An interactive decision framework for multiple objective production planning

The formation of realistic implementable medium-range production plans requires explicit recognition of the multiple conflicting objectives of production planning. However, suggested applications of multiobjective optimization to production planning have been limited to goal programming procedures which fail to capitalize on the intrinsic flexibility of a multiobjective model. Alternatively, interactive multiobjective solution techniques could be used to allow planners to enhance decision making without excessive computational effort. This study describes an interactive multiple objective decision framework and evaluates its effectiveness via a multiobjective capacitated lot sizing model based on a real manufacturing facility. The results suggest that this approach is an effective solution strategy and useful decision aid for complex production planning problems.     

A synchronous reference point-based interactive method for stochastic multiobjective programming

Stochastic multiobjective programming models are highly complex problems, due to the presence of random parameters, together with several conflicting criteria that have to be optimized simultaneously. Even the widely used concept of efficiency has to be redefined for these problems. The use of interactive procedures can somehow ease this complexity, allowing the decision maker to learn about the problem itself, and to look for his most preferred solution. Reference point schemes can be adapted to stochastic problem, by asking the decision maker to provide, not only desirable levels for the objectives, but also the desired probability to achieve these values. In this paper, we analyze the different kinds of achievement scalarizing functions that can be used in this environment, and we study the efficiency (in the stochastic sense) of the different solutions obtained. As a result, a synchronous interactive method is proposed for a class of stochastic multiobjective problems, where only the objective functions are random. Several solutions can be generated by this new method, making use of the same preferential information, using the different achievement scalarizing functions. The preferential information (levels and probabilities for the objectives) is incorporated into the achievement scalarizing functions in a novel way to generate the new solutions. The special case of linear normal problems is addressed separately. The performance of the algorithm is illustrated with a numerical example.     

Using Interactive Multiobjective Optimization in Continuous Casting of Steel

We discuss some pros and cons of using different types of multiobjective optimization methods for demanding real-life problems like continuous casting of steel. In particular, we compare evolutionary approaches that are used for approximating the set of Pareto-optimal solutions to interactive methods where a decision maker actively takes part and can direct the solution process to such Pareto-optimal solutions that are interesting to her/him. Among the latter type of methods, we describe an interactive classification-based multiobjective optimization method: NIMBUS. NIMBUS converts the original objective functions together with preference information coming from the decision maker into scalar-valued optimization problems. These problems can be solved using any appropriate underlying solvers, like evolutionary algorithms. We also introduce an implementation of NIMBUS, called IND-NIMBUS, for solving demanding multiobjective optimization problems defined with different modelling and simulation tools. We apply NIMBUS and IND-NIMBUS in an optimal control problem related to the secondary cooling process in the continuous casting of steel. As an underlying solver we use a real-coded genetic algorithm. The aim in our problem is to find a control resulting with steel of the best possible quality, that is, minimizing the defects in the final product. Since the constraints describing technological and metallurgical requirements are so conflicting that they form an empty feasible set, we formulate the problem as a multiobjective optimization problem where constraint violations are minimized.     

Pareto navigator for interactive nonlinear multiobjective optimization

We describe a new interactive learning-oriented method called Pareto navigator for nonlinear multiobjective optimization. In the method, first a polyhedral approximation of the Pareto optimal set is formed in the objective function space using a relatively small set of Pareto optimal solutions representing the Pareto optimal set. Then the decision maker can navigate around the polyhedral approximation and direct the search for promising regions where the most preferred solution could be located. In this way, the decision maker can learn about the interdependencies between the conflicting objectives and possibly adjust one’s preferences. Once an interesting region has been identified, the polyhedral approximation can be made more accurate in that region or the decision maker can ask for the closest counterpart in the actual Pareto optimal set. If desired, (s)he can continue with another interactive method from the solution obtained. Pareto navigator can be seen as a nonlinear extension of the linear Pareto race method. After the representative set of Pareto optimal solutions has been generated, Pareto navigator is computationally efficient because the computations are performed in the polyhedral approximation and for that reason function evaluations of the actual objective functions are not needed. Thus, the method is well suited especially for problems with computationally costly functions. Furthermore, thanks to the visualization technique used, the method is applicable also for problems with three or more objective functions, and in fact it is best suited for such problems. After introducing the method in more detail, we illustrate it and the underlying ideas with an example.     

