Multiobjective Nonlinear Programming Problems Research Articles

Multi-objective non-linear programming problem with rough interval parameters: an application in municipal solid waste management

In dealing with the real-world optimization problems, a decision-maker has to frequently face the ambiguity and hesitancy due to various uncontrollable circumstances. Rough set theory has emerged as an indispensable tool for representing this ambiguity because of its characteristic of incorporating agreement and understanding of all the involved specialists and producing more realistic conclusions. This paper studies an application of the rough set theory for a multi-objective non-linear programming problem that originates for the management of solid wastes. Municipal solid waste management is a global problem that affects every country. Because of the poor waste management system in many nations, the bulk of municipal solid waste is disposed of in open landfills with no recovery mechanism. Hence, an effective and long term waste management strategy is the demand of the day. This research offers an incinerating, composting, recycling, and disposing system for the long-term management of the municipal solid waste. A model for the municipal solid waste management with the goal of minimizing the cost of waste transportation, cost of waste treatment and maximizing the revenue generated from various treatment facilities is developed under rough interval environment. To tackle the conflicting nature of different objectives, an approach is proposed that gives the optimistic and pessimistic views of the decision-maker for optimizing the proposed model. Also, the biasness/preference of the decision-maker for a specific objective is handled by establishing the respective non-linear membership and non-membership functions instead of the linear ones. Finally, to demonstrates the practicality of the proposed methodology, a case study is solved and the obtained Pareto-optimal solution has been compared to those obtained by the existing approaches.

A product family-based supply chain hypernetwork resilience optimization strategy

In product manufacturing, the supply chain can be adversely affected by the occurrence of uncertain risk events. Optimizing supply chain resilience is a challenging multi-objective nonlinear programming problem. Due to the development of mass customization production modes, manufacturing enterprises are more concerned with the service supply capability for a given product family. However, optimizing supply chain resilience while considering the product family is neglected in existing works. To tackle it, this paper first establishes a product family-based supply chain hypernetwork, mapping the product bill of materials to the potential suppliers. Second, a product family-based supply chain hypernetwork resilience optimization strategy (PF-SCHROS) is proposed using an improved genetic algorithm. Experiments are carried out to compare the proposed strategy with the traditional methods including interior point method, sequential quadratic programming, and genetic algorithm. Results show that our approach greatly reduces the loss originating from supplier chain disruption and can adequately improve supply chain service capability in various disruption scenarios.

Multi-Objective Non-Linear Programming Problems in Linear Diophantine Fuzzy Environment

Due to various unpredictable factors, a decision maker frequently experiences uncertainty and hesitation when dealing with real-world practical optimization problems. At times, it’s necessary to simultaneously optimize a number of non-linear and competing objectives. Linear Diophantine fuzzy numbers are used to address the uncertain parameters that arise in these circumstances. The objective of this manuscript is to present a method for solving a linear Diophantine fuzzy multi-objective nonlinear programming problem (LDFMONLPP). All the coefficients of the nonlinear multi-objective functions and the constraints are linear Diophantine fuzzy numbers (LDFNs). Here we find the solution of the nonlinear programming problem by using Karush-Kuhn-Tucker condition. A numerical example is presented.

An Investigation of Linear Diophantine Fuzzy Nonlinear Fractional Programming Problems

The linear Diophantine fuzzy set notion is the main foundation of the interactive method of tackling nonlinear fractional programming problems that is presented in this research. When the decision maker (DM) defines the degree α of α level sets, the max-min problem is solved in this interactive technique using Zimmermann’s min operator method. By using the updating technique of degree α, we can solve DM from the set of α-cut optimal solutions based on the membership function and non-membership function. Fuzzy numbers based on α-cut analysis bestowing the degree α given by DM can first be used to classify fuzzy Diophantine inside the coefficients. After this, a crisp multi-objective non-linear fractional programming problem (MONLFPP) is created from a Diophantine fuzzy nonlinear programming problem (DFNLFPP). Additionally, the MONLFPP can be reduced to a single-objective nonlinear programming problem (NLPP) using the idea of fuzzy mathematical programming, which can then be solved using any suitable NLPP algorithm. The suggested approach is demonstrated using a numerical example.

