Fuzzy Multi-objective Optimization Problem Research Articles

Using Genetic Algorithms and Core Values of Cooperative Games to Solve Fuzzy Multiobjective Optimization Problems

A new methodology for solving the fuzzy multiobjective optimization problems is proposed in this paper by considering the fusion of cooperative game theory and genetic algorithm. The original fuzzy multiobjective optimization problem needs to be transformed into a scalar optimization problem, which is a conventional optimization problem. Usually, the assignments of suitable coefficients to the corresponding scalar optimization problem are subjectively determined by the decision makers. However, these assignments may cause some biases by their subjectivity. Therefore, this paper proposes a mechanical procedure to avoid this subjective biases. We are going to formulate a cooperative game using the α-level functions of the multiple fuzzy objective functions. Under this setting, the suitable coefficients can be determined mechanically by involving the core values of the cooperative game, which is formulated using the multiple fuzzy objective functions. We shall prove that the optimal solutions of the transformed scalar optimization problem are indeed the nondominated solutions of fuzzy multiobjective optimization problem. Since the core-nondominated solutions will depend on the coefficients that are determined by the core values of cooperative game, there will be a lot of core-nondominated solutions that will also depend on the corresponding coefficients. In order to obtain the best core-nondominated solution, we shall invoke the genetic algorithms by evolving the coefficients.

A note on “A new method for intuitionistic fuzzy multi-objective linear fractional optimization problem and its application in agricultural land allocation problem”

Moges et al. (Inf. Sci. 625 (2023) 457–475) proposed a method to find a Pareto-optimal solution (POS) of intuitionistic fuzzy multi-objective linear fractional optimization problem (MOLFOP). In this method, firstly, an intuitionistic fuzzy MOLFOP is transformed into a crisp MOLFOP with the help of a ranking function. Then, using the existing method (Fuzzy Sets Syst. 125 (2002) 335–342), the transformed crisp MOLFOP is transformed into a crisp multi-objective linear optimization problem (MOLOP). Finally, it is assumed that to find a POS of the considered intuitionistic fuzzy MOLFOP is equivalent to find a POS of the transformed crisp MOLOP. It is pertinent to mention that this assumption is valid only if the transformed crisp MOLFOP is equivalent to the considered intuitionistic fuzzy MOLFOP as well as the transformed crisp MOLOP is equivalent to the transformed crisp MOLFOP. In this note, a numerical example is considered to show that the transformed crisp MOLFOP is not equivalent to the considered intuitionistic fuzzy MOLFOP. Also, on the basis of the existing results (Fuzzy Sets Syst. 246 (2014) 156–159), it is pointed out that the transformed crisp MOLOP is not equivalent to the transformed crisp MOLFOP. Therefore, Moges et al.’s method is not valid and hence, the results of agricultural planning problem of Ethopia for cultivation of different crops, obtained by Moges et al. with their proposed method, are not correct.

New Approach to Solving Fuzzy Multiobjective Linear Fractional Optimization Problems

In this paper, an iterative approach based on the use of fuzzy parametric functions is proposed to find the best preferred optimal solution to a fuzzy multiobjective linear fractional optimization problem. From this approach, the decision-maker imposes tolerance values or termination conditions for each parametric objective function. Indeed, the fuzzy parametric values are computed iteratively, and each fuzzy fractional objective is transformed into a fuzzy non-fractional parametric function using these values of parameters. The core value of fuzzy numbers is used to transform the fuzzy multiobjective non-fractional problem into a deterministic multiobjective non-fractional problem, and the ε-constraint approach is employed to obtain a linear single objective optimization problem. Finally, by setting the value of parameter ε, the Dangtzig simplex method is used to obtain an optimal solution. Therefore, the number of solutions is equal to the number of used values, and the optimal solution is chosen according to the preference of the decision-maker. We have provided a didactic example to highlight the step of our approach and its numerical performances.

