Fuzzy Multi-objective Optimization Problem Research Articles

An interactive satisficing approach for solving fuzzy multiobjective Linear optimization problems based on the Attainable Reference Point Method

An interactive satisficing approach for solving fuzzy multiobjective Linear optimization problems based on the Attainable Reference Point Method

An imprecise fuzzy risk approach for water quality management of a river system

Uncertainty plays an important role in water quality management problems. The major sources of uncertainty in a water quality management problem are the random nature of hydrologic variables and imprecision (fuzziness) associated with goals of the dischargers and pollution control agencies (PCA). Many Waste Load Allocation (WLA) problems are solved by considering these two sources of uncertainty. Apart from randomness and fuzziness, missing data in the time series of a hydrologic variable may result in additional uncertainty due to partial ignorance. These uncertainties render the input parameters as imprecise parameters in water quality decision making. In this paper an Imprecise Fuzzy Waste Load Allocation Model (IFWLAM) is developed for water quality management of a river system subject to uncertainty arising from partial ignorance. In a WLA problem, both randomness and imprecision can be addressed simultaneously by fuzzy risk of low water quality. A methodology is developed for the computation of imprecise fuzzy risk of low water quality, when the parameters are characterized by uncertainty due to partial ignorance. A Monte-Carlo simulation is performed to evaluate the imprecise fuzzy risk of low water quality by considering the input variables as imprecise. Fuzzy multiobjective optimization is used to formulate the multiobjective model. The model developed is based on a fuzzy multiobjective optimization problem with max–min as the operator. This usually does not result in a unique solution but gives multiple solutions. Two optimization models are developed to capture all the decision alternatives or multiple solutions. The objective of the two optimization models is to obtain a range of fractional removal levels for the dischargers, such that the resultant fuzzy risk will be within acceptable limits. Specification of a range for fractional removal levels enhances flexibility in decision making. The methodology is demonstrated with a case study of the Tunga–Bhadra river system in India.

A novel algorithm for solving optimal path planning problems based on parametrization method and fuzzy aggregation

In this Letter a new approach for solving optimal path planning problems for a single rigid and free moving object in a two and three dimensional space in the presence of stationary or moving obstacles is presented. In this approach the path planning problems have some incompatible objectives such as the length of path that must be minimized, the distance between the path and obstacles that must be maximized and etc., then a multi-objective dynamic optimization problem (MODOP) is achieved. Considering the imprecise nature of decision maker's (DM) judgment, these multiple objectives are viewed as fuzzy variables. By determining intervals for the values of these fuzzy variables, flexible monotonic decreasing or increasing membership functions are determined as the degrees of satisfaction of these fuzzy variables on their intervals. Then, the optimal path planning policy is searched by maximizing the aggregated fuzzy decision values, resulting in a fuzzy multi-objective dynamic optimization problem (FMODOP). Using a suitable t-norm, the FMODOP is converted into a non-linear dynamic optimization problem (NLDOP). By using parametrization method and some calculations, the NLDOP is converted into the sequence of conventional non-linear programming problems (NLPP). It is proved that the solution of this sequence of the NLPPs tends to a Pareto optimal solution which, among other Pareto optimal solutions, has the best satisfaction of DM for the MODOP. Finally, the above procedure as a novel algorithm integrating parametrization method and fuzzy aggregation to solve the MODOP is proposed. Efficiency of our approach is confirmed by some numerical examples.

Interactive fuzzy multiobjective reliability optimization using NSGA-II

In many practical situations where reliability enhancement is involved, the decision making is complicated because of the presence of several mutually conflicting objectives. Presence of multiple objectives in a problem, in principle, gives rise to a set of optimal solutions (largely known as Pareto-optimal solutions), instead of a single optimal solution. This type of problem is known as multiobjective optimization problem (MOOP). In general, a MOOP can be solved using weighted sums or decision-making schemes. An alternative way is to look for the Pareto-optimal front. Many evolutionary algorithms (EAs) like genetic algorithm (GA) have been suggested to solve MOOP, hence termed as multiobjective evolutionary algorithms (MOEAs). Nondominated sorting genetic algorithm (NSGA-II) is one such MOEA which demonstrates the ability to identify a Pareto-optimal front efficiently. Thus, it provides the decision maker (DM) a complete picture of the optimal solution space. This paper presents the reliability optimization of a life-support system in a space capsule where reliability of the system is maximized while minimizing the cost. An interactive fuzzy satisficing method for deriving a Pareto-optimal solution preferred by the DM is presented here. Prior preference of the DM has been taken into account here. Using the concept of fuzzy sets and convex fuzzy decision making a multiobjective fuzzy optimization problem is formulated from the original crisp optimization problem. Different nonlinear membership functions based on the DM’s preference have been employed for the fuzzification. Then, NSGA-II is applied to solve the resulting fuzzified MOOP. Resulting Pareto-optimal solution gives the DM variety of alternatives to seek an appropriate solution by modifying parameters interactively according to his/her preference again. Various Pareto-optimal fronts under different preferences of DM have been reported.

