Jay Yellen

A decomposition approach to unit maintenance scheduling

The authors present the results of duality theory-based decomposition of the thermal generator unit maintenance scheduling problem. In the first stage, a master problem is solved to determine a trial solution for the maintenance schedule decision variables. In the second stage, a subproblem calculates the minimum operating cost while satisfying the reliability constraints for each week of the study period. New constraints for the master problem, similar to Benders cuts, are generated from the solution of the subproblem using duality theory, so that an improved maintenance schedule can be obtained. The procedure continues until an optimal or near-optimal solution is found. >

Weighted graphs and university course timetabling

Abstract A weighted graph is used to model the problem of scheduling university courses minimizing conflict. A variety of vertex section criteria are introduced, and are used in a heuristic algorithm for finding approximate solutions in the form of least-cost k-colorings. The trade-off between these approximate algorithms and exact integer programming methods is discussed, and computational results are presented.

Unit maintenance scheduling with fuel constraints

In the unit maintenance scheduling (UMS) problem, fuel usage limitations at a unit affect the loading of the unit and hence the maintenance schedule of units. Accounting for this interaction of fuel limits at a unit with unit maintenance and system constraints greatly increases the complexity of the problem. Using duality theory, the UMS problem with fuel constraints (UMS-FCs) is decomposed into a master problem and subproblems. The definition and the coordination of master and subproblems give rise to alternate algorithms to solve the original problem. Results of one such decomposition are presented. In this approach, the master problem solves for a trial maintenance schedule to satisfy unit maintenance constraints and unit fuel limits over all time intervals of the study. Given the maintenance schedule from the master problem, a subproblem calculates the minimum operating cost subject to system constraints for each interval of the study period. If one or more subproblems are infeasible, additional constraints are generated and then added to the master problem so that an improved maintenance schedule that satisfies the fuel and system constraints is obtained. The iteration between master and subproblems is continued until an optimal or near-optimal solution is found. >

Bulk input queues with quorum and multiple vacations

The authors study a single-server queueing system with bulk arrivals and batch service in accordance to the general quorum discipline: a batch taken for service is not less than r and not greater than R(≥r). The server takes vacations each time the queue level falls below r(≥1) in accordance with the multiple vacation discipline. The input to the system is assumed to be a compound Poisson process. The analysis of the system is based on the theory of first excess processes developed by the first author. A preliminary analysis of such processes enabled the authors to obtain all major characteristics for the queueing process in an analytically tractable form. Some examples and applications are given.

Linear combinations of heuristics for examination timetabling

Although they are simple techniques from the early days of timetabling research, graph colouring heuristics are still attracting significant research interest in the timetabling research community. These heuristics involve simple ordering strategies to first select and colour those vertices that are most likely to cause trouble if deferred until later. Most of this work used a single heuristic to measure the difficulty of a vertex. Relatively less attention has been paid to select an appropriate colour for the selected vertex. Some recent work has demonstrated the superiority of combining a number of different heuristics for vertex and colour selection. In this paper, we explore this direction and introduce a new strategy of using linear combinations of heuristics for weighted graphs which model the timetabling problems under consideration. The weights of the heuristic combinations define specific roles that each simple heuristic contributes to the process of ordering vertices. We include specific explanations for the design of our strategy and present the experimental results on a set of benchmark real world examination timetabling problem instances. New best results for several instances have been obtained using this method when compared with other constructive methods applied to this benchmark dataset.

Handbook of Graph Theory, Second Edition

Connectivity is one of the central concepts of graph theory, from both a theoret- ical and a practical point of view. Its theoretical implications are mainly based on the existence of nice max-min characterization results, such as Menger’s theorems. In these theorems, one condition which is clearly necessary also turns out to be sufficient. Moreover, these results are closely related to some other key theorems in graph theory: Ford and Fulkerson’s theorem about flows and Hall’s theorem on perfect matchings. With respect to the applications, the study of connectivity parameters of graphs and digraphs is of great interest in the design of reliable and fault-tolerant interconnection or communication networks. Since graph connectivity has been so widely studied, we limit ourselves here to the presentation of some of the key results dealing with finite simple graphs and digraphs. For results about infinite graphs and connectivity algorithms the reader can consult, for instance, Aharoni and Diestel [AhDi94], Gibbons [Gi85], Halin [Ha00], Henzinger, Rao, and Gabow [HeRaGa00], Wigderson [Wi92]. For further details, we refer the reader to some of the good textbooks and surveys available on the subject: Berge [Be76], Bermond, Homobono, and Peyrat [BeHoPe89], Frank [Fr90, Fr94, Fr95], Gross and Yellen [GrYe06], Hellwig and Volkmann [HeVo08], Lov ´asz [Lo93], Mader [Ma79], Oellermann [Oe96], Tutte [Tu66].