Multi-objective optimisation for solving cooperative continuous static games using Karush-Kuhn-Tucker conditions

This paper introduces cooperative continuous static games (CCSG) with parameters in the cost functions of the players and in the right-hand side of the constraints. The CCSG is converted into the corresponding multi-objective nonlinear programming problem. The resulted nonlinear programming problem is converted into the single objective nonlinear programming problem through the use of the weighted sum method. A solution method for obtaining the stability set of the second kind without differentiability for the CCSG is presented using Karush-Kuhn-Tucker conditions. A numerical example is given for the illustration.

Multi-objective non-linear solid transportation problem with fixed charge, budget constraints under uncertain environments

ABSTRACT Transportation-based models are widely used to solve real-life problems such as supply chain problems, inventory level problems, business models, management problems, etc. The parameters of these problems have been conventionally measured as deterministic. However, these parameters have vagueness due to the uncertainty in the data sets. Such uncertainty of the parameters is generally measured using randomness (probability theory) or fuzziness (fuzzy theory), with certain assumptions and limitations. Closed intervals interpretation is another concept to deal with the uncertainties of real-world issues in recent decades, avoiding the limitations in probability theory and fuzzy set theory. This study formulates a fixed-charged multi-objective non-linear solid transportation with parameters such as availability of items, conveyance capacity, transportation cost, requirement, and budget for destinations in the form of closed intervals, where objectives are minimization of total transportation cost and action of time under the restriction over budget. An efficient solution procedure is developed with the help of interval analysis based on the parametric perception of intervals. Initially, the objective functions are converted into a crisp function using multiple integrations, and as a result, the problem is converted into a deterministic multi-objective non-linear programming problem. In addition, a real-world numerical problem is considered for testing the developed solution process for greater comprehension and clarity.

A study of uncertainty multi-objective nonlinear programming problems for rough intervals

In real conditions, the parameters of multi-objective nonlinear programming (MONLP) problem models can’t be determined exactly. Hence in this paper, we concerned with studying the uncertainty of MONLP problems. We propose algorithms to solve rough and fully-rough-interval multi-objective nonlinear programming (RIMONLP and FRIMONLP) problems, to determine optimal rough solutions value and rough decision variables, where all coefficients and decision variables in the objective functions and constraints are rough intervals (RIs). For the RIMONLP and FRIMONLP problems solving methodology are presented using the weighting method and slice-sum method with Kuhn-Tucker conditions, We will structure two nonlinear programming (NLP) problems. In the first one of this NLP problem, all of its variables and coefficients are the lower approximation (LAI) it’s RIs. The second NLP problems are upper approximation intervals (UAI) of RIs. Subsequently, both NLP problems are sliced into two crisp nonlinear problems. NLP is utilized because numerous real systems are inherently nonlinear. Also, rough intervals are so important for dealing with uncertainty and inaccurate data in decision-making (DM) problems. The suggested algorithms enable us to the optimal solutions in the largest range of possible solution. Finally, Illustrative examples of the results are given.

Grey-based approach for estimating software reliability under nonhomogeneous Poisson process

Due to the randomness and time dependence of the factors affecting software reliability, most software reliability models are treated as stochastic processes, and the non-homogeneous Poisson process (NHPP) is the most used one. However, the failure behavior of software does not follow the NHPP in a statistically rigorous manner, and the pure random method might be not enough to describe the software failure behavior. To solve these problems, this paper proposes a new integrated approach that combines stochastic process and grey system theory to describe the failure behavior of software. A grey NHPP software reliability model is put forward in a discrete form, and a grey-based approach for estimating software reliability under the NHPP is proposed as a nonlinear multi-objective programming problem. Finally, four grey NHPP software Reliability models are applied to four real datasets, the dynamic R-square and predictive relative error are calculated. Comparing with the original single NHPP software reliability model, it is found that the modeling using the integrated approach has a higher prediction accuracy of software reliability. Therefore, there is the characteristics of grey uncertain information in the NHPP software reliability models, and exploiting the latent grey uncertain information might lead to more accurate software reliability estimation.

