An extension of the directed search domain algorithm to bilevel optimization

A method is developed for generating a well-distributed Pareto set for the upper level in bilevel multiobjective optimization. The approach is based on the Directed Search Domain (DSD) algorithm, which is a classical approach for generation of a quasi-evenly distributed Pareto set in multiobjective optimization. The approach contains a double-layer optimizer designed in a specific way under the framework of the DSD method. The double-layer optimizer is based on bilevel single-objective optimization and aims to find a unique optimal Pareto solution rather than generate the whole Pareto frontier on the lower level in order to improve the optimization efficiency. The proposed bilevel DSD approach is verified on several test cases, and a relevant comparison against another classical approach is made. It is shown that the approach can generate a quasi-evenly distributed Pareto set for the upper level with relatively low time consumption.     

Multi-objective genetic algorithm for solving N-version program design problem

N-version programming (NVP) is a programming approach for constructing fault tolerant software systems. Generally, an optimization model utilized in NVP selects the optimal set of versions for each module to maximize the system reliability and to constrain the total cost to remain within a given budget. In such a model, while the number of versions included in the obtained solution is generally reduced, the budget restriction may be so rigid that it may fail to find the optimal solution. In order to ameliorate this problem, this paper proposes a novel bi-objective optimization model that maximizes the system reliability and minimizes the system total cost for designing N-version software systems. When solving multi-objective optimization problem, it is crucial to find Pareto solutions. It is, however, not easy to obtain them. In this paper, we propose a novel bi-objective optimization model that obtains many Pareto solutions efficiently.We formulate the optimal design problem of NVP as a bi-objective 0–1 nonlinear integer programming problem. In order to overcome this problem, we propose a Multi-objective genetic algorithm (MOGA), which is a powerful, though time-consuming, method to solve multi-objective optimization problems. When implementing genetic algorithm (GA), the use of an appropriate genetic representation scheme is one of the most important issues to obtain good performance. We employ random-key representation in our MOGA to find many Pareto solutions spaced as evenly as possible along the Pareto frontier. To pursue improve further performance, we introduce elitism, the Pareto-insertion and the Pareto-deletion operations based on distance between Pareto solutions in the selection process.The proposed MOGA obtains many Pareto solutions along the Pareto frontier evenly. The user of the MOGA can select the best compromise solution among the candidates by controlling the balance between the system reliability and the total cost.     

Multiresponse Optimization and Pareto Frontiers

Methods that can capture evenly distributed solutions along the Pareto frontier are useful for multiresponse optimization problems because they provide a large variety of alternative solutions to the decision maker from among a set of nondominated solutions. However, methods often used for optimizing dual and multiple dual response problems have been rarely evaluated in terms of their ability to capture those solutions. This article provides this information by evaluating a global criterion–based method and the popular weighted mean square error method. Convex and nonconvex response surfaces were considered, and results of the methods were compared with those of a lexicographic approach on the basis of two examples from the literature. Regarding the results, it is shown that the user can be successful in capturing Pareto solutions in convex and nonconvex regions using the global criterion–based method. Moreover, it is shown that the starting point affects the distribution of solutions along the Pareto frontier but is not pivotal to obtain a complete representation of the Pareto frontier. For this purpose, it is necessary to decrease the weight increment and to compute for more solutions. Copyright © 2011 John Wiley & Sons, Ltd.     

A MULTIOBJECTIVE INTERACTIVE SEQUENTIAL HYBRID OPTIMIZATION TECHNIQUE FOR DESIGN DECISION MAKING

The paper proposes an interactive method for obtaining a solution to a multiple objective design decision making problem. The focus is on generating Pareto solutions including those that are in the non-convex region, and are desirable to obtain in an engineering design context. After the generation of a small subset of the Pareto solutions, the designer's feedback is elicited in order to eliminate part of the subset. The process is repeated until il iteratively narrows down the Pareto solution set to a size small enough so that the designer is able to easily select a final solution. The advantage of this approach is that the designer can view a few sample points from the Pareto set before zooming into the region preferred and without expending computation time in generating a complete Pareto set. The process has been demonstrated with the help of an example, the design of a fleet of ships, that has mixed-discrete variables and hence a genetic algorithm is used as the optimizer.     

