An approach to generate comprehensive piecewise linear interpolation of pareto outcomes to aid decision making

Multiple criteria decision making is a well established field encompassing aspects of search for solutions and selection of solutions in presence of more than one conflicting objectives. In this paper, we discuss an approach aimed towards the latter. The decision maker is presented with a limited number of Pareto optimal outcomes and is required to identify regions of interest for further investigation. The inherent sparsity of the given Pareto optimal outcomes in high dimensional space makes it an arduous task for the decision maker. To address this problem, an existing line of thought in literature is to generate a set of approximated Pareto optimal outcomes using piecewise linear interpolation. We present an approach within this paradigm, but one that delivers a comprehensive linearly interpolated set as opposed to its subset delivered by existing methods. We illustrate the advantage in doing so in comparison to stricter non-dominance conditions imposed in existing PAreto INTerpolation method. The interpolated set of outcomes delivered by the proposed approach are non-dominated with respect to the given Pareto optimal outcomes, and additionally the interpolated outcomes along uniformly distributed reference directions are presented to the decision maker. The errors in the given interpolations are also estimated in order to further aid decision making by establishing confidence in achieving true Pareto outcomes in their vicinity. The proposed approach for interpolation is computationally less demanding (for higher number of objectives) and also further amenable to parallelization. We illustrate the performance of the approach using six well established tri-objective test problems and two real-life examples. The problems span different types of fronts, such as convex, concave, mixed, degenerate, highlighting the wide applicability of the approach.     

Enumeration of Pareto optimal multi-criteria spanning trees – a proof of the incorrectness of Zhou and Gen's proposed algorithm

The minimum spanning tree (MST) problem is a well-known optimization problem of major significance in operational research. In the multi-criteria MST (mc-MST) problem, the scalar edge weights of the MST problem are replaced by vectors, and the aim is to find the complete set of Pareto optimal minimum-weight spanning trees. This problem is NP-hard and so approximate methods must be used if one is to tackle it efficiently. In an article previously published in this journal, a genetic algorithm (GA) was put forward for the mc-MST. To evaluate the GA, the solution sets generated by it were compared with solution sets from a proposed (exponential time) algorithm for enumerating all Pareto optimal spanning trees. However, the proposed enumeration algorithm that was used is not correct for two reasons: (1) It does not guarantee that all Pareto optimal minimum-weight spanning trees are returned. (2) It does not guarantee that those trees that are returned are Pareto optimal. In this short paper we prove these two theorems.     

Generalized Homotopy Approach to Multiobjective Optimization

This paper proposes a new generalized homotopy algorithm for the solution of multiobjective optimization problems with equality constraints. We consider the set of Pareto candidates as a differentiable manifold and construct a local chart which is fitted to the local geometry of this Pareto manifold. New Pareto candidates are generated by evaluating the local chart numerically. The method is capable of solving multiobjective optimization problems with an arbitrary number k of objectives, makes it possible to generate all types of Pareto optimal solutions, and is able to produce a homogeneous discretization of the Pareto set. The paper gives a necessary and sufficient condition for the set of Pareto candidates to form a (k-1)-dimensional differentiable manifold, provides the numerical details of the proposed algorithm, and applies the method to two multiobjective sample problems.     

Tradeoff directions in multiobjective optimization problems

Multiobjective optimization problems (MOP) typically have conflicting objectives wherein a gain in one objective is at the expense of another. Tradeoff directions, which measure the change in some objectives relative to changes in others, provide important information as to the best direction of improvement from the current solution. In this paper we present a general definition of tradeoffs as a cone of directions and provide a general method of calculating tradeoffs at every Pareto optimal point of a convex MOP. This extends current definitions of tradeoffs which assume certain conditions on the feasible set and the objective functions. Two comprehensive numerical examples are provided to illustrate the tradeoff directions and the methods used to calculate them.     

