Evaluation of mathematical models for flexible job-shop scheduling problems

With the rapid development in computer technologies, mathematical programming-based technique to solve scheduling problems is significantly receiving attention from researchers. Although, it is not efficient solution method due to the NP-hard structure of these problems, mathematical programming formulation is the first step to develop an effective heuristic. Numerous comparative studies for variety scheduling problems have appeared over the years. But in our search in literature there is not an entirely review for mathematical formulations of flexible job shop scheduling problems (FJSP). In this paper, four the most widely used formulations of the FJSP are compiled from literature and a time-indexed model for FJSP is proposed. These formulations are evaluated under three categories that are distinguished by the type of binary variable that they rely on for using of sequencing operations on machines. All five formulations compared and results are presented.     

Perturbed solutions of variational inequality problems over polyhedral sets

This paper is concerned with variational inequality problems defined over polyhedral sets, which provide a generalization of many diverse problems of mathematical programming, complementarity, and mathematical economics. Differentiability properties of locally unique perturbed solutions to such problems are studied. It is shown that, if a simple sufficient condition is satisfied, then the perturbed solution is locally unique, continuous, and directionally differentiable. Furthermore, under an additional regularity assumption, the perturbed solution is also continuously differentiable.     

An historical example of mathematical modelling: the trajectory of a cannonball

Classroom considerations of the concept and processes of mathematical modelling can do much to strengthen students’ problem solving skills. A systematic exposure to the techniques of mathematical modelling helps students formulate problems, re‐think those problems in mathematical terms, appreciate possible solution constraints and seek solutions that are realistic within the scope and conditions of the problem. While many mathematical modelling situations can be found in today's world, there are special pedagogical values in examining existing mathematical models that have an historical basis. Such an examination should reveal the mechanics of a modelling situation and how a model evolves or is refined to meet ever increasing human demands for accuracy or practicality. The trajectory of a cannonball provides such a modelling example. This topic captures the imagination of students and supplies the basis for a variety of classroom discussions and problem solving encounters.     

Learning to Solve Mathematical Application Problems: A Design Experiment With Fifth Graders

Recent research has shown that many upper elementary school children do not master the skill of solving mathematical application problems. In this design experiment, a learning environment for teaching and learning how to model and solve mathematical application problems was developed and tested in 4 classes of 5th graders. Pupils were taught a series of heuristics embedded in an overall metacognitive strategy for solving mathematical application problems. Meanwhile, pupils of 7 control classes followed regular mathematics classes. The implementation and effectiveness of the experimental learning environment were tested in a study with a pretest-posttest-retention test design with an experimental and a control group. The results indicate that the intervention had a positive effect on different aspects of pupils' mathematical modeling and problem-solving abilities.     

Learning to Solve Mathematical Application Problems: A Design Experiment With Fifth Graders

Recent research has shown that many upper elementary school children do not master the skill of solving mathematical application problems. In this design experiment, a learning environment for teaching and learning how to model and solve mathematical application problems was developed and tested in 4 classes of 5th graders. Pupils were taught a series of heuristics embedded in an overall metacognitive strategy for solving mathematical application problems. Meanwhile, pupils of 7 control classes followed regular mathematics classes. The implementation and effectiveness of the experimental learning environment were tested in a study with a pretest-posttest-retention test design with an experimental and a control group. The results indicate that the intervention had a positive effect on different aspects of pupils' mathematical modeling and problem-solving abilities.     

Applications of symmetry to problem solving

Symmetry is an important mathematical concept which plays an extremely important role as a problem-solving technique. Nevertheless, symmetry is rarely used in secondary school in solving mathematical problems. Several investigations demonstrate that secondary school mathematics teachers are not aware enough of the importance of this elegant problem-solving tool. In this paper we present examples of problems from several branches of mathematics that can be solved using different types of symmetry. Teachers' attitudes and beliefs regarding the use of symmetry in the solutions of these problems are discussed.     

Mathematical Habits of the Mind for Preservice Teachers

This article provides an overview of mathematical habits of the mind and discusses how the concept relates to Polya's problem solving principles as well as exemplification. Specific problems are discussed as a means to assist preservice elementary school teachers' in their development of mathematical habits of the mind. Without a technique to begin solving these rich problems, preservice teachers may have difficulty getting started. The process of preservice teachers outlining their thinking as they progress through Polya's process is discussed. Students' reflections from this technique are discussed to explore the outcomes that may be expected from establishing an environment where students are encouraged to develop mathematical habits of the mind.     

Constraint propagation,relational arithmetic in AI systems and mathematical programs

This paper explores the interrelationships between methods developed in mathematical programming to discover the structure of constraint (feasibility) sets and constraint propagation over networks used by some AI systems to perform inferences about quantities. It is shown that some constraint set problems in mathematical programming are equivalent to inferencing problems for constraint networks with interval labels. This makes the inference and query capabilities associated with AI systems that use logic programming, directly accessible to mathematical programming systems. On the other hand, traditional and newer methods which mathematical programming uses to obtain information about its associated feasibility set can be used to determine the propagation of constraints in a network of nodes of an AI system. When viewed from this point of view, AI problems can access additional mathematical programming analytical tools including new ways to incorporate qualitative data into constraint sets via interval and fuzzy arithmetic.This work was partially supported by the Industrial Consortium to Develop an Intelligent Mathematical Programming System — Amoco Oil Company, General Research Corporation, Ketron Management Science, Shell Oil Company, MathPro, and US West Advanced Technologies.     