Refinery units maintenance scheduling using integer programming

Abstract The refinery unit maintenance scheduling problem is to determine the schedule for preventive maintenance of refinery units over a specified operational planning horizon so that the unit utilization level is maximized and maintenance constraints are satisfied. This paper presents an integer programming approach to model and solve the problem. Results of a test example are given to show the applicability of the approach.

The design and implementation of an interactive course-timetabling system

We describe our design and implementation of a dual-objective course-timetabling system for the Science Division at Rollins College, and we compare the results of our system with the actual timetable that was manually constructed for the Fall 2009 school term. The course timetables at Rollins, as at most colleges in the U.S., must be created before students enroll in classes, and our “wish list” of pairs of classes that we would like to offer in non-overlapping timeslots is considerably larger than if we were to consider only those that absolutely must be in non-overlapping timeslots. This necessitates assigning different levels of conflict severity for the class pairs and setting our objective to minimize total conflict severity. Our second objective is to create timetables that result in relatively compact schedules for the instructors and students.In addition to our automatic construction, a second, equally important component of our system is a graphical user interface (GUI) that enables the user to participate in the input, construction, and modification of a timetable. In the input phase, course incompatibility, instructor and student preferences, and desire for compact schedules all require subjective judgments. The GUI allows the user to quantify and convert this information to the weighted-graph model. In the construction and modification phase, the GUI enables the user to directly assign or reassign courses to timeslots while guided by heuristics.

On the solvability of groups of central type

Let G be a finite group with center Z and irreducible complex character x so that x(l)2= [G: Z]. If the 2-Sylow subgroup of G/Z has order 16 or less then G is solvable. Introduction. Let G be a finite group with center Z. If G has an irreducible complex character X with X(1)2 = [G: Z], then G is called a group of central type [4]. It was conjectured in [8] that groups of central type are solvable and several authors have given partial results in this direction [2][4], [6], [9], and [12]. For example, in [4] it is proved that if G is a group of central type and if for any prime p, ptmI[G :Z] implies m < 2 then G is solvable. In [6], the integer 2 in this result was replaced by 4. Here we show that if 2m I [G: Z] implies m < 4 and G is of central type, then G is solvable. The proof employs the well-known characterizations of simple groups with small 2-Sylow subgroups and information on possible homomorphic images of groups of central type given in [6]. All unexplained notation and terminology is as in [7]. Lemma 1. Let G be a finite group having a nonabelian composition factor S appearing exactly n times. Then there exist a homomorphic image X of G and an integer m such that 1 < m < n and S X ... XSm <X< Aut(S x ... x Sm) where Si eS for all i = 1, 2,. . .m. Proof. Use induction on IGI. Let T be a minimal nontrivial normal subgroup of G. If G is simple then G = S and S < G < Aut(S). So we may assume tIl < T < G. We haveT = T1 x ... x Tk where the Ti are isomorphic simple groups, i= 1, 2, ..., k. Case 1. If Ti S for all i= 1, 2, ..., k, then consider G =G/T. By the inductive hypothesis there exists a homomorphic image X of G and an integer m such that 1 < m < n and SI x ... x Sm < X < Aut(SI x ... xSm)I where S. S. This is the desired homomorphic image X of G. Case2. If T. Sforall i=1,2,...,k,then TSI k...xS, k< n and S. c S. Since T A G, CG(T) A G and we have G/CG(T) < Aut(T). Since T is minimal normal, CG(T) n T = I1 } and thus Received by the editors July 31, 1974. AMS (MOS) subject classifications (1970). Primary 20C159 20D10; Secondary 20D059 20D45.

Irregular Colorings of Regular Graphs

Abstract An irregular coloring of a graph is a proper vertex coloring that distinguishes vertices in the graph either by their own colors or by the colors of their neighbors. In algebraic graph theory, graphs with a certain amount of symmetry can sometimes be specified in terms of a group and a smaller graph called a voltage graph . In Radcliffe and Zhang (2007) [3] , Radcliffe and Zhang found a bound for the irregular chromatic number of a graph on n vertices. In this paper we use voltage graphs to construct graphs achieving that bound.