A new mixed integer linear programming model for product development using quality function deployment

Quality function deployment (QFD) is a product development process performed to maximize customer satisfaction. In the QFD, the design requirements (DRs) affecting the product performance are primarily identified, and product performance is improved to optimize customer needs (CNs). For product development, determining the fulfillment levels of design requirements (DRs) is crucial during QFD optimization. However, in real world applications, the values of DRs are often discrete instead of continuous. To the best of our knowledge, there is no mixed integer linear programming (MILP) model in which the discrete DRs values are considered. Therefore, in this paper, a new QFD optimization approach combining MILP model and Kano model is suggested to acquire the optimized solution from a limited number of alternative DRs, the values of which can be discrete. The proposed model can be used not only to optimize the product development but also in other applications of QFD such as quality management, planning, design, engineering and decision-making, on the condition that DR values are discrete. Additionally, the problem of lack of solutions in integer and linear programming in the QFD optimization is overcome. Finally, the model is illustrated through an example.     

The progressive party problem: Integer linear programming and constraint programming compared

Many discrete optimization problems can be formulated as either integer linear programming problems or constraint satisfaction problems. Although ILP methods appear to be more powerful, sometimes constraint programming can solve these problems more quickly. This paper describes a problem in which the difference in performance between the two approaches was particularly marked, since a solution could not be found using ILP.The problem arose in the context of organizing a progressive party at a yachting rally. Some yachts were to be designated hosts; the crews of the remaining yachts would then visit the hosts for six successive half-hour periods. A guest crew could not revisit the same host, and two guest crews could not meet more than once. Additional constraints were imposed by the capacities of the host yachts and the crew sizes of the guests.Integer linear programming formulations which included all the constraints resulted in very large models, and despite trying several different strategies, all attempts to find a solution failed. Constraint programming was tried instead and solved the problem very quickly, with a little manual assistance. Reasons for the success of constraint programming in this problem are identified and discussed.     

一种求解整数规划与混合整数规划非线性罚函数方法

证明了任何一个变量有界的整数规划问题（IP）和混合整数规划问题（MIP）都可以转化为一个等价的非整数（或连续化）规划问题（NIP），并给出一个用非线性精确罚函数法来求解该等价NIP的方法，从而达到求解IP或MIP的目的，数值实验表明了算法的可行性。该方法可广泛用于各应用领域里IP和MIP的求解，特别是为非线性IP和MIP问题提供了一条通用 的求解途径，对解决许多实际优化问题具有重要意义。     

An exact algorithm for the bilevel mixed integer linear programming problem under three simplifying assumptions

We present an exact algorithm for the bilevel mixed integer linear programming (BMILP) problem under three simplifying assumptions. Although BMILP has been studied for decades and widely applied to various real world problems, there are only a few BMILP algorithms. Compared to these existing ones, our new algorithm relies on fewer and weaker assumptions, explicitly considers finite optimal, infeasible, and unbounded cases, and is proved to terminate finitely and correctly. We report results of our computational experiments on a small library of BMILP test instances, which we created and made publicly available online.     

高效求解整数线性规划问题的分支算法

为了提高求解一般整数线性规划问题的效率,提出了一种基于目标函数超平面移动的分支算法。对于给定的目标函数整数值,首先利用线性规划松弛问题的最优单纯形表确定变量的上、下界,然后将变量的上、下界条件加入约束条件中对相应的目标函数超平面进行切割,最后应用分支定界算法中的分支方法来搜寻目标函数超平面上的可行解。通过对一些经典的数值例子的求解计算并与经典的分支定界算法进行比较,结果表明,该算法减少了分支数和单纯形迭代数,具有较大的实用价值。     

Improving the efficiency of a mixed integer linear programming based approach for multi-class classification problem

Data classification is one of the fundamental issues in data mining and machine learning. A great deal of effort has been done for reducing the time required to learn a classification model. In this research, a new model and algorithm is proposed to improve the work of Xu and Papageorgiou (2009). Computational comparisons on real and simulated patterns with different characteristics (including dimension, high overlap or heterogeneity in the attributes) confirm that, the improved method considerably reduces the training time in comparison to the primary model, whereas it generally maintains the accuracy. Particularly, this speed-increase is significant in the case of high overlap. In addition, the rate of increase in training time of the proposed model is much less than that of the primary model, as the set-size or the number of overlapping samples is increased.     

