*2.1. Fundamental Analysis*

One of the most used sources of information in the managemen<sup>t</sup> of stock portfolios comes from the so-called fundamental analysis. The fundamental indicators provided by this analysis allow the practitioner to evaluate stocks from multiple perspectives. Such indicators are constructed from the financial statements that the companies (underlying the stocks) present publicly on a regular basis.

Fundamental indicators provide information that is often exploited in the literature to forecast future stock performance and to select the most competitive stocks. These indicators can be used both qualitatively and quantitatively. Regarding the latter, the financial information published by companies is synthesized in the form of ratios that shed light on the current state of the company, providing remarkable information on what can be expected from the financial health of the company and the possible future price of its stock. When this analysis is used in the literature, the fundamental indicators are usually aggregated in an overall assessment value that requires subjective preferences from the practitioner (cf. e.g., [13]); however, the aggregation procedure is not straightforward and represents an important challenge.

On the other hand, different fundamental indicators could be more convenient for companies with different types of activities ([14]). Some fundamental indicators that can be used for trans-business companies are described in Section 4.1 (cf. [10,11,13,14]).

#### *2.2. Artificial Neural Networks*

Artificial neural networks are nowadays very popular among techniques from computational intelligence that have been used for many applications, such as classification, clustering, pattern recognition and prediction in diverse scientific and technological disciplines ([15,16]). Similarly to other computational intelligence techniques, applications of ANN are very diversified due to its capability to model systems and phenomena from the fields of sciences, engineering and social sciences.

Analogously to a nervous system, an ANN is built from neurons, which are the basic elements for processing signals. Neurons are interconnected to form a network, with additional connections (synaptic relations) for input and output signals. Weights are assigned to each of these and other connection. The computing of suitable values for these weights is performed by training algorithms. An ANN needs to be trained before it can be used by using data from the system or phenomenon to model. Neurons are configured to form layers, in which neurons have parallel connections for inputs and outputs. ANN complexity varies from a network with a single layer of a single neuron to networks with several layers, each having several neurons. Networks with only forward connections are known as feedforward networks. Networks with forward and backward connections are known as feedbackward networks ([15]). The term deep learning refers to ANN with complex multilayers ([17]). Roughly speaking, deep learning has more complex connections between layers and also more neurons than previous types of networks. Some neural networks that form deep learning networks are convolutional networks, recursive networks and recurrent networks.

#### *2.3. Multi-Objective Optimization Problem*

Without loss of generality, a multi-objective optimization problem (MOP) can be defined in terms of maximization (although minimization is also common) as follows:

$$\text{maximize } F(\mathbf{x}) = [f\_1(\mathbf{x}), f\_2(\mathbf{x}), \dots, f\_k(\mathbf{x})]^\top$$

$$\text{subject to } \mathbf{x} \in \Omega$$

where Ω is the set of decision variable vectors *x* = [*<sup>x</sup>*1, *x*2, ... , *xm*)] that fulfill the set of constraints of the problem, and then *F* : Ω → <sup>R</sup>*k*, where R*<sup>k</sup>* is the so-called objective space.

It is evident that the notation used here states that all functions *fi* (objectives) should be maximized; however, it is also possible that one requires some functions *fi* to be minimized instead. To keep standard notation, we assume that the latter can be simply achieved by multiplying the minimizing function by −1.

In the context of stock portfolio management, the functions *fi* are usually in conflict with each other. This means that improving *fj* deteriorates *fk* for some *j* = *k*. Therefore, there is no solution *x* ∈ Ω that maximizes all the *k* objectives simultaneously. Nevertheless, it is still possible to define some solutions *x* that poses the best characteristics in terms of their impact on the objectives; this is commonly carried out through Pareto optimality (cf. [18]).

Let *u*, vs. ∈ R*<sup>k</sup>* denote impacts of solutions *x* and *y*, respectively. *u* dominates *v* if and only if *ui* ≥ *vi* for all *i* = 1, ... , *k*, and *uj* > *vj* for at least one *j* = 1, ... , *k*. Then, a solution *x*<sup>∗</sup> ∈ Ω is Pareto optimal if there is no solution *y* ∈ Ω such that *<sup>F</sup>*(*y*) dominates *<sup>F</sup>*(*x*<sup>∗</sup>). Note that there can be more than one Pareto optimal solution. The set of all the Pareto optimal solutions is called the Pareto set (PS), and the set of all their corresponding objective vectors is the Pareto front (PF).