Minimizing the area of a Pareto confidence region

A constrained optimization problem is formulated and solved in order to determine the smallest confidence region for the parameters of the Pareto distribution in a proposed family of sets. The objective function is the area of the region, whereas the constraints are related to the required confidence level. Explicit expressions for the area and confidence level of a given region are first deduced. An efficient procedure based on minimizing the corresponding Lagrangian function is then presented to solve the nonlinear programming problem. The process is valid when some of the smallest and largest observations have been discarded or censored, i.e., both single (right or left) and double censoring are allowed. The optimal Pareto confidence region is derived by simultaneously solving three (four) nonlinear equations in the right (double) censoring case. In most practical situations, Newton’s method with the balanced set as the starting point only needs a few iterations to find the global solution. In general, the reduction in area of the optimal Pareto region with respect to the balanced set is considerable if the sample size, n, is small or moderately large, which is usual in practice. This reduction is sometimes impressive when n is quite small and the censoring degree is fairly high. Two numerical examples regarding component lifetimes and fire claims are included for illustrative and comparative purposes.     

Cone contraction and reference point methods for multi-criteria mixed integer optimization

Interactive approaches employing cone contraction for multi-criteria mixed integer optimization are introduced. In each iteration, the decision maker (DM) is asked to give a reference point (new aspiration levels). The subsequent Pareto optimal point is the reference point projected on the set of admissible objective vectors using a suitable scalarizing function. Thereby, the procedures solve a sequence of optimization problems with integer variables. In such a process, the DM provides additional preference information via pair-wise comparisons of Pareto optimal points identified. Using such preference information and assuming a quasiconcave and non-decreasing value function of the DM we restrict the set of admissible objective vectors by excluding subsets, which cannot improve over the solutions already found. The procedures terminate if all Pareto optimal solutions have been either generated or excluded. In this case, the best Pareto point found is an optimal solution. Such convergence is expected in the special case of pure integer optimization; indeed, numerical simulation tests with multi-criteria facility location models and knapsack problems indicate reasonably fast convergence, in particular, under a linear value function. We also propose a procedure to test whether or not a solution is a supported Pareto point (optimal under some linear value function).     

On extensions of the core and the anticore of transferable utility games

We consider several related set extensions of the core and the anticore of games with transferable utility. An efficient allocation is undominated if it cannot be improved, in a specific way, by sidepayments changing the allocation or the game. The set of all such allocations is called the undominated set, and we show that it consists of finitely many polytopes with a core-like structure. One of these polytopes is the $L_1$ -center, consisting of all efficient allocations that minimize the sum of the absolute values of the excesses. The excess Pareto optimal set contains the allocations that are Pareto optimal in the set obtained by ordering the sums of the absolute values of the excesses of coalitions and the absolute values of the excesses of their complements. The $L_1$ -center is contained in the excess Pareto optimal set, which in turn is contained in the undominated set. For three-person games all these sets coincide. These three sets also coincide with the core for balanced games and with the anticore for antibalanced games. We study properties of these sets and provide characterizations in terms of balanced collections of coalitions. We also propose a single-valued selection from the excess Pareto optimal set, the min-prenucleolus, which is defined as the prenucleolus of the minimum of a game and its dual.     

Interactive Polyhedral Outer Approximation (IPOA) strategy for general multiobjective optimization problems

We propose an interactive polyhedral outer approximation (IPOA) method to solve a broad class of multiobjective optimization problems (MOP) with, possibly, nonlinear and nondifferentiable objective and constraint functions, and with continuous or discrete decision variables. During the interactive optimization phase, the method progressively constructs a polyhedral approximation of the decision-maker’s (DM’s) unknown preference structure and a polyhedral outer-approximation of the feasible set of MOP. The piecewise linear approximation of the DM’s preferences also provides a mechanism for testing the consistency of the DM’s assessments and removing inconsistencies; it also allows post-optimality analysis. All the feasible trial solutions are non-dominated (efficient, or Pareto-optimal) so preference assessments are made in the context of non-dominated alternatives only. Upper and lower bounds on the yet unknown optimal value are produced at every iteration, allowing terminating the search prematurely at a good-enough solution and providing information about the closeness of this solution to the optimal solution. The IPOA method includes a preliminary phase in which a limited probe of the efficient set is conducted in order to find a good initial trial solution for the interactive phase. The computational requirements of the algorithm are relatively simple. The results of an extensive computational study are reported.     