Multiobjective optimization with an adaptive weight determination scheme using the concept of hyperplane

Finding an optimum design that satisfies all performances in a design problem is very challenging. To overcome this problem, multiobjective optimization methods have been researched to obtain Pareto optimum solutions. Among the different methods, the weighted sum method is widely used for its convenience. However, since the different weights do not always guarantee evenly distributed solutions on the Pareto front, the weights need to be determined systematically. Therefore, this paper presents a multiobjective optimization using a new adaptive weight determination scheme. Solutions on the Pareto front are gradually found with different weights, and the values of these weights are adaptively determined by using information from the previously obtained solutions' positions. For an n-objective problem, a hyperplane is constructed in n -dimensional space, and new weights are calculated to find the next solutions. To confirm the effectiveness of the proposed method, benchmarking problems that have different types of Pareto front are tested, and a topology optimization problem is performed as an engineering problem. A hypervolume indicator is used to quantitatively evaluate the proposed method, and it is confirmed that optimized solutions that are evenly distributed on the Pareto front can be obtained by using the proposed method.     

Evolutionary engineering design synthesis of on-board traffic monitoring sensors

In this paper, a formal engineering design synthesis methodology based on evolutionary computation is presented, with special emphasis on the design and optimization of distributed independent systems. A case study concerned with design of a sensory system for traffic monitoring purposes is presented, along with simulations of traffic scenarios at several levels of abstraction. It is shown how the methodology introduced is able to deal with the engineering design challenges present in the case study, and effectively synthesize novel design solutions of good quality. Moreover, when the fitness function is formulated as an aggregation of design preferences with different weights and trade-off strategies, the complete Pareto optimal frontier can be determined by the evolutionary synthesis methodology. The results of this study suggest that the approach can be useful for designers to solve challenging engineering design synthesis problems.     

Pareto layer: Its formulation and search by way of evolutionary multi-objective optimization

This article examines multi-objective problems where a solution (product) is related to a cluster of performance vectors within a multi-objective space. Here the origin of such a cluster is not uncertainty, as is typical, but rather the range of performances attainable by the product. It is shown that, in such cases, comparison of a solution to other solutions should be based on its best performance vectors, which are extracted from the cluster. The result of solving the introduced problem is a set of Pareto optimal solutions and their representation in the objective space, which is referred to here as the Pareto layer. The authors claim that the introduced Pareto layer is a previously unattended novel representation. In order to search for these optimal solutions, an evolutionary multi-objective algorithm is suggested. The article also treats the selection of a solution from the obtained optimal set.     

Practical solutions for multi-objective optimization: An application to system reliability design problems

For multiple-objective optimization problems, a common solution methodology is to determine a Pareto optimal set. Unfortunately, these sets are often large and can become difficult to comprehend and consider. Two methods are presented as practical approaches to reduce the size of the Pareto optimal set for multiple-objective system reliability design problems. The first method is a pseudo-ranking scheme that helps the decision maker select solutions that reflect his/her objective function priorities. In the second approach, we used data mining clustering techniques to group the data by using the k-means algorithm to find clusters of similar solutions. This provides the decision maker with just k general solutions to choose from. With this second method, from the clustered Pareto optimal set, we attempted to find solutions which are likely to be more relevant to the decision maker. These are solutions where a small improvement in one objective would lead to a large deterioration in at least one other objective. To demonstrate how these methods work, the well-known redundancy allocation problem was solved as a multiple objective problem by using the NSGA genetic algorithm to initially find the Pareto optimal solutions, and then, the two proposed methods are applied to prune the Pareto set.     

Generating Well-Distributed Sets of Pareto Points for Engineering Design Using Physical Programming

Engineering design generally involves two, possibly integrated, phases: (i) generating design options, and (ii) choosing the most satisfactory option on the basis of some determined criteria. The depth, or lack, of integration between these two phases defines different design approaches, and differing philosophical views from the part of researchers in the field of computational design. Optimization-Based Design (OBD) covers the spectrum of this depth of integration. While most OBD approaches strongly integrate these two phases, some employ computational optimization only in the first or second phase. Regardless of where a method or researcher lies in this philosophical spectrum, some requisite characteristics are fundamental to the effectiveness of OBD methods. In particular, (i) the Aggregate Objective Function (AOF) used in the optimization must have the ability to generate all Pareto solutions, (ii) the generation of any existing Pareto solutions must be possible with reasonable ease, and (iii) even changes in the AOF parameters should yield a well distributed set of Pareto solutions. This paper examines the effectiveness of physical programming (PP) with respect to the latter, yielding favorable conclusions. Previous papers have led to similarly positive conclusions with respect to the former two. This paper also presents a comparative study featuring PP and other popular methods, where PP is shown to perform favorably. A PP-based method for generating the Pareto frontier is presented.     

