天课时分配下的多阶段新高考走班制排课算法

伴随新高考改革,高中排课过程需要考虑学生的科目选择。潜在的学生上课时间冲突提高了排出可行课表的难度,排课过程中对课表的复杂要求也更难得到满足。针对这些挑战提出一种多阶段优化算法解决高中“走班制”教学课程时间表优化问题。优化侧重点从课表时段分配转为天课时分配,即对每个课程班每天的课时数目进行决策。除需要满足课时不冲突的约束条件外,主要优化目标为“课时分布均匀”“教案平齐”“同时上课”。根据问题特点设计了三种课表变换算子用于在教学班天课时分配阶段提升新设计的爬山算法的寻优能力。在三组不同难度和规模的实验数据上,多阶段优化算法以高于85%的概率排出可行课表。相较大规模真实案例,人工生成案例和中规模真实案例在目标函数上得到较为理想的优化。整体课表的教案平齐违反主要源于行政班课表。发现同时上课的设置具有指导其他目标函数优化的能力。     

Model-based problem solving for university timetable validation and improvement

Constraint satisfaction problems can be expressed very elegantly in state-based formal methods such as B. But can such specifications be directly used for solving real-life problems? In other words, can a formal model be more than a design artefact but also be used at runtime for inference and problem solving? We will try and answer this important question in the present paper with regard to the university timetabling problem. We report on an ongoing project to build a curriculum timetable validation tool where we use a formal model as the basis to validate timetables from a student’s perspective and to support incremental modification of timetables. In this article we describe the problem domain, the formalization in B and our approach to execute the formal model in a production system using ProB.     

Computing delay resistant railway timetables

In the past, much research has been dedicated to compute optimum railway timetables. A typical objective has been the minimization of passenger waiting times. But only the planned nominal waiting times have been addressed, whereas delays as they occur in daily operations have been neglected. Delays have been rather treated mainly in an online context and solved as a separate optimization problem, called delay management.We provide the first computational study which aims at computing delay resistant periodic timetables. In particular we assess the delay resistance of a timetable by evaluating it subject to several delay scenarios to which optimum delay management will be applied.We arrive at computing delay resistant timetables by selecting a new objective function which we design to be somehow in the middle of the traditional simple timetabling objective and the sophisticated delay management objective. This is a slight extension of the concept of “light robustness” (LR) as it has been proposed by Fischetti and Monaci [2006. Robust optimization through branch-and-price. In: Proceedings of AIRO]. Moreover, in our application we are able to provide accurate interpretations for the ingredients of LR. We apply this new technique to real-world data of a part of the German railway network of Deutsche Bahn AG. Our computational results suggest that a significant decrease of passenger delays can be obtained at a relatively small price of robustness, i.e. by increasing the nominal travel times of the passengers.     

Multiobjective multiproduct parcel distribution timetabling: a real‐world application

Multiobjective multiproduct parcel distribution timetabling problem is concerned with generating effective timetables for parcel distribution companies that provide interdependent services (products) and have more than one objective. A parcel distribution timetabling problem is inherently multiobjective because of the multitude of criteria that can measure the performance of a timetable. This paper provides the mathematical formulation of the problem and applies the model to a real‐world case study. The application shows that without a common ground with the practitioners, it would be impossible to define the actual requirements and objectives of the company; problem definition is as important as model construction and solution method.     

Scalable and exact sampling method for probabilistic generative graph models

Interest in modeling complex networks has fueled the development of multiple probabilistic generative graph models (PGGMs). PGGMs are statistical methods that model the network distribution and match common characteristics of real world networks. Recently, scalable sampling algorithms for well known PGGMs, made the analysis of large-scale, sparse networks feasible for the first time. However, it has been demonstrated that these scalable sampling algorithms do not sample from the original underlying distribution, and sometimes produce very unlikely graphs. To address this, we extend the algorithm proposed in Moreno et al. (in: IEEE 14th international conference on data mining, pp 440–449, 2014) for a single model and develop a general solution for a broad class of PGGMs. Our approach exploits the fact that PGGMs are typically parameterized by a small set of unique probability values—this enables fast generation via independent sampling of groups of edges with the same probability value. By sampling within groups, we remove bias due to conditional sampling and probability reallocation. We show that our grouped sampling methods are both provably correct and efficient. Our new algorithm reduces time complexity by avoiding the expensive rejection sampling step previously necessary, and we demonstrate its generality, by outlining implementations for six different PGGMs. We conduct theoretical analysis and empirical evaluation to demonstrate the strengths of our algorithms. We conclude by sampling a network with over a billion edges in 95 s on a single processor.     