#### *2.4. Evolutionary Multi-Objective Optimization*

Multi-objective evolutionary algorithms (MOEAs) are high-level procedures designed to discover good enough solutions to MOPs (solutions that are close to the global optimum). They are especially useful with incomplete or imperfect information or a limited computing capacity ([19]).

MOEAs address MOPs using principles from biological evolution. They use a population of individuals, each representing a solution to the MOP. The individuals in the population reproduce among them, using so-called evolutionary operators (selection, crossover, mutation), to produce a new generation of individuals. Often, this new generation of individuals is composed of both parents and children that posses the best fitness; this fitness represents the impact on the objectives of the MOP. Since each individual encodes a solution to the MOP, MOEAs can approximate a set of trade-off alternatives simultaneously

The performance of MOEAs has been assessed in different fields (e.g., [20,21]). They have been widely accepted as convenient tools for addressing the problem of stock portfolio managemen<sup>t</sup> ([10–12]). The main goal of MOEAs is to find a set of solutions that approximate the true Pareto front in terms of convergence and diversity. Convergence refers to determining the solutions that belong to the PF, while diversity refers to determining the solutions that best represent all the PF. Thus, the intervention of the decision maker is not traditionally used in the process. Thus, rather little interest has been paid in the literature to choosing one of the efficient solutions as the final one in contrast to the interest paid in approximating the whole Pareto front.

Usually, two types of MOEAs are highlighted in the literature: differential evolution and genetic algorithms. Differential evolution (DE) has been found to be very simple and effective ([22]), particularly when addressing non-linear single-objective optimization problems ([23,24]). On the other hand, in a genetic algorithm (GA), solutions to a problem are sought in the form of strings of characters (the best representations are usually those that reflect something about the problem that is being addressed), virtually always applying recombination operators such as crossing, selection and mutation operators. GAs compose one of the most popular meta-heuristics applied to the Portfolio Optimization Problem ([12]).

As a very effective and efficient way to address MOPs, the authors of ([25]) exploited the idea of creating subproblems underlying the original optimization problem. This way, addressing these subproblems the algorithm proposed in ([25]) indirectly addresses the original problem. In that work, the so-called Multiobjective Evolutionary Algorithm Based on Decomposition (MOEA/D) was presented. The goal of MOEA/D is to create subproblems such that, for each subproblem, a simpler optimization problem can be more effectively and efficiently addressed; each subproblem consists on the aggregation of all the objectives through a scalar function. MOEA/D was extended to the context of interval numbers in [12].

#### **3. Literature Review**

There are many contributions to portfolio managemen<sup>t</sup> literature in recent years. In this section, we give an overview of some recent and relevant works on the following subjects: price forecasting, stock selection and portfolio optimization, as well as works on portfolio managemen<sup>t</sup> by algorithms that exploits both uptrends and downtrends in stock prices.

Due to the non-linearity of stock data, a model developed using traditional approaches with single intelligent techniques may not use the resources in an effective way. Therefore, there is a need for developing a hybridization of intelligent techniques for an effective predictive model [26].

#### *3.1. Portfolio Management: Price Forecasting, Stock Selection and Portfolio Optimization*

In recent years, there have been plenty of contributions on price forecasting based on either statistical or computational intelligence methods (see [10,27]). The stock market is characterized by extreme fluctuations, non-linearity, and shifts in internal and external environmental variables. Artificial intelligence techniques can detect such non-linearity, resulting in much-improved forecast results [28].

Among the computational intelligence methods used for price forecasting are deep learning (e.g., [29–32]) and machine learning (e.g., [33–35]). In [10], a hybrid stock selection model with a stock prediction stage based on an artificial neural network (ANN) trained with the extreme learning machine (ELM) training algorithm ([6,36]) was proposed. The ELM algorithm has been tested for financial market prediction in other works (see [7–9]).

There are important works on methods for stock selection, which have several different fundamental theories, from operations research methods (e.g., [37,38]) to approaches originating in modern portfolio theory (Mean-variance model) (e.g., [38,39]) and soft computing methods (e.g., [40,41]), including hybrid approaches (e.g., [10,42,43]).

