# Modeling and Optimizing the Multi-Objective Portfolio Optimization Problem with Trapezoidal Fuzzy Parameters

^{*}

## Abstract

**:**

## 1. Introduction

_{F}is the space of feasible portfolios, usually determined by the available budget and other constraints that the Decision Maker (DM) wants to impose (e.g., budget limits on types, geographic areas, social roles of projects, etc.).

## 2. Elements of Fuzzy Theory

#### 2.1. Fuzzy Sets

**(x))/x є X} where μ**

_{A}**(x) is called the membership function or grade of membership of x in A which maps X to the real membership subspace M [17]. The range of the membership function is a subset of the nonnegative real numbers whose supremum is finite. Elements with a zero degree of membership usually are not listed.**

_{A}#### 2.2. Generalized Fuzzy Numbers

- ${\mathsf{\mu}}_{\mathrm{A}}\left(\mathrm{x}\right)$ is a continuous function from $R$ to the closed interval $\left[0,1\right]$
- ${\mathsf{\mu}}_{\mathrm{A}}\left(\mathrm{x}\right)=0,-\infty <\mathrm{x}\mathrm{a}$
- ${\mathsf{\mu}}_{\mathrm{A}}\left(\mathrm{x}\right)=\mathrm{L}\left(\mathrm{x}\right)$, is strictly increasing on $\left[\mathrm{a},\mathrm{b}\right]$
- ${\mathsf{\mu}}_{\mathrm{A}}\left(\mathrm{x}\right)=\mathrm{w}$, for $\mathrm{b}<\mathrm{x}<\mathsf{\alpha}$
- ${\mathsf{\mu}}_{\mathrm{A}}\left(\mathrm{x}\right)=\mathrm{R}\left(\mathrm{x}\right)$ is strictly decreasing on $\left[\mathsf{\alpha},\mathsf{\beta}\right]$
- ${\mathsf{\mu}}_{\mathrm{A}}\left(\mathrm{x}\right)=0$, for $\mathsf{\beta}<\mathrm{x}<\infty $

#### 2.3. Trapezoidal Addition Operator

#### 2.4. Graded Mean Integration (GMI)

#### 2.5. Order Relation in the Set of the Trapezoidal Fuzzy Numbers

- ${A}_{1}<{A}_{2}$ if only if $P\left({A}_{1}\right)<P\left({A}_{2}\right)$
- ${A}_{1}>{A}_{2}$ if only if $P\left({A}_{1}\right)>P\left({A}_{2}\right)$
- ${A}_{1}={A}_{2}$ if only if $P\left({A}_{1}\right)=P\left({A}_{2}\right)$

#### 2.6. Pareto Dominance

## 3. Multi-Objective Portfolio Optimization Problem with Trapezoidal Fuzzy Parameters

#### 3.1. Mathematical Model

**C**the total available budget, O the number of objectives, ${c}_{i}$ the cost of the project i,

**b**

_{ij}the produced benefit with the execution of the project i in objective j, K the number of areas to consider, M the number of regions, ${\mathit{A}}_{k}^{min}$ and ${\mathit{A}}_{k}^{max}$ the lower and upper limits in the available budget for the area k, and ${\mathit{R}}_{m}^{min}$ and ${\mathit{R}}_{m}^{max}$ the lower and upper limits in the available budget for the region m. The arrays ${\mathrm{a}}_{\mathrm{i}}$and ${b}_{i}$ contain the area and region assigned to the project i. $\widehat{x}=\left({x}_{1},{x}_{2},\dots \dots .,{x}_{n}\right)$ is a binary vector that specifies the selected projects included in the portfolio. If x

_{i}= 1 then the project i is selected, otherwise it is not. Now we define the MOPOP with Fuzzy Trapezoidal parameters as follows:

**bold**and italic are trapezoidal fuzzy numbers.

^{n}. Then the solution algorithms must search across this space to find the Pareto optimal solutions. On the other hand, given that the well-known NP-hard Knapsack problem can be easily reduced to MOPOP, the latter is also NP-hard [21].

#### 3.2. Strategy to Generate the Fuzzy Trapezoidal Instances

_{1}, c

_{2}), and objectives (m

_{1}, m

_{2}). Then to generate a fuzzy interval instance the following interval type values must be determined:

- $\left[B,{B}^{\prime}\right]$ ← Budget as interval
- [a
_{i}, a′_{i}] ← Limits of each area I = 1, 2, …, a - [r
_{i}, r′_{i}] ← Limits of each region r = 1, 2, …, r - [b
_{ij}, b′_{ij}] ← Benefit from the objective I = 1, 2, ..., m and for each project j = 1, 2, …, p - {C
_{i}, A_{i}, R_{i}} ← Real values of the cost, area and region for each project i = 1, 2, …, p.

_{l}= (0.7 * B)/(1.7ª + 0.1a

^{2}), a′

_{l}= (1.27 * B)/(1.7ª + 0.1a

^{2})]

_{u}= ((1.02 + 0.06r) * B)/r), a′

_{u}= ((2.635 + 0.155a) * B)/a

_{i}= a

_{l}+ Random (a′

_{l}− a

_{l}) for i = 1,2,…,a

_{i}= a

_{u}+ Random (a′

_{u}− a

_{l}) for i = 1,2,…,a

_{l}= (0.7 * B)/(1.7a + 0.1a

^{2}), r′

_{l}= (1.27 * B)/(1.7a + 0.1a

^{2})]

_{u}= ((1.02 + 0.06r) *B)/r), r′

_{u}= ((2.635 + 0.155a) * B)/a

_{i}= r

_{l}+ Random (r′

_{l}− r

_{l}) for i = 1,2,…,r

_{I}= r

_{u}+ Random (r′

_{u}− r

_{l}) for i = 1,2,…,r

_{i}= Random(a) i = 1,2,…,p

_{i}= Random(r) i = 1,2,…,p

_{1}+ Random (m

_{2}− m

_{1}), b

_{ij}= 0.8*o,

_{ij}= 1.1*o for i = 1, 2, …, p and i = 1, 2, …, m

Algorithm 1. o2p25_0I fuzzy interval type instance |

// Fuzzy interval type value of the total available budget |

[76800, 83200] |

// Number of objectives |

2 |

// Number of areas |

3 |

// Fuzzy interval type values of the upper and lower bounds of the available budget // in each area, a row for each area. |

