Solving electronic spreadsheet models by interactive multicriterion optimization: a budgeting illustration

In previous papers a connection between general spreadsheet models and interactive multicriterion optimization has been discussed. The purpose of the present paper is to provide proof of the concept with a detailed example. The main idea is that interactive multicriterion optimization can be viewed as a new way of interacting with spreadsheet models. It offers advantages over the two prevailing ones, namely what-if and goal-seeking. A simplified budgeting example is given to clarify this connection. Directions of improvement are assumed to be provided by the decision maker using one of the several existing methods. One improvement iteration is illustrated and analyzed following the method of Loganathan and Sherali.     

Interactive Optimization Aspects of Electronic Spreadsheet Models for Design and Planning

This paper extends the discussion of the relationship between optimization and electronic spreadsheet modelling. Previous authors have described two connections. These are, first, the use of spreadsheets to facilitate optimization projects and, second, the use of optimization within the spreadsheet model itself such as for cell value computation. In this paper we argue that use of the spreadsheet model for planning and design problems often may actually be viewed as interactive multi-criterion optimization per se. This provides a third connection between optimization and spreadsheet models. We base our arguments on a simplified example arising in insurance consulting work and note other examples as well. We also propose a method for incorporating this observation into existing spreadsheet models and indicate desirable innovations in spreadsheet software.     

Metric properties of quality criteria space for multicriteria optimization problems

The problem of the construction of an object functioning in the regime of optimum performance at the design stage is reduced to the solution of the problem of multicriterion optimization, where the quality criteria are chosen to be its most essential characteristics (parameters). At the same time in all methods of multicriterion optimization the vector quality criterion is considered basically in the linear Euclidean space. Actually, in most cases, the criterion space is non-Euclidean - it is curved. Therefore, such setting cannot give results adequately reflecting the processes running in real systems.In order for the design system to really satisfy the optimality requirements the authors of the given paper offer an absolutely new approach to the solution of the problems of multicriterion optimization based on the definition of the quality criteria space and on finding an invariant corresponding to the distance between any two points of that space.The idea of the study of the metric properties of the quality criteria space and their use in solving problems of multicriterion optimization was offered in the work [1]. But that idea, due to its complexity, has not been completely realized until now. When solving such problems the quality criteria space was automatically identified with the Euclidean space with corresponding metrics. In the general case this could not give results adequately reflecting the processes occurring in real systems.In the present paper metric properties of space criteria are studied for the first time, using as the main instrument the mathematical apparatus of tensor analysis, Riemannian geometry, differential equations in partial derivatives etc. Boundary problems relative to the components of the metric tensor of the n-dimensional space of the phenomenon states enabling to determine its metric properties are posed. The knowledge of the metric tensor furthers the objective appraisal of the phenomenon state and the definition of the optimal state.     

An Algorithm for Solving Multicriterion Linear Programming Problems with Examples

The purpose of this paper is to develop a useful technique for solving linear programmes involving more than one objective function. Motivation for solving multicriterion linear programmes is given along with the inherent difficulty associated with obtaining a satisfactory solution set. By applying a linear programming approach for the solution of two person–zero sum games with mixed strategies, it is shown that a linear optimization problem with multiple objective functions can be formulated in this fashion in order to obtain a solution set satisfying all the requirements for an efficient solution of the problem. The solution method is then refined to take into account disparities between the magnitude of the values generated by each of the objective functions and solution preferences as determined by a decision-maker. A summary of the technique is then given along with several examples in order to demonstrate its applicability.     

Interactive Compromise Programming

The purpose of this paper is to introduce a solution method for multiple objective linear programming (MOLP) problems. The method, called interactive compromise programming (ICP), offers a practical solution to MOLP problems by combining judgement with an automatic optimization technique in decision-making. This is realised by using the method of compromise programming and the method of a two-person zero-sum game in an iterative way. The method is illustrated by a numerical example.     