A Case Study in Periodic Timetabling

In the overwhelming majority of public transportation companies, designing a periodic timetable is even nowadays largely performed manually. Software tools only support the planners in evaluating a periodic timetable, or by letting them comfortably shift sets of trips by some minutes, but they rarely use optimization methods. One of the main arguments against optimization is that there is no clear objective in practice, but that many criteria such as amount of rolling stock required, average passenger changing time, average speed of the trains, and the number of cross-wise correspondences have to be considered.This case study will demonstrate on the example of the Berlin underground (BVG) that all these goals can be met if carefully modeled, and that timetables constructed by optimization lead to considerable improvements.Our approach uses the Periodic Event Scheduling Problem (PESP) with several add-ons concerning problem reduction and strengthening. The resulting integer linear programs are solved with the CPLEX MIP-Solver. We have been able to construct periodic timetables that improve the current timetable considerably. For any of the above criteria, we have been able to identify global lower and upper bounds. Our favorite timetable improves the current BVG timetable in each of these criteria.     

Increasing the effectiveness of associative classification in terms of class imbalance by using a novel pruning algorithm

Having received considerable interest in recent years, associative classification has focused on developing a class classifier, with lesser attention paid to the probability classifier used in direct marketing. While contributing to this integrated framework, this work attempts to increase the prediction accuracy of associative classification on class imbalance by adapting the scoring based on associations (SBA) algorithm. The SBA algorithm is modified by coupling it with the pruning strategy of association rules in the probabilistic classification based on associations (PCBA) algorithm, which is adjusted from the CBA for use in the structure of the probability classifier. PCBA is adjusted from CBA by increasing the confidence through under-sampling, setting different minimum supports (minsups) and minimum confidences (minconfs) for rules of different classes based on each distribution, and removing the pruning rules of the lowest error rate. Experimental results based on benchmark datasets and real-life application datasets indicate that the proposed method performs better than C5.0 and the original SBA do, and the number of rules required for scoring is significantly reduced.     

Urban rail transit planning using a two-stage simulation-based optimization approach

In urban metro systems, stochastic disturbances occur repeatedly as a result of an increment of demands or travel time variations, therefore, improving the service quality and robustness through minimizing the passengers waiting time is a real challenge. To deal with dwell time variability, travel time and demand uncertainty, a two-stage GA-based simulation optimization approach is proposed in order to minimize the expected passenger waiting times. The proposed method here has the capability of generating robust timetables for a daily operation of a single-loop urban transit rail system. The first stage of the algorithm includes the evaluation of even-headway timetables through simulation experiments. In the second stage, the search space is limited to the uneven-headway patterns in such a manner where the algorithm keeps the average of headways close to the best even-headway timetable, obtained from the first stage. The optimization is intended to adjust headways through simulation experiments. Computational experiments are conducted on Tehran Metropolitan Railway (IRAN) and the outcomes of optimized timetable obtained by this proposed method are demonstrated. This newly proposed two-stage search approach could achieve to a more efficient solution and speed up the algorithm convergence.     

Scatter search technique for exam timetabling

At universities where students enjoy flexibility in selecting courses, the Registrar’s office aims to generate an appropriate exam timetable for numerous courses and large number of students. An appropriate, real-world exam timetable should show fairness towards all students, respecting the following constraints: (a) eliminating or minimizing the number of simultaneous exams; (b) minimizing the number of consecutive exams; (c) minimizing the number of students with two or three exams per day (d) eliminating the possibility of more than three exams per day (e) exams should fit in rooms with predefined capacity; and (f) the number of exam periods is limited. These constraints are conflicting, which makes exam timetabling intractable. Hence, solving this problem in realistic time requires the use of heuristic approaches. In this work, we develop an evolutionary heuristic technique based on the scatter search approach for finding good suboptimal solutions for exam timetabling. This approach is based on maintaining and evolving a population of solutions. We evaluate our suggested technique on real-world university data and compare our results with the registrar’s manual timetable in addition to the timetables of other heuristic optimization algorithms. The experimental results show that our adapted scatter search technique generates better timetables than those produced by the registrar, manually, and by other meta-heuristics.     