[13060, 16560] [46245, 49745] |

[13810, 15810] [47895, 48095] |

[13210, 16410] [46545, 49445] |

// Number of regions. |

2 |

// Fuzzy interval type values of the upper and lower bounds of the available budget // in each region, a row for each region. |

[22775, 24275] [67950, 68050] |

[23325, 23725] [67900, 68100] |

// Number of projects |

25 |

// For each project, there is a row that includes the following: fuzzy interval type |

// value of the project cost, project area, project region, and the fuzzy interval type |

// value of the benefits obtained with each objective. (only 5 of the 25 projects are |

// showed) |

[9308, 10082] [1] [1] [7642, 8278] [231, 249] |

[8290, 8980] [2] [1] [8506, 9214] [404, 436] |

[5895, 6385] [3] [1] [3831, 4149] [111, 119] |

[9053, 9807] [1] [2] [3908, 4232] [399, 431] |

[6058, 6562] [1] [2] [5760, 6240] [418, 452] |

Algorithm 2. o2p25_0T fuzzy trapezoidal instance |

// Fuzzy trapezoidal value of the total available budget |

[76800, 83200, 0.5, 0.5] |

// Number of objectives |

2 |

// Number of areas |

3 |

// Fuzzy trapezoidal values of the upper and lower bounds for the available budget // in each area, a row for each area. |

[13060, 16560, 0.5, 0.5] [46245, 49745, 0.5, 0.5] |

[13810, 15810, 0.5, 0.5] [47895, 48095, 0.5, 0.5] |

[13210, 16410, 0.5, 0.5] [46545, 49445, 0.5, 0.5] |

// Number of regions. |

2 |

// Fuzzy trapezoidal values of the upper and lower bounds for the available budget |

// in each region, a row for each region. |

[22775, 24275, 0.5, 0.5] [67950, 68050, 0.5, 0.5] |

[23325, 23725, 0.5, 0.5] [67900, 68100, 0.5, 0.5] |

// Number of projects |

25 |

// For each project, there is a row that includes the following: fuzzy trapezoidal value // of the project cost, project area, project region, and the fuzzy trapezoidal values of |

// the benefits obtained with each objective. (only 5 of the 25 projects are showed) |

[9308, 10082, 0.5, 0.5] [1] [1] [7642, 8278, 0.5, 0.5] [231, 249, 0.5, 0.5] |

[8290, 8980, 0.5, 0.5] [2] [1] [8506, 9214, 0.5, 0.5] [404, 436, 0.5, 0.5] |

[5895, 6385, 0.5, 0.5] [3] [1] [3831, 4149, 0.5, 0.5] [111, 119, 0.5, 0.5] |

[9053, 9807, 0.5, 0.5] [1] [2] [3908, 4232, 0.5, 0.5] [399, 431, 0.5, 0.5] |

[6058, 6562, 0.5, 0.5] [1] [2] [5760, 6240, 0.5, 0.5] [418, 452, 0.5, 0.5] |

#### 3.3. Evaluating the Solutions and Verifying the Feasibility

**z**

_{1}and

**z**

_{2}are the benefits generated by the projects selected in the binary vector

**x**. The constraint verifies that the cost of that project is not higher than the available budget (

**C**).

**x**, we have:

## 4. Steady-State T-NSGA-II Algorithm

#### 4.1. Representation of the Solutions

_{i}= 1 represents the inclusion of project i in the portfolio. The first element in the vector is s

_{0}, and the last is s

_{n}

_{–1}. Figure 1 shows an example of this representation.

#### 4.2. One-Point Crossover Operator

_{i}contains its values {s

_{0}, …, s

_{cp}}, and the right

_{i}contains its values {s

_{cp}

_{+1}, …, s

_{n}

_{–1}}. Finally, it mixes the split sections to generate two new offsprings h

_{1}, h

_{2}, where h

_{1}uses left

_{1}and right

_{2}, and h

_{2}uses left

_{2}and right

_{1}. The parents are chosen at random. The steady-state approach only utilizes the first offspring h

_{1}. The number of crossovers that are done is a defined parameter. Figure 2 shows an example of this operator.

#### 4.3. Uniform Mutation Operator

_{0}, s

_{1}, …, s

_{n}

_{–1}} [24]. The process generates for each index i, for 0 ≤ i ≤ n − 1, a random number u in the range [0, 1], and if u < mut then the value of s

_{i}changes from 1 to 0 or vice versa, otherwise the value s

_{i}remains intact. The parameter mut is the mutation probability used by the operator. Figure 3 shows an example of the use of this mutation.

#### 4.4. Initial Population

#### 4.5. Population Sorting

#### 4.6. Non-Dominated Sorting

#### 4.7. Calculating the Crowding Distance

#### 4.8. Calculating the Spatial Spread Deviation (SSD)

#### 4.9. Pseudocode of the T-NSGA-II Algorithm

Algorithm 3. T-NSGA-II pseudocode |

INPUT: Instance with the trapezoidal parameters of the portfolio problem. OUTPUT: Approximated Pareto Front NOTE: The algorithm is called T-NSGA-II-CD when the Crowding Distance is used, and T-NSGA-II-SSD when is used the Spatial Spread Deviation. *************************************** 1. Create the initial population pop 2. Evaluate all the solutions in pop 3. Order pop using no-dominated Sorting 4. For all solutions in pop calculate Spatial Spread Deviation/Crowding distance |

