A note on “Solving linear programming problems under fuzziness and randomness environment using attainment values”

This paper is an amendment to Hop’s paper [N.V. Hop, Solving linear programming problems under fuzziness and randomness environment using attainment values, Information Sciences 177 (2007) 2971-2984], in solving linear programming problems under fuzziness and randomness environments. Hop introduced a new characterization of relationship, attainment values, to enable the conversion of fuzzy (stochastic) linear programming models into corresponding deterministic linear programming models. The purpose of this paper is to provide a correction and an improvement of Hop’s analytical work through rationalization and simplification. More importantly, it is shown that Hop’s analysis does not support his demonstration or the solution-finding mechanism; the attainment values approach as he had proposed does not result in superior performance as compared to other existing approaches because it neglects some relevant and inevitable theoretical essentials. Two numerical examples from Hop’s paper are also employed to show that his approach, in the conversion of fuzzy (stochastic) linear programming problems to corresponding problems, is questionable and can neither find the maximum nor the minimum in the examples. The models of the examples are subsequently amended in order to derive the correct optimal solutions.     

A new mathematical programming approach to multi-group classification problems

In this paper we introduce a goal programming formulation for the multi-group classification problem. Although a great number of mathematical programming models for two-group classification problems have been proposed in the literature, there are few mathematical programming models for multi-group classification problems. Newly proposed multi-group mathematical programming model is compared with other conventional multi-group methods by using different real data sets taken from the literature and simulation data. A comparative analysis on the real data sets and simulation data shows that our goal programming formulation may suggest efficient alternative to traditional statistical methods and mathematical programming formulations for the multi-group classification problem.     

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.     

Several multi-criteria programming methods for classification

In this paper, we propose a general optimization-based model for classification. Then we show that some well-known optimization-based methods for classification, which were developed by Shi et al. [Data mining in credit card portfolio management: a multiple criteria decision making approac. In: Koksalan M, Zionts S, editors. Multiple criteria decision making in the new millennium. Berlin: Springer; 2001. p. 427–36] and Freed and Glover [A linear programming approach to the discriminant problem. Decision Sciences 1981; 12: 68–79; Simple but powerful goal programming models for discriminant problems. European Journal of Operational Research 1981; 7: 44–60], are special cases of our model. Moreover, three new models, MCQP (multi-criteria indefinite quadratic programming), MCCQP (multi-criteria concave quadratic programming) and MCVQP (multi-criteria convex programming), are developed based on the general model. We also propose algorithms for MCQP and MCCQP, respectively. Then we apply these models to three real-life problems: credit card accounts, VIP mail-box and social endowment insurance classification. Extensive experiments are done to compare the efficiency of these methods.     

多下层分式双层规划问题的改进遗传算法

针对一类具有多个线性下层问题的分式双层规划, 提出一种基于新编码方式的遗传算法。 首先,利用对偶理论,将问题化为单层非线性规划;接着,利用下层对偶问题的可行基编码,针对任意编码个体,解出对偶变量值,使得单层规划变为线性分式规划;最后,求解产生的线性分式规划,其目标值作为个体的适应度值。 这种编码方式及适应度的计算有效提高了遗传算法的效率。 通过对4个算例的计算,验证了算法的有效性。     

Efficient integer programming formulations for optimum sink location and routing in heterogeneous wireless sensor networks

Sensors spend most of their limited battery energy on communicating the collected environmental information to sinks. Therefore, the determination of the optimal sink locations and sensor-to-sink information flow routes becomes important for the survivability of sensor networks. In this work, we address these important design issues using an integrated approach and propose new mixed-integer linear programming models to determine the optimal sink locations and information flow paths between sensors and sinks when sensor locations are given. The first group of proposed models is energy-aware and tries to minimize total routing energy, whereas the second group is financially driven with the objective of minimizing total cost. We do not only report computational results providing information on the solution efficiency of the new formulations, and the accuracy of their linear programming relaxations, but also propose and test new heuristics and lower bounding approaches for the most efficient formulation.     

求解支付值为区间直觉模糊数的矩阵对策的线性规划方法

提出了支付值为区间直觉模糊集的矩阵对策定义及其解的概念, 将求解局中人的极大-极小与极小-极大策略问题转化为求解一对辅助的非线性多目标规划, 进而转化为一对易于求解的原始-对偶线性规划. 数值实例表明了所提方法的有效性和实用性. 所提出的区间直觉模糊集矩阵对策理论与方法既是对经典矩阵对策理论的发展, 又可为解决其他带有区间直觉模糊信息的对策问题提供新的途径.