Generalizations of Tikhonov’s regularized method of least squares to non-Euclidean vector norms

Tikhonov’s regularized method of least squares and its generalizations to non-Euclidean norms, including polyhedral, are considered. The regularized method of least squares is reduced to mathematical programming problems obtained by “instrumental” generalizations of the Tikhonov lemma on the minimal (in a certain norm) solution of a system of linear algebraic equations with respect to an unknown matrix. Further studies are needed for problems concerning the development of methods and algorithms for solving reduced mathematical programming problems in which the objective functions and admissible domains are constructed using polyhedral vector norms.     

Two-Stage Approach for Quantitative Policy Analysis Using Bilevel Programming

Quantitative policy analysis problems with hierarchical decision-making can be modeled as bilevel mathematical programming problems. In general, the solution of these models is very difficult; however, special cases exist in which an optimal solution can be obtained by ordinary mathematical programming techniques. In this paper, a two-stage approach for the formulation, construction, solution, and usage of bilevel policy problem is presented. An outline of an example for analyzing Israel's public expenditure policy is also given.     

Extreme point programming with nonlinear constraints

This paper summarizes previous results obtained by the authors on methods of solving extreme point mathematical programming problems with linear constraints. It is also shown how these results can be extended to yield an algorithm for solving extreme point mathematical programming problems with nonlinear constraints. Numerical examples to illustrate the algorithms are included.     

关于工科高等数学教育改革的一些思考

分析了当前工科高等数学教育存在的问题以及产生的原因,提出了工科院校高等数学教育改革的定位和措施.这些措施将对高等数学教学起到积极影响.     

矩形图元下料问题的优化模型

本对下料问题中的一刀切原则给出了一个数学描述，从而给出了一个关于矩形料板约束二维一刀切下料问题的优化模型。     

中国数学规划学科发展概述

数学规划又称数学优化, 是运筹学的一个重要分支. 它主要研究在一定约束条件下, 如何求一个实数或者整数变量的实函数的最大值或者最小值. 它是运筹学和管理科学中最常用的一种建模工具和求解问题的方法, 在工程、经济和金融等领域有非常广泛的应用. 首先简单介绍数学规划的发展历史、应用领域及其主要研究方向; 然后简述数学规划的发展现状和在中国的发展进程; 最后, 讨论数学规划若干研究前沿问题与研究展望.     

An approach to find redundant objective function(s) and redundant constraint(s) in multi-objective nonlinear stochastic fractional programming problems

Structural redundancies in mathematical programming models are nothing uncommon and nonlinear programming problems are no exception. Over the past few decades numerous papers have been written on redundancy. Redundancy in constraints and variables are usually studied in a class of mathematical programming problems. However, main emphasis has so far been given only to linear programming problems. In this paper, an algorithm that identifies redundant objective function(s) and redundant constraint(s) simultaneously in multi-objective nonlinear stochastic fractional programming problems is provided. A solution procedure is also illustrated with numerical examples. The proposed algorithm reduces the number of nonlinear fractional objective functions and constraints in cases where redundancy exists.     

Airport noise modelling and aircraft scheduling so as to minimize community annoyance

A mathematical model of the annoyance created at an airport by aircraft operations is developed. The model incorporates population distribution considerations around an airport and the annoyance caused by aircraft noise. The objective function of this model corresponds to seeking to minimize total population annoyance created by all aircraft operations in a 24-hour period. Several factors are included in this model as constraint relationships. Aircraft operations by type and time period are upper bounded. Demand for flight services is incorporated by including lower bounds on the number of operations by type of aircraft, runway used and time period. Also upper bounds on the number of operations for each runway are included. The mathematical model as formulated is recognized as corresponding to a nonlinear integer mathematical programming problem.The solution technique selected makes use of a successive linear approximation optimization algorithm. An especially attractive feature of this solution algorithm is that it is capable of obtaining solutions to large problems. For example, it would be feasible to attempt the solution of problems involving several thousand variables and over 500 linear constraints. This suggested solution algorithm was implemented on a computer and computational results obtained for example problems.     

Contact problems for bodies with complex coatings

The paper deals with various statements and mathematical models of contact and contact-wear problems for bodies with coatings. It is shown that the mathematical models for a number of such problems can be represented as a mixed integral equation or a system of mixed integral equations with additional conditions. It is also shown that these equations contain rapidly varying or even discontinuous functions in the case of interaction between bodies of complex shape and with some surface properties. Therefore, it is necessary to use a special approach for constructing efficient analytic solutions. Its implementation is demonstrated by an example.     

基于逻辑关系的数学模型—逻辑模型的理论与分析

用数学模型研究实际问题是现代科学研究的常用方法.通常采用的数学模型是各种方程.但是使用方程作为研究手段也存在着许多问题,例如无法应用于不可计算的或者不具有数量概念的实际情况中,这样许多问题无法加以讨论.以命题为基础,通过数理逻辑的概念和方法,建立了具有实际意义的逻辑模型的一般理论,分析了逻辑模型的一些基本性质.逻辑模型可以看成传统模型的一种推广.