Design Optimization of Electric Motors by Multiobjective Fuzzy Genetic Algorithms

This paper presents a multiobjective fuzzy genetic algorithm optimization approach to design the submersible induction motor with two objective functions: the full load torque and the manufacturing cost. A multiobjective fuzzy optimization problem is formulated and solved using a genetic algorithm. The optimally designed motor is compared with an industrial motor having the same ratings. The results of optimal design show the reduction in the manufacturing cost, and the improvement in the full load torque of the motor.

Hybrid organization of functional departments and manufacturing cells in the presence of imprecise data

Although the benefits of forming cells are widely recognized, certain industrial decision makers do not wish an entire transformation because of possible changes that may occur, for example, in the demand, mix or routing of products. Therefore, hybrid organization of functional departments and manufacturing cells in the same shop may appear quite relevant. The problem addressed here is how to allocate machines and products so as to design such systems that take advantages of both types of layout. We suggest an approach that favours cells for the parts that are regularly manufactured over time (i.e. stable parts), and functional departments for the parts that require more flexibility. We also consider the difficulties arising in the collection of precise data in reality. Hybrid shop design problem is formulated as a constrained fuzzy multi-objective optimization problem and an evolutionary programming algorithm is devised to solve it. The approach is illustrated in test examples.

Fuzzy multi-stage De-Novo programming problem

The De-Novo programming problem proposed by Zeleny is well-known for its value on designing an optimal system by extending existed resources instead of finding the optimum in a given system with fixed resources. Since few papers are dedicated to explore the De-Novo programming problem with multiple stages and its resolution approach, the De-Novo programming problem is innovatively extended to a fuzzy dynamic programming problem in this paper, which is solved by the evolving algorithm so as to reflect greater realities. First, a traditional De-Novo programming problem is modified to a De-Novo programming problem with multiple fuzzy goals, fuzzy constraints and multiple stages. Second, we regard this fuzzy multi-stage De-Novo programming problem as a fuzzy dynamic programming problem, which is identical to a fuzzy multi-objective combinatorial optimization problem. Finally, we actually validate the feasibility of genetic algorithm on such a problem.

Mixed-Discrete Fuzzy Multiobjective Programming for Engineering Optimization Using Hybrid Genetic Algorithm

Although much attention has been focused on the development and applications of fuzzy optimization, multiobjective programming, and mixed-discrete optimization methods separately, fuzzy multiobjective optimization problems in mixed-discrete design space have not been addressed in the literature. It is mainly because of the lack of mature and robust theories of mixed-discrete and multiobjective optimization. In most practical applications, designers often encounter problems involving imprecise or fuzzy information, multiple objectives, and mixed-discrete design variables. A new method is presented in which the fuzzy λ formulation and game theory techniques are combined with a mixed-discrete hybrid genetic algorithm for solving mixed-dixcrete fuzzy multiobjective programming problems. Three example problems, dealing with the optimal designs of a two-bar truss, a conical convective spine, and a 25-bar truss, demonstrate that the method can be flexibly and effectively applied to various kinds of engineering design problems to obtain more realistic and satisfactory results in an imprecise environment. Nomenclature D = fuzzy feasible solution domain d =f avorable search direction Fiti(X) = fitness function fi(X) = ith objective function f max i = upper bound of ith objective function f min i =l ower bound of ith objective function G j = allowable interval of the constraint function g j g j(X) = jth inequality constraint function k = number of objective functions M = size of the population m = number of inequality constraints n = total number of design variables nd = number of discrete design variables nq = number of discrete design variables with equal spacing X = design vector X c = centroid of the composite polygon Xw =v ertex with the lowest fitness value xi = ith design variable µD(X) = membership function of the design vector µfi(X) = membership function of the ith objective µgj(X) = membership function of the jth constraint

Fuzzy regression approach to process modelling and optimization of epoxy dispensing