5. pop sorting due to fronts and Spatial Spread Deviation/CD6. Main loop, until stopping condition is met*** Steady state approach: only one generated individual is considered to include in popc 7. Create popc using crossover operator *********************************************************************** 8. Create popm using mutation operator 9. Join popc and popm to create popj 10. Evaluate solutions in popj and put feasibles in popf 11. Add popf to pop, and calculate objective functions 12. Order pop using no-dominated sorting 13. Calculate Spatial Spread Deviation/Crowding distance 14. pop sorting due to the front ranking and Spatial Spread Deviation/CD15. Truncate pop to keep a population of original size 16. No-dominated sorting 17. Calculate Spatial Spread Deviation/Crowding distance of the individuals in pop 18. pop sorting due to front ranking and Spatial Spread Deviation/CD19. End Main loop20. Return (Front 0). ***Approximated Pareto Front |

## 5. T-FAME Algorithm

#### 5.1. Fuzzy Controller

Algorithm 4. Fuzzy controller structure. |

[System] |

Name=‘FuzzyController ‘ |

Type=‘mamdani’ |

Version=2.0 |

NumInputs=2 |

NumOutputs=1 |

NumRules=9 |

AndMethod=‘min’ |

OrMethod=‘max’ |

ImpMethod=‘min’ |

AggMethod=‘max’ |

DefuzzMethod=‘centroid’ |

[Input1] |

Name=‘Stagnation’ |

Range=[0 1] |

NumMFs=3 |

MF1=‘Low’:’trimf’,[−0.4 0 0.4] |

MF2=‘Mid’:’trimf’,[0.1 0.5 0.9] |

MF3=‘High’:’trimf’,[0.6 1 1.4] |

[Input2] |

Name=‘UseOp’ |

Range=[0 1] |

NumMFs=3 |

MF1=‘Low’:’trimf’,[−0.4 0 0.4] |

MF2=‘Mid’:’trimf’,[0.1 0.5 0.9] |

MF3=‘High’:’trimf’,[0.6 1 1.4] |

[Output1] |

Name=‘ProbOp’ |

Range=[0 1] |

NumMFs=3 |

MF1=‘Low’:’trimf’,[−0.4 0 0.4] |

MF2=‘Mid’:’trimf’,[0.1 0.5 0.9] |

MF3=‘High’:’trimf’,[0.6 1 1.4] |

[Rules] |

3 3, 2 (1) : 1 |

3 2, 1 (1) : 1 |

3 1, 2 (1) : 1 |

2 3, 2 (1) : 1 |

2 2, 1 (1) : 1 |

2 1, 2 (1) : 1 |

1 3, 3 (1) : 1 |

1 2, 2 (1) : 1 |

1 1, 1 (1) : 1 |

#### 5.2. Additional Genetic Operators

#### 5.3. Used Structures to Store the Population and the Approximated Pareto Front

**pop**to maintain a solutions population, which contains the following information for each solution

**i**:

- V(i): vector binary associated to the solution i.
- O
_{1}(i) and O_{2}(i): values of the two objectives of the solution i, converted to GMI values. - r(i): ranking of the solution i is the number of the front in which is located.
- Dominated(i): solutions dominated by the solution i.
- Domines(i): solutions that dominates to solution i.
- CD (i): Crowding Distance value of the solution i.
- SSD(i): Spatial Spread Deviation value of solution i.

**i**:

- V(i): vector binary associated to the solution i.
- O(i): real vector of the graded mean values of the fuzzy triangular objectives of the solution V(i).
- r(i): ranking of the solution i is the number of the front in which is located.
- Dominated(i): solutions dominated by the solution i.
- Domines(i): solutions that dominates to solution i.
- SSD (i): Spatial Spread Deviation value of the solution i.

#### 5.4. T-FAME Algorithm Pseudocode

Algorithm 5. T-FAME pseudocode |

INPUT: Instance with the trapezoidal parameters of the portfolio problem. OUTPUT: Approximated Pareto front Variables pop: Population of solutions (binary vectors)Front: Limited sized set were no-dominated solutions are keptOperator: Vector of size SizeOP that contains the index of the available operators Parents: Vector of size NParents that contains the chosen parents ProbOp(i): Probability that operator i has of being chosen, it has values between 0 and 1 UseOp(i): Normalized Indicator of how much operator i has been used, it has values between 0 and 1 Stagnation: Normalized indicator of the number of generated solutions that couldn’t be inserted into Front, because they were either dominated solutions or there was not space available for them, it can have values between 0 and 1. MAXEVAL: Maximum number of evaluations of the objective function (stopping criterion) Window: Size of the time window. eval: Accumulator of the evaluations of the objective function v: Counter of the time windows that have elapsed Functions |

CreateaSon(Operator(i), Parents): Generates one solution using the previous chosen operator i with the chosen parents (Steady state) Evaluate(Son): Calculates the objective values of Son and verify feasibility FuzzyController(Stagnation, UseOp(i)): Function that invokes the fuzzy controller with Stagnation and UseOp(i) as input values and returns the probability of selection of all the operators no-dominated_sortingSSD(NewPop): Sorts the fronts of NewPop by dominance and uses as ranking the SSD values of the solutions. EliminateWorstSolutionSSD(NewPop): Eliminates from the last front of NewPop the solution with the worst SSD, and assign NewPop to pop. |