Implementation aspects of interactive multiobjective optimization for modeling environments: the case of GAMS-NIMBUS

Interactive multiobjective optimization methods have provided promising results in the literature but still their implementations are rare. Here we introduce a core structure of interactive methods to enable their convenient implementation. We also demonstrate how this core structure can be applied when implementing an interactive method using a modeling environment. Many modeling environments contain tools for single objective optimization but not for interactive multiobjective optimization. Furthermore, as a concrete example, we present GAMS-NIMBUS Tool which is an implementation of the classification-based NIMBUS method for the GAMS modeling environment. So far, interactive methods have not been available in the GAMS environment, but with the GAMS-NIMBUS Tool we open up the possibility of solving multiobjective optimization problems modeled in the GAMS modeling environment. Finally, we give some examples of the benefits of applying an interactive method by using the GAMS-NIMBUS Tool for solving multiobjective optimization problems modeled in the GAMS environment.     

Stochastic Programming with Multivariate Second Order Stochastic Dominance Constraints with Applications in Portfolio Optimization

In this paper we study optimization problems with multivariate stochastic dominance constraints where the underlying functions are not necessarily linear. These problems are important in multicriterion decision making, since each component of vectors can be interpreted as the uncertain outcome of a given criterion. We propose a penalization scheme for the multivariate second order stochastic dominance constraints. We solve the penalized problem by the level function methods, and a modified cutting plane method and compare them to the cutting surface method proposed in the literature. The proposed numerical schemes are applied to a generic budget allocation problem and a real world portfolio optimization problem.     

Q-analysis for modeling and decision making

This paper provides an overview of existing and potential applications of a system-theoretic approach called Q-analysis, using the examples of design and analysis of expert systems in medical image processing and analysis: namely the organization of a histopathologic knowledge base. Q-analysis is also applied to a multicriterion decision-making (MCDM) problem using a method called multicriterion Q-analysis (MCQA). A brief discussion of the advantages and limitations of Q-analysis is given, with suggestions for further applications.     

Multicriterion reliability-oriented optimization of structural systems by stochastic programming

Papers deals with multicriterion reliability-oriented optimization of truss structures by stochastic programming. Deterministic approach to structural optimization appears to be insufficient when loads acting upon a structure and material properties of the structure elements have a random nature. The aim of this paper is to show importance of random modelling of the structure and influence of random parameters on an optimal solution. Usually, quality of engineering structure design is considered in terms of displacements, total cost and reliability. Therefore, optimization problem has been formulated and then solved in order to show interaction between displacement and a total cost objective function. The examples of 4-bar and 25-bar truss structures illustrate our considerations. The results of optimization are presented in the form of diagrams.     

Interactive algorithm for multiobjective optimization

A man-machine interactive algorithm is given for solving multiobjective optimization problems involving one decision maker. The algorithm, a modification of the Frank-Wolfe steepest ascent method, gives at each iteration a significant freedom and ease for the decision-maker's self-expression, and requires a minimal information on his local estimate of the steepest-ascent direction. The convergence of the iterative algorithm is proved under natural assumptions on the convergence and stability of the basic Frank-Wolfe algorithm.     

“Value” of additional information in multicriterion analysis under uncertainty

The expected value of information in classical (monocriterion) decision analysis has been well covered in the literature. One cannot say the same thing about the multicriterion analysis, particularly when one is in the presence of multicriterion aggregation procedures based on outranking relations for a ranking problematic. The objective of this paper is to try to extend the Bayesian approach to a multicriterion analysis in the context of uncertainty. After illustrating the a posteriori analysis, we shall mention some difficulties associated with the pre a posteriori analysis and the concepts of the “expected value” of perfect or imperfect information.     

一种改进的GDF多目标优化算法

本文提出一种交互式非线性多目标优化算法，该算法是ＧＤＦ多目标优化算法的改进，具有这样的特点：算法采用了既约设计空间策略，具有良好的收敛性；算法生成的迭代点是有效解；算法具有多种一维搜索准则；对于线性多目标问题，算法只需一次交互迭代即可示出多目标问题的最优解。     

Decomposition methods in optimization

Decomposition methods are proposed for global and multicriterion optimization, combining computational models of efficiency criteria with different complexity and accuracy.Translated from Programmno-apparatnye Sredstva i Matematicheskoe Obespechenie Vychislitel'nykh Sistem, pp. 148–155, 1994.     