The fundamental theory for portfolio optimization is Markowitz's mean-variance model ([44]). Its formulation marked the beginning of Modern portfolio theory ([45]). However, Markowitz's original model is considered too basic since it neglects real-world issues related to investors, trading limitations, portfolio size and others ([43]). For evaluating a portfolio's performance, the model is based on measuring the expected return and the risk; the latter is represented by the variance in the portfolio's historical returns. Since the variance takes into account both negative and positive deviations, other risk measures have been proposed, such as the Conditional Value at Risk (CVaR) ([46,47]). As a result, numerous works have improved the model, creating more risk measures and proposing restrictions that bring them closer to practical aspects of stock market trading ([27]). Consequently, many optimization methods based on exact algorithms (e.g., [48–55]) and heuristic and hybrid optimization (e.g., [29,56–65]) have been proposed to solve the emerging portfolio optimization models ([27,40,45]).

According to [12], the investor or decision maker in the portfolio selection problem manages a multiple criteria problem in which, along with the objective of return maximization, he/she faces the uncertainty of risk. Different attitudes assumed by decision makers may lead them to select different alternatives. A way of modeling both risk and subjectivity of the decision maker in terms of significant confidence intervals was first proposed in [12]. The probabilistic confidence intervals of the portfolio returns characterize the portfolios during the optimization. The optimization is performed by means of a widely accepted decomposition-based evolutionary algorithm, the MOEA/D ([25,66]). This approach is inspired on the independent works of ([67,68]) on interval analysis theory.

#### *3.2. Exploiting Uptrends and Downtrends in Strategies for Stock Investment*

Regarding alternative strategies to the known buy-and-hold approach for stock investment, in ([69]), the authors propose two new trading strategies to outperform the buy-and-hold approach, which is based on the efficient market hypothesis. The proposed strategies are based on a generalized time-dependent strategy proposed in ([70]) but propose different timing for changing the buying/selling position. According to ([71]), the decision to adopt a long or short position in an asset requires a view of its immediate future price movements. A typical short seller would have to assess the potential future behavior of the asset price by means of evaluating several factors, such as past returns and market effects as well as and technical indicators, such as market ratios ([71]). There are a few works published in the literature to address the problem of trading strategies for the short position. An interesting work that considers not only the short position but both the short and long position is ([72]), in which a simultaneous long-short trading strategy (SLS) is proposed. Such a strategy is based partially on the property that a positive gain with zero initial investment is expected, which holds for all discrete and continuous price processes with independent multiplicative growth and a constant trend. Other works based on SLS

are ([73–75]). However, these works show the results of the algorithm on a previously defined stock portfolio, unlike the proposed approach that comprehensively performs price forecasting, stock selection and portfolio optimization in the presence of both uptrends and downtrends.

#### **4. Methods and Materials**

The procedure followed here consists of applying several techniques from the so-called computational intelligence to address the complexity of stock investments in the presence of both increasing and decreasing prices. Future stock prices are forecasted using an ANN, as well as the tendency that such prices will show. These estimations are then combined with certain indicators from the fundamental analysis to define the stocks that will likely receive resources (the selected stocks). Finally, another evolutionary algorithm is used to optimize portfolios, i.e., to define the proportions of resources to be allocated to each stock.

#### *4.1. An Artificial Neural Network to Estimate Future Prices*

In this work, the immediate next period price of the considered stocks are estimated by means of an ANN. Following the recommendations of ([6,10,36]), we use a single-layer feedforward network (whose setting is created once per each stock) and train the ANN by means of the so-called extreme learning machine algorithm because of its superior capacities in similar problems to the one addressed here (cf. [7–9]).

The ANN works independently per stock to estimate its price in the subsequent immediate period. The return of each stock is used as the target variable, while thirteen variables are used as input to train the ANN. Let *rt* denote the stock return for a given period *t*. *rt* is calculated from the stock price for that period (*pt*) and the immediate previous one (*pt*−1), as defined by Equation (1).

$$r\_t = \frac{p\_t - p\_{t-1}}{p\_{t-1}} \tag{1}$$

The high complexity involved in forecasting future stock prices requires one to consider a variety of transaction data as explanatory variables. Therefore, we followed the recommendations provided in ([10,76,77]) to determine sixteen transaction data as explanatory variables to the forecasting model used here. The sixteen input variables are described as follows:

Close price. Last transacted price of the stock before the market officially closes.

Open Price. First price of the stock at which it was traded at the open of the period's trading.

High. Highest price of the stock in the period's trading.

Low. Lowest price of the stock in the period's trading.

Average Price. Average price of the stock in the period's trading.