Simulation and Optimization of Wiped-Film Poly-Ethylene Terephthalate (PET) Reactor Using Multiobjective Differential Evolution (MODE)

The multiobjective differential evolution (MODE), which is an extension of the Differential Evolution (DE), is applied to solve the multiobjective optimization problem (MOOP) of wet film Poly-Ethylene Terephthalate (PET) reactor considering minimization of acid end group and vinyl end group as the main objectives. The objective function is modified to solve five different possible cases. The results show that a Pareto set (a set of equally good solutions) is obtained for the cases when two of the parameters (residence time of the polymeric reaction mass, θ, and the speed of the wiped-film agitator, N) are considered as decision variables, unlike a unique solution obtained using Nondominated Sorting Genetic Algorithm (NSGA). The Pareto optimal front provides wide-ranging optimal sets of operating conditions. And an appropriate set of operating conditions can be selected based on the requirements of the user.     

Trade-off analysis approach for interactive nonlinear multiobjective optimization

When solving multiobjective optimization problems, there is typically a decision maker (DM) who is responsible for determining the most preferred Pareto optimal solution based on his preferences. To gain confidence that the decisions to be made are the right ones for the DM, it is important to understand the trade-offs related to different Pareto optimal solutions. We first propose a trade-off analysis approach that can be connected to various multiobjective optimization methods utilizing a certain type of scalarization to produce Pareto optimal solutions. With this approach, the DM can conveniently learn about local trade-offs between the conflicting objectives and judge whether they are acceptable. The approach is based on an idea where the DM is able to make small changes in the components of a selected Pareto optimal objective vector. The resulting vector is treated as a reference point which is then projected to the tangent hyperplane of the Pareto optimal set located at the Pareto optimal solution selected. The obtained approximate Pareto optimal solutions can be used to study trade-off information. The approach is especially useful when trade-off analysis must be carried out without increasing computation workload. We demonstrate the usage of the approach through an academic example problem.     

Local Pareto approximation for multi-objective optimization

The design process of complex systems often resorts to solving an optimization problem, which involves different disciplines and where all design criteria have to be optimized simultaneously. Mathematically, this problem can be reduced to a vector optimization problem. The solution of this problem is not unique and is represented by a Pareto surface in the objective function space. Once a Pareto solution is obtained, it may be very useful for the decision-maker to be able to perform a quick local approximation in the vicinity of this Pareto solution for sensitivity analysis. In this article, new linear and quadratic local approximations of the Pareto surface are derived and compared to existing formulas. The case of non-differentiable Pareto points (solutions) in the objective space is also analysed. The concept of a local quick Pareto analyser based on local sensitivity analysis is proposed. This Pareto analysis provides a quantitative insight into the relation between variations of the different objective functions under constraints. A few examples are considered to illustrate the concept and its advantages.     

Multiobjective optimization by genetic algorithms: application to safety systems

When attempting to optimize the design of engineered systems, the analyst is frequently faced with the demand of achieving several targets (e.g. low costs, high revenues, high reliability, low accident risks), some of which may very well be in conflict. At the same time, several requirements (e.g. maximum allowable weight, volume etc.) should also be satisfied. This kind of problem is usually tackled by focusing the optimization on a single objective which may be a weighed combination of some of the targets of the design problem and imposing some constraints to satisfy the other targets and requirements. This approach, however, introduces a strong arbitrariness in the definition of the weights and constraints levels and a criticizable homogenization of physically different targets, usually all translated in monetary terms.The purpose of this paper is to present an approach to optimization in which every target is considered as a separate objective to be optimized. For an efficient search through the solution space we use a multiobjective genetic algorithm which allows us to identify a set of Pareto optimal solutions providing the decision maker with the complete spectrum of optimal solutions with respect to the various targets. Based on this information, the decision maker can select the best compromise among these objectives, without a priori introducing arbitrary weights.     

Level Diagrams analysis of Pareto Front for multiobjective system redundancy allocation

Reliability-based and risk-informed design, operation, maintenance and regulation lead to multiobjective (multicriteria) optimization problems. In this context, the Pareto Front and Set found in a multiobjective optimality search provide a family of solutions among which the decision maker has to look for the best choice according to his or her preferences. Efficient visualization techniques for Pareto Front and Set analyses are needed for helping decision makers in the selection task.In this paper, we consider the multiobjective optimization of system redundancy allocation and use the recently introduced Level Diagrams technique for graphically representing the resulting Pareto Front and Set. Each objective and decision variable is represented on separate diagrams where the points of the Pareto Front and Set are positioned according to their proximity to ideally optimal points, as measured by a metric of normalized objective values. All diagrams are synchronized across all objectives and decision variables. On the basis of the analysis of the Level Diagrams, we introduce a procedure for reducing the number of solutions in the Pareto Front; from the reduced set of solutions, the decision maker can more easily identify his or her preferred solution.     