Optimization of time based fuzzy multi-objective reliability redundancy allocation problem for [formula omitted] system using tuning and neighborhood based fuzzy MOPSO algorithm

In this paper, a time based fuzzy multi objective reliability redundancy allocation problem (FMORRAP) is proposed for the xj−out−of−m series–parallel system. The main objective is to maximize the system reliability with minimization of system cost and system repairing cost by optimizing the number of redundant components at each stage. The objectives are achieved by satisfying entropy constraints with limited redundant components at each stage and same for the whole system. Here the uncertainty of reliability, cost and repairing cost of each component is maintained by using triangular fuzzy number (TFN). The decreasing factor of component reliability and cost follows the change in the length of radius for the inverse logarithmic spiral with respect to time. Similarly the increasing factor of component repairing cost follows the same for the logarithmic spiral. Tuning and Neighborhood based Fuzzy Multi-Objective Particle Swarm Optimization (TNF-MOPSO) algorithm is proposed to solve fuzzy multi-objective optimization problems (MOOP). The proposed problem and proposed algorithm are illustrated by using a bench mark problem. The proposed algorithm produced better membership of objective functions for most of the time values and high satisfaction level of reliability for all values of the time parameter. It also outperforms the standard approaches MOPSO and NF-MOPSO.

A new method for intuitionistic fuzzy multi-objective linear fractional optimization problem and its application in agricultural land allocation problem

This paper presents a new method for solving an intuitionistic fuzzy multi-objective linear fractional optimization (IFMOLFO) problem with crisp and intuitionistic fuzzy constraints. Here, all uncertain parameters are represented as triangular intuitionistic fuzzy numbers. We used an accuracy ranking function and variable transformation in the proposed method to convert an IFMOLFO problem into a crisp multi-objective linear optimization problem. Then, we formulated the first phase of the weighted intuitionistic fuzzy goal programming (WIFGP) model to obtain an intuitionistic fuzzy non-dominant (IFND) solution for the IFMOLFO problem. Several strategies for obtaining an IFND solution to the IFMOLFO problem have been proposed in the literature. However, in addition to constructing the phase-I WIFGP model, this study shows that the IFND solution may not be Pareto-optimal when some of the under-deviation variables are zero. As a result, the second phase of the WIFGP model is applied to address this issue. The benefits of both models are merged to provide a novel method, unlike any other method in the literature, for producing optimal solutions that satisfy both IFND and Pareto-optimal requirements. The suggested algorithm’s efficiency and reliability are demonstrated by addressing a real-life case study of an agricultural production planning problem and followed by solving a numerical example from literature.

A deterministic and nature-inspired algorithm for the fuzzy multi-objective path optimization problem

Increasing evaluation indexes have been involved in the network modeling, and some parameters cannot be described precisely. Fuzzy set theory becomes a promising mathematical method to characterize such uncertain parameters. This study investigates the fuzzy multi-objective path optimization problem (FMOPOP), in which each arc has multiple crisp and fuzzy weights simultaneously. Fuzzy weights are characterized by triangular fuzzy numbers or trapezoidal fuzzy numbers. We adopt two fuzzy number ranking methods based on their fuzzy graded mean values and distances from the fuzzy minimum number. Motivated by the ripple spreading patterns on the natural water surface, we propose a novel ripple-spreading algorithm (RSA) to solve the FMOPOP. Theoretical analyses prove that the RSA can find all Pareto optimal paths from the source node to all other nodes within a single run. Numerical examples and comparative experiments demonstrate the efficiency and robustness of the newly proposed RSA. Moreover, in the first numerical example, the processes of the RSA are illustrated using metaphor-based language and ripple spreading phenomena to be more comprehensible. To the best of our knowledge, the RSA is the first algorithm for the FMOPOP that can adopt various fuzzy numbers and ranking methods while maintaining optimality.

Generalized techniques for solving intuitionistic fuzzy multi-objective non-linear optimization problems

This paper focuses on the methods for the efficient solution of multi-objective non-linear optimization problems with uncertain parameters represented as intuitionistic fuzzy numbers. In most of the existing techniques for such problems, generally linear membership (satisfaction) functions have been used. But every real life problem cannot be justified and modeled using the linear functions, so the efficient solution methodologies from the literature such as Zimmermann’s technique, Maximum additive operator technique, γ-operator technique have been extended in this paper by defining the non-linear membership functions in place of the linear ones. Unlike the classical versions of these techniques, the non-linear non-membership (dissatisfaction) functions have also been incorporated along with the memberships. Intuitionistic fuzzy number with non-linear grade functions has been introduced. Appropriate theorems have been proved to support the claims. Two numerical examples in the intuitionistic fuzzy environment from the field of manufacturing and transportation have been considered for the illustration of the proposed technique. The obtained results show the applicability and reliability of the suggested extensions and their comparison with the results obtained from that of the non-modified (traditional) techniques reflects its effectiveness.

