A Goal Programming Method for Generating Priority Vectors

The generation of priority vectors from pairwise comparison information is an integral part of the Analytic Hierarchy Process (AHP). Traditionally, either the right eigenvector method or the logarithmic least squares method have been used. In this paper, a goal programming method (GPM) is presented that has, as its objective, the generation of the priority vector whose associated comparison values are, on average, the closest to the pairwise comparison information provided by the evaluator. The GPM possesses the properties of correctness in the consistent case, comparison order invariance, smoothness, and power invariance. Unlike other methods, it also possesses the additional property that the presence of a single outlier cannot prevent the identification of the correct priority vector. The GPM also has a pair of naturally meaningful consistency indicators that offer the opportunity for empowering the decision maker. The GPM is thus an attractive alternative to other proposed methods.     

模糊择近原则在多目标容差设计中的应用

利用模糊数学中的择近原则,以容差-成本模型为基础,通过引入模糊数学中的隶属度函数和贴近度原则,结合实验并设计方法提出了一种模糊容差稳健优化设计方法,较好地解决了多目标容差设计的全局最优问题.最后通过一个实例验证了该方法的合理、有效性.     

An Algorithm for Solving Optimization Problems with One Linear Objective Function and Finitely Many Constraints of Fuzzy Relation Inequalities

An optimization model with one linear objective function and fuzzy relation equation constraints was presented by Fang and Li (1999) as well as an efficient solution procedure was designed by them for solving such a problem. A more general case of the problem, an optimization model with one linear objective function and finitely many constraints of fuzzy relation inequalities, is investigated in this paper. A new approach for solving this problem is proposed based on a necessary condition of optimality given in the paper. Compared with the known methods, the proposed algorithm shrinks the searching region and hence obtains an optimal solution fast. For some special cases, the proposed algorithm reaches an optimal solution very fast since there is only one minimum solution in the shrunk searching region. At the end of the paper, two numerical examples are given to illustrate this difference between the proposed algorithm and the known ones.     

Finding multiple solutions to general integer linear programs

Integer linear programming (ILP) problems occur frequently in many applications. In practice, alternative optima are useful since they allow the decision maker to choose from multiple solutions without experiencing any deterioration in the objective function. This study proposes a general integer cut to exclude the previous solution and presents an algorithm to identify all alternative optimal solutions of an ILP problem. Numerical examples in real applications are presented to demonstrate the usefulness of the proposed method.     

Approximating Sampled Sinusoids and Multiband Signals Using Multiband Modulated DPSS Dictionaries

Many signal processing problems—such as analysis, compression, denoising, and reconstruction—can be facilitated by expressing the signal as a linear combination of atoms from a well-chosen dictionary. In this paper, we study possible dictionaries for representing the discrete vector one obtains when collecting a finite set of uniform samples from a multiband analog signal. By analyzing the spectrum of combined time- and multiband-limiting operations in the discrete-time domain, we conclude that the information level of the sampled multiband vectors is essentially equal to the time–frequency area. For representing these vectors, we consider a dictionary formed by concatenating a collection of modulated discrete prolate spheroidal sequences (DPSS’s). We study the angle between the subspaces spanned by this dictionary and an optimal dictionary, and we conclude that the multiband modulated DPSS dictionary—which is simple to construct and more flexible than the optimal dictionary in practical applications—is nearly optimal for representing multiband sample vectors. We also show that the multiband modulated DPSS dictionary not only provides a very high degree of approximation accuracy in an MSE sense for multiband sample vectors (using a number of atoms comparable to the information level), but also that it can provide high-quality approximations of all sampled sinusoids within the bands of interest.     

Optimal operating strategy for a long-haul liner service route

This paper proposes an optimal operating strategy problem arising in liner shipping industry that aims to determine service frequency, containership fleet deployment plan, and sailing speed for a long-haul liner service route. The problem is formulated as a mixed-integer nonlinear programming model that cannot be solved efficiently by the existing solution algorithms. In view of some unique characteristics of the liner shipping operations, this paper proposes an efficient and exact branch-and-bound based ε-optimal algorithm. In particular, a mixed-integer nonlinear model is first developed for a given service frequency and ship type; two linearization techniques are subsequently presented to approximate this model with a mixed-integer linear program; and the branch-and-bound approach controls the approximation error below a specified tolerance. This paper further demonstrates that the branch-and-bound based ε-optimal algorithm obtains a globally optimal solution with the predetermined relative optimality tolerance ε in a finite number of iterations. The case study based on an existing long-haul liner service route shows the effectiveness and efficiency of the proposed solution method.     