Concurrent optimal allocation of distributed manufacturing resources using extended Teaching-Learning-Based Optimization

The optimal allocation of distributed manufacturing resources is a challenging task for supply chain deployment in the current competitive and dynamic manufacturing environments, and is characterised by multiple objectives including time, cost, quality and risk that require simultaneous considerations. This paper presents an improved variant of the Teaching-Learning-Based Optimisation (TLBO) algorithm to concurrently evaluate, select and sequence the candidate distributed manufacturing resources allocated to subtasks comprising the supply chain, while dealing with the trade-offs among multiple objectives. Several algorithm-specific improvements are suggested to extend the standard form of TLBO algorithm, which is only well suited for the one-dimensional continuous numerical optimisation problem well, to solve the two-dimensional (i.e. both resource selection and resource sequencing) discrete combinatorial optimisation problem for concurrent allocation of distributed manufacturing resources through a focused trade-off within the constrained set of Pareto optimal solutions. The experimental simulation results showed that the proposed approach can obtain a better manufacturing resource allocation plan than the current standard meta-heuristic algorithms such as Genetic Algorithm, Particle Swarm Optimisation and Harmony Search. Moreover, a near optimal resource allocation plan can be obtained with linear algorithmic complexity as the problem scale increases greatly.     

Formulation and application of weight-function-based physical programming

Physical programming is effective in multi-objective optimization since it assists the designer to find the most preferred solution. Preference-function-based physical programming (PFPP) abandons the weighted-sum approach and its performance in generating Pareto solutions is susceptible to the transformation of pseudo-preferences. With the aim of integrating a weighted-sum approach into physical programming and generating well-distributed Pareto solutions, a weight-function-based physical programming (WFPP) method has been proposed. The approach forms a weight function for each normalized criterion and uses the variable weighted sum of all criteria as the aggregate objective function. Implementation for numerical and engineering design problems indicates that WFPP works as well as PFPP. The design process of generating Pareto solutions by WFPP is further presented, where the pseudo-preferences are allowed to transform in different ranges. Examples and results demonstrate that solutions generated by WFPP have better diversity performance than those of PFPP, especially when the pseudo-preferences are far from the true Pareto front.     

Determination of Pareto frontier in multi-objective maintenance optimization

The objective of a maintenance policy generally is the global maintenance cost minimization that involves not only the direct costs for both the maintenance actions and the spare parts, but also those ones due to the system stop for preventive maintenance and the downtime for failure. For some operating systems, the failure event can be dangerous so that they are asked to operate assuring a very high reliability level between two consecutive fixed stops. The present paper attempts to individuate the set of elements on which performing maintenance actions so that the system can assure the required reliability level until the next fixed stop for maintenance, minimizing both the global maintenance cost and the total maintenance time. In order to solve the previous constrained multi-objective optimization problem, an effective approach is proposed to obtain the best solutions (that is the Pareto optimal frontier) among which the decision maker will choose the more suitable one. As well known, describing the whole Pareto optimal frontier generally is a troublesome task. The paper proposes an algorithm able to rapidly overcome this problem and its effectiveness is shown by an application to a case study regarding a complex series-parallel system.     

Microgenetic multiobjective reconfiguration algorithm considering power losses and reliability indices for medium voltage distribution network

This study proposes and applies an evolutionary-based approach for multiobjective reconfiguration in electrical power distribution networks. In this model, two types of indicators of power quality are minimised: (i) power system's losses and (ii) reliability indices. Four types of reliability indices are considered. A microgenetic algorithm ('GA) is used to handle the reconfiguration problem as a multiobjective optimisation problem with competing and non-commensurable objectives. In this context, experiments have been conducted on two standard test systems and a real network. Such problems characterise typical distribution systems taking into consideration several factors associated with the practical operation of medium voltage electrical power networks. The results show the ability of the proposed approach to generate well-distributed Pareto optimal solutions to the multiobjective reconfiguration problem. In the systems adopted for assessment purposes, our proposed approach was able to find the entire Pareto front. Furthermore, better performance indexes were found in comparison to the Pareto envelope-based selection algorithm 2 (PESA 2) technique, which is another well-known multiobjective evolutionary algorithm available in the specialised literature. From a practical point of view, the results established, in general, that a compact trade-off region exists between the power losses and the reliability indices. This means that the proposed approach can recommend to the decision maker a small set of possible solutions in order to select from them the most suitable radial topology.