**************************************************************** 1. Create(pop) **Create random population 2. Front=NoDominated(pop) **Insert in Front the no-dominated solutions of pop 3. $\forall i\u03f5\left\{1,2,\dots .,SizeOP\right\}ProbOp\left(i\right)=$1, UseOp(i)=0 4. v =0; Stagnation = 0; eval=0; 5. while (eval<MAXEVAL) do. **** Stop condition ** Chose |NParents| ** With a probability $\beta $ each parent is taken from Front to intensify) and with 1-$\beta $ from pop to diversify. 6. $\forall i\u03f5\left\{1,2,..\left|NParents\right|\right\}$ do 7. if (RandomDouble(0,1) $\le \beta $) then **The parent is chosen from Front8. Parents[i] ← TournamentSSD( Front) 9. Else **The parent is chosen from pop10. Parents[i] ← TournamentSSD( pop)*** Roulette to choose an operator with the selection probabilities of the operators 11. sum=0 12. $i=Random\left(1,2,\dots ,NParents\right)$ 13. sum=sum+ProbOp(i) 14. while (sum>0) do 15. $i=Random\left(1,2,\dots ,NParents\right)$ 16. sum=sum+ProbOp(i) ********** ***** The chosen operator is associated with the last value of i ** ***Steady state approach 17. Son ← CreateaSon(Operator(i), Parents) **** Get the objective vector values corresponding to Son and verify feasibility. 18. Evaluate(Son) 19. eval=eval+1 20. UseOp(Operator(i)) = UseOp(Operator(i))+ 1 . 0/ Window 21. v=v+1 ******************************* 22. If (Son dominates a set S of solutions in Front) 23. then { Front=Front\S; Front=Front ${{\displaystyle \cup}}^{}$Son}24. else If ($\exists $ s Front such that s dominates Son)25. then (Stagnation= Stagnation+1.0/ Window) 26. else if (Sizeof( Front)<100)27. then (Front=Front ${{\displaystyle \cup}}^{}$Son)28. else { 29. Front=Front ${{\displaystyle \cup}}^{}$Son ** Front[1 00]=Son30. Calculate SSD for all the solutions in Front31. Sort the solutions in Front in ascending order by SSD32. Eliminate the solution in Front with worst SSD:Front[100]33. If (Son Front) 34. then Stagnation= Stagnation+1.0/ Window 35. } 36. If (v == Window) then **** The Fuzzy Controller is used to update the selection probability ****of all the operators 37. $\forall i\u03f5\left\{1,2,\dots SizeOP\right\}$ 38. $ProbOp\left(i\right)=\mathrm{Fuzzy}Controller\left(Stagnation,UseOp\left(i\right)\right)$ 39. v =0; Stagnation = 0; 40. End if 41. pop=pop ${{\displaystyle \cup}}^{}$Son42. NewPop= pop 43. no-dominated_sortingSSD(NewPop) 44. pop ← EliminateWorstSolutionSSD(NewPop)45. End while 46. Return( Front) *** Approximated Pareto front generated |

## 6. Experimental Results

#### 6.1. Performance Metrics Used

#### 6.2. Experimental Setup

#### 6.3. Experiment 1. Validating the Implemented Algorithms

#### 6.4. Experiment 2. Evaluating the Performance of the Algorithms with Instances of 25 Projects

_{I}

_{RQ}. In the result tables, for each instance the best and second-best values are marked with solid or light black, respectively. In order to indicate if the observed differences in the performance of the algorithms are significant or not, for each algorithm the symbol $\bigwedge $ indicates that the performance of T-FAME is significantly better that the algorithm which it is being compared. The symbol $\bigvee $ indicates the opposite, and the symbol $=$ indicates that the difference is not significant. These symbols are marked with an asterisk when the t student test was applied. To confirm the results obtained with the paired tests, a global evaluation is done with the three algorithms. This evaluation was done by applying a Friedman test with 95% confidence.

#### 6.5. Experiment 3. Evaluation of the Algorithm’ Perfomances Using Instances with 100 Projects

^{25}. For this experiment, 9 instances of 2 objectives and 100 projects were used, these instances represented a greater complexity for the algorithms because the solution space increased to 2

^{100}. The experiment conditions were just as in the previous one, using the same metrics but in a scenario of greater complexity scenario. For each instance, the reference set contains the non-dominated solutions obtained from the combination of the 30 generated fronts. The computation of the metrics uses the reference set as an approximation to the optimum Pareto Front. The computation of the median value and interquartile ranges uses the metric values of the 30 instances sorted in ascending order. With the sorted array, the median value was the average of the metric values from positions 15 and 16. At the same time, the interquartile ranges correspond to those in positions 23 and 8, corresponding to the 75% and 25% of the metrics values, respectively. The experiment performs a hypothesis test to validate the obtained results. The hypothesis was proven using the parametric t student test on those data sets that passed the normality and homoscedasticity tests and using the non-parametric Wilcoxon signed-rank test on those that do not. Both tests apply a confidence level of 95%, pairing T-FAME with each of the other two algorithms. Table 6, Table 7, Table 8 and Table 9 shows the results of the normality homoscedasticity tests done for all the instances used in this work (25 and 100 projects) and the metrics of hypervolume and generalized spread.

_{I}

_{Rq}. In the result tables, for each instance the best and second best values are marked with solid or light black, respectively. In order to indicate if the observed differences in the performance of the algorithms are significant or not, for each algorithm the symbol $\bigwedge $ indicates that the performance of T-FAME is significantly better that the algorithm which it is being compared. The symbol $\bigvee $ indicates the opposite, and the symbol $=$ indicates that the difference is not significant. These symbols are marked with an asterisk where the t student test was applied. To confirm the results obtained with the paired tests, a global evaluation is done with the three algorithms. This evaluation was done by applying a Friedman test with 95% confidence.