Market Capitalization. Price per share multiplied by the number of outstanding shares of a publicly held company.

Return Rate. Profit on an investment over a period, expressed as a proportion of the original investment.

Volume. Number of shares traded (or their equivalent in money) of a stock in a given period.

Total asset turnover. Net sales over the average value of total assets on the company's balance sheet between the beginning and the end of the period.

Fixed asset turnover. Net sales over the average value of fixed assets.

Volatility. Standard deviation of prices.

General Capital. Number of preferred and common shares that a company is authorized to issue.

Price to Earnings. Market value per share over earnings per share.

Price to Book. Market price per share over book value per share.

Price to Sales. Market price per share over revenue per share. Price to Cash Flow. Market price per share over operating cash flow per share.

The training process consists of taking sixty historical values for these sixteen variables randomly out of a set of ninety historical periods and leaving the rest of values to test the ANN. After the ANN is trained, two errors are computed: training error and testing error. The lower the testing error, the better the predictive capacity the ANN has. Nevertheless, since the extreme learning machine algorithm uses a random procedure to compute the weights and bias of the network, we do not always obtain the same results. Therefore, we run the algorithm *na* times and chose the one with better results. It is important to highlight that each input variable is normalized taking into account the sixty periods of the training data (the target variable is not normalized).

As mentioned before, our approach seeks to take advantage of market downtrends. To achieve this, we use the ANN's forecast. A long or a short position will be chosen according to the forecasted value of the return; that is, if the forecasted value for a stock return is positive, a long position is chosen, otherwise a short position is chosen

#### *4.2. Evolutionary Algorithms to Select Stocks*

It is common that practitioners use indicators from the so-called fundamental analysis to assess the financial health of stocks. Besides these indicators, here, we use the stock prices and tendencies forecasted by the ANN to define which stocks should be further considered for investment. To ponder all these values, we establish an optimization problem following the recommendations in ([78]) and use an evolutionary algorithm to address it as recommended in ([10]).

Let *S* = {*<sup>s</sup>*1,*s*2, ... , *card*(*S*)} be the set of considered stocks, *vj*(*si*) be the evaluation of stock *si* on the *j*th indicator, *j* = 1, ... , *N* (for the sake of simplicity, assume that *v*1 is the forecasted return as calculated by Equation (1)), and *wi* be the relative importance of each indicator and forecasted return (the latter is denoted by *w*1). The score of stock *si* can be calculated as follows (cf. [10,79]):

$$score(s\_i) = \sum\_{j=1}^{N} w\_j v\_j(s\_i) \tag{2}$$

Since increasing *vj*(*si*) for *j* = 1, ... , *N* indicates the convenience of the stock, determining the most appropriate values for *wj* becomes crucial to determining the most plausible stocks as those that maximize Equation (2).

If we want to take advantage of market downtrends, sometimes we will be interested in obtaining the more negative returns to invest in a short position. To implement this idea, the value of each factor *vj* is taken as positive or negative according to the prediction given by the ANN model on the previous stage. Namely, if the ANN model predicts a positive stock return, a long position will be chosen for this stock and the factor values are taken as they are. However, if the ANN model predicts a negative stock return, a short position is chosen for this stock and the return and each factor value are multiplied by −1, so Equation (2) is still valid.

To determine the most convenient values for *wj* (*j* = 1, ... , *<sup>N</sup>*), we use the function recommended in ([78]). Let us define this function.

For a given historical period *t*, a set of predefined weights will allow one to determine the score of each stock; thus, the top, say, 5% of the stocks can be selected. These top stocks constitute the set of "selected" stocks, and the rest constitute the set of "non-selected" stocks for period *t*. Let *Rtselected* and *<sup>R</sup>tnon*−*selected* be the average returns of the stocks in these sets (as calculated by Equation (1)), respectively. The convenience of the predefined weights is then calculated as the arithmetic difference between the average returns of the selected and non-selected stocks that they produce, that is:

$$\text{Maximize}\,\xi(W) = \frac{1}{T}\sum\_{t=1}^{T} (R\_{selected}^t - R\_{non-selected}^t) \tag{3}$$

where *T* is the number of historical returns used to assess the weights in *W*, *W* = [*<sup>w</sup>*1, *w*2,..., *w N*] and *ξ*(*W*) is the convenience of the weights in *W*.

