Timetabling is a typical real world scheduling activity that arises at least once a year at every educational institution. The three most common edu- cational timetablingproblem categories are examination timetabling, course timetabling and class/teacher timetabling, as they are commonly called. The basic distinction occurs in terms of the elements to be scheduled, that is, ex- ams, course options or regular lessons. The examination timetablingproblem consists in scheduling the exams for a set of courses, over a limited time period, while avoiding the overlapping of exams for each student . For a detailed description of this problem see the surveys made by Carter and Laporte  or Burke et al. . In course timetabling problems, for each student a set of lectures is previously defined. Then all lectures included within the institu- tion’s set of courses must be scheduled in such a way that the overlapping of lectures for courses with students in common is minimized (vide, for instance, Downsland ; Kiaer and Yellen ). This problem arises in universities or other educational institutions with flexible curricula.
Abstract: We consider the well known NP–hard teacher/class timetablingproblem. Variable neighborhood search and tabu search heuristics are developed based on idea of the Formulation Space Search approach. Two types of solution representation are used in the heuristics. For each representation we consider two families of neighborhoods. The first family uses swapping of time periods for teacher (class) timetable. The second family bases on the idea of large Kernighan-Lin neighborhoods. Computation results for difficult random test instances show high efficiency of the proposed approach.
The High School TimetablingProblem consists in assigning timeslots and re- sources to events, satisfying constraints which heavily depend on the specific institu- tion. This work deals with the problem of the ongoing III International Timetabling Competition (ITC), which includes a diverse set of instances from many educational institutions around the world. We proposed an approach based on Simulated An- nealing and Variable Neighborhood Search metaheuristics. One important struc- tural feature of our approach is the use of the Kingston’s High School Timetabling Engine (KHE) to generate initial solutions combined with the multi-neighborhood search. Such approach led us to the finals of the ongoing competition.
The school timetablingproblem can be described as scheduling a set of lessons (combination of classes, teachers, subjects and rooms) in a weekly timetable. This paper presents a novel way to generate timetables for high schools. The algorithm has three phases. Pre-scheduling, initial phase and optimization through tabu search. In the first phase, a graph based algorithm used to create groups of lessons to be scheduled simultaneously; then an initial solution is built by a sequential greedy heuristic. Finally, the solution is optimized using tabu search algorithm based on frequency based diversification. The algorithm has been tested on a set of real problems gathered from Iranian high schools. Experiments show that the proposed algorithm can effectively build acceptable timetables.
The High School TimetablingProblem is faced by many educa- tional institutions around the world. The basic search version consists in assigning teacher class activities to timeslots and rooms in such a way that no teacher, class or room is involved with more than one event at time. Generally, this assignment is repeated weekly until the end of the semester. Many other constraints are considered in real problems, like availability of teachers, to avoid idle times and to limit the number of lessons of the same subject taught to a class in a day. Beyond its practical importance, this problem was proven to be NP Hard [1,2]. Progress in heuristic and exact approaches for tackling these problems is a major goal of current research in Operations Research and Artiﬁcial Intelligence.
The High School TimetablingProblem remains subject of many research in Operational Research and Artificial Intelligence fields because of its hardness to solve and practical importance. A solution for this problem basically consists in the schedule of lessons to timeslots and the assignment of resources for these lessons. This allocation should satisfy many a priori specified constraints. This work considers the solution of the problem of the Third International Timetabling Competition (ITC2011), which includes a diverse set of instances from many educational institutions around the world. This work presents many local search methods to solve the problem. The format for specifying instances considered was XHSTT, allowing any instance specified in that format to be manipulated by the proposed algorithms. One important structural feature of our approach is the use of the KHE engine to generate initial solutions combined with a multi-neighborhood search approach. The achieved results include the development of the algorithm winner of competition. Moreover, we found feasible solutions to thirteen out of eighteen instances and we improved the best known solution to fifteen out of sixteen instances.
Abstract The application of the Late Acceptance Hill- Climbing (LAHC) to solve the High School TimetablingProblem is the subject of this manuscript. The original algo- rithm and two variants proposed here are tested jointly with other state-of-art methods to solve the instances proposed in the Third International Timetabling Competition. Following the same rules of the competition, the LAHC-based algo- rithms noticeably outperformed the winning methods. These results, and reports from the literature, suggest that the LAHC is a reliable method that can compete with the most employed local search algorithms.
Problem 2 is the optimal approximation problem of Problem 1. It occurs frequently in experimental design . Here the matrix X ∗ may be a matrix obtained from experiments, but it may not satisfy the structural requirement (generalized anti-reflexive matrices with respect to matrix pairs (P, Q)) and/or matrix equations (B X = C, X D = E). The optimal estimate ˆ X is the matrix that satisfies both restrictions and is the optimal approximation of X ∗ . See for instance [14, 15].
For the interesting generalized shuffling problems, some research problems are open. If some constraints are added to the move sequences, then the problem become complicated. For example, we can add a restriction to a move that only two adjacent coins of different colors can be moved. For this constraint shuffling problem, what is its optimal solution? Can we find an efficient algorithm to generate optimal solutions for the constraint shuffling problem in the time proportional to the output size? This is also an open problem. We will investigate the these problems further.