A complexity analysis and an algorithmic approach to student sectioning in existing timetables

We show that the following fundamental question in student sectioning can be efficiently decided in polynomial time: Is it possible to assign \(m\) students to \(k\) sectioned courses with respect to a given timetable with \(l\) timeslots such that the individual capacities of all sections are not exceeded and no student has more than one appointment per timeslot? Our result opens the possibility of efficiently checking the feasibility of candidate timetables with respect to this question as part of timetabling algorithms. Based on a succinct representation of solutions, we also provide an algorithm to compute optimal assignments in \(O( k^2l^2 \log (\mathrm{sum}_{A}))\), where \(\mathrm{sum}_{A}\) is the sum of all specified section capacities. On the other hand, we prove that adding any single of the following constraints turns the above question into an NP-complete problem:

• Course-selection constraint: Student’s course-selections must be respected.
• Timeslot constraint: Students have individual timeslot restrictions.
• Multiple-event constraint: Sections may have multiple events in the timetable, and there must be no timeslot clashes between all section-events for each student.

Hence our investigation gives insight into the location of the borderline between efficiently solvable and computationally hard problem variations.

     

NBCC:一种数据流上变化的挖掘算法

针对数据流上变化的挖掘问题,提出了算法NBCC,首先利用精确抽样的方法对数据流构建概要数据结构,然后借鉴经典朴素贝叶斯分类方法的思想,将训练样本集分成Ci类,i=1,2,…,m。对测试样本集设定一个阈值!:当P(Ci｜X)相似文献   

Optimal flight scheduling models for cargo airlines under alliances

In this paper we develop several coordinated scheduling models combining airport selection, fleet routing and timetable setting, in order to help airlines solve for the most satisfactory cargo fleet routes and timetables when they enter into alliances. We employ network flow techniques to construct the models, which are formulated as a multiple commodity network flow problem and can be solved using a mathematical programming solver. To evaluate the models, we perform numerical tests based on real operating data from two Taiwan airlines. The preliminary results are good, showing that the models would be useful for airline alliances.     

The Equivalence of Sampling and Searching

In a sampling problem, we are given an input x∈{0,1} ⁿ , and asked to sample approximately from a probability distribution \(\mathcal{D}_{x}\) over \(\operatorname{poly} ( n ) \) -bit strings. In a search problem, we are given an input x∈{0,1} ⁿ , and asked to find a member of a nonempty set A _x with high probability. (An example is finding a Nash equilibrium.) In this paper, we use tools from Kolmogorov complexity to show that sampling and search problems are “essentially equivalent.” More precisely, for any sampling problem S, there exists a search problem R _S such that, if \(\mathcal{C}\) is any “reasonable” complexity class, then R _S is in the search version of \(\mathcal{C}\) if and only if S is in the sampling version. What makes this nontrivial is that the same R _S works for every  \(\mathcal{C}\) . As an application, we prove the surprising result that SampP=SampBQP if and only if FBPP=FBQP. In other words, classical computers can efficiently sample the output distribution of every quantum circuit, if and only if they can efficiently solve every search problem that quantum computers can solve.     