As was stated in Section 2.4, the differential evolution (DE) algorithm has been found to be highly effective in non-linear mono-objective optimization problems, especially in problems related to financial problems ([11,23,24]); therefore, this type of algorithm represents serious advantages over other optimization algorithms, particularly over other meta-heuristics. We use here a basic version of the DE algorithm as presented by Algoritm 1 in ([80]). Let us describe this algorithm.

To determine the best values for *wj* (*i* = 1, ... , *N*), the decision variables considered by the DE will be the values *wj* such that each individual in the DE will contain the values for *wj*fulfilling the constraints of the problem: *wj*≥ 0 and ∑ *N j*=1*wj*= 1.

 the

 Lines 1–8 of Algorithm 1 randomly initialize the population of the DE; that is, the lines initialize feasible individuals by placing them in a random position within the search space. To ensure feasibility, the values for *wj* in each individual are normalized in Lines 5 and 6.

The parameters used by the DE algorithm consist of a crossover probability, *CR* ∈ [0, 1], a differential weight, *F* ∈ [0, 2], and a number of individuals in the population, *populationsize* ≥ 4. Each individual in the population is represented by a real-valued vector *z* = [*<sup>z</sup>*1, *z*2, ... , *zN*] , where *zj* is the value assigned to the *j*th decision variable and *N* is the number of decision variables (in Problem (2), the decision variables are the *N* weights). The termination criterion used here for the search procedure consists of a predefined number of iterations (generations). The evolutionary process is performed in Lines 9–22. Here, for each generation of the DE, the solutions in the population are evolved such that the new population is composed of the best solutions found so far. Finally, the best solution found overall is selected in Line 23.

#### **Algorithm 1** Differential evolution used to address Problem (3).

**Require:** *Niterations*, *CR*, *F*, *populationsize* **Ensure:** The values *w*1, *w*2,..., *w N* found that best solves Problem (3) 1: *P* ← ∅ 2: *i* ← 1 3: **while** (*i* ≤ *populationsize*) **do** 4: Randomly, define *zk* ∈ [0, 1] for *z* = [*<sup>z</sup>*1, *z*2,..., *zN*] 5: *sum* ← ∑ *N k*=1 *zk* 6: *zk* ← *zk*/*sum* (*k* = 1, 2, . . . , *N*) 7: *P* ← *P* ∪ {*z*} 8: **end while** 9: *j* ← 1 10: **while** (*j* ≤ *Niterations*) **do** 11: **for all** (*z* ∈ *P*) **do** 12: Randomly, define *a*, *b*, *c* ∈ *P*, such that *z*, *a*, *b*, *c* are all different 13: Randomly, define *r* ∈ {1, . . . , *N*} 14: **for all** (*i* ∈ {1, . . . , *N*}) **do** 15: Randomly, define *u* ∈ [0, 1] 16: If *u* < *CR* or *i* = *r*, set *yi* = *ai* + *F* · (*bi* − *ci*), otherwise set *yi* = *zi* 17: **end for** 18: *sum* ← ∑ *N k*=1 *yk* 19: *yk* ← *yk*/*sum* (*k* = 1, 2, . . . , *N*) 20: If *ξ*(*z*) ≤ *ξ*(*y*), then replace *z* for *y* in *P* (see Equation (3)) 21: **end for** 22: **end while** 23: Select the individual *z* ∈ *P* with the highest value *ξ*(*z*); this individual representsbest set of weights *w*1, *w*2,..., *w N*for Equation (2).

Different fundamental indicators could be more convenient for companies with different types of activities (see, e.g., [14]). We use here some fundamental indicators that can be

used for trans-business companies following the works in ([10,11,13,14]). In this work, we use *N* = 13 factors to define the score of each stock as described below.

Forecasted return: Output of the ANN.

Return on equity: Net income over average shareholder's equity.

Return on asset: Net income over total assets.

Operating income margin: Operating earnings over revenue.

Net income margin: Total liabilities over total shareholder's equity.

Levered free cash flow: Amount of money the company left over after paying its financial debts.

Current ratio: Current assets over current liabilities.

Quick ratio: (Cash and equivalents + marketable securities + accounts receivable) over current liabilities.

Inventory turnover ratio: Net sales over ending inventory.

Receivable turnover ratio: Net credit sales over average accounts receivable.

Operating income growth rate: (Operating income in the current quarter − operating income at the previous quarter) over operating income in the previous quarter.

Net income growth rate: (Net income after tax in the current quarter − net income after tax at the previous quarter) over net income after tax in the previous quarter.

