation in probability - based multi - objective optimization methodology (Zheng et al, 2022a), f₁ exhibits the bigger the better, which therefore belongs to the beneficial objective, and f₂ manifests the smaller the better, which thus belongs to the unbeneficial objective, respectively.

Therefore, the answer to the portfolio problem is the optimization of a bi-objective problem. Furthermore, probability - based multi - objective optimization methodology can be used to assess it reasonably. Of course, all the evaluations in probability - based multi - objective optimization methodology can be applied rationally.

In this section, the problem of "portfolio" is analyzed based on probability - based multi - objective optimization methodology. This is a typical bi-objective optimization problem.

Let x₁, x₂, x₃ and x₄ be the investment percentages of four securities, A, B, C and D, respectively. There is actually a constraint condition for this problem, i.e., x₁ + x₂ + x₃ + x₄ = 1; therefore, it has actually three independent variables, namely x₁, x₂ and x₃ .

Since the sampling points of this bi-objective optimization problem are positioned in the 4 – dimensional space, it is necessary to include at least 23 sampling points with the characteristics of the "good lattice point" in the effective region for the discretization of data processing (Yu et al, 2022; Zheng et al, 2022a; Zheng et al, 2022b).

According to Fang et al (2018), this is a "uniform design for experiments with mixtures" problem due to the constraint condition of the four variables. Let us take the uniform table U*₂₃(23⁷) as the initial table to construct a uniform test design table UM₂₃(23⁴) with mixtures, as shown in Table 1.

The uniform test table UM₂₃(23⁴) with the mixtures of Table 1 is based on the uniform design table U*₂₃(23⁷). Because here the number of variables s equals to 4, and n equals to 23, from the above rules, x_k1 = 1 – c_k1 ^1/3, x_k2 = c_k1 ^1/3×(1 – c_k2 ^1/2), x_k3 = c_k1 ^1/3×c_k2 ^1/2×(1-c_k3), x_k4 = c_k1 ^1/3×c_k2 ^1/2×c_k3 (Fang et al, 2018).

Furthermore, we can get the values of the rate of return function f₁ and the risk function f₂ , the distribution of their partial preferable probability and their total preferable probability, as well as the ranking at the sampling points (alternative scenario), which are shown in Table 2.

Fig. 2 shows the variation of the return rate with respect to risk at the discrete sampling points. The results reflect that the 2nd discrete sampling point gives the maximum total preferable probability closely followed by the 6th sampling point; therefore, they could be taken as the optimal solution to this portfolio problem.

Regarding the 2nd sampling point, the corresponding investment ratio is x₁"= 0.0222, x₂"= 0.3494, x₃"= 0.2596, x₄" = 0.3688, which leads to the rate of return of 12.98% and the risk of 9.09%.

As for the 6th sampling point, its investment ratio is at x₁' = 0.0871, x₂' = 0.4665, x₃' = 0.1068, x₄' = 0.3397, and its obtained rate of return is 14.23% with the risk of 10.73%.

Table 1

Таблица 1 – Таблица унифицированного теста UM₂₃(23⁴) с вариантами, основанными на таблице унифицированной модели U*₂₃(23⁷)

Табела 1 –Табела униформног теста UM₂₃(23⁴) са варијацијама заснованим на табели униформног дизајна U*₂₃(23⁷)

Table 2

Таблица 2 – Результаты оценки ставки доходности f₁, риска f₂, предпочтительной вероятности и ранжирования по точкам выборки

Табела 2 – Резултати процене стопе приносa f₁, ризикa f₂, пожељне вероватноће и рангирање по тачкама узорковања

Figure 2

Рис. 2 – Варианты ставки доходности по сравнению с риском в дискретных точках выборки

Слика 2 – Варијација стопе приноса у односу на ризик у тачкама дискретног узорковања