Paired domination on interval and circular-arc graphs

We study the paired-domination problem on interval graphs and circular-arc graphs. Given an interval model with endpoints sorted, we give an O(m+n) time algorithm to solve the paired-domination problem on interval graphs. The result is extended to solve the paired-domination problem on circular-arc graphs in O(m(m+n)) time.     

Interval Branch and Bound with Local Sampling for Constrained Global Optimization

In this article, we introduce a global optimization algorithm that integrates the basic idea of interval branch and bound, and new local sampling strategies along with an efficient data structure. Also included in the algorithm are procedures that handle constraints. The algorithm is shown to be able to find all the global optimal solutions under mild conditions. It can be used to solve various optimization problems. The local sampling (even if done stochastically) is used only to speed up the convergence and does not affect the fact that a complete search is done. Results on several examples of various dimensions ranging from 1 to 100 are also presented to illustrate numerical performance of the algorithm along with comparison with another interval method without the new local sampling and several noninterval methods. The new algorithm is seen as the best performer among those tested for solving multi-dimensional problems.     

Genetic algorithm for asymmetric traveling salesman problem with imprecise travel times

This paper presents a variant of the asymmetric traveling salesman problem (ATSP) in which the traveling time between each pair of cities is represented by an interval of values (wherein the actual travel time is expected to lie) instead of a fixed (deterministic) value as in the classical ATSP. Here the ATSP (with interval objective) is formulated using the usual interval arithmetic. To solve the interval ATSP (I-ATSP), a genetic algorithm with interval valued fitness function is proposed. For this purpose, the existing revised definition of order relations between interval numbers for the case of pessimistic decision making is used. The proposed algorithm is based on a previously published work and includes some new features of the basic genetic operators. To analyze the performance and effectiveness of the proposed algorithm and different genetic operators, computational studies of the proposed algorithm on some randomly generated test problems are reported.     

An interval entropy method for equality constrained multiobjective optimization problems

Based on the maximum entropy principle and the idea of a penalty function, an evaluation function is derived to solve multiobjective optimization problems with equality constraints. Combining with interval analysis method, we define a generalized Krawczyk operator, design interval iteration with constrained functions and new region deletion test rules, present an interval algorithm for equality constrained multiobjective optimization problems, and also prove relevant properties. A theoretical analysis and numerical results indicate that the algorithm constructed is effective and reliable.     

一种基于区间评分的评价算法及其实现

本文针对评审工作中打分专家对各指标熟悉度不同的实际问题，提出一种基于区间评分的评价算法。因该算法通过专家给出评分区间长度反映出专家对各评审指标的熟悉度，为此定义区间的加法、减法、数乘运算及区间值的概念，这样可解决区间无法直接排序的问题。最后给出了该算法的具体步骤和实际算例。     

The selection of the optimal parameter in the modulus-based matrix splitting algorithm for linear complementarity problems

The modulus-based matrix splitting (MMS) algorithm is effective to solve linear complementarity problems (Bai in Numer Linear Algebra Appl 17: 917–933, 2010). This algorithm is parameter dependent, and previous studies mainly focus on giving the convergence interval of the iteration parameter. Yet the specific selection approach of the optimal parameter has not been systematically studied due to the nonlinearity of the algorithm. In this work, we first propose a novel and simple strategy for obtaining the optimal parameter of the MMS algorithm by merely solving two quadratic equations in each iteration. Further, we figure out the interval of optimal parameter which is iteration independent and give a practical choice of optimal parameter to avoid iteration-based computations. Compared with the experimental optimal parameter, the numerical results from three problems, including the Signorini problem of the Laplacian, show the feasibility, effectiveness and efficiency of the proposed strategy.

     

Elitist genetic algorithm for assignment problem with imprecise goal

The objective of this research paper is to solve a generalized assignment problem with imprecise cost(s)/time(s) instead of precise one by elitist genetic algorithm (GA). Here, the impreciseness of cost(s)/time(s) has been represented by interval valued numbers, as interval valued numbers are the best representation than others like random variable representation with a known probability distribution and fuzzy representation. To solve these types of problems, an elitist GA has been developed with interval valued fitness function. In this developed GA, the existing ideas about the order relations of interval valued numbers have been modified from the point of view of two types of decision making viz., optimistic decision making and pessimistic decision making. This modified approach has been used in the selection process for selecting better chromosomes/individuals for the next generation and in finding the best as well as the worst chromosomes/individuals in each generation. Here two new crossover schemes and two new mutation schemes have been introduced. In order to maintain the feasibility with crossover operations, a repair algorithm has been suggested. Extensive comparative computational studies based on different parameters of our developed algorithm on one illustrative example have also been reported.     