#### *4.3. Optimizing Stock Portfolios*

The final activity to perform stock investments consists of determining how the resources should be allocated. A given distribution of resources among the selected stocks is known as the stock portfolio. Defining the most convenient distribution of resources is known as portfolio optimization. In this final activity, the decision alternatives are no longer individual stocks but complete portfolios. Thus, it is necessary to determine multiple criteria to comprehensively assess portfolios.

Formally, a stock portfolio is a vector *x* = [*<sup>x</sup>*1, *x*2, ... , *xm*] such that *xi* is the proportion of the total investment that is allocated to the *i*th stock. Let *ri* be the return of the *i*th stock calculated according to Equation (1); the return of a given portfolio *x* is defined as follows:

$$\mathcal{R}(\mathbf{x}) = \sum\_{i=1}^{m} x\_i r\_i \tag{4}$$

Of course, if we knew the *t* + 1 return of the stocks, we could allocate resources that maximize *<sup>R</sup>*(*x*) without uncertainty; however, since this is impossible, the multiple criteria used to assess portfolios are estimations of *<sup>R</sup>*(*x*). These estimations usually come from probability theory.

According to ([12]), the most convenient portfolio *x* can be determined by optimizing a set of confidence intervals that describe the probabilistic distribution of the portfolio's return:

$$\underset{\mathbf{x}\in\Omega}{\text{Maximize}}\{\boldsymbol{\theta}(\mathbf{x})=(\boldsymbol{\theta}\_{\boldsymbol{\beta}\_{1}}(\mathbf{x}),\boldsymbol{\theta}\_{\boldsymbol{\beta}\_{2}}(\mathbf{x}),\ldots,\boldsymbol{\theta}\_{\boldsymbol{\beta}\_{k}}(\mathbf{x}))\}\tag{5}$$

where *θβi* (*x*) = {[*ci*, *di*] : *<sup>P</sup>*(*ci* ≤ *<sup>E</sup>*(*R*(*x*)) ≤ *di*) = *βi*}, *<sup>E</sup>*(*R*(*x*)) is the expected return of portfolio *x*, *<sup>P</sup>*(*ω*) is the probability that event *ω* occurs and Ω is the set of feasible portfolios.

Maximizing confidence intervals as conducted in Equation (5) does not mean increasing the wideness of the intervals; rather, it refers to the intuition that rightmost returns in the probability distribution are desired. We use the so-called interval theory ([68]) to measure the possibility that a confidence interval is greater than another one. In interval theory, an interval number allows one to encompass the uncertainty involved in the definition of a quantity.

Since we are trying to find the best portfolios in terms of confidence intervals around their expected return, intervals further to the right are better (rather than comparing intervals in terms of their width). Therefore, the comparison method used must provide this feature. There are several works in the literature describing methods that possess

this property (e.g., [81,82]); however, the method proposed in [83] is the most broadly mentioned in the literature [84].

The authors of [83] presented a possibility function to define the order between two interval numbers that has been increasingly used in the literature (e.g., [12,85–87]). Let *I* = [*<sup>i</sup>*<sup>−</sup>, *i*+] and *J* = [*j*<sup>−</sup>, *j*+] be two interval numbers, and the possibility function presented in [83] is defined as follows:

$$popability(I \ge J) := \begin{cases} -1, & \text{if } \ p(I, f) > 1 \\ 0, & \text{if } \ p(I, f) < 0 \\ \ p(I, f), & \text{otherwise} \end{cases}$$

where *p*(*<sup>I</sup>*, *J*) = *<sup>i</sup>*+−*j*<sup>−</sup> (*i*+−*i*−)+(*j*<sup>+</sup>−*j*−). Moreover, if *i* = *i*<sup>+</sup> = *j*− and *j* = *j*+ = *j*<sup>−</sup>,

$$\text{possible}(I \ge J) := \begin{cases} \begin{array}{c} 1, \text{ if } i \ge j \\ 0, \text{ otherwise} \end{array} \end{cases}$$

 then

Since Problem (5) can potentially have many objectives defined as interval numbers as well as multiple constraints, we use MOEA/D (see Section 2.4), as advised by ([12]). In ([12]), MOEA/D was adapted to deal with these types of objectives; the adaptation has been proven to provide good results in contexts related to stock investments. For reasons of space in this paper, the reader is referred to ([12]) for specific details about this improvement to MOEA/D.