Stability radius of an efficient solution of a vector problem of integer linear programming in the Gölder metric

A vector (multicriterion) problem of integer linear programming is considered on a finite set of feasible solutions. A metric l_p, 1 ≤ p ≤ ∞, is defined on the parameter space of the problem. A formula of the maximum permissible level of perturbations is obtained for the parameters that preserve the efficiency (Pareto optimality) of a given solution. Necessary and sufficient conditions of two types of stability of the problem are obtained as corollaries. This work has been carried out with financial support from the Belgosuniversity within the framework of the Intercollegiate Program “Fundamental and Applied Investigations” (project No. 492/28). __________ Translated from Kibernetika I Sistemnyi Analiz, No. 4, pp. 175–181, July–August 2006.     

Fractional programming formulation for the vertex coloring problem

We devise a new formulation for the vertex coloring problem. Different from other formulations, decision variables are associated with pairs of vertices. Consequently, colors will be distinguishable. Although the objective function is fractional, it can be replaced by a piece-wise linear convex function. Numerical experiments show that our formulation has significantly good performance for dense graphs.     

多周期混合整数规划购油模型及应用

采用PIMS软件中的多周期混合整数规划技术建立炼油企业购油计划模型，使优化结果与实际购油方式相吻合；采用虚拟周期方法解决原油期末库存质量控制问题；采用滚动处理方式解决炼厂月、季原油选购计划的衔接和全局优化问题。文中还给出了多周期MIP模型技术在某炼厂中的应用以及不同方案的效益对比。     

Applying software engineering techniques in the development and management of linear and integer programming applications

This work addresses characteristics of software environments for mathematical modeling and proposes a system for developing and managing models of linear and integer programming (IP) problems. The main features of this modeling environment are: version control of models and data; client‐server architecture, which allows the interaction among modelers and decision makers; the use of a database to store information about the models and data scenarios; and the use of remote servers of optimization, which allows the optimization problems to be solved on different machines. The modeling environment proposed in this work was validated using mathematical programming models that exploit different characteristics, such as the treatment of conditions for generating variables and constraints, the use of calculated parameters derived from other parameters, and the use of integer and continuous variables in mixed IP models among others. This validation showed that the proposed environment is able to treat models found in various application areas of operations research and to solve problems with tens of thousands of variables and constraints.     

A Hybrid approach for integer programming combining genetic algorithms, linear programming and ordinal optimization

Hybrid methods are promising tools in integer programming, as they combine the best features of different methods in a complementary fashion. This paper presents such a framework, integrating the notions of genetic algorithm, linear programming, and ordinal optimization in an effort to shorten computation times for large and/or difficult integer programming problems. Capitalizing on the central idea of ordinal optimization and on the learning capability of genetic algorithms to quickly generate good feasible solutions, and then using linear programming to solve the problem that results from fixing the integer part of the solution, one may be able to obtain solutions that are close to optimal. Indeed ordinal optimization guarantees the quality of the solutions found. Numerical testing on a real-life complex scheduling problem demonstrates the effectiveness and efficiency of this approach.     

Regularization of a lexicographic vector problem of integer programming

A stability criterion for a vector integer linear problem of lexicographic optimization is obtained. A regularization method is proposed that allows us to reduce a possible unstable output problem to a sequence of perturbed stable equivalent problems. Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 125–130, November–December, 1999.     

Inter‐sample avoidance in trajectory optimizers using mixed‐integer linear programming

This paper proposes an extension to trajectory optimization using mixed‐integer linear programming. The purpose of the extension is to ensure that avoidance constraints are respected at all times between discrete samples, not just at the sampling times themselves. The method is very simple and involves applying the same switched constraints at adjacent time steps. This requires fewer additional constraints than the existing approach and is shown to reduce computation time. A key benefit of efficient inter‐sample avoidance is the facility to reduce the number of time steps without having to compensate by enlarging the obstacles. A further extension to the principle is presented to account for curved paths between samples, proving useful in cases where narrow passageways are traversed. Copyright © 2013 John Wiley & Sons, Ltd.     

Integer linear programming for a cutting problem in the wood-processing industry: a case study

In this paper, the authors present a case study from the wood-processing industry. It focuses on a cutting process in which material from stock is cut down in order to provide the items required by the customers in the desired qualities, sizes, and quantities. In particular, two aspects make this cutting process special. Firstly, the cutting process is strongly interdependent, with a preceding handling process, which, consequently, cannot be planned independently. Secondly, if the trim loss is of a certain minimum size, it can be returned into stock and used as input to subsequent cutting processes. In order to reduce the cost of the cutting process, a decision support tool has been developed that incorporates an integer linear programming model as a central feature. The model is described in detail, and experience from the application of the tool is reported.     