Http://pu.edu.pk/images/journal/maths/PDF/Paper_3_53_9_2021.pdf

Decision-making is one of the contemporary issues in this modern era due to the interaction of risk and uncertainties in every aspect of the daily lives of human beings. Accordingly, solving practical optimization problems tends to be more challenging. The fundamental reason for this investigation is to explore an effective solution technique for multi-objective optimization problems (MOOPs) in an intuitionistic fuzzy environment (IFE) addressing the issue of determining proper violation parameters and tolerances to the objectives and constraints. The other significant characteristic of this study is the consideration of the decisionmaker’s perspective, namely, optimistic, pessimistic and mixed views in the solution procedure. In the proposed method, compared to the existing study, the required number of iterations and stages are considerably reduced in solving intuitionistic fuzzy multi-objective optimization problems (IFMOOPs). Hence it has imperative advantages in solving complex real-world problems without much difficulty. One problem is solved to demonstrate the competency of the planned approach. A comparative analysis is also undertaken to ascertain the efficiency of the technique.

Robust neutrosophic programming approach for solving intuitionistic fuzzy multiobjective optimization problems

This study presents the modeling of the multiobjective optimization problem in an intuitionistic fuzzy environment. The uncertain parameters are depicted as intuitionistic fuzzy numbers, and the crisp version is obtained using the ranking function method. Also, we have developed a novel interactive neutrosophic programming approach to solve multiobjective optimization problems. The proposed method involves neutral thoughts while making decisions. Furthermore, various sorts of membership functions are also depicted for the marginal evaluation of each objective simultaneously. The different numerical examples are presented to show the performances of the proposed solution approach. A case study of the cloud computing pricing problem is also addressed to reveal the real-life applications. The practical implication of the current study is also discussed efficiently. Finally, conclusions and future research scope are suggested based on the proposed work.

Multi-objective Optimization for Roadway Maintenance Engineering Based on ev-MOGA

The performance of roadway maintenance is a highly correlated factor on national socio-economic issues. Roadway maintenance work is a complex task, involving market game, engineering, planning and scheduling. Taking such aspects as quality, schedule and cost into account, this paper develops a multi-objective roadway maintenance system, in which ev-MOGA is adopted to optimize the decisions concerning roadway maintenance. The experimental results can provide a references for fuzzy multi-objective optimization (MOP) problems.

FDMOABC: Fuzzy Discrete Multi-Objective Artificial Bee Colony approach for solving the non-deterministic QoS-driven web service composition problem

The multi-objective quality of service (QoS)-driven web service composition problem (MOQWSCP) aims to find the best combinations of atomic web services (i.e. composite service) to answer high quality of the optimized QoS criteria in a way that maximize benefit QoS parameters such as availability and reliability and minimize the negative ones like price and response time, where the users’ requirements should be satisfied. Due to the dynamic environments in which the elementary services are invoked, some services’ QoS parameters are often ambiguous and uncertain, so, it is inappropriate to express them by fixed values. Hence, The QoS parameters are represented by trapezoidal fuzzy numbers. Thus, we formulate MOQWSCP as a fuzzy multi-objective optimization problem (FMOQWSCP). A fuzzy discrete multi-objective artificial bee colony (FDMOABC) approach is provided to solve the formulated FMOQWSCP, for which we have integrated a new fuzzy ranking method to cope with solutions sorting and a new fuzzy distance measure that is used to control and keep the diversity of FDMOABC’s solutions. Furthermore, a fuzzy multi-criteria decision-making method (FMCDMM) is provided to determine the best composite service among the Pareto-optimal solutions generated by FDMOABC. Finally, two kinds of comparisons are performed to validate the performance and the effectiveness of FDMOABC and FMCDMM methods. In the former, the combined FDMOABC and FMCDMM methods is compared against the fuzzy single objective optimization approaches TGA and EFPA, whereas in the later, a multi-objective optimization comparison is performed among FDMOABC and the fuzzy-extended versions of NSGA-II and SPEA2 algorithms.