Mathematical Modeling of Multilayer Filters for Telecommunication

The article presents an interactive approach to the design of Wavelength Division Multiplexing (WDM) filters that combines classical design methods for multi-resonator systems with a semi-heuristic integer optimization method. This approach proves to be exceptionally efficient from the computational point of view and generates a whole range of different filter designs with virtually equivalent spectral properties. The article also considers a method for analyzing the sensitivity of WDM filters to manufacturing errors. The sensitivity analysis relies on computer simulation of thin-layer coating with optical monitoring of layer thickness by the turning point method. This analysis enables the designer to assess alternative filter designs with the purpose of improving manufacturing efficiency.     

Tentative guidelines to help choosing an appropriate MCDA method

Despite the development of a large number of refined multicriterion decision aid (MCDA) methods, none can be considered as the `super method' appropriate to all decision making situations. Hence, how can one choose an appropriate method to a specific decision situation? Recent experimental studies in psychology and behaviour have revealed, on the one hand, that the human thinking is not to be modelled by logical rules and calculations, and, on the other hand, that the response mode affects the preference formation as well as the use of compensatory or noncompensatory strategies. The aim of this paper is to draw a conceptual framework for articulating tentative guidelines to choose an appropriate MCDA method. This paper also presents the results of the comparison of well known multicriterion aggregation procedures (MCAP) on the basis of these guidelines. In our opinion this study can constitute a first step for proposing a methodological approach to select an appropriate MCDA method to a specific decision making situation. Such an approach should be validated and may be integrated into a decision support system. Moreover, the framework suggested is helpful to develop useful methods and to address neglected issues within the field.     

LMI优化的一种原对偶中心路径算法

系统和控制理论中许多重要的问题,都可转化为具有线性目标函数、线性矩阵不等式约束的LMI优化问题,从而使其在数值上易于求解.本文给出一种求解LMI优化问题的原对偶中心路径算法,该算法利用牛顿方法求解中心路径方程得到牛顿系统,并将该牛顿系统对称化以避免得到非对称化的搜索方向.文章详细分析了算法的计算复杂性.     

Multi-level optimization for multi-objective problems

A multi-level solution method is presented for multi-objective optimization of large-scale systems associated with the hierarchical structure of decision-making. The method, consisting of a multi-level problem formulation and an interactive algorithm, has distinct advantages in handling the difficulties which are often experienced in engineering. The method is illustrated by its application to an optimal design of a processing system.     

A Newton-type univariate optimization algorithm for locating the nearest extremum

This paper introduces an algorithm for univariate optimization using linear lower bounding functions (LLBF's). An LLBF over an interval is a linear function which lies below the given function over an interval and matches the given function at one end point of the interval. We first present an algorithm using LLBF's for finding the nearest root of a function in a search direction. When the root-finding method is applied to the derivative of an objective function, it is an optimization algorithm which guarantees to locate the nearest extremum along a search direction. For univariate optimization, we show that this approach is a Newton-type method, which is globally convergent with superlinear convergence rate. The applications of this algorithm to global optimization and other optimization problems are also discussed.     

Improving the computational efficiency in a global formulation (GLIDE) for interactive multiobjective optimization

In this paper, we present a new general formulation for multiobjective optimization that can accommodate several interactive methods of different types (regarding various types of preference information required from the decision maker). This formulation provides a comfortable implementation framework for a general interactive system and allows the decision maker to conveniently apply several interactive methods in one solution process. In other words, the decision maker can at each iteration of the solution process choose how to give preference information to direct the interactive solution process, and the formulation enables changing the type of preferences, that is, the method used, whenever desired. The first general formulation, GLIDE, included eight interactive methods utilizing four types of preferences. Here we present an improved version where we pay special attention to the computational efficiency (especially significant for large and complex problems), by eliminating some constraints and parameters of the original formulation. To be more specific, we propose two new formulations, depending on whether the multiobjective optimization problem to be considered is differentiable or not. Some computational tests are reported showing improvements in all cases. The generality of the new improved formulations is supported by the fact that they can accommodate six interactive methods more, that is, a total of fourteen interactive methods, just by adjusting parameter values.     

Interactive Solution Approach to a Multiobjective Optimization Problem in a Paper Machine Headbox Design

A successful application of the interactive multiobjective optimization method NIMBUS to a design problem in papermaking technology is described. Namely, an optimal shape design problem related to the paper machine headbox is studied. First, the NIMBUS method, the numerical headbox model, and the associated multiobjective optimization problem are described. Then, the results of numerical experiments are presented.