PORTFOLIO OPTIMIZATION WITH LINEAR PROGRAMMING BASED ON TRAPEZOIDAL FUZZY NUMBERS

Öz In today's developing financial markets, various complex techniques are used in the creation of portfolios that will provide the best return to the investors. In this study, a portfolio selection model that includes investment data and expert opinions is proposed. This model consists of two stages. In the first stage, the weight of the criteria in the portfolio selection problem was determined by the Constrained Fuzzy Analytic Hierarchy Process method proposed by Enea and Piazza. In the second stage, the model proposed by Lai and Hwang was used to solve the problem of fuzzy linear programming to be formed by using the determined criteria weights. These two methods in the literature use triangular fuzzy numbers (TFNs) in the solutions process of the problem. The methods used in this study were developed for trapezoidal fuzzy numbers (TrFNs) and an alternative method for portfolio selection problems was proposed.


INTRODUCTION
In today's developing and liberalized financial markets, many different and complex techniques are used in the creation of portfolios that will provide the best return according to the risk levels that investors can take. The classical mean-variance model, which is considered as the basis of modern portfolio theory, was developed by Markowitz (Markowitz, 1952). According to Markowitz, it is not possible to reduce the risk by merely diversifying. The relationship between the returns is taken into account when making portfolio diversification with the Markowitz model. Fuzzy Logic approach, which is started by Zadeh and has many applications, is one of the techniques widely used in optimal portfolio selection. There are many studies in the literature that use the portfolio selection process and the fuzzy logic approach. Sadjadi et al. address fuzzy linear programming method, which determines the amount of investment in different time periods. They expressed the rate of return and borrowing rates as triangular fuzzy numbers (TFNs). Using fuzzy set theory, they developed a model for the cash amount and profits of investors (Sadjadi et al., 2011). Lukovac et al. proposed a new model for developing a human resources portfolio based on a neuro-fuzzy approach. The purpose of their model is to enable insight into the existing potential and plan assets to improve and promote the employees' potential in a company (Lukovac et al., 2017). In their study, Devran and Deniz compared differences between Markowitz's Modern Portfolio Theory and Traditional Portfolio Theory . Deniz et al. researched diversification benefit of gold, platinum and silver for stock portfolio in Istanbul Stock Exchange (BIST) between April 1999-April 2018 periods . Jafarzadeh et al. proposed a new method in their paper. This method combines Quality Function Development (QFD), fuzzy logic, and Data Envelopment Analysis (DEA) to accounts for prioritisation, uncertainty and interdependency (Jafarzadeh et al., 2018). Wang et al. introduced the Sharpe ratio in fuzzy environments and proposed a fuzzy Value-at-Risk ratio in their study. They built a multi-objective model based on these to ratios, to evaluate their joint impact on portfolio selection. Finally, they justified the superiority of algorithm by comparing with existing solvers on benchmark problems and exemplified the model effectiveness by using three case studies on portfolio selection (Wang et al., 2018). In their work, Kim and Kim developed a new model for the optimal Liquefied Natural Gas (LNG) import portfolio. Their model consists of a two-step portfolio model combining the mean-variance (MV) portfolio and the linear programming (LP) model (Kim and Kim, 2018). The purpose of Topaloglu's study is revealing the relationship between financial risks and firm value. He used panel data analysis method in his study (Topaloglu, 2018). Liagkouras proposed a new algorithm for the solution of portfolio optimization problem. He tested the performance of the proposed algorithm to the optimal allocation of limited resources to a number of competing investment opportunities for optimizing the objectives (Liagkouras, 2019). In their work, Bolos et al. developed a modern and innovative management tool based on the artificial intelligence technique and the use of systems with fuzzy logic for companies to substantiate investment decisions in assets purchased from the market (Bolos et al., 2019).
Another technique used effectively in portfolio selection is the Analytic Hierarchy Process (AHP). This method was developed by Saaty to solve many complex problems. All the criteria that affect the stock prices affect the selection of investors in the optimal portfolio selection. Therefore, it is a complex and multicriteria decision problem for investors to choose the appropriate stocks and decide on which ratios they will form the portfolio. In the AHP, many criteria affecting stock prices are handled in a hierarchy, making the complex structure more regular. In this method, the relative importance of the criteria is determined by the decision makers. Decision makers use linguistic expressions when comparing. The linguistic variables under the uncertain evaluations in their verbal judgments can be expressed in more rational terms with fuzzy numbers. Fuzzy AHP method has been developed by using fuzzy numbers in comparisons due to this feature of fuzzy numbers. Tanaka and Asai used objective function coefficients and right-hand side coefficients of constraints as fuzzy functions in their studies (Tanaka and Asai, 1984). Nakamura solved the multi-objective linear programming models, which are represented by triangular membership functions, by transforming them into fuzzy linear programming models with partial membership functions (Nakamura, 1984). Chang defined a new approach for handling fuzzy AHP, with the use of TFNs for pairwise comparison scale of fuzzy AHP (Chang, 1996). Tiryaki and Ahlatcioglu proposed a new method for group decision making in fuzzy environment. In their paper, they used TFNs for the rating of each stock and the weight of each criterion. They aimed to provide investors with information about ranking and weighting (Tiryaki and Ahlatcioglu, 2005). Chen and Cheng proposed a fuzzy multi criteria decision making methodology in ranking portfolios of information sourcing projects under uncertainty conditions (Chen and Cheng, 2009). Yucel and Guneri expressed the linguistic values as TrFNs to assess the weights of the factors. They obtained the weights by calculated the distances of each factor between Fuzzy Positive Ideal Rating and Fuzzy Negative Ideal Rating (Yucel and Guneri, 2011). In their study, Xiao and Fu proposed a greycorrelation multi-attribute decision-making method based on intuitionistic TrFNs to solve the problem that the attribute weight depends on the various statuses and the attribute values offer multi-attribute decision making in the form of intuitionistic TrFNs (Xiao and Fu, 2015). In their article, Solimanpur et al. presented a new model for optimal portfolio selection using the genetic algorithm and AHP (Solimanpur et al., 2015). Piasecki and Siwek focused on describing the imprecision risk for the portfolio rather than describing its uncertainty. They used the present values of portfolio assets as TrFNs in their study (Piasecki and Siwek, 2018). In their paper, Chatterje et al. used fuzzy AHP for project prioritization in portfolio management (Chatterje et al., 2018).
With the developing technology, investors started to direct their investments by using computer software programs. However, since these programs do not consider the investor's opinions, they provide the same expected return on the same risk level for all investors. With this drawback, investment related existing software packages can be considered somewhat inefficient. In this study, a model including expert opinions has been proposed. The model consists of two stages. In the first stage, the Constrained Fuzzy AHP method proposed by Enea and Piazza was developed for TrFNs and the weights of the criteria were determined. In the second stage, linear programming problem has been established in which the weight of the criteria obtained as TrFNs is used as price variables in the objective function. The model proposed by Lai and Hwang has been developed for TrFNs in order to solve the linear programming problem created by using TrFNs. In this way, as an alternative to the methods in which the expert opinions are used as triangular fuzzy numbers in the literature, a new method using trapezoidal fuzzy numbers is proposed. In order to examine the effectiveness of the model, a sample portfolio selection problem in the literature is discussed. The results obtained from the existing methods and the results obtained from the proposed model were compared.