Fuzzy Goal Programming with an Imprecise Intuitionistic Fuzzy Preference Relations

Fuzzy goal programming (FGP) is applied to solve fuzzy multi-objective optimization problems. In FGP, the weights are associated with fuzzy goals for the preference among them. However, the hierarchy within the fuzzy goals depends on several uncertain criteria, decided by experts, so the preference relations are not always easy to associate with weight. Therefore, the preference relations are provided by the decision-makers in terms of linguistic relationships, i.e., goal A is slightly or moderately or significantly more important than goal B. Due to the vagueness and ambiguity associated with the linguistic preference relations, intuitionistic fuzzy sets (IFSs) are most efficient and suitable to handle them. Thus, in this paper, a new fuzzy goal programming with intuitionistic fuzzy preference relations (FGP-IFPR) approach is proposed. In the proposed FGP-IFPR model, an achievement function has been developed via the convex combination of the sum of individual grades of fuzzy objectives and amount of the score function of IFPRs among the fuzzy goals. As an extension, we presented the linear and non-linear, namely, exponential and hyperbolic functions for the intuitionistic fuzzy preference relations (IFPRs). A study has been made to compare and analyze the three FGP-IFPR models with intuitionistic fuzzy linear, exponential, and hyperbolic membership and non-membership functions. For solving all three FGP-IFPR models, the solution approach is developed that established the corresponding crisp formulations, and the optimal solution are obtained. The validations of the proposed FGP-IFPR models have been presented with an experimental investigation of a numerical problem and a banking financial statement problem. A newly developed distance measure is applied to compare the efficiency of proposed models. The minimum value of the distance function represents a better and efficient model. Finally, it has been found that for the first illustrative problem considered, the exponential FGP-IFPR model performs best, whereas for the second problem, the hyperbolic FGP-IFPR model performs best and the linear FGP-IFPR model shows worst in both cases.

New optimality conditions for multiobjective fuzzy programming problems

In this paper we study fuzzy multiobjective optimization problems defined for $n$ variables. Based on a new $p$-dimensional fuzzy stationary-point definition, necessary efficiency conditions are obtained. And we prove that these conditions are also sufficient under new fuzzy generalized convexity notions. Furthermore, the results are obtained under general differentiability hypothesis.

Hybrid NSGA-II based decision-making in fuzzy multi-objective reliability optimization problem

This paper suggests a hybrid NSGA-II based decision-making method in a fuzzy multi-objective reliability optimization problem. Multi-objective evolutionary algorithms (MOEAs) are popular techniques to be solved various kinds of multi-objective optimization problems efficiently. NSGA-II is one of the elitist MOEAs, which is largely used in engineering design problems. The reliability-based system design problem comprises various kinds of uncertainties such as expert’s information character, qualitative statements, vagueness, incompleteness, unclear system boundaries, etc. Fuzzy optimization techniques can be useful during the initial stage of the conceptual design of a system. In many complex problems, it is not possible to produce the entire Pareto-optimal set in one simulation run. Apart from this, getting a well diverse solution set is another important phenomenon in this field. The proposed approach finds the optimal system design by resolving these issues in a fuzzy multi-objective reliability optimization problem. A numerical example of the over-speed protection system consisting of three mutually conflicting objectives such as system reliability, system cost, and system weight is considered with several design constraints to illustrate the method. Finally, the results obtained by the proposed approach are discussed with the existing approach.

An approach for solving fully fuzzy multi-objective linear fractional optimization problems

This article presents an algorithm for solving fully fuzzy multi-objective linear fractional (FFMOLF) optimization problem. Some computational algorithms have been developed for the solution of fully fuzzy single-objective linear fractional optimization problems. Veeramani and Sumathi (Appl Math Model 40:6148–6164, 2016) pointed out that no algorithm is available for solving a single-objective fully fuzzy optimization problem. Das et al. (RAIRO-Oper Res 51:285–297, 2017) proposed a method for solving single-objective linear fractional programming problem using multi-objective programming. Moreover, it is the fact that no method/algorithm is available for solving a FFMOLF optimization problem. In this article, a fully fuzzy MOLF optimization problem is considered, where all the coefficients and variables are assumed to be the triangular fuzzy numbers (TFNs). So, we are proposing an algorithm for solving FFMOLF optimization problem with the help of the ranking function and the weighted approach. To validate the proposed fuzzy intelligent algorithm, three existing classical numerical problems are converted into FFMOLF optimization problem using approximate TFNs. Then, the proposed algorithm is applied in an asymmetric way. Since there is no algorithm available in the existing literature for solving this difficult problem, we compare the obtained efficient solutions with corresponding existing methods for deterministic problems.