O presente trabalho apresenta um estudo algorítmico do Multicast Packing Problem le- vando em consideração uma abordagem multiobjetivo. Para tal, faz-se uma extensa revisão sobre o problema em questão. Esta revisão serviu como ponto de referência para definição de um modelo matemático multiobjetivo, tendo em vista que não há na literatura nenhum trabalho que tenha tratado o tema neste aspecto. Em seguida, define-se os casos de teste utilizados no processo de experimentação dos algoritmos. Uma vez que tanto modelo matemático multiob- jetivo quanto os casos de teste foram criados, então desenvolve-se vários algoritmos com base nas abordagens clássicas para problemas de otimização multiobjetivo: NSGA2 (3 versões) e SPEA2 (3 versões). Além disso, adaptou-se a metaheurística GRASP (2 versões) para aplica- ção considerando o modelo proposto. Estes algoritmos foram compostos por três operadores de recombinação (C1, C2, C3), dois operadores de construção de solução, um operador de mutação e um operador de busca local. Por fim, um extenso processo de experimentação dos algoritmos é realizado. Este processo possui três etapas: a primeira etapa consistiu de ajustar os parâmetros que cada algoritmo necessita, neste caso o ajuste de parâmetro foi realizado para todas as versões do SPEA2, NSGA2 e GRASP; A segunda etapa consistiu de verificar, para cada algoritmo, qual a melhor versão. Por fim, as melhores versões de cada algoritmo, no total 3 versões, foram comparadas entre si visando identificar qual o melhor algoritmo dentre todos. Os algoritmos foram avaliados com base nos indicadores de qualidade Hypervolume e Epsilon Multiplicativo. Os resultados dos experimentos foram avaliados através de testes estatísticos não-paramétricos (teste de Mann-Whitney e teste de Friedman). A avaliação dos resultados foi favoráravel ao NSGA2-C2 segundo a metodologia de avaliação utilizada.
We argued that if sale is rare, then the problem of equilibrium delay is mitigated. This suggests the use of commitment devices by developers to make rare sale more credible. When players compete at point of sale, the delay problem becomes worse. Each player prefers to conduct their sale without the other present, in order to mitigate competition. In contrast, cooperation at point of sale eliminates the delay problem, provided sellers discount the future.
Using formulas (1) and (2), k-th ant may either follows the most favorable path already established or may randomly select a path to follow based on a probability distribution established by distance and pheromone accumulation. Furthermore, each time the ant k visits the depot, number of circuits which are constructed by ant k adds by one. Meanwhile, this selection process continues until at least 3(p-t) vertices remain to visit, where t is the number of circuits which are constructed by ant k. This is for constructing a feasible solution of the HpMP problem. In this case, the ant k ends its circuit at the depot and makes a new circuit by starting from it. This process is continued until each customer is visited and the circuits are completed.
In the sequel we always denote by F (T ) the set of fixed points of the nonexpansive semi-group T , VI(H,B,M) the set of solutions to the variational inequality (1.2) and MEP(F ) the set of solutions to the following auxiliary problem for a system of mixed equilibrium problems:
Control systems of robots can allow quick use of motor primitives. The problem of learning of motor primitives is well studied. It is possible to rapidly learn motor primitives for many different complex behaviors, tennis-like swings , T-ball batting , drumming , biped locomotion , ball-in-a-cup , industrial applications . How- ever, ability to quickly and reliably use of different motor primitives is insufficient for fast adaptation to variations of the situation. Robots need an ability to select proper sequences of motor primitives. In particular, robots need to generalize motor primitives to a different behavior by trial and error without re-learning the task. In some cases, motor primitives can be adapted both spatially and temporally without changing the overall shape of the motion . In particular, we can define meta-parameters as some small set of parameters that adapt the global motion behavior. A generalization of behaviors can be achieved by adapting these meta-parameters. For instance, the end position can be considered as a meta-parameter. Such approach was con- sidered in the context of supervised learning for tennis-like swings with static ball targets , object manipulation , minigolf , drumming . Also, supervised learning used to generalize meta-parameters in real-time . Also, such approach was used in the context of reinforcement learning . A prediction of a trajectory gives us another example of an adaptation of movement to situations. In particular, a prediction of a trajectory from a previously demonstrated set and refinement of this trajectory by motion planning is used for an adaptation of movement . It should be noted that meta-parameters can be used to adapt to changes in the behavior of the robot itself. For instance, quadrotor vehicles are inherently unstable nonlinear systems. They exhibit exceedingly complex behavior at high speeds. An algorithm that exploits data from previous repetitions in order to learn to precisely follow a predefined trajectory was presented for quadrotor vehicles .
For real world problems, additional information about the problems we are solving are available, and so we may not have to solve them from scratch. One of the approaches is making use of apriori information, which can be a solution to a smaller input instance of a problem to solve a larger instance of it. This approach is called reoptimization. The idea was first mentioned in . Reoptimization may help to improve the approximability of the problem or the running time of the solution to it. In fact, we can obtain a PTAS for a reoptimization variant of a problem given that the unmodified problem is approximable . The formal definition of reoptimization is as follows.
When being asked about the opinion on the outline of our predelivery problem management process model and the flow of problem reports and corrections, six out of eight companies agreed upon it. The seventh company pointed out that the testing time and cost may be substantially multiplied with our model. The eighth company responded that for simple problems that can be resolved right away, the flow might be redundant. Still however, all the companies studied believe that formal predelivery problem management process can help improve the testing process in the following way: • It leads to a fast solution;
The satisfiability problem (SAT) was the first known NP- complete problem. The problem SAT is the problem of determining if the variables of a given boolean function in conjunctive normal form (CNF) can be assigned in such a way as to make the formula evaluate to true. Different variants of SAT were considered. In particular, the problem 3SAT is the problem of determining if the variables of a given 3-CNF can be assigned in such a way as to make the formula evaluate to true. Encoding problems as Boolean satisfiability (see e.g. , –) and solving them with very efficient satisfiability algorithms (see e.g. –) has recently caused considerable interest. In this paper we consider reductions from c-AHP-M and c-AHP-D to SAT and 3SAT.