Optimal Dispatching Strategy of Active Distribution Network for Promoting Local Consumption of Renewable Energy

Large-scale renewable energy sources (RESs) have been integrated into the active distribution network (ADN). For promoting the local consumption of RESs within ADN, an optimal dispatching strategy was proposed with two-stage hierarchical energy management framework. On the spatial boundary, a two-layer energy management framework was designed with the local optimization layer and the global optimization layer. The local optimization layer was for optimal power flow in the branch feeder with the objective functions of minimizing operation costs and maximizing the consumption of RESs. The global optimization layer was for optimal power flow in the main feeder with the objective functions of minimizing power loss and the voltage deviation of nodes. On the time scale, two-stage optimal dispatching models were established, including the day-ahead optimal models and intra-day optimal models. The day-ahead optimal models identified the operation status of the controllable units, and then the intra-day optimal models were updated with the ultra-short-term forecast results. A risk indicator was introduced to quantify the uncertainty of RES, and a non-dominated sorting genetic algorithm with elite strategy was adopted to solve the multi-objective nonlinear programming problem. An actual project in northern China was used as the testing system. The results of case studies verify that the proposed strategy can effectively realize the maximum local consumption of RESs and support the economic operation of ADN.

Study on multi-objective nonlinear programming problem with rough parameters

Rough set theory, introduced by Pawlak in 1981, is one of the important theories to express the vagueness not by means of membership but employing a boundary region of a set, i.e., an object is approximately determined based on some knowledge. In our real-life, there exists several parameters which impact simultaneously on each other and hence dealing with such different parameters and their conflictness create a multi-objective nonlinear programming problem (MONLPP). The objective of the paper is to deal with a MONLPP with rough parameters in the constraint set. The considered MONLPP with rough parameters are converted into the two-single objective problems namely, lower and upper approximate problems by using the weighted averaging and the ɛ- constraints methods and hence discussed their efficient solutions. The Karush-Kuhn-Tucker’s optimality conditions are applied to solve these two lower and upper approximate problems. In addition, the rough weights and the rough parameter ɛ are determined by the lower and upper the approximations corresponding each efficient solution. Finally, two numerical examples are considered to demonstrate the stated approach and discuss their advantages over the existing ones.

Research on the Coordination Model of Passenger Transportation Mode in the Intercity Comprehensive Transportation Corridor

At present, the study of comprehensive transportation corridors primarily focuses on the planning and construction of transportation corridors. There are few studies on the coordinated operation of all modes of transportation after the construction of the transportation corridor is completed. Comprehensive transportation corridor takes on the characteristics that the volume of transport demand and supply is large, the number of transportation modes is not only one, and competition among different transportation modes is fierce. In order to prevent unhealthy competition among different transportation modes from disrupting the transportation market, this paper takes the passenger transportation modes within the intercity comprehensive transportation corridor as the studied object and establishes a coordination model for them. The model introduces the disaggregate model as the research theory, takes the quantizable attributes of transportation modes as the decision variables, takes the reasonable distribution of passenger flow among different transportation modes as the objective function, and takes the supply capacity of different transportation modes and the reasonable value range of decision variables as the constraints. This model is a multiobjective nonlinear programming problem. The multiobjective genetic algorithm is designed to solve the model. The real number encoding method is adopted to encode the decision variables; the penalty function method is used to eliminate the solutions that do not meet the nonlinear constraints after crossover and mutation; the repairment algorithm is used to convert the solutions that do not satisfy the linear constraints and bound constraints after crossover and mutation into the feasible region. Pareto optimal solution set is obtained through continuous selection, crossover, and mutation. Finally, a numeric example is made to demonstrate that the method proposed in this paper is effective and feasible.