(1901-4964) New optimality conditions for multiobjective fuzzy programming problems

In this paper we study fuzzy multiobjective optimization problems defi ned for n variables. Based on a new p-dimensionalfuzzy stationary-point de finition, necessary efficiency conditions are obtained. And we prove that these conditions arealso sufficient under new fuzzy generalized convexity notions. Furthermore, the results are obtained under generaldifferentiability hypothesis.

Variational-like inequalities for n-dimensional fuzzy-vector-valued functions and fuzzy optimization

Abstract The existing results on the variational inequality problems for fuzzy mappings and their applications were based on Zadeh’s decomposition theorem and were formally characterized by the precise sets which are the fuzzy mappings’ cut sets directly. That is, the existence of the fuzzy variational inequality problems in essence has not been solved. In this paper, the fuzzy variational-like inequality problems is incorporated into the framework of n-dimensional fuzzy number space by means of the new ordering of two n-dimensional fuzzy-number-valued functions we proposed [Fuzzy Sets and Systems 295 (2016) 19-36]. As a theoretical basis, the existence and the basic properties of the fuzzy variational inequality problems are discussed. Furthermore, the relationship between the variational-like inequality problems and the fuzzy optimization problems is discussed. Finally, we investigate the optimality conditions for the fuzzy multiobjective optimization problems.

A Newton method for capturing Pareto optimal solutions of fuzzy multiobjective optimization problems

In this study, a Newton method is developed to obtain (weak) Pareto optimal solutions of an unconstrained multiobjective optimization problem (MOP) with fuzzy objective functions. For this purpose, the generalized Hukuhara differentiability of fuzzy vector functions and fuzzy max-order relation on the set of fuzzy vectors are employed. It is assumed that the objective functions of the fuzzy MOP are twice continuously generalized Hukuhara differentiable. Under this assumption, the relationship between weakly Pareto optimal solutions of a fuzzy MOP and critical points of the related crisp problem is discussed. Numerical examples are provided to demonstrate the efficiency of the proposed methodology. Finally, the convergence analysis of the method under investigation is discussed.

Software requirement optimization using a fuzzy artificial chemical reaction optimization algorithm

In agile methods, software products are developed in several releases. In each release, a new set of requirements for development is proposed. Due to technical and non-technical problems, it is almost impossible to develop all the proposed requirements in the next release. The select of an optimal subset from among all the proposed requirements has become an important problem for the developer team. The aim of this problem is to select an optimal subset from among the requirements for product development in the next release so that it has the highest satisfaction to the clients and the lowest cost for the manufacturing company. Since this problem faces two conflicting objectives and several constraints, it is placed in the NP-hard problems category. In this paper, we intend to formulate this problem for the first time as a fuzzy multi-objective optimization problem. We intend to use an artificial chemical reaction optimization algorithm to solve this problem. In the implementation stage, we make use of five interactions between requirements as one of the constraints of the problem for the first time. Two randomized fuzzy synthetic datasets are used to do the experiments. The results of the proposed algorithm are evaluated using three criteria of multi-objective problems. The results and diagrams of the proposed algorithm are very reliable and can help the developer team to make a decision.

A Novel Fuzzy QOS Based Improved Honey Bee Behavior Algorithm For Efficient Load Balancing In Cloud

Nature inspired algorithm based Load balancing of tasks on virtual machines (VMs) has become an area of greater research interest. Honey Bee Behavior Based Load Balancing (HBB-LB) was introduced to balance the load with a maximum throughput. This approach also balances the priorities of the tasks on the VM to minimize the waiting time of the tasks. However, HBB-LB considers only the VM load for balancing the load, which might not be sufficiently effective. This paper proposes an Improved Honey Bee Behavior Based Load Balancing (IHBB-LB), taking into consideration a few more QoS parameters of VM, such as service response time, availability, reliability, cost and throughput to enhance load balancing. Response time is vital in determining the instant activity of a VM while availability determines available resource and state of VM (idle or active) and Reliability determines the level of trust in a VM. Most importantly, Cost for utilizing a VM and Throughput (capability of VM) are also essential in determining the VM efficiency. But, the inclusion of multiple QoS parameters results in multi-objective optimization problem. As a number of QoS parameters are computed, the Fuzzification of the QoS values was performed through the generated fuzzy rules and multi-objective optimization problem was eliminated. The experiments were performed in terms of makespan, response time, degree of imbalance and the number of tasks migrated and results indicate that the IHBB-LB provides a better level of performance.