CONSTRAINED FUZZY AHP
The Constrained Fuzzy AHP method focuses on the constraints within the fuzzy AHP in order to take for all available information into consideration. This method is also used to calculate the weights of alternatives in the portfolio selection process. The weights of the alternatives are calculated with the Constrained Fuzzy AHP method using TFNs . The formulas used in the calculations are given in Equation (1-3). Let be the fuzzy score for the criterion of triangular fuzzy pairwise comparison matrix, where the indices and denote its lower, medium and upper respectively. According to the Constrained Fuzzy AHP method proposed by Enea and Piazza, the center value of the fuzzy score related to criterion ( ) calculated by Equation (1) (Enea and Piazza, 2004).
can be evaluated using the crisp mathematical programming model: and similarly, can be evaluated using the crisp mathematical programming model: Tiryaki and Ahlatcioğlu used the Fuzzy Analytic Hierarchy Process in the problem of portfolio selection. They are intended to decide the content of the portfolio that will be created. To do this, they handled the fuzzy AHP method given by Enea and Piazza. And they proposed Revised Constrained Fuzzy AHP method by revising some mistakes in this method (Tiryaki and Ahlatcioglu, 2009). Ahari et al. planned to allocate a limited funds among the stocks of some pharmaceutical companies in the Tehran stock market, in their study. They used two fuzzy AHP method which proposed by Enea -Piazza and Van Laarhoven -Pedrycz (Ghazanfar Ahari et al., 2011). In his study, Yaghoobi apply the fuzzy analytic hierarchy process (AHP) to the issue of ordering key success factors (KSFs) for software development projects. To do this, Yaghoobi first simplified the constrained fuzzy AHP method, and then from systematic literature reviews, a preliminary list of potential KSFs that influences software development projects was identified and compiled (Yaghoobi, 2018).