Multi-objective optimization of dive trajectory for morphing unmanned aerial-underwater vehicle

A well-planned dive trajectory is the key for the trans-media maneuver and flight control of morphing unmanned aerial-underwater vehicles (MUAUVs). The main focus of this paper is the application of a watch-based multi-objective grey wolf optimizer (MOGWO) to dive trajectory optimization problems. A multi-objective trajectory optimization model considering the change of sweep angle and static moment is developed and parameterized by the Gauss pseudospectral method, and a multi-objective nonlinear programming problem is constructed. To overcome the defect that, in the MOGWO, the rest wolves blindly follow the best wolves, the watch strategy is introduced, which provides the wolves with the ability of independent exploration. Subsequently, the watch-based MOGWO is employed to generate the Pareto front, which is compared with that obtained through other multi-objective techniques. The simulation results demonstrated that the proposed method is more reliable and can obtain more widely distributed non-dominated solutions, indicating that the watch-based MOGWO is effective and feasible in dealing with multi-objective trajectory optimization problems with complex constraints. In addition, comparative studies on optimal trajectories demonstrated that the maneuverability and gliding ability of the MUAUV are improved through cooperation between the angle-of-attack, bank angle, and sweep angle.

Using Multi-Objective Bat Algorithm for Solving Multi-Objective Non-linear Programming Problem

Human beings are greatly inspired by nature. Nature has the ability to solve very complex problems in its own distinctive way. The problems around us are becoming more and more complex in the real time and at the same instance our mother nature is guiding us to solve these natural problems. Nature gives some of the logical and effective ways to find solutions to these problems. Nature acts as an optimized source for solving the complex problems. Decomposition is a basic strategy in traditional multi-objective optimization. However, it has not yet been widely used in multi-objective evolutionary optimization. 
 Although computational strategies for taking care of Multi-objective Optimization Problems (MOPs) have been accessible for a long time, the ongoing utilization of Evolutionary Algorithm (EAs) to such issues gives a vehicle to tackle extremely enormous scope MOPs.
 MOBATD is a multi-objective bat algorithm that incorporates the dominance concept with the decomposition approach. Whilst decomposition simplifies the MOP by rewriting it as a set of Tchebycheff Approach, solving these problems simultaneously, within the BAT framework, might lead to premature convergence because of the leader selection process which uses the Tchebycheff Approach as a criterion. Dominance plays a major role in building the leaders archive, allowing the selected leaders to cover less dense regions while avoiding local optima and resulting in a more diverse approximated Pareto front. The results from 5 standard MOPs show that the MOBATD outperforms some developmental methods based on decomposition. All the results were achieved by MATLAB (R2017b).

Approaches for Solving Fully Fuzzy Rough Multi-Objective Nonlinear Programming Problems

Practical nonlinear programming problem often encounters uncertainty and indecision due to various factors that cannot be controlled. To overcome these limitations, fully fuzzy rough approaches are applied to such a problem. In this paper, an effective two approaches are proposed to solve fully fuzzy rough multi-objective nonlinear programming problems (FFRMONLP) where all the variables and parameters are fuzzy rough triangular numbers. The first, based on a slice sum technique, a fully fuzzy rough multi-objective nonlinear problem has turned into five equivalent multi-objective nonlinear programming (FFMONLP) problems. The second proposed method for solving FFRMONLP problems is α-cut approach, where the triangular fuzzy rough variables and parameters of the FFRMONLP problem are converted into rough interval variables and parameters by α-level cut, moreover the rough MONLP problem turns into four MONLP problems. Furthermore, the weighted sum method is used in both proposed approaches to convert multi-objective nonlinear problems into an equivalent nonlinear programming problem. Finally, the effectiveness of the proposed procedure is demonstrated by numerical examples.

Intuitionistic Fuzzy Sets and Dynamic Programming for Multi-objective Non-linear Programming Problems

The practical applications of engineering designs are often concerned by the presence of interrelated decisions. Naturally, these decisions are induced by solving many conflicting and incommensurable objectives. To deal with the interlinked decisions as well as the paradox natures among objectives, this paper presents an integrated approach based on dynamic programming approach (DPA) and intuitionistic fuzzy set (IFS) denoted as DPA-IFS for solving multi-objective optimization problem (MOOP). In DPA-IFS, the principle of DPA aims to generate an efficient solution of MOOP. In contrast, IFS aims to handle conflicting natures among the objective functions by means of the satisfaction (maximization the degree of membership) and dissatisfaction (minimization the degree of non-membership) concepts. The illustration is investigated by numerical illustrations taken from the literature. Furthermore, a new closeness strategy-based distance function is introduced to measure the worth of a satisfactory solution. The proposed methodology is validated on the IEEE 30-bus with six generators as a real test paradigm in the electrical power system. The results obtained by the DPA-IFS show the superior results than those obtained by different approaches.