Usage scenarios for design space exploration with a dynamic multiobjective optimization formulation

In a recent publication, we presented a new strategy for engineering design and optimization, which we termed formulation space exploration. The formulation space for an optimization problem is the union of all variable and design objective spaces identified by the designer as being valid and pragmatic problem formulations. By extending a computational search into this new space, the solution to any optimization problem is no longer predefined by the optimization problem formulation. This method allows a designer to both diverge the design space during conceptual design and converge onto a solution as more information about the design objectives and constraints becomes available. Additionally, we introduced a new way to formulate multiobjective optimization problems, allowing the designer to change and update design objectives, constraints, and variables in a simple, fluid manner that promotes exploration. In this paper, we investigate three usage scenarios where formulation space exploration can be utilized in the early stages of design when it is possible to make the greatest contributions to development projects. Specifically, we look at formulation space boundary exploration, Pareto frontier generation for multiple concepts in the formulation space, and a new way to perform targeted boundary expansion. The benefits of these methods are illustrated with the conceptual design of an impact driver.     

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.     

Reliability and redundancy apportionment using crisp and fuzzy multiobjective optimization approaches

The reliability of a multistage system with several components in each stage can be improved either by using more reliable components, or by adding redundant components in parallel in any stage. In many practical situations where reliability enhancement is involved, the decision making is complicated because of the presence of several mutually conflicting goals. For example, in the reliability based design of a system, the designer may be required to maximize the reliability and minimize the cost, weight or volume. This work considers the problem of reliability allocation for multistage systems with components having time-dependent reliability. Two multiobjective optimization techniques are presented, coupled with heuristic procedures, to solve the mixed integer nonlinear programming problems. A generalization of the problem in the presence of vague information results in an ill-structured reliability apportionment problem. The solution of such multiobjective problems is also presented in the present work using the techniques of fuzzy optimization.     

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.     

Design optimization using multiple dominance relations

A challenge in engineering design is to choose suitable objectives and constraints from many quantities of interest, while ensuring an optimization is both meaningful and computationally tractable. We propose an optimization formulation that can take account of more quantities of interest than existing formulations, without reducing the tractability of the problem. This formulation searches for designs that are optimal with respect to a binary relation within the set of designs that are optimal with respect to another binary relation. We then propose a method of finding such designs in a single optimization by defining an overall ranking function to use in optimizers, reducing the cost required to solve this formulation. In a design under uncertainty problem, our method obtains the most robust design that is not stochastically dominated faster than a multiobjective optimization. In a car suspension design problem, our method obtains superior designs according to a k-optimality condition than previously suggested multiobjective approaches to this problem. In an airfoil design problem, our method obtains designs closer to the true lift/drag Pareto front using the same computational budget as a multiobjective optimization.     

MULTIOBJECTIVE DESIGN OPTIMIZATION BASED ON SATISFACTION METRICS

A formal multiobjective optimization method based on satisfaction metrics is presented for designing an engineering system with mathematical rigour. Three satisfaction-based design models with different tradeoff strategies are developed to facilitate the incorporation of satisfaction metrics into the context of design formulations. These models are derived from different combinations of satisfaction-incorporated design objectives, enabling the conversion of the original multiple objectives appropriately to a single unified goal. This makes it easy to apply any available single-objective mathematical programming solver for the resulting problem solving. Not only does the method generate a Pareto-optimal solution, but also it allows for the generation of many design alternatives in a feasible design space. A computational procedure is also suggested to guide design implementations. For illustration, an example is worked out to show the computational details and the utility of the newly developed design models.     

Global formulation for interactive multiobjective optimization

Interactive methods are useful and realistic multiobjective optimization techniques and, thus, many such methods exist. However, they have two important drawbacks when using them in real applications. Firstly, the question of which method should be chosen is not trivial. Secondly, there are rather few practical implementations of the methods. We introduce a general formulation that can accommodate several interactive methods. This provides a comfortable implementation framework for a general interactive system. Besides, this implementation allows the decision maker to choose how to give preference information to the system, and enables changing it anytime during the solution process. This change-of-method option provides a very flexible framework for the decision maker.     