Epoxy dispensing is one of the popular processes to perform microchip encapsulation for chip-on-board (COB) packages. However, determination of proper process parameters setting for optimal quality of the encapsulation is difficult due to the complex behaviour of the encapsulant during dispensing and the uncertainties caused by fuzziness of epoxy dispensing systems. In conventional regression models, deviations between the observed values and the estimated values are supposed to be in probability distribution. However, when data is irregular, the obtained regression model has an unnaturally wide possibility range. In fact, these deviations in some processes such as epoxy dispensing can be regarded as system fuzziness that can be dealt with properly using fuzzy regression method. In this paper, a fuzzy regression approach with fuzzy intervals to process modelling of epoxy dispensing for microchip encapsulation is described. Two fuzzy regression models relating three process parameters and two quality characteristics respectively for epoxy dispensing were developed. They were then introduced to formulate a fuzzy multi-objective optimization problem. A fuzzy linear programming technique was employed to formulate the optimization model. By solving the model, an optimal setting of process parameters can be obtained. Validation experiments were conducted to evaluate the effectiveness of the proposed approach to process modelling and optimization of epoxy dispensing for microchip encapsulation.

Fuzzy Optimization for an Integrated Process of Extractive Fermentation Including Cell Recycle for Lactic Acid Production

Abstract In this study, we consider a bioprocess optimization problem for a multistage extractive fermentation with cell recycling to produce lactic acid. A generalized mathematical model is first introduced. The aim of the optimization problem is to simultaneously obtain the maximum productivity and conversion so that the optimization problem is formulated as a fuzzy multiobjective optimization problem. The fuzzy decision is introduced to convert the multiobjective fuzzy optimization problem into the fuzzy goal attainment problem. Hybrid differential evolution is applied to solve the fuzzy goal attainment problem to obtain a Pareto optimal solution.

Performance analysis and fuzzy optimization of a two-stage fermentation process with cell recycling including an extractor for lactic acid production

In this study, we consider a two-stage fermenter with cell recycling including an extractor to produce lactic acid. When the extractor was omitted, the proposed process was simplified to two special processes. Under the same operating conditions, we compared the overall lactic acid productivity and glucose conversion for the three processes. The proposed fermentation process was more efficient than the other processes. To simultaneously obtain the maximum productivity and conversion, the problem is formulated as a fuzzy multiobjective optimization problem. The fuzzy decision is introduced to convert the multiobjective fuzzy optimization problem into the fuzzy goal attainment problem. Hybrid differential evolution is applied to solve the fuzzy goal attainment problem to obtain a global Pareto solution.

Fuzzy global optimization of complex system reliability

The problem of optimizing the reliability of complex systems has been modeled as a fuzzy multi-objective optimization problem. Three different kinds of optimization problems: reliability optimization of a complex system with constraints on cost and weight; optimal redundancy allocation in a multistage mixed system with constraints on cost and weight; and optimal reliability allocation in a multistage mixed system with constraints on cost, weight, and volume have been solved. Four numerical examples have been solved to demonstrate the effectiveness of the present methodology. The influence of various kinds of aggregators is also studied. The inefficiency of the noncompensatory min operator has been demonstrated. One of the well-known global optimization meta-heuristics, the threshold accepting, has been invoked to take care of the optimization part of the model. A software has been developed to implement the above model. The results obtained are encouraging.

Application of fuzzy probability in water quality management of a river system

Water quality management of a river system is addressed in a fuzzy and probabilistic framework. Two types of uncertainty, namely randomness and vagueness, are treated simultaneously in the management problem. A fuzzy-set-based definition that is a more general case of the existing crisp-set-based definition of low water quality is introduced. The event of low water quality at a check-point in the river system is considered as a fuzzy event. The risk of low water quality is then defined as the probability of the fuzzy event of low water quality. Instead of constraining the risk of violation of water quality standard by a finite value of risk through a chance constraint, a fuzzy set of low risk that considers a range of risk levels with appropriate membership values is introduced. Different goals associated with the management problem are expressed as fuzzy sets. The resulting management problem is formulated as a fuzzy multiobjective optimization problem.

Pareto Simulated Annealing for Fuzzy Multi-Objective Combinatorial Optimization

The paper presents a metaheuristic method for solving fuzzy multi-objective combinatorial optimization problems. It extends the Pareto simulated annealing (PSA) method proposed originally for the crisp multi-objective combinatorial (MOCO) problems and is called fuzzy Pareto simulated annealing (FPSA). The method does not transform the original fuzzy MOCO problem to an auxiliary deterministic problem but works in the original fuzzy objective space. Its goal is to find a set of approximately efficient solutions being a good approximation of the whole set of efficient solutions defined in the fuzzy objective space. The extension of PSA to FPSA requires the definition of the dominance in the fuzzy objective space, modification of rules for calculating probability of accepting a new solution and application of a defuzzification operator for updating the average position of a solution in the objective space. The use of the FPSA method is illustrated by its application to an agricultural multi-objective project scheduling problem.