Solution of some multi-objective nonlinear programming problems*

Solution of some multi-objective nonlinear programming problems*

On optimistic, pessimistic and mixed approaches under different membership functions for fully intuitionistic fuzzy multiobjective nonlinear programming problems

The concept of measuring the membership degree along with a non-membership degree, gives rise to the intuitionistic fuzzy set theory. The present study formulates a nonlinear programming problem with multiple objectives, including all the parameters and decision variables as intuitionistic fuzzy numbers. The problem is further investigated subject to optimistic, pessimistic and mixed approaches under linear, exponential and hyperbolic membership functions. The article redefines pessimistic and mixed point of view to be in the true spirit compatible with the definition of an intuitionistic fuzzy number. Accuracy function is used to reduce the problem to an equivalent crisp multiobjective nonlinear programming problem and then optimal compromise solution is obtained under different approaches using various membership/non-membership functions. At appropriate places, theorems have also been proved to establish the equivalence between the original formulation and its crisp counterparts under each approach. Further, practical applications in production planning and transportation problem are illustrated to explain the optimistic, pessimistic and mixed approaches using the proposed algorithm and finally a comparison is also drawn.

Regional Integrated Energy Site Layout Optimization Based on Improved Artificial Immune Algorithm

Regional integrated energy site layout optimization involves multi-energy coupling, multi-data processing and multi-objective decision making, among other things. It is essentially a kind of non-convex multi-objective nonlinear programming problem, which is very difficult to solve by traditional methods. This paper proposes a decentralized optimization and comprehensive decision-making planning strategy and preprocesses the data information, so as to reduce the difficulty of solving the problem and improve operational efficiency. Three objective functions, namely the number of energy stations to be built, the coverage rate and the transmission load capacity of pipeline network, are constructed, normalized by linear weighting method, and solved by the improved p-median model to obtain the optimal value of comprehensive benefits. The artificial immune algorithm was improved from the three aspects of the initial population screening mechanism, population updating and bidirectional crossover-mutation, and its performance was preliminarily verified by test function. Finally, an improved artificial immune algorithm is used to solve and optimize the regional integrated energy site layout model. The results show that the strategies, models and methods presented in this paper are feasible and can meet the interest needs and planning objectives of different decision-makers.

Solving matrix games with hesitant fuzzy pay-offs

The objective of this paper is to develop matrix games with pay-offs of triangular hesitant fuzzy elements (THFEs). To solve such games, a new methodology has been derived based on the notion of weighted average operator and score function of THFEs. Firstly, we formulate two non-linear programming problems with THFEs. Then applying the score function of THFEs, we transform these two problems into two non-linear multi-objective programming problems with triangular fuzzy numbers (TFNs). Finally, the Lexicographic method is used to solve these two multi-objective programming problems. A market share problem is considered to show the validity and applicability of the proposed methodology.

Optimization for multi-objective sum of linear and linear fractional programming problem: fuzzy nonlinear programming approach

Multi-objective linear plus linear fractional programming problem is an emerging tool for solving problems in different environments such as production planning, financial and corporate planning and healthcare and hospital planning which has attracted many researchers in recent years. This paper presents a method to find a Pareto optimal solution for the multi-objective linear plus linear fractional programming problem through nonlinear membership function. The proposed approach defines a fuzzy goal for each objective through a nonlinear membership function. By means of nonlinear membership function, the multi-objective linear plus linear fractional programming problem transformed into a multi-objective nonlinear programming problem. Applying the linear approximation method, the nonlinear objectives are converted into linear. In order to solve the multi-objective linear programming problem, the fuzzy goal programming model is formulated by minimizing the negative deviational variables. The proposed procedure is illustrated through numerical examples and a real-life application problem. Further, it is compared with the existing methods. Finally, the Euclidean distance function has been used to prove the efficiency of the proposed method.