Two issues in using mixtures of polynomials for inference in hybrid Bayesian networks

We discuss two issues in using mixtures of polynomials (MOPs) for inference in hybrid Bayesian networks. MOPs were proposed by Shenoy and West for mitigating the problem of integration in inference in hybrid Bayesian networks. First, in defining MOP for multi-dimensional functions, one requirement is that the pieces where the polynomials are defined are hypercubes. In this paper, we discuss relaxing this condition so that each piece is defined on regions called hyper-rhombuses. This relaxation means that MOPs are closed under transformations required for multi-dimensional linear deterministic conditionals, such as Z = X + Y, etc. Also, this relaxation allows us to construct MOP approximations of the probability density functions (PDFs) of the multi-dimensional conditional linear Gaussian distributions using a MOP approximation of the PDF of the univariate standard normal distribution. Second, Shenoy and West suggest using the Taylor series expansion of differentiable functions for finding MOP approximations of PDFs. In this paper, we describe a new method for finding MOP approximations based on Lagrange interpolating polynomials (LIP) with Chebyshev points. We describe how the LIP method can be used to find efficient MOP approximations of PDFs. We illustrate our methods using conditional linear Gaussian PDFs in one, two, and three dimensions, and conditional log-normal PDFs in one and two dimensions. We compare the efficiencies of the hyper-rhombus condition with the hypercube condition. Also, we compare the LIP method with the Taylor series method.     

Mean-of-order <Emphasis Type="Italic">p</Emphasis> reduced-bias extreme value index estimation under a third-order framework

Reduced-bias versions of a very simple generalization of the ‘classical’ Hill estimator of a positive extreme value index (EVI) are put forward. The Hill estimator can be regarded as the logarithm of the mean-of-order-0 of a certain set of statistics. Instead of such a geometric mean, it is sensible to consider the mean-of-order-p (MOP) of those statistics, with p real. Under a third-order framework, the asymptotic behaviour of the MOP, optimal MOP and associated reduced-bias classes of EVI-estimators is derived. Information on the dominant non-null asymptotic bias is also provided so that we can deal with an asymptotic comparison at optimal levels of some of those classes. Large-scale Monte-Carlo simulation experiments are undertaken to provide finite sample comparisons.     

Finding preferred subsets of Pareto optimal solutions

Multi-objective optimization algorithms can generate large sets of Pareto optimal (non-dominated) solutions. Identifying the best solutions across a very large number of Pareto optimal solutions can be a challenge. Therefore it is useful for the decision-maker to be able to obtain a small set of preferred Pareto optimal solutions. This paper analyzes a discrete optimization problem introduced to obtain optimal subsets of solutions from large sets of Pareto optimal solutions. This discrete optimization problem is proven to be NP-hard. Two exact algorithms and five heuristics are presented to address this problem. Five test problems are used to compare the performances of these algorithms and heuristics. The results suggest that preferred subset of Pareto optimal solutions can be efficiently obtained using the heuristics, while for smaller problems, exact algorithms can be applied.     