NAUTILUS method: An interactive technique in multiobjective optimization based on the nadir point

Most interactive methods developed for solving multiobjective optimization problems sequentially generate Pareto optimal or nondominated vectors and the decision maker must always allow impairment in at least one objective function to get a new solution. The NAUTILUS method proposed is based on the assumptions that past experiences affect decision makers’ hopes and that people do not react symmetrically to gains and losses. Therefore, some decision makers may prefer to start from the worst possible objective values and to improve every objective step by step according to their preferences. In NAUTILUS, starting from the nadir point, a solution is obtained at each iteration which dominates the previous one. Although only the last solution will be Pareto optimal, the decision maker never looses sight of the Pareto optimal set, and the search is oriented so that (s)he progressively focusses on the preferred part of the Pareto optimal set. Each new solution is obtained by minimizing an achievement scalarizing function including preferences about desired improvements in objective function values. NAUTILUS is specially suitable for avoiding undesired anchoring effects, for example in negotiation support problems, or just as a means of finding an initial Pareto optimal solution for any interactive procedure. An illustrative example demonstrates how this new method iterates.     

利用近似能量极小构造平面C1三次Hermite插值曲线

研究了利用近似能量极小构造平面$C^1$三次Hermite插值曲线的方法.该方法的主要的目是求出$C^1$三次Hermite插值曲线的最佳切矢.通过将应变能、曲率变化能和组合能的近似函数极小化,得到了求解最佳切矢的线性方程组.通过求解发现,近似曲率变化能极小不存在唯一解, 而近似应变能极小和近似组合能极小由于方程系统的系数矩阵为严格对角占优故都存在唯一解.最后, 通过实例表明了本文方法构造平面$C^1$三次Hermite插值曲线的有效性.     

Parametric aggregation in ordered weighted averaging

Incorporating further information into the ordered weighted averaging (OWA) operator weights is investigated in this paper. We first prove that for a constant orness the minimax disparity model [13] has unique optimal solution while the modified minimax disparity model [16] has alternative optimal OWA weights. Multiple optimal solutions in modified minimax disparity model provide us opportunity to define a parametric aggregation OWA which gives flexibility to decision makers in the process of aggregation and selecting the best alternative. Finally, the usefulness of the proposed parametric aggregation method is illustrated with an application in metasearch engine.     

A semidefinite programming approach to the generalized problem of moments

We consider the generalized problem of moments (GPM) from a computational point of view and provide a hierarchy of semidefinite programming relaxations whose sequence of optimal values converges to the optimal value of the GPM. We then investigate in detail various examples of applications in optimization, probability, financial economics and optimal control, which all can be viewed as particular instances of the GPM. This work was supported by french ANR-grant NT05-3-41612, and part of it was completed in January 2006 while the author was a member of IMS, the Institute for Mathematical Sciences of NUS (The National University of Singapore).     

基于二元语义多属性群决策的投影法

针对解决具有语言评价信息的多属性群决策问题,提出了一种基于二元语义信息处理的群决策方法。该方法采用近年来最新发展的二元语义概念对语言评价信息进行处理和运算,并依据传统投影分析方法的基本思想,通过计算备选方案对正理想方案和负理想方案的投影值,进而计算备选方案对正理想方案相对贴近度,最终确定最优方案。该方法具有对语言信息处理较为精确的特点,避免了以往采用的语言信息处理方法所带来的信息扭臼和损失。最后给出了实例分析。结果表明方法简单,有效和易于计算。     

Optimal input system identification for homogeneous and nonhomogeneous boundary conditions

A recent paper by Mehra has considered the design of optimal inputs for linear system identification. The method proposed involves the solution of homogeneous linear differential equations with homogeneous boundary conditions. In this paper, a method of solution is considered for similar-type problems with nonhomogeneous boundary conditions. The methods of solution are compared for the homogeneous and nonhomogeneous cases, and it is shown that, for a simple numerical example, the optimal input for the nonhomogeneous case is almost identical to the homogeneous optimal input when the former has a small initial condition, terminal time near the critical length, and energy input the same as for the homogeneous case. Thus tentatively, solving the nonhomogeneous problem appears to offer an attractive alternative to solving Mehra's homogeneous problem.     