A direct solution framework for structural optimization problems with interval uncertainties

A direct solution framework based on multi-objective evolutionary algorithm is developed to solve the structural optimization problems with interval uncertainties. The midpoint and radius of the uncertain original objective are treated as two equally important objectives, which are solved by a multi-objective evolutionary algorithm. The satisfaction value of interval possibility degree model is utilized to deal with nonlinear uncertain constraints and then the degree of constraint violation based on this model is calculated to judge the design vector individuals which one is feasible or infeasible. Subsequently, a selection strategy based on interval constrained-domination rule is utilized to realize the ranking of different design vectors. Finally, two numerical examples and the structural design of augmented reality glasses are investigated to verify the applicability and effectiveness of the proposed method.     

Research on the vehicle routing problem with interval demands

In this paper, a vehicle routing problem with interval demands is investigated based on the motivation of dispatching vehicles to deliver perishable products in practice. A nonlinear interval-based programming method is used to build a model for the vehicle routing problem with interval demands, which assumes that demands of customers are uncertain but fall in given intervals and actual demand of a customer becomes known only when the vehicle visited the customer. A vehicle-coordinated strategy was designed to solve the service failure problem. A hybrid algorithm based on the artificial immune system is also proposed to solve the model for vehicle routing problem with interval demands. The validity of methods and sensitivity analysis are illustrated by conducting some numerical examples. We find that the tolerant possibility degree of interval number has significant impacts on the distances. The planned distance strictly increased, while the additional distance strictly decreased and the total distance after coordinated transport has a U-typed relationship with the tolerant possibility degree of interval number.     

一类求解非线性等式和不等式约束优化问题的区间算法

在Moore二分法的基础上,通过构造的区间列L中标志矢量R的分量取值来删除部分不满足约束条件的区域,将非线性约束优化问题转化为初始域子域上的无约束优化问题,该算法可利用极大熵方法求解多目标优化问题,理论分析和数值结果均表明,这种算法是稳定且可靠的.     

市场约束条件下的非寿险费率厘定

在非寿险费率厘定中,经常遇到的一个实际问题是某些风险类别的费率不能过高或不能过低。在这种约束条件下,传统的广义线性模型将不能直接用于费率厘定。本文给出了一种在一般线性约束条件下,如何应用迭代算法对常用的广义线性模型进行调整,从而得到满足特定约束条件的费率厘定结果。本文的实证研究结果表明,该方法具有灵活性和现实可行性,能够解决非寿险费率厘定中常见的市场约束问题。     

区间数的一种改进的排序方法

为了解决具有不确定性区间数的多属性决策问题 ,本文给出了区间数的一种简洁有效的排序方法 .我们首先引入了一个区间数大于另一个区间数的可能度概念 ,进而推出其有关性质 ,并对排序起着重要作用的结论给出了证明 .在此基础上本文还指出了区间数排序的一种简洁算法 ,使得对多个区间数的排序更为快捷有效 ,并通过实例加以验证 .     

Uncertain constrained optimization by interval-oriented algorithm

This paper deals with an interval-oriented approach to solve general interval constrained optimization problems. Generally, this type of problems has infinitely many compromise solutions. The aim of this approach is to obtain one of such solutions with higher accuracy and lower computational cost. The proposed algorithm is nothing but a different kind of branch and bound algorithm with multi-section division criterion of the search region (or box). In the proposed technique, the prescribed/accepted region is divided into several distinct subregions and in each feasible subregion the interval objective function value is computed. Then the subregion containing the best objective value is found by applying a specific interval ranking rule defined with respect to the pessimistic decision makers’ point of view. The process is continued until the interval width for each variable in the accepted subregion is negligible. Finally, the algorithm converges to a compromise solution in interval form. To illustrate the method and also to test the efficiency as well as the effectiveness of the proposed algorithm, we have solved some numerical examples.     