     

Multi-coefficients goal programming

In today’s increasingly competitive business environment, maintaining profit margins or quantities of goods sold is an important issue for businesses. Accordingly, more and more industries use group pricing discrimination strategy to attract potential customers in order to increase competitive advantage. Hence, to find ways maximize profit and to minimize total cost, group pricing discrimination strategy has become an important issue for decision makers. Unfortunately, these types of problems cannot be solved by any current goal programming models. The objective of this study is to deduce a new method, which we call the multi-coefficients goal programming, for group pricing discrimination problems. In addition, an example is given to illustrate the correctness and usefulness of the proposed model.     

Dependent-chance programming: A class of stochastic optimization

This paper provides a theoretical framework of dependent-chance programming, as well as dependent-chance multiobjective programming and dependent-chance goal programming which are new types of stochastic optimization. A stochastic simulation based genetic algorithm is also designed for solving dependent-chance programming models.     

利用产生式编程构建低耦合的软件模块--AOP的原理和实践

分析了面向对象理论遇到的难以解决的问题。针对此类问题提出了利用产生式编程构建通用领域模型和低耦合的模块的思想。以AspectOrientedProgramming(AOP)为例,列举了其主要实现手段,分析了它们的利弊,对比了传统OO方法的Observer模式实现和利用AOP的Observer模式实现。     

A STEP-compliant Industrial Robot Data Model for robot off-line programming systems

Recently, various robot off-line programming systems have promoted their own robot data models, resulting in a plethora of robot representation methods and unchangeable data files among CAx and robot off-line programming systems. The current paper represents a STEP-compliant Industrial Robot Data Model (IRDM) for data exchange between CAx systems and robot off-line programming systems. Using this novel representation method, most resources involved in a robot manufacturing system can be represented. The geometric and mathematic aspects of industrial robots have been defined in IRDM, so that the robot off-line programming system could have abundant information to represent robots’ kinematic and dynamic behaviors. In order to validate the proposed models and approaches, a prototype robot off-line programming system with 3D virtual environment is presented. The functionalities of IRDM not only have significant meaning for providing a unified data platform for robot simulation systems, but also have the potential capability to represent robot language using the object-oriented concept.     

A matheuristic for the firefighter problem on graphs

In this paper, we describe a computational study conducted on The Firefighter Problem (FFP ), which models fire spreading and containment in a graph. Once the fire breaks out on a set of vertices, the goal is to save as many vertices as possible with limited resources. Practical applications of the FFP occur in areas such as disease control and network security. The FFP is NP‐hard and heuristics have been proposed earlier. Our main contributions include improvements to an existing integer linear programming formulation that led to an average speedup of two to compute exact solutions. Moreover, we developed a novel matheuristic, a technique based on the interoperation between metaheuristics and mathematical programming. We performed extensive experiments on public benchmarks both for parameter tuning and for comparison of our results with those from the literature. A rigorous statistical analysis shows that our new matheuristic outperforms the existing approaches.     

In the determination of the most efficient decision making unit in data envelopment analysis

In recent years, several mixed integer linear programming (MILP) models have been proposed for determining the most efficient decision making unit (DMU) in data envelopment analysis. However, most of these models do not determine the most efficient DMU directly; instead, they make use of other less related objectives. This paper introduces a new MILP model that has an objective similar to that of the super-efficiency model. Unlike previous models, the new model’s objective is to directly discover the most efficient DMU. Similar to the super-efficiency model, the aim is to choose the most efficient DMU. However, unlike the super-efficiency model, which requires the solution of a linear programming problem for each DMU, the new model requires that only a single MILP problem be solved. Consequently, additional terms in the objective function and more constraints can be easily added to the new model. For example, decision makers can more easily incorporate a secondary objective such as adherence to a publicly stated preference or add assurance region constraints when determining the most efficient DMU. Furthermore, the proposed model is more accurate than two recently proposed models, as shown in two computational examples.     

Solving multiple objective quasi-convex goal programming problems by linear programming

This study presents a novel means of resolving multiple objective goal programming (GP) problems with quasi-convex linear penalty functions. The proposed method initially expresses a quasi-convex function by the maximum operator of two convex functions, then solves it via a linear programming technique. The proposed method does not contain any zero–one variables; nor does it require dividing the multi-objective quasi-convex GP problem into large sub-problems as in conventional methods. Some illustrative examples are provided.     

Gene expression programming based due date assignment in a simulated job shop

In this paper, a new approach for due date assignment in a multi-stage job shop is proposed and evaluated. The proposed approach is based on a genetic programming technique which is known as gene expression programming (GEP). GEP is a relatively new member of the genetic programming family. The primary objective of this research is to compare the performance of the proposed due date assignment model with several previously proposed conventional due date assignment models. For this purpose, simulation models are developed and comparisons of the due date assignment models are made mainly in terms of the mean absolute percent error (MAPE), mean percent error (MPE) and mean tardiness (MT). Some additional performance measurements are also given. Simulation experiments revealed that for many test conditions the proposed due date assignment method dominates all other compared due date assignment methods.     