Effective implementation of the ε-constraint method in Multi-Objective Mathematical Programming problems

As indicated by the most widely accepted classification, the Multi-Objective Mathematical Programming (MOMP) methods can be classified as a priori, interactive and a posteriori, according to the decision stage in which the decision maker expresses his/her preferences. Although the a priori methods are the most popular, the interactive and the a posteriori methods convey much more information to the decision maker. Especially, the a posteriori (or generation) methods give the whole picture (i.e. the Pareto set) to the decision maker, before his/her final choice, reinforcing thus, his/her confidence to the final decision. However, the generation methods are the less popular due to their computational effort and the lack of widely available software. The present work is an effort to effectively implement the ε-constraint method for producing the Pareto optimal solutions in a MOMP. We propose a novel version of the method (augmented ε-constraint method - AUGMECON) that avoids the production of weakly Pareto optimal solutions and accelerates the whole process by avoiding redundant iterations. The method AUGMECON has been implemented in GAMS, a widely used modelling language, and has already been used in some applications. Finally, an interactive approach that is based on AUGMECON and eventually results in the most preferred Pareto optimal solution is also proposed in the paper.     

基于种群自适应调整的多目标差分进化算法

为提高已有多目标进化算法在求解复杂多目标优化问题上的收敛性和解集分布性,提出一种基于种群自适应调整的多目标差分进化算法。该算法设计一个种群扩增策略,它在决策空间生成一些新个体帮助搜索更优的非支配解;设计了一个种群收缩策略,它依据对非支配解集的贡献程度淘汰较差的个体以减少计算负荷,并预留一些空间给新的带有种群多样性的扰动个体;引入精英学习策略,防止算法陷入局部收敛。通过典型的多目标优化函数对算法进行测试验证,结果表明所提算法相对于其他算法具有明显的优势,其性能优越,能够在保证良好收敛性的同时,使获得的Pareto最优解集具有更均匀的分布性和更广的覆盖范围,尤其适合于高维复杂多目标优化问题的求解。     

An interactive fuzzy satisficing method for structured multiobjective linear fractional programs with fuzzy numbers

In this paper, by considering the experts' vague or fuzzy understanding of the nature of the parameters in the problem formulation process, multiobjective linear fractional programming problems with block angular structure involving fuzzy numbers are formulated. Using the a-level sets of fuzzy numbers, the corresponding nonfuzzy a-multiobjective linear fractional programming problem is introduced. The fuzzy goals of the decision maker for the objective functions are quantified by eliciting the corresponding membership functions including nonlinear ones. Through the introduction of extended Pareto optimality concepts, if the decision maker specifies the degree a and the reference membership values, the corresponding extended Pareto optimal solution can be obtained by solving the minimax problems for which the Dantzig-Wolfe decomposition method and Ritter's partitioning procedure are applicable. Then a linear programming-based interactive fuzzy satisficing method with decomposition procedures for deriving a satisficing solution for the decision maker efficiently from an extended Pareto optimal solution set is presented. An illustrative numerical example is provided to demonstrate the feasibility of the proposed method.     

An improved algorithm for solving biobjective integer programs

A parametric algorithm for identifying the Pareto set of a biobjective integer program is proposed. The algorithm is based on the weighted Chebyshev (Tchebycheff) scalarization, and its running time is asymptotically optimal. A number of extensions are described, including: a technique for handling weakly dominated outcomes, a Pareto set approximation scheme, and an interactive version that provides access to all Pareto outcomes. Extensive computational tests on instances of the biobjective knapsack problem and a capacitated network routing problem are presented.     

An interactive fuzzy satisficing method for multiobjective 0–1 programming problems with fuzzy numbers through genetic algorithms with double strings

In this paper, by considering the experts' vague or fuzzy understanding of the nature of the parameters in the problem-formulation process, multiobjective 0–1 programming problems involving fuzzy numbers are formulated. Using the a-level sets of fuzzy numbers, the corresponding nonfuzzy α-programming problem is introduced. The fuzzy goals of the decision maker (DM) for the objective functions are quantified by eliciting the corresponding linear membership functions. Through the introduction of an extended Pareto optimality concept, if the DM specifies the degree α and the reference membership values, the corresponding extended Pareto optimal solution can be obtained by solving the augmented minimax problems through genetic algorithms with double strings. Then an interactive fuzzy satisficing method for deriving a satisficing solution for the DM efficiently from an extended Pareto optimal solution set is presented. An illustrative numerical example is provided to demonstrate the feasibility and efficiency of the proposed method.     