## 7. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

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Work | Algorithm | Instances | Metrics | Preferences | E/D | Parameters | Steady State |
---|---|---|---|---|---|---|---|

[11] Social projects | HHGA-SPPv1 HHGA-SPPv2 | (3,4,20) (3,9,100) | No-dominated Solutions | Yes | E | Real | NA |

[12] Interdependent social projects, several objectives | NO-ACO | (10,4,25) (10,9,100) | No-dominated Solutions, ROI solutions | Yes | E | Real | NA |

[13] Social projects with priorities and sinergy | ACO-SPRI ACO-SOP ACO-SOP sinergy | (1,ND,25) (1,ND,40) (1,ND,100) | No-dominated Solutions | Yes | E | Real | NA |

[14] Social projects, several objectives | H-MCSGA I-MCSGA | (3,9,100) (2,9,150) (1,16,500) | No-dominated solutions, higher net flow | Yes | E | Real | No |

[10] Portfolio selection with interval parameters | I-NSGA-II-CD | (1,2,100) (1,9,100) | Cardinality | Yes | E | Intervals | No |

[15] Dynamic portfolio selection and several objectives | D-NSGA-II-FF AbYSS-FF D-MOEA\D--FF | (30,2,100) (30,3,100) (30,9,100) | Hypervolume modified, Spread modified, inverted generational distance modified | Yes | D | Real | No |

This work Portfolio selection with trapezoidal fuzzy numbers | T-NSGA-II-CD T-NSGA-II-SSD T-FAME | (12,2,25) (9,2,100) | Hypervolume, Generalized Spread | No | E | Trapezoidal fuzzy numbers | Yes |

AND Antecedents | Consequent | |
---|---|---|

Stagnation | Utilization | ProbOp |

High | High | Mid |

High | Mid | Low |

High | Low | Mid |

Mid | High | Mid |

Mid | Mid | Low |

Mid | Low | Mid |

Low | High | High |

Low | Mid | Mid |

Low | Low | Low |

Parameter | Value |
---|---|

Evaluation of the objective function | 5000 |

Population Size | 50 |

Crossover population % | 70 |

Mutation population % | 40 |

Mutation % | 5 |

Parameter | Value |
---|---|

Evaluation of the objective function | 5000 |

Population Size | 25 |

Front Size | 100 |

Tournament Size | 5 |

Number of parents | 4 |

Window Size | 13 |

Differential Evolution Crossover % | 10 |

Number of mutations in FM | 2 |

Front choice probability ($\beta )$ | 0.9 |

Pareto Optimal Front | T-NSGA-II-CD | T-NSGA-II-SSD | T-FAME | |||
---|---|---|---|---|---|---|

O2 | O1 | O2 | O1 | O2 | O1 | O2 |

3530 | 78,510 | 3465 | 81,155 | 3425 | 81,285 | 3530 |

3805 | 62,350 | 4245 | 66,240 | 4400 | 77,480 | 3715 |

3825 | 76,360 | 3840 | 75,650 | 3860 | 74,485 | 3750 |

3840 | 70,035 | 3870 | 68,610 | 4240 | 73,425 | 3775 |

3865 | 77,020 | 3490 | 70,350 | 4005 | ||

3965 | 66,605 | 4070 | 66,850 | 4375 | ||

3980 | 62,755 | 4090 | 59,865 | 4385 | ||

4000 | 77,900 | 3490 | ||||

4025 | 77,920 | 3485 | ||||

4035 | ||||||

4060 | ||||||

4065 | ||||||

4120 | ||||||

4150 | ||||||

4215 | ||||||

4235 | ||||||

4240 | ||||||

4260 | ||||||

4310 | ||||||

4375 | ||||||

4400 | ||||||

4435 | ||||||

4460 |

**Table 6.**Hypervolume normality test, the null hypothesis is that the samples follow a normal distribution which is accepted (a) when p-value < 0.05 and rejected (r) otherwise.

T-NSGA-II-CD | T-NSGA-II-SSD | T-FAME | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

Instance | Statistic | p-Value | R | Statistic | p-Value | R | Statistic | p-Value | R | Tests |

o2p25_0T | 0.9429 | 0.1089 | a | 0.83756 | 0.00034 | r | 0.96919 | 0.51737 | a | t,W |

o2p25_1T | 0.93655 | 0.07348 | a | 0.92817 | 0.04391 | r | 0.97408 | 0.65561 | a | t,W |

o2p25_2T | 0.92141 | 0.02918 | r | 0.95491 | 0.22837 | a | 0.96987 | 0.53551 | a | W,t |

o2p25_3T | 0.94311 | 0.11035 | a | 0.90566 | 0.01159 | r | 0.94528 | 0.12625 | a | t,W |

o2p25_4T | 0.95413 | 0.21782 | a | 0.93505 | 0.06696 | a | 0.89022 | 0.00488 | r | W,W |

o2p25_5T | 0.86113 | 0.00107 | r | 0.89584 | 0.00665 | r | 0.94768 | 0.14643 | a | W,W |

o2p25_6T | 0.9023 | 0.00956 | r | 0.89233 | 0.00548 | r | 0.96519 | 0.41715 | a | W,W |

o2p25_7T | 0.94961 | 0.16508 | a | 0.86559 | 0.00134 | r | 0.92644 | 0.03953 | r | W,W |

o2p25_8T | 0.92385 | 0.0338 | r | 0.91474 | 0.01963 | r | 0.85737 | 0.00089 | r | W,W |

o2p25_9T | 0.94965 | 0.16541 | a | 0.89673 | 0.00699 | r | 0.97209 | 0.59792 | a | t,W |

o2p25_10T | 0.92989 | 0.04877 | r | 0.78913 | 0.00004 | r | 0.97575 | 0.70469 | a | W,W |

o2p25_11T | 0.93191 | 0.05518 | a | 0.95357 | 0.21047 | a | 0.96642 | 0.44633 | a | t,t |

o2p25_12T | 0.94626 | 0.13411 | a | 0.95055 | 0.17491 | a | 0.98323 | 0.9033 | a | t,t |

o2p100_1T | 0.96346 | 0.37847 | a | 0.96637 | 0.44525 | a | 0.98333 | 0.90552 | a | t,t |

o2p100_2T | 0.95885 | 0.28944 | a | 0.98951 | 0.98844 | a | 0.9737 | 0.64441 | a | t,t |

o2p100_3T | 0.93272 | 0.05801 | a | 0.9821 | 0.87827 | a | 0.94779 | 0.14745 | a | t,t |