Methods of pairwise comparisons and fuzzy global criterion for multiobjective optimization in structural engineering

The method of pairwise comparison inherently contains information of ambiguity, fuzziness and conflict in design goals for a multiobjective structural design, This paper applies the principle of paired comparison so that the vaguely formulated problem can be modified and a set of numerically acceptable weight would reflect the relatively important degree of multiple objectives. This paper also presents a fuzzy global criterion method (FGCM(lambda)) included fuzzy constraints that coupled with the objective weighting rank obtained from the modified pairwise comparisons for fuzzy multiobjective optimization problems. Descriptions in sequence of this combined method and problem solving experiences are given in the current article. Multiobjective design examples of truss and mechanical spring structures illustrate this optimization process containing the revising judgement techniques.

Generalized hybrid method for fuzzy multiobjective optimization of engineering systems

A generalized hybrid approach is presented for the multiobjective optimization of engineering systems in the presence of objectives and constraints that are partly fuzzy and partly crisp. The methodology is based on both fuzzy-set and Dempster-Shafer theories to capture the features of incomplete, imprecise, uncertain, or vague information that is often present in real-world engineering systems. The original partly fuzzy multiobjective optimization problem is first defuzzified into a crisp generalized multiobjective optimization problem using fuzzy-set theory. The resulting multiobjective problem is then transformed into an equivalent single-objective optimization problem using a modified Dempster-Shafer theory. The computational details of the approach are illustrated with a structural design example.

On Multi-Objective Fuzzy Optimization of Ship Structures

In order to solve the multi-objective fuzzy optimization problems of ship structures, two approaches, known as the maximum level method and the linear weighted sum method, based on the different forms of fuzzy decision-making are presented. In these methods, the membership functions are constructed for the subsets of the fuzzy goals subject to the fuzzy constraints, and then the multi-objective fuzzy optimization problem is transformed into the single-objective ordinary optimization problem, which can be solved by using nonlinear programming techniques. The proposed approaches are illustrated by two numerical examples in the application of ship structural design.

Multi-objective fuzzy optimization of structures based on generalized fuzzy decision-making

In order to find a satisfactory solution for the multi-objective fuzzy optimization problem of structures, the membership functions are constructed for the subsets of the fuzzy objectives subject to the fuzzy constraints, and three approaches are presented by using the different types of generalized fuzzy decision-making. Two illustrative numerical examples given show that in creating the user-oriented design environment in computer-aided design systems the application of generalized fuzzy decision-making to multi-objective fuzzy optimization of structures looks promising.

Optimal apportionment of reliability and redundancy in series systems under multiple objectives

A multiobjective reliability apportionment problem for a series system with time-dependent reliability is presented. The resulting mathematical programming formulation determines the optimal level of component reliability and the number of redundant components at each stage. The problem is a multiobjective, nonlinear, mixed-integer mathematical programming problem, subject to several design constraints. Sequential unconstrained minimization techniques in conjunction with heuristic algorithms are used to find an optimum solution. A generalization of the problem in view of inherent vagueness in the objective and the constraint functions results in an ill-structured reliability apportionment problem. This multiobjective fuzzy optimization problem is solved using nonlinear programming. The computational procedure is illustrated through a numerical example. The fuzzy optimization techniques can be useful during initial stages of the conceptual design of engineering systems where the design goals and design constraints have not been clearly identified or stated, and for decision making problems in ill-structured situations.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Bus Timetabling as a Fuzzy Multiobjective Optimization Problem Using Preference-based Genetic Algorithm

Transportation plays a vital role in the development of a country and the car is the most commonly used means. However, in third world countries long waiting time for public buses is a common problem, especially when people need to switch buses. The problem becomes critical when one considers buses joining different villages and cities. Theoretically this problem can be solved by assigning more buses on the route, which is not possible due to economical problem. Another option is to schedule the buses so that customers who want to switch buses at junction cities need not have to wait long. This paper discusses how to model single frequency routes bus timetabling as a fuzzy multiobjective optimization problem and how to solve it using preference-based genetic algorithm by assigning appropriate fuzzy preference to the need of the customers. The idea will be elaborated with an example.