Pareto Frontier Based Concept Selection Under Uncertainty,with Visualization

In a recent publication, we presented a new multiobjective decision-making tool for use in conceptual engineering design. In the present paper, we provide important developments that support the next phase in the evolution of the tool. These developments, together with those of our previous work, provide a concept selection approach that capitalizes on the benefits of computational optimization. Specifically, the new approach uses the efficiency and effectiveness of optimization to rapidly compare numerous designs, and characterize the tradeoff properties within the multiobjective design space. As such, the new approach differs significantly from traditional (non-optimization based) concept selection approaches where, comparatively speaking, significant time is often spent evaluating only a few points in the design space. Under the new approach, design concepts are evaluated using a so-calleds-Pareto frontier; this frontier originates from the Pareto frontiers of various concepts, and is the Pareto frontier for thesetof design concepts. An important characteristic of the s-Pareto frontier is that it provides a foundation for analyzing tradeoffs between design objectives and the tradeoffs between design concepts. The new developments presented in this paper include; (i) the notion ofminimally representingthe s-Pareto frontier, (ii) the quantification of concept goodness using s-Pareto frontiers, (iii) the development of an interactive design space exploration approach that can be used to visualizen-dimensional s-Pareto frontiers, and (iv) s-Pareto frontier-based approaches for considering uncertainty in concept selection. Simple structural examples are presented that illustrate representative applications of the proposed method.     

Efficient Pareto Frontier Exploration using Surrogate Approximations

In this paper we present an efficient and effective method of using surrogate approximations to explore the design space and capture the Pareto frontier during multiobjective optimization. The method employs design of experiments and metamodeling techniques (e.g., response surfaces and kriging models) to sample the design space, construct global approximations from the sample data, and quickly explore the design space to obtain the Pareto frontier without specifying weights for the objectives or using any optimization. To demonstrate the method, two mathematical example problems are presented. The results indicate that the proposed method is effective at capturing convex and concave Pareto frontiers even when discontinuities are present. After validating the method on the two mathematical examples, a design application involving the multiobjective optimization of a piezoelectric bimorph grasper is presented. The method facilitates multiobjective optimization by enabling us to efficiently and effectively obtain the Pareto frontier and identify candidate designs for the given design requirements.     

Integrated optimization of machine product design and process design

In order to attain the true integration of computer-aided design and computer-aided manufacturing not only is a smooth flow of information required, but also decision making for both product design and process design must be synthesized. In this paper an integrated design process is proposed in which decisions concerning both product design and process design are simultaneously made. According to the proposed design procedures, an integrated optimization problem is formulated. This optimization is expressed as a multiobjective optimization problem which produces many Pareto optimum solution sets corresponding to combinations of materials used for parts. The algorithm for solving the problem is also presented. The proposed method is applied to designing a cylindrical co-ordinate robot, thereby demonstrating the effectiveness of conducting a simultaneous process through product design and process design.     

Computer-aided design by optimization in architecture

This paper shows that optimization concepts are particularly useful in design because of their direct assistance in decision making. In this they subsume evaluation or appraisal techniques. One approach based on dynamic programming is presented as being directly applicable in computer-aided architectural design. Multi-attribute objectives in design can be handled using optimization concepts. Finally, multi-objective design, including multi-attribute multi-objective design, can be handled via the use of Pareto optimality approaches. The result of such processes is a solution database which the designer searches. The solution database contains information about the design decisions themselves as well as the performance of each solution in its various objectives. The designer still assumes responsibility for selecting particular solutions since there is no unique solution produced. It is suggested that any problem which can be manipulated quantitatively can be solved using these concepts.     