o2p100_4T | 0.78768 | 0.00004 | r | 0.78085 | 0.00003 | r | 0.89022 | 0.00488 | r | W,W |

o2p100_5T | 0.95289 | 0.20189 | a | 0.94588 | 0.13101 | a | 0.93478 | 0.06586 | a | t,t |

o2p100_6T | 0.94043 | 0.09341 | a | 0.93788 | 0.07976 | a | 0.95224 | 0.194 | a | t,t |

o2p100_7T | 0.97249 | 0.60937 | a | 0.99025 | 0.99229 | a | 0.94017 | 0.0919 | a | t,t |

o2p100_8T | 0.96892 | 0.51019 | a | 0.98362 | 0.9115 | a | 0.96805 | 0.48728 | a | t,t |

o2p100_9T | 0.57553 | 0 | r | 0.52513 | 0 | r | 0.71502 | 0 | r | W,W |

**Table 7.**Hypervolume homoscedasticity test, the null hypothesis is that all the input populations come from populations with equal variances, which is accepted (a) when p-value < 0.05 and rejected (r) otherwise. We can observe that the null hypothesis is accepted (a) for all the instances. The parametric t student test can be applied for all the instances that accept the null hypothesis in the normality tests.

Instance | Statistic | p-Value | R |
---|---|---|---|

o2p25_0T | 8.46563 | 0.00044 | a |

o2p25_1T | 17.23159 | 0 | a |

o2p25_2T | 8.53517 | 0.00041 | a |

o2p25_3T | 11.87763 | 0.00003 | a |

o2p25_4T | 7.1698 | 0.00131 | a |

o2p25_5T | 7.60431 | 0.0009 | a |

o2p25_6T | 7.19194 | 0.00129 | a |

o2p25_7T | 2.20562 | 0.11631 | a |

o2p25_8T | 8.18222 | 0.00055 | a |

o2p25_9T | 4.45024 | 0.01445 | a |

o2p25_10T | 3.63843 | 0.03037 | a |

o2p25_11T | 3.98587 | 0.02207 | a |

o2p25_12T | 9.90574 | 0.00013 | a |

o2p100_1T | 0.27401 | 0.76098 | a |

o2p100_2T | 2.14347 | 0.1234 | a |

o2p100_3T | 0.29369 | 0.74624 | a |

o2p100_4T | 1.79147 | 0.17281 | a |

o2p100_5T | 5.98972 | 0.00365 | a |

o2p100_6T | 1.09354 | 0.33959 | a |

o2p100_7T | 2.30064 | 0.10626 | a |

o2p100_8T | 4.20117 | 0.01812 | a |

o2p100_9T | 1.39539 | 0.25322 | A |

**Table 8.**Generalized Spread normality test, the null hypothesis is that the samples follow a normal distribution which is accepted (a) when p-value < 0.05 and rejected (r) otherwise.

T-NSGA-II-CD | T-NSGA-II-SSD | T-FAME | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

Instance | Statistic | p-Value | R | Statistic | p-Value | R | Statistic | p-Value | R | Tests |

o2p25_0T | 0.92895 | 0.04606 | r | 0.97607 | 0.71429 | a | 0.9784 | 0.78164 | a | W,t |

o2p25_1T | 0.98376 | 0.91432 | a | 0.95618 | 0.24658 | a | 0.97193 | 0.59314 | a | t,t |

o2p25_2T | 0.98074 | 0.84479 | a | 0.97925 | 0.8053 | a | 0.96813 | 0.48946 | a | t,t |

o2p25_3T | 0.9215 | 0.02934 | r | 0.9225 | 0.03116 | r | 0.96419 | 0.39452 | a | W,W |

o2p25_4T | 0.95187 | 0.18969 | a | 0.96214 | 0.35091 | a | 0.68255 | 0 | r | W,W |

o2p25_5T | 0.96913 | 0.51555 | a | 0.95677 | 0.25552 | a | 0.92403 | 0.03416 | r | W,W |

o2p25_6T | 0.87495 | 0.00216 | r | 0.97296 | 0.62306 | a | 0.958 | 0.27513 | a | W,t |

o2p25_7T | 0.94053 | 0.094 | a | 0.95631 | 0.24864 | a | 0.94784 | 0.14792 | a | t,t |

o2p25_8T | 0.9648 | 0.40819 | a | 0.95561 | 0.23827 | a | 0.94282 | 0.10833 | a | t,t |

o2p25_9T | 0.97001 | 0.53934 | a | 0.97168 | 0.58607 | a | 0.9686 | 0.50171 | a | t,t |

o2p25_10T | 0.92765 | 0.04254 | r | 0.96999 | 0.53902 | a | 0.97623 | 0.71907 | a | W,t |

o2p25_11T | 0.91446 | 0.01932 | r | 0.96986 | 0.53537 | a | 0.95816 | 0.27785 | a | W,t |

o2p25_12T | 0.95492 | 0.22856 | a | 0.98402 | 0.91939 | a | 0.95432 | 0.22029 | a | t,t |

o2p100_1T | 0.92495 | 0.03611 | r | 0.92054 | 0.02771 | r | 0.94295 | 0.10926 | a | W,W |

o2p100_2T | 0.9812 | 0.85642 | a | 0.95454 | 0.22326 | a | 0.95353 | 0.21003 | a | t,t |

o2p100_3T | 0.92278 | 0.03169 | r | 0.86033 | 0.00103 | r | 0.96482 | 0.40857 | a | W,W |

o2p100_4T | 0.65395 | 0 | r | 0.79925 | 0.00006 | r | 0.68255 | 0 | r | W,W |

o2p100_5T | 0.91266 | 0.01738 | r | 0.86347 | 0.0012 | r | 0.96541 | 0.4223 | a | W,W |

o2p100_6T | 0.90797 | 0.01323 | r | 0.91912 | 0.02544 | r | 0.90857 | 0.01369 | r | W,W |

o2p100_7T | 0.89328 | 0.00578 | r | 0.89889 | 0.00789 | r | 0.96516 | 0.41655 | a | W,W |

o2p100_8T | 0.94824 | 0.15169 | a | 0.96578 | 0.43096 | a | 0.96071 | 0.32297 | a | t,t |

o2p100_9T | 0.49141 | 0 | r | 0.68971 | 0 | r | 0.68313 | 0 | r | W,W |

**Table 9.**Generalized Spread homoscedasticy test, the null hypothesis is that all the input populations come from populations with equal variances, which is accepted (a) when p-value < 0.05 and rejected (r) otherwise. Observe that the null hypothesis is accepted (a) for all the instances. The parametric t student test can be applied for all the instances that accept the null hypothesis in the normality tests.