THE CASE THAT THE PRICE VARIABLES IN THE OBJECTIVE FUNCTION ARE FUZZY
An efficient method to handle the uncertain parameters of a LP is to express the uncertain parameters by fuzzy numbers which are more realistic and create a conceptual and theoretical framework for dealing with imprecision and vagueness.
In their study Lai and Hwang discussed the situation where the coefficients of the price variables in the objective function of the linear programming problem are uncertain (Lai and Hwang, 1992). In their work, they aim to maximize the highest possible value of the uncertain profit, minimize the risk of lower profits, and maximize the possibility of higher profits. If the c_i parameter corresponding to the prices in the objective function is a fuzzy number, the linear programming problem is modeled as given in Equation (4).
Here, the objective function coefficients are TFNs, expressed as ̃ and have a triangular probability distribution as given in Figure 1. is the vector of three objective functions, and In order to keep the triangular shape of the possibility distribution, it is necessary to make a little change. Instead of maximizing these three objectives simultaneously, we are going to maximize minimize and maximize , where the last two objective functions are actually relative measures from the first objective function (see Figure 2). The three new objectives also guarantee the previous argument of pushing the triangular possibility distribution in direction of the right-hand side. .

(6)
The crisp multi objective linear programming Equation (6) is equivalent to maximizing the most possible value of the imprecise profit. At the same time, we have minimized the inferior side of the possibility distribution. This means minimizing (I) which, in our opinion, is equivalent to "the risk of obtaining lower profit". And, we have also maximized the region (II) of the possibility distribution, which is equivalent to "the possibility of obtaining higher profit". This normalization has also been applied by Seo and Sakawa's study (Seo and Sakawa, 1988). Finally, we solve Zimmerman's following equivalent single-objective linear programming model: Liu and Gao point out limitations of the existing method to solve fully fuzzy linear programming (FFLP) problem and proposed a modified method to overcome these limitations (Liu and Gao, 2016). The aim of Ebrahimnejad's article is to introduce a formulation of FLP problems involving interval-valued TrFNs for the decision variables and the right-hand-side of the constraints. He proposed a new method for solving this kind of FLP problems based on comparison of interval-valued fuzzy numbers by the help of signed distance ranking (Ebrahimnejad, 2018). Taleshian and Fathali investigated the p-median problem with fuzzy variables and weights of vertices. They showed that the fuzzy objective function also can be replaced by crisp functions in their study (Taleshian and Fathali, 2016). In their study, Dong and Wan developed a new method for the fuzzy linear program in which all the objective coefficients, technological coefficients and resources are TrFNs (Dong and Wan, 2018). Arik and Toksari investigated a multi-objective parallel machine scheduling problem under fully fuzzy environment with fuzzy job deterioration effect, fuzzy learning effect and fuzzy processing times (Arik and Toksari, 2018). In their work, Yu et al. studied on the dual-channel (the traditional channel and the E-commerce channel) supply chain network design (SCND) for Fresh Agri-Product (FAP) under information uncertainty (Yu et al., 2018).

AN ALGORITHM BASED ON FUZZY NUMBERS FOR PORTFOLIO SELECTION
Step 1: being the number of criteria and being the number of decision makers involved, comparison values ( ) relative to each criterion are determined by the decision makers as in Table 1.
The new TrFNs, which express the decision maker's opinions, are obtained by repeating the same process times for each paired comparison, using geometric mean. The obtained new TrFNs are given Table  2.

Table 2. The comparison matrix of each criterion, which consists of the new trapezoid numbers obtained
Step 3: The weights of the alternatives or criteria are calculated with the proposed method using TrFNs. The formulas used in the calculations are given in . Let be the fuzzy score for the criterion of trapezoidal fuzzy pairwise comparison matrix, where the indices and denote its lower, medium1,medium2 and upper respectively. and can be evaluated using the crisp mathematical programming model, The weights obtained from Step 3 of the algorithm are shown in Table 3. Step 4: By using the obtained weights, the linear programming problem given by Equation (21)  In Equation (21), and are the weights coefficients that present the fuzzy goals and fuzzy constraints obtained from TrFNs. and represents the fuzzy goals and the fuzzy constraint parameters.
Step 5: The four critical parameters of the trapezoid fuzzy weights obtained are pushed to the right as shown in Figure 3, so that the fuzzy goal is maximized. The new four objective functions for solving the model are created as in Equation (22): Step 6: In order to solve the equation (22), Zimmerman's fuzzy programming method was used in the normalization process. Firstly, the Positive Ideal Solution (PIS) and the Negative Ideal Solution (NIS) of the four objective functions are obtained by Equation (23-26).
Step 7: Then the linear membership function of these objective functions is calculated: Step 8: Finally, the results of the distribution of the portfolio are obtained by solving the linear programming model given by Equation (29). (29)