Consecutive Ones Matrices for Multi-dimensional Orthogonal Packing Problems

The multi-dimensional orthogonal packing problem (OPP) is a well studied decisional problem. Given a set of items with rectangular shapes, the problem is to decide whether there is a non-overlapping packing of these items in a rectangular bin. The rotation of items is not allowed. A powerful caracterization of packing configurations by means of interval graphs was recently introduced. In this paper, we propose a new algorithm using consecutive ones matrices as data structure. This new algorithm is then used to solve the two-dimensional orthogonal knapsack problem. Computational results are reported, which show its effectiveness.     

Preemptive scheduling of interval orders is polynomial

In 1979, Papadimitriou and Yannakakis gave a polynomial time algorithm for the scheduling of jobs requiring unit completion times when the precedence constraints form an interval order. The authors solve here the corresponding problem, for preemptive scheduling (a job can be interrupted to work on more important tasks, and completed at a later time, subject to the usual scheduling constraints.) The m-machine preemptive scheduling problem is shown to have a polynomial algorithm, for both unit time and variable execution times as well, when the precedence constraints are given by an interval order.     

基于标准区间灰数的发展带离散DDGM预测模型

由于区间灰数运算体系尚不完善,灰数间的代数运算将导致结果灰度增加,难以有效构建基于"区间灰数"的灰色发展带预测模型.对此,通过将区间灰数进行标准化处理,分解成基于实数形式的"白部"和"灰部"两个部分;然后分别对"白部"和"灰部"建立发展带预测模型,再推导并还原得到区间灰数的发展带预测模型;最后,将模型用于摆动幅度大且整体趋势增长的区间灰数在未来时刻的预测,预测效果验证了所提出模型的有效性.     

Proportion-based robust optimization and team orienteering problem with interval data

In this paper, a proportion-based robust optimization approach is developed to deal with uncertain combinatorial optimization problems. This approach assumes that a certain proportion of uncertain coefficients in each solution are allowed to change and optimizes a deterministic model so as to achieve a trade-off between optimality and feasibility when the coefficients change. We apply this approach on team orienteering problem with interval data (TOPID), a variant of vehicle routing problem, which has not yet been studied before. A branch and price algorithm is proposed to solve the robust counterpart by using two novel dominance relations. Finally, numerical study is performed. The results show the usefulness of the proposed robust optimization approach and the effectiveness of our algorithm.     

2-approximation algorithm for minmax absolute maximum lateness scheduling-location problem

The problem of minmax absolute scheduling-location is investigated on trees with interval edge length. Jobs are located at vertices and must travel to the machine. The goal is to find a machine location and simultaneously a schedule to minimize the maximum lateness in the worst-case. We derive a result that could reduce the robust versions to deterministic problems. An efficient algorithm is developed to solve special cases. A 2-approximation algorithm is proposed for the robust problem on the underlying tree.     

Interval differential evolution with dimension-reduction interval analysis method for uncertain optimization problems

A constrained nonlinear interval optimization method under the framework of differential evolution algorithm is developed to solve the uncertain structural optimization problems with interval uncertainties. The proposed method is a direct optimization method based on the interval differential evolution and dimension-reduction interval analysis. The interval preferential rule based on the satisfaction value of interval possibility degree model is used to realize the direct interval ranking of different design vectors. At each evolutionary generation, the outer optimizer by differential evolution optimizer searches for the best solution within the design space. The dimension-reduction interval analysis is employed to calculate the intervals of objective and constraints for each design vector in the inner layer. This operation transforms the original nesting optimization problem into a single loop one which improves the computational efficiency of the proposed method. Finally, the effectiveness of the presented direct method is verified by two numerical examples and an engineering application.     

A nonlinear interval number programming method for uncertain optimization problems

In this paper, a method is suggested to solve the nonlinear interval number programming problem with uncertain coefficients both in nonlinear objective function and nonlinear constraints. Based on an order relation of interval number, the uncertain objective function is transformed into two deterministic objective functions, in which the robustness of design is considered. Through a modified possibility degree, the uncertain inequality and equality constraints are changed to deterministic inequality constraints. The two objective functions are converted into a single-objective problem through the linear combination method, and the deterministic inequality constraints are treated with the penalty function method. The intergeneration projection genetic algorithm is employed to solve the finally obtained deterministic and non-constraint optimization problem. Two numerical examples are investigated to demonstrate the effectiveness of the present method.