Instance | Statistic | p-Value | R |
---|---|---|---|

o2p25_0T | 0.33509 | 0.71619 | a |

o2p25_1T | 3.11548 | 0.04934 | a |

o2p25_2T | 5.44373 | 0.00592 | a |

o2p25_3T | 7.81001 | 0.00076 | a |

o2p25_4T | 0.38001 | 0.68498 | a |

o2p25_5T | 3.01271 | 0.05431 | a |

o2p25_6T | 1.58378 | 0.21106 | a |

o2p25_7T | 10.87966 | 0.00006 | a |

o2p25_8T | 1.51668 | 0.22518 | a |

o2p25_9T | 19.54345 | 0 | a |

o2p25_10T | 5.78604 | 0.00437 | a |

o2p25_11T | 7.0285 | 0.00148 | a |

o2p25_12T | 15.29209 | 0 | a |

o2p100_1T | 8.48884 | 0.00043 | a |

o2p100_2T | 9.53401 | 0.00018 | a |

o2p100_3T | 3.46674 | 0.0356 | a |

o2p100_4T | 1.42075 | 0.24708 | a |

o2p100_5T | 3.96176 | 0.02256 | a |

o2p100_6T | 4.19408 | 0.01824 | a |

o2p100_7T | 4.62372 | 0.01235 | a |

o2p100_8T | 5.30008 | 0.00673 | a |

o2p100_9T | 0.90643 | 0.40774 | a |

Hypervolume | |||
---|---|---|---|

Instance | T-NSGA-II-CD | T-NSGA-II-SSD | T-FAME |

o2p25_0T | 0.4747_{0.0858}_{⋁}_{*} | 0.3183_{0.3853}_{⋁} | 0.2024_{0.2491} |

o2p25_1T | 0.3807_{0.0510}_{⋁}_{*} | 0.2460_{0.2325}_{⋁} | 0.2003_{0.2876} |

o2p25_2T | 0.3591_{0.0614}_{⋁} | 0.2467_{0.2042}_{⋁}_{*} | 0.1613_{0.1526} |

o2p25_3T | 0.2832_{0.0549}_{⋁}_{*} | 0.2770_{0.2311}_{⋁} | 0.1345_{0.1646} |

o2p25_4T | 0.3510_{0.0812}_{⋁} | 0.2836_{0.1489}_{⋁} | 0.1875_{0.1673} |

o2p25_5T | 0.2635_{0.0383}_{⋁} | 0.1529_{0.1495}_{⋁} | 0.1070_{0.1048} |

o2p25_6T | 0.3797_{0.0609}_{⋁} | 0.2465_{0.1870}_{⋁} | 0.1380_{0.2060} |

o2p25_7T | 0.2348_{0.2446}_{⋁} | 0.2816_{0.3644}_{⋁} | 0.1427_{0.1694} |

o2p25_8T | 0.2574_{0.0664}_{⋁} | 0.1838_{0.2259}_{⋁} | 0.1630_{0.1747} |

o2p25_9T | 0.4026_{0.1184}_{⋁}_{*} | 0.2449_{0.2455}_{⋁} | 0.1539_{0.1615} |

o2p25_10T | 0.2580_{0.0710}_{⋁} | 0.1451_{0.1566}_{⋁} | 0.1126_{0.1070} |

o2p25_11T | 0.3918_{0.0946}_{⋁}_{*} | 0.2327_{0.1687}_{=*} | 0.1876_{0.1657} |

o2p25_12T | 0.2934_{0.0708}_{⋁}_{*} | 0.2621_{0.2174}_{=*} | 0.2352_{0.1969} |

Generalized Spread | |||
---|---|---|---|

Instance | T-NSGA-II-CD | T-NSGA-II-SSD | T-FAME |

o2p25_0T | 0.6178_{0.1985}_{⋀} | 0.4190_{0.1534 = *} | 0.4154_{0.2064} |

o2p25_1T | 0.7344_{0.1685}_{⋀}_{*} | 0.4477_{0.1289 = *} | 0.4661_{0.1128} |

o2p25_2T | 0.6065_{0.2078}_{⋀}_{*} | 0.3929_{0.1025 = *} | 0.3983_{0.1047} |

o2p25_3T | 0.7276_{0.2387}_{⋀} | 0.5181_{0.1370 =} | 0.5225_{0.0790} |

o2p25_4T | 0.6475_{0.3031}_{⋀} | 0.4646_{0.1432}_{⋁} | 0.5511_{0.1078} |

o2p25_5T | 0.7228_{0.1715}_{⋀} | 0.4204_{0.0925}_{⋀} | 0.4168_{0.1293} |

o2p25_6T | 0.6258_{0.1539}_{⋀} | 0.4026_{0.0982}_{⋁}_{*} | 0.4629_{0.1703} |

o2p25_7T | 0.8314_{0.5343}_{⋀}_{*} | 0.4995_{0.2457}_{⋁}_{*} | 0.5833_{0.2388} |

o2p25_8T | 0.7546_{0.1739}_{⋀}_{*} | 0.4693_{0.1447 = *} | 0.4646_{0.1059} |

o2p25_9T | 0.6534_{0.3432}_{⋀}_{*} | 0.4825_{0.1435 = *} | 0.4726_{0.0690} |

o2p25_10T | 0.6542_{0.2697}_{⋀} | 0.4793_{0.1031 = *} | 0.4779_{0.0891} |

o2p25_11T | 0.6540_{0.3103}_{⋀} | 0.4369_{0.1073 = *} | 0.4784_{0.1629} |

o2p25_12T | 07079_{0.2465}_{⋀}_{*} | 0.4684_{0.0953 = *} | 0.4654_{0.0793} |

Hypervolume (p-Value = 5.68 × 10^{−6}) | Generalized Spread (p-Value = 5.71 × 10^{−5}) | ||
---|---|---|---|