求解整数线性规划的一种高效隐数搜寻

提出了一种求解整数线性规划的新的隐数算法。首先,该算法引入了一组线性变换,将线性松弛问题的最优非基变量变换到一组新变量,使新变量有更小的取值范围。然后,在目标函数超平面上对非基变量和新变量进行隐数计算,从而大大提高了隐数搜寻的效率。     

Regression test suite minimization using integer linear programming model

Software testers always face the dilemma of whether to retest the software with all the test cases or select a few of them on the basis of their fault detection ability. This paper introduces a novel approach to minimizing the test suite as an integer linear programming problem with optimal results. The minimization method uses the cohesion values of the program parts affected by the changes made to the program. The hypothesis is that the program parts with low cohesion values are more prone to errors. This assumption is validated on the mutation fault detection ability of the test cases. The experimental study carried out on 30 programs evaluates the effectiveness and usefulness of the proposed framework. The experimental results show that the minimized test suite can efficiently reveal the errors and ensure acceptable software quality. Copyright © 2017 John Wiley & Sons, Ltd.     

New integer linear programming approaches for course timetabling

The most complete form of academic timetabling problem is the population and course timetabling problem. In this problem, there may be multiple classes of each subject, and the decision on which students are to constitute each class is made in concert with the decision on the timetable for each class. In order to solve this problem, it is normally simplified or decomposed in some fashion. One simplification commonly used in practice is known as blocking: it is assumed that the classes can be partitioned into sets of classes (or blocks) that will be timetabled in parallel. This restricts clashing to occur only between classes in the same block, and essentially removes the timetabling aspect of the problem, which can be carried out once the blocks are constituted and the classes populated. The problem of constituting the blocks and populating the classes, known as the course blocking and population problem, is nevertheless a challenging problem, and provides the focus of this paper. We demonstrate, using data provided by a local high school, that integer linear programming approaches can solve the problem in a matter of seconds. Key features include remodelling to remove symmetry caused by students with identical subject selection, and the observation that in practice, only integrality of the block composition variables needs to be enforced; the class population aspects of the model have strong integrality properties.     

Target setting with minimum improving costs in data envelopment analysis: A mixed integer linear programming approach

This study addresses a problem called cost‐minimizing target setting in data envelopment analysis (DEA) methodology. The problem is how to make an inefficient decision‐making unit efficient by allocating to it as few organizational resources as possible, assuming that the marginal costs of reducing inputs or increasing outputs are known and available, which is different from previous furthest and closest DEA targets setting methods. In this study, an existed cost minimizing target setting heuristics approach based on input‐oriented model is examined to show that there exist some limitations. This study develops a simple mixed integer linear programming to determine the desired targets on the strongly efficient frontier based on non‐oriented DEA model considering the situation in the presence of known marginal costs of reducing inputs and increasing outputs simultaneously. Some experiments with the simulated datasets are conducted, and results show that the proposed model can obtain more accurate projections with lower costs compared with those from furthest and closest target setting approaches. Besides, the proposed model can be realistic and efficient in solving cost‐minimizing target setting problem.     

A genetic algorithm for interval nonlinear integer programming problem

In this paper, we formulate an optimal design of system reliability problem as a nonlinear integer programming problem with interval coefficients, transform it into a single objective nonlinear integer programming problem without interval coefficients, and solve it directly with keeping nonlinearity of the objective function by using Genetic Algorithms (GA). Also, we demonstrate the efficiency of this method with incomplete Fault Detecting and Switching (FDS) for allocating redundant units.     

求解整数线性规划问题的定界阻止算法的改进*

在现有求解整数线性规划问题的定界阻止算法的基础上提出了一种改进。该算法通过目标函数超平面截线性规划松弛问题的有效约束锥而形成一个单纯形;然后,引入一串平行片来切割该单纯形产生更低维的凸多面体;最后,在片上的这些凸多面体上执行阻止搜寻程序。由于单纯形和片上凸多面体的极顶点可以直接通过公式计算,且变量在片上凸多面体上的取值区间更窄,改进的定界阻止算法既方便又高效,这得到了一些经典算例和随机产生的算例的验证。