Multiple-criteria decision-making in two-sided assembly line balancing: A goal programming and a fuzzy goal programming models

Two-sided assembly lines are especially used at the assembly of large-sized products, such as trucks and buses. In this type of a production line, both sides of the line are used in parallel. In practice, it may be necessary to optimize more than one conflicting objectives simultaneously to obtain effective and realistic solutions. This paper presents a mathematical model, a pre-emptive goal programming model for precise goals and a fuzzy goal programming model for imprecise goals for two-sided assembly line balancing. The mathematical model minimizes the number of mated-stations as the primary objective and it minimizes the number of stations as a secondary objective for a given cycle time. The zoning constraints are also considered in this model, and a set of test problems taken from literature is solved. The proposed goal programming models are the first multiple-criteria decision-making approaches for two-sided assembly line balancing problem with multiple objectives. The number of mated-stations, cycle time and the number of tasks assigned per station are considered as goals. An example problem is solved and a computational study is conducted to illustrate the flexibility and the efficiency of the proposed goal programming models. Based on the decision maker's preferences, the proposed models are capable of improving the value of goals.     

An interval programming approach for the bilevel linear programming problem under fuzzy random environments

In this paper, we address a class of bilevel linear programming problems with fuzzy random variable coefficients in objective functions. To deal with such problems, we apply an interval programming approach based on the $\alpha $ -level set to construct a pair of bilevel mathematical programming models called the best and worst optimal models. Through expectation optimization model, the best and worst optimal problems are transformed into the deterministic problems. By means of the Kth best algorithm, we obtain the best and worst optimal solutions as well as the corresponding range of the objective function values. In this way, more information can be provided to the decision makers under fuzzy random circumstances. Finally, experiments on two examples are carried out, and the comparisons with two existing approaches are made. The results indicate the proposed approaches can get not only the best optimal solution (ideal solution) but also the worst optimal solution, and is more reasonable than the existing approaches which can only get a single solution (ideal solution).     

Solving the bi‐objective capacitated p‐median problem with multilevel capacities using compromise programming and VNS

A bi‐objective optimisation using a compromise programming (CP) approach is proposed for the capacitated p‐median problem (CPMP) in the presence of the fixed cost of opening facility and several possible capacities that can be used by potential facilities. As the sum of distances between customers and their facilities and the total fixed cost for opening facilities are important aspects, the model is proposed to deal with those conflicting objectives. We develop a mathematical model using integer linear programming (ILP) to determine the optimal location of open facilities with their optimal capacity. Two approaches are designed to deal with the bi‐objective CPMP, namely CP with an exact method and with a variable neighbourhood search (VNS) based matheuristic. New sets of generated instances are used to evaluate the performance of the proposed approaches. The computational experiments show that the proposed approaches produce interesting results.     

动态规划算法在基于子区间消除的随机点定位问题中的应用

针对原有基于判决方程的子区间消除算法中所存在的判决结果与决策表不相符,以及当子区间划分规模增大时,运行时间呈平方次增长的问题,本文提出了一种全新的基于动态规划的子区间消除算法。新算法充分利用动态规划在多阶段决策问题中的卓越性能,将子区间的消除问题划分为合理性判断和新区间生成两部分,这两个部分均可以利用动态规划中子问题分割的思想来解决。文中证明了通过解决这些子问题可以构造得到原问题的最优解,分析了算法的时间复杂度和空间复杂度。为了检验新算法的性能,本文从理论和实验两种维度,进行了新旧两种算法的对比。实验结果表明,该方法大大降低了算法的时间复杂度,有效克服了子区间规模增大所导致的问题,提高了算法的灵活性和运行速度。     

Control applications of nonlinear convex programming

Since 1984 there has been a concentrated effort to develop efficient interior-point methods for linear programming (LP). In the last few years researchers have begun to appreciate a very important property of these interior-point methods (beyond their efficiency for LP): they extend gracefully to nonlinear convex optimization problems. New interior-point algorithms for problem classes such as semidefinite programming (SDP) or second-order cone programming (SOCP) are now approaching the extreme efficiency of modern linear programming codes. In this paper we discuss three examples of areas of control where our ability to efficiently solve nonlinear convex optimization problems opens up new applications. In the first example we show how SOCP can be used to solve robust open-loop optimal control problems. In the second example, we show how SOCP can be used to simultaneously design the set-point and feedback gains for a controller, and compare this method with the more standard approach. Our final application concerns analysis and synthesis via linear matrix inequalities and SDP.