Allocation of risks and equilibrium in markets with finitely many traders

The optimal risk allocation problem, equivalently the optimal risk sharing problem, in a market with n traders endowed with risk measures ?₁,…,?_n is a classical problem in insurance and mathematical finance. This problem however only makes sense under a condition motivated from game theory which is called Pareto equilibrium. There are many situations of practical interest, where this condition does not hold. This is the case if the risk measures are based on essential different views towards risk. In this paper we introduce and analyze a meaningful extension of the optimal risk allocation (risk sharing) problem without assuming the equilibrium condition. The main point of this is to introduce a suitable and well motivated restriction on the class of admissible allocations which prevents effects of artificial ‘risk arbitrage’. As a result we obtain a new coherent risk measure which describes the inherent risk which remains after using admissible risk exchange in an optimal way.     

Two-Sided Pareto Front Approximations

A new approach to derive Pareto front approximations with evolutionary computations is proposed here. At present, evolutionary multiobjective optimization algorithms derive a discrete approximation of the Pareto front (the set of objective maps of efficient solutions) by selecting feasible solutions such that their objective maps are close to the Pareto front. However, accuracy of such approximations is known only if the Pareto front is known, which makes their usefulness questionable. Here we propose to exploit also elements outside feasible sets to derive pairs of such Pareto front approximations that for each approximation pair the corresponding Pareto front lies, in a certain sense, in-between. Accuracies of Pareto front approximations by such pairs can be measured and controlled with respect to distance between elements of a pair. A rudimentary algorithm to derive pairs of Pareto front approximations is presented and the viability of the idea is verified on a limited number of test problems.     

Hierarchical generation of α-Pareto optimal solutions in large-scale multi-objective non-linear systems with fuzzy parameters

This paper proposes a decomposition method for hierarchical generation of α-Pareto optimal solutions in large-scale multi-objective non-linear programming (MONLP) problems with fuzzy parameters in the objective functions and in the constraints (FMONLP). These fuzzy parameters are characterized by fuzzy numbers. For such problems, the concept of α-Pareto optimality introduced by extending the ordinary Pareto optimality based on the α-level sets of fuzzy numbers. The decomposition method is based on the principle of decompose the original problem into interdependent sub-problems. In this method, the global multi-objective non-linear problem is decomposed into smaller multi-objective sub-problems. The smaller sub-problems, which obtained solved separately by using the weighting method and through an operative procedure. All these solution are coordinates in such a way that an optimal solution for the global problem achieved. In addition, an interactive fuzzy decision-making algorithm for hierarchical generation of α-Pareto optimal solution through the decomposition method is developed. Finally, two numerical examples given to illustrate the results developed in this paper.     

Eigenvalue analysis of constrained minimization problem for homogeneous polynomial

In this paper, the concepts of Pareto H-eigenvalue and Pareto Z-eigenvalue are introduced for studying constrained minimization problem and the necessary and sufficient conditions of such eigenvalues are given. It is proved that a symmetric tensor has at least one Pareto H-eigenvalue (Pareto Z-eigenvalue). Furthermore, the minimum Pareto H-eigenvalue (or Pareto Z-eigenvalue) of a symmetric tensor is exactly equal to the minimum value of constrained minimization problem of homogeneous polynomial deduced by such a tensor, which gives an alternative methods for solving the minimum value of constrained minimization problem. In particular, a symmetric tensor \({\mathcal {A}}\) is strictly copositive if and only if every Pareto H-eigenvalue (Z-eigenvalue) of \({\mathcal {A}}\) is positive, and \({\mathcal {A}}\) is copositive if and only if every Pareto H-eigenvalue (Z-eigenvalue) of \({\mathcal {A}}\) is non-negative.