On the application of graph colouring techniques in round-robin sports scheduling

The purpose of this paper is twofold. First, it explores the issue of producing valid, compact round-robin sports schedules by considering the problem as one of graph colouring. Using this model, which can also be extended to incorporate additional constraints, the difficulty of such problems is then gauged by considering the performance of a number of different graph colouring algorithms. Second, neighbourhood operators are then proposed that can be derived from the underlying graph colouring model and, in an example application, we show how these operators can be used in conjunction with multi-objective optimisation techniques to produce high-quality solutions to a real-world sports league scheduling problem encountered at the Welsh Rugby Union in Cardiff, Wales

Planning, Scheduling, and Timetabling in a University Setting

Methods and procedures for modeling university student populations, predicting course enrollment, allocating course seats, and timetabling final examinations are studied and proposed. The university enrollment model presented uses a multi-dimensional state space based on student demographics and the Markov property, rather than longitudinal data to model student movement. The procedure for creating adaptive course prediction models uses student characteristics to identify groups of undergraduates whose specific course enrollment rates are significantly different than the rest of the university population. Historical enrollment rates and current semester information complete the model for predicting enrollment for the coming semester. The course prediction model aids in the system for reserving course seats for new students during summer registration sessions. The seat release model addresses how to estimate seat need each session, how to release seats among multiple course sections, and how to predict seat shortages and surpluses. Finally, procedures for creating reusable university final examination timetables are developed and compared. Course times, rather than individual courses, are used as the assignment elements because the demand for course times remains relatively constant despite changes in course schedules. Our heuristic procedures split the problem into two phases: a clustering phase--to minimize conflicts--and a sequencing phase--to distribute exams throughout finals week while minimizing the occurrence of consecutive exams. Results for all methods are compared using enrollment data from Clemson University

Automated examination timetabling with application to Stellenbosch University

Thesis (MCom)--Stellenbosch University, 2018.ENGLISH SUMMARY : The examination timetabling problem (ETP) consists of scheduling the examination papers of students in such a way that no student is required to write two or more examination papers at the same time in an examination period. It is well known that this problem is NP–complete, which makes it an interesting research topic for researcher. There exist many different variants of the ETP, and the one focused on in this project is the variant that can be applied to Stellenbosch University. The purpose of this project is to find examination timetables that will spread most students’ examination papers more or less equally over the full duration of the examination period. Stellenbosch University is used as case study. The primary objective of any algorithm is to satisfy the hard constraints provided by Stellenbosch University. For example, one hard constraint is that certain examination papers must be scheduled on fixed timeslots. Graph colouring is used in a two–phase heuristic algorithm to obtain a feasible initial solution. In the first phