The bi-objective uncertain rebalancing model is formulated as follows:

Huang [29] introduced so-called “9999 Method” in order to calculate expected value and variance in mean-variance model when securities returns are described by uncertain variables with different uncertainty distributions.

Assume ξi is an uncertain variable with uncertainty distribution Φi, and ki a positive number for i = 1, 2, ..., n, respectively. Let Ѱi represent the uncertainty distributions of kiξi, i=1, 2, ..., n, respectively. Then, we have:

Now considering Ѱ to be the uncertainty distribution of kiξi+k2ξ2 + ⋯ + knξn, we have:

This means the uncertainty distribution Ѱ of kiξi+k2ξ2 + ⋯ + knξn can be represented on a computer as shown in Table 11.

Suppose ξ = kiξi+k2ξ2 + ⋯ + knξn is an uncertain variable. According to Lemma 4 and 9999 Method, the expected value and variance of uncertain variable ξ are as follows:

As a result, the objectives of the proposed rebalancing model can be replaced by followings:

There are various approaches for solving a multi-objective mathematical programming (MOMP) problem. Miettinen [283] classified them into four categories:

1. No-preferences methods 2. A priori methods 3. A posteriori methods 4. Interactive methods

While in no-preferences methods the decision maker (DM) has no participation in the solution process, a priori methods ask for the DM preferences and opinions before the solution process. In a posteriori methods, which are also called generation methods, first the Pareto optimal set (or a representation of it) is generated, and then the DM selects the most preferred solution. In interactive methods, the DM gets involved in the solution process by correcting his/her preferences in each iteration and after being presented only part of the Pareto optimal points.

Considering selection problems, we generally search for every possible combination of assets that generates different efficiencies with different combinations of risk—expected return according to the investors’ preferences. These different efficiencies will form the efficient frontier [284]. Afterwards, each individual can find his/her own optimal efficient frontier according to their risk preferences. Thus, solving the bi-objective rebalancing problem falls into the a posteriori methods category in which the Pareto optimal set is represented by the efficient frontier, and then the DM (here the investor) will choose the optimal solution according to his/her preferences. Miettinen [283] introduced two basic a posteriori methods: the weighting method and the ε-constraint method. While the ε-constraint method can find every Pareto optimal solution of any MOMP regardless of convexity of the problem, the weighting method fails to find all of the Pareto optimal solutions when the problem is non-convex. Accordingly, we utilize the ε-constraint method in order to solve the bi-objective uncertain rebalancing problem. In this method, one of the objective functions is optimized by formulating other objective functions as constraints and transferring them to the constraint part of the model [285]. Assume the following MOMP problem:

Then, using ε-constraint method, we will have the following single-objective problem:

The Pareto optimal solutions of the problem are obtained by initialization and parametric variation in the RHS of the constrained objective functions (i.e., ξi) and then solving the model for these different parameters. Consequently, using ε-constraint method, the objective function corresponding to return in our proposed model is formulated as:

where λ represents the minimum expected return required by the investor. By solving the model for different values of λ, the solutions obtained will form an efficient frontier. In addition, the minimum expected return required by the investor is obviously greater than the return on risk-free asset. Thus:

Finally, the uncertain rebalancing model is formulated as follows:

Considering different levels of belief degrees confirmed changing in model solutions and efficient frontier diagrams. Therefore, considering uncertain variables affects rebalancing model and wider belief degree ranges make more conservative results. In order to improve the performance of the algorithm, future researches will consider other meta-heuristics such as artificial bee colony algorithm.