APPLICATION
In this section, a portfolio selection problem which is present in the literature is discussed. In addition, a solution was obtained by the method based on the TrFNs proposed in this study. In the study, 5 companies were determined using past price movements obtained from BIST; Anadolu Cam, Trakya Cam, Mardin Cimento, Eregli Demir Celik and Izmir Demir Celik. In order to determine the distribution of the portfolio on these five companies, seven criteria were determined as Price/Earnings (P/E), Net Profit/Stockholder's Equity ( (MV/BV), Net Profit/Sales Revenue (NP/SR) and Net Profit/Total Assets (NP/TA). The hierarchical structure of the problem is given in Figure 4 (Ahlatcioglu, 2005).  Table 4. In the study of the literature, the importance degrees of the criteria were determined as TFNs by four decision makers and is given in Table 5 (Ahlatcioglu, 2005). Table 5. Importance degrees or criteria determined by the four decision makers DM 1 (5 7 9) (7 9 10) (5 7 9) (5 7 9) (7 9 10) (5 7 9) (3 5 7) M 2 (7 9 10) (5 7 9) (5 7 9) (7 9 10) (7 9 10) (5 7 9) (3 5 7) M 3 (9 10 10) (7 9 10) (5 7 9) (9 10 10) (7 9 10) (5 7 9) (5 7 9) M 4 (5 7 9) (9 10 10) (5 7 9) (7 9 10) (9 10 10) (5 7 9) (3 5 7) In this study, these importance degrees are converted to TrFNs as shown in Figure 5. The importance degrees which are TFNs have been converted into TrFNs by keeping the upper and lower bounds of the fuzzy numbers constant and spreading the center to a certain range. The geometric averages of the decision-makers' views, which are converted into TrFNs, are taken. The importance degrees of the criteria obtained are given in Table 6. (5.00 6.50 7.50 9.00) C 4 (6.85 8.22 8.95 9.74) C 5 (7.45 8.79 9.43 10.00) C 6 (5.00 6.50 7.50 9.00) C 7 (3.41 4.93 5.94 7.45) The seven criteria for the problem are compared with each other according to the purpose of "portfolio selection". The importance degrees of the criteria given in Table 6 were compared with each other in pairs and thus the comparison results are obtained. The same procedure was repeated for each criterion to obtain a fuzzy binary comparison matrix, which was given in Table 7. The fuzzy weight of each criterion is calculated by the proposed algorithm and given in Table 8. The linear programming model was created by using the obtained fuzzy weights.
, The maximum and minimum values for each objective function were determined under the constraints of the model by using WinQSB software. Solutions for each of 6 objective functions are given in Table 9. A fuzzy multi objective linear programming model (P 1 ) is generated using fuzzy weights obtained from the Constrained Fuzzy AHP method using TrFNs.
The proposed algorithm is used to solve the P 1 problem, where the price variables in the objective function are TrFNs. The new objective functions such as Equality (31) are created; (31) PIS and NIS of each objective function given by the Equation (31) are obtained and the results are given in Table 10. The P 2 model is created by using the membership values obtained. P 2 : The results obtained by solving the P 2 model using the WinQSB program are given in Table 11.

CONCLUSION
In the case of uncertainty, unplanned investments can cause unexpected losses to the investor. In addition, many portfolio selection methods do not consider expert opinions. Because these methods use only financial data without considering the investor's views, they provide the same expected return for all investors at the same risk level. To handle this problem, adding both financial ratios and experience based expert opinions in to the model has made the proposed method more effective. In the proposed model, the weights of criteria were determined by using the decision makers' opinions. When the studies in the literature were examined, it was seen that TFNs have been widely used in the calculation of the weights of criteria. Instead of using TFNs, this study focused on TrFNs to characterize fuzzy measures of linguistic values. The reason for using the TrFNs was that it was more representative to linguistic estimations in portfolio selection. In terms of optimal solution, it was observed that TrFNs gave more flexible results than TFNs. As a result of the study, it has been determined that investors' funds should be allocated to the stocks Anadolu Cam, Trakya Cam, Mardin Cimento, Eregli Demir Celik and Izmir Demir Celik by 0%, 0%, 22.2%, 0% and 77.8%, respectively (Table 11), using the recommended model for optimal portfolio distribution.
The proposed method can be applied to larger-scale portfolio selection problems in which more criteria and expert opinions are used, and the optimal allocation of stocks can be done out of a variety of sectors.