固定支付下养老基金资产组合CEV模型及最优投资策略

在固定支付水平的条件之下,就养老基金资产组合问题建立常方差弹性(CEV)模型,应用随机控制原理求出了相应的非线性Hamilton-Jacobi-Bellman偏微方程,再用Legendre变换将其转化为线性偏微方程,建立对偶问题.通过对偶问题的求解,从而求得原问题的精确解析解,确定风险资产和无风险资产的最优投资比例,实现了满足养老基金既定支出水平下总资产的对数效用最大化,从实际市场的角度改进发展了经典的Merton模型结果.     

An objective hyperplane search procedure for solving the general all-integer linear programming (ILP) problem

We describe an objective hyperplane search method for solving a class of integer linear programming (ILP) problems. We formulate the search as a bounded knapsack problem and develop requisite theory for formulating knapsack problems with composite constraints and composite objective functions that facilitate convergence to an ILP solution. A heuristic solution algorithm was developed and used to solve a variety of test problems found in the literature. The method obtains optimal or near-optimal solutions in acceptable ranges of computational effort.     

空军消耗性航材储备优化模型研究

针对我军装备改型多以及部分消耗性航材长期储存容易变质失效的实际借鉴EOQ模型的思想,构建了消耗性航材储备模型.模型以总储备经费最小为目标,综合考虑了航材使用优先级以及航材储备中产生的各项费用,并从理论上证明了模型具有全局最优解.最后通过算例说明方法可行.     

Iteration in a subspace for solving matrix games

A new method for solving symmetric matrix games is proposed and justified. It is based on iterating vectors in the null space of a specially constructed matrix. According to the numerical tests performed, the efficiency of the proposed method is comparable with that of the available iterative algorithms having about the same computational complexity. This approach is also applicable to more complicated problems than the calculation of particular optimal strategies. For instance, it can be used for finding the unique minimum length solution.     

An exact algorithm for solving the bilevel facility interdiction and fortification problem

We present an exact approach for solving the $r$-interdiction median problem with fortification. Our approach consists of solving a greedy heuristic and a set cover problem iteratively that guarantees to find an optimal solution upon termination. The greedy heuristic obtains a feasible solution to the problem, and the set cover problem is solved to verify optimality of the solution and to provide a direction for improvement if not optimal. We demonstrate the performance of the algorithm in a computational study.     

On optimal algorithms for the joint replenishment problem

In this paper, we demonstrate that the algorithm for determining the optimal strict cyclic policy for the Joint Replenishment Problem suggested by Fung and Ma¹ does not guarantee an optimal solution. We propose a modification that will ensure that the algorithm obtains the optimal strict cyclic policy. We then perform a comprehensive computational study to compare the modified Fung and Ma algorithm with other optimal algorithms for the problem. The study reveals that the optimal algorithm of Viswanathan² is computationally more efficient than other methods.     

Fractional Order Systems Time-Optimal Control and Its Application

This paper deals with the time-optimal control problem for a class of fractional order systems. An analytic solution of the time-optimal problem is proposed, and the optimal transfer route is provided. Considering it is usually adopted in the discrete situation for actual control system, the sampling date may induce chattering phenomenon, an alternative sub-optimal solution is constructed. Additionally, the special and meaningful application of fractional order tracking differentiator is introduced to explain our main results. The effectiveness and advantages of the proposed method have been illustrated by numerical examples.     

线形分式规划消耗系数矩阵灵敏度分析及应用

基于解非线形规划的凸单纯形法,对一类线形分式规划的消耗系数矩阵进行灵敏度分析.求出使最优解或最优基保持最优的消耗系数矩阵中列向量和行向量的可变范围.并进行了应用计算.