Algorithm | Ranking | Algorithm | Ranking |

T-NSGA-II-CD | 14 | T-NSGA-II-SSD | 19 |

T-NSGA-II-SSD | 25 | T-FAME | 20 |

T-FAME | 39 | T-NSGA-II-CD | 39 |

Hypervolume | |||
---|---|---|---|

Instance | T-NSGA-II-CD | T-NSGA-II-SSD | T-FAME |

o2p100_1T | 0.4681_{0.1948}_{⋀}_{*} | 0.5064_{0.1804}_{⋀}_{*} | 0.6214_{0.2130} |

o2p100_2T | 0.4094_{0.1613}_{⋀}_{*} | 0.5475_{0.2357}_{=*} | 0.5107_{0.2107} |

o2p100_3T | 0.5524_{0.2781=*} | 0.6366_{0.3261}_{=*} | 0.5947_{0.2887} |

o2p100_4T | 0.7738_{0.3543}_{⋀} | 0.9261_{0.5476}_{⋀} | 0.9395_{0.4006} |

o2p100_5T | 0.2893_{0.1453}_{⋀}_{*} | 0.3519_{0.2193}_{⋀}_{*} | 0.4611_{0.2668} |

o2p100_6T | 0.5359_{0.3131}_{=*} | 0.5422_{0.4082}_{=*} | 0.6163_{0.5234} |

o2p100_7T | 0.2713_{0.1066}_{⋀}_{*} | 0.3477_{0.1816}_{⋀}_{*} | 0.4896_{0.2093} |

o2p100_8T | 0.3550_{0.1282}_{=*} | 0.5173_{0.2759}_{⋁}_{*} | 0.3894_{0.2611} |

o2p100_9T | 0.9142_{0.3142}_{⋀} | 1_{0.1428}_{⋁} | 1_{0.0285} |

Generalized Spread | |||
---|---|---|---|

Instance | T-NSGA-II-CD | T-NSGA-II-SSD | T-FAME |

o2p100_1T | 0.5209_{0.3128}_{⋀} | 0.3210_{0.1922}_{⋀} | 0.3039_{0.1152} |

o2p100_2T | 0.5360_{0.2984}_{⋀}_{*} | 0.3105_{0.1349}_{⋁}_{*} | 0.3995_{0.2272} |

o2p100_3T | 0.4849_{0.1753}_{⋀} | 0.3791_{0.1171}_{⋀} | 0.3777_{0.2171} |

o2p100_4T | 0.2828_{0.0915}_{⋀} | 0.2555_{0.0661}_{⋁} | 0.2651_{0.0746} |

o2p100_5T | 0.6008_{0.2320}_{⋀} | 0.3796_{0.2193}_{⋀} | 0.2977_{0.1051} |

o2p100_6T | 0.3729_{0.2967}_{⋀} | 0.3457_{0.1845}_{⋀} | 0.2876_{0.1838} |

o2p100_7T | 0.5056_{0.2843}_{⋀} | 0.3221_{0.1803}_{⋀} | 0.3185_{0.1463} |

o2p100_8T | 0.5424_{0.2142}_{⋀}_{*} | 0.3154_{0.1280}_{⋁}_{*} | 0.3338_{0.1274} |

o2p100_9T | 0.4084_{0.0670}_{⋀} | 0.3681_{0.0604=} | 0.3718_{0.0489} |

Hypervolume (p-Value = 0.00104) | Generalized Spread (p-Value = 0.00113) | ||
---|---|---|---|

Algorithm | Ranking | Algorithm | Ranking |

T-FAME | 12.5 | T-FAME | 13 |

T-NSGA-II-SSD | 14.5 | T-NSGA-II-SSD | 14 |

T-NSGA-II-CD | 27 | T-NSGA-II-CD | 27 |

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Estrada-Padilla, A.; Lopez-Garcia, D.; Gómez-Santillán, C.; Fraire-Huacuja, H.J.; Cruz-Reyes, L.; Rangel-Valdez, N.; Morales-Rodríguez, M.L.
Modeling and Optimizing the Multi-Objective Portfolio Optimization Problem with Trapezoidal Fuzzy Parameters. *Math. Comput. Appl.* **2021**, *26*, 36.
https://doi.org/10.3390/mca26020036

**AMA Style**

Estrada-Padilla A, Lopez-Garcia D, Gómez-Santillán C, Fraire-Huacuja HJ, Cruz-Reyes L, Rangel-Valdez N, Morales-Rodríguez ML.
Modeling and Optimizing the Multi-Objective Portfolio Optimization Problem with Trapezoidal Fuzzy Parameters. *Mathematical and Computational Applications*. 2021; 26(2):36.
https://doi.org/10.3390/mca26020036

**Chicago/Turabian Style**

Estrada-Padilla, Alejandro, Daniela Lopez-Garcia, Claudia Gómez-Santillán, Héctor Joaquín Fraire-Huacuja, Laura Cruz-Reyes, Nelson Rangel-Valdez, and María Lucila Morales-Rodríguez.
2021. "Modeling and Optimizing the Multi-Objective Portfolio Optimization Problem with Trapezoidal Fuzzy Parameters" *Mathematical and Computational Applications* 26, no. 2: 36.
https://doi.org/10.3390/mca26020036