An Overview of Optimization Formulations for Multiresponse Surface Problems

Many industrial applications involve more than one quality characteristic. For example, in robust parameter design, the quality characteristics include the process mean and process variance. Such applications lead to multiresponse surface problems in which it is necessary to determine optimal operating conditions according to some specified optimization criterion involving the quality characteristics. The purpose of this article is to address this problem from a multiobjective decision‐making framework. The foremost approaches in multiresponse optimization are categorized and integrated. Guidelines are presented to help select appropriate formulations. Moreover, the applicability and computational aspects of the methods in various decision‐making contexts are discussed. Numerical examples are also provided. Copyright © 2012 John Wiley & Sons, Ltd.     

Solving complex design problems through multiobjective optimisation taking into account judgements of users

In the design process of products or systems, a current trend consists in taking into account judgments of users. In this context, a multiobjective optimisation method taking into account judgments of a panel of subjects is proposed. It is aimed at identifying the best trade-offs between quantitative objectives and judgments of users. The method is divided in two steps: (1) judgment data acquisition and (2) integration of the judgment data into the multiobjective optimisation process. The method is based on a stochastic Pareto-based evolutionary algorithm for optimisation and on a multilinear interpolation for judgment modelling. The combination of these techniques makes it possible to solve complex problems, with up to eight decision variables and up to at least eight objectives. Relevant applications of the method include optimisation with judgments about various aspects of the product or system, identification of the best trade-offs satisfying at the same time several groups with different judgments, and analysis of the interest of market segmentation. For illustration purpose, a pilot study about an individual office lighting design problem is processed.     

Selecting a near‐optimal design for multiple criteria with improved robustness to different user priorities

In a decision‐making process, relying on only one objective can often lead to oversimplified decisions that ignore important considerations. Incorporating multiple, and likely competing, objectives is critical for balancing trade‐offs on different aspects of performance. When multiple objectives are considered, it is often hard to make a precise decision on how to weight the different objectives when combining their performance for ranking and selecting designs. We show that there are situations when selecting a design with near‐optimality for a broad range of weight combinations of the criteria is a better test selection strategy compared with choosing a design that is strictly optimal under very restricted conditions. We propose a new design selection strategy that identifies several top‐ranked solutions across broad weight combinations using layered Pareto fronts and then selects the final design that offers the best robustness to different user priorities. This method involves identifying multiple leading solutions based on the primary objectives and comparing the alternatives using secondary objectives to make the final decision. We focus on the selection of screening designs because they are widely used both in industrial research, development, and operational testing. The method is illustrated with an example of selecting a single design from a catalog of designs of a fixed size. However, the method can be adapted to more general designed experiment selection problems that involve searching through a large design space.     

Multiobjective robust optimization method for drawbead design in sheet metal forming

It is recognized that fracture and wrinkling in sheet metal forming can be eliminated via an appropriate drawbead design. Although deterministic multiobjective optimization algorithms and finite element analysis (FEA) have been applied in this respect to improve formability and shorten design cycle, the design could become less meaningful or even unacceptable when considering practical variation in design variables and noises of system parameters. To tackle this problem, we present a multiobjective robust optimization methodology to address the effects of parametric uncertainties on drawbead design, where the six sigma principle is adopted to measure the variations, a dual response surface method is used to construct surrogate model and a multiobjective particle swarm optimization is developed to generate robust Pareto solutions. In this paper, the procedure of drawbead design is divided into two stages: firstly, equivalent drawbead restraining forces (DBRF) are obtained by developing a multiobjective robust particle swarm optimization, and secondly the DBRF model is integrated into a single-objective particle swarm optimization (PSO) to optimize geometric parameters of drawbead. The optimal design showed a good agreement with the physical drawbead geometry and remarkably improve the formability and robust. Thus, the presented method provides an effective solution to geometric design of drawbead for improving product quality.     

INTEREST: a reference-point-based interactive procedure for stochastic multiobjective programming problems

This paper presents a reference point-based interactive algorithm, which has been specifically designed to deal with stochastic multiobjective programming problems. This algorithm combines the classical information used in this kind of methods, i.e. values that the decision maker regards as desirable for each objective, with information about the probabilities the decision maker wishes to accept. This novel aspect allows the method to fully take into account the randomness of the final outcome throughout the whole solution process. These two pieces of information have been introduced in an adapted achievement-scalarizing function, which assures each solution obtained to be probability efficient.