Multidimensional hierarchical classification

Hierarchical classification can be seen as a multidimensional classification problem where the objective is to predict a class, or set of classes, according to a taxonomy. There have been different proposals for hierarchical classification, including local and global approaches. Local approaches can suffer from the inconsistency problem, that is, if a local classifier has a wrong prediction, the error propagates down the hierarchy. Global approaches tend to produce more complex models. In this paper, we propose an alternative approach inspired in multidimensional classification. It starts by building a multi-class classifier per each parent node in the hierarchy. In the classification phase, all the local classifiers are applied simultaneously to each instance, providing a probability for each class in the taxonomy. Then the probability of the subset of classes, for each path in the hierarchy, is obtained by combining the local classifiers results. The path with highest probability is returned as the result for all the levels in the hierarchy. As an extension of the proposal method, we also developed a new technique, based on information gain, to classifies at different levels in the hierarchy. The proposed method was tested on different hierarchical classification data sets and was compared against state-of-the-art methods, resulting in superior predictive performance and/or efficiency to the other approaches in all the datasets.     

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.     

A one-dimensional analysis for the probability of error of linear classifiers for normally distributed classes

Computing the probability of error is an important problem in evaluating classifiers. When dealing with normally distributed classes, this problem becomes intricate due to the fact that there is no closed-form expression for integrating the probability density function. In this paper, we derive lower and upper bounds for the probability of error for a linear classifier, where the random vectors representing the underlying classes obey the multivariate normal distribution. The expression of the error is derived in the one-dimensional space, independently of the dimensionality of the original problem. Based on the two bounds, we propose an approximating expression for the error of a generic linear classifier. In particular, we derive the corresponding bounds and the expression for approximating the error of Fisher's classifier. Our empirical results on synthetic data, including up to two-hundred-dimensional featured samples, show that the computations for the error are extremely fast and quite accurate; it differs from the actual error in at most ε=0.0184340683. The scheme has also been successfully tested on real-life data sets drawn from the UCI machine learning repository.     

Augmenting learning function to Bayesian network inferences with maximum likelihood parameters

Computing the posterior probability distribution for a set of query variables by search result is an important task of inferences with a Bayesian network. Starting from real applications, it is also necessary to make inferences when the evidence is not contained in training data. In this paper, we are to augment the learning function to Bayesian network inferences, and extend the classical “search”-based inferences to “search + learning”-based inferences. Based on the support vector machine, we use a class of hyperplanes to construct the hypothesis space. Then we use the method of solving an optimal hyperplane to find a maximum likelihood hypothesis for the value not contained in training data. Further, we give a convergent Gibbs sampling algorithm for approximate probabilistic inference with the presence of maximum likelihood parameters. Preliminary experiments show the feasibility of our proposed methods.     

A two-phase heuristic evolutionary algorithm for personalizing course timetables: a case study in a Spanish university

This paper presents, as a case study, the application of a two-phase heuristic evolutionary algorithm to obtain personalized timetables in a Spanish university. The algorithm consists of a two-phase heuristic, which, starting from an initial ordering of the students, allocates students into groups, taking into account the student's preferences as a primal factor for the assignment. An evolutionary algorithm is then used in order to select the ordering of students which provides the best assignment.The algorithm has been tested in a real problem, the timetable of the Telecommunication Engineering School at Universidade de Vigo (Spain), and has shown good performance in terms of the number of constraints fulfilled and groups assigned to students.     

Computer generation of negative binomial variates by envelope rejection

This paper proposes an envelope rejection method for computer sampling from the negative binomial distribution, with index parameter α and probability parameter p, based on a simple probability distribution inequality. Comparative timings with existing methods of generating NB variates show that the proposed method is comparable or faster when α is not too large (α<11), and it is fastest for all α when p>0.7.     

Improving the modulo simplex algorithm for large-scale periodic timetabling

The periodic event scheduling problem (PESP), in which events have to be scheduled repeatedly over a given period, is a complex and well-known discrete problem with numerous real-world applications. The most prominent of them is to find periodic timetables in public transport. Although even finding a feasible solution to the PESP is NP-hard, recent achievements demonstrate the applicability and practicability of the periodic event scheduling model. In this paper we propose different approaches to improve the modulo network simplex algorithm (Nachtigall and Opitz, 2008 [17]), which is a powerful heuristic for the PESP problem, by exploiting improved search methods in the modulo simplex tableau and larger classes of cuts to escape from the many local optima. Numerical experiments on large-scale railway instances show that our algorithms not only perform better than the original method, but even outperform a state-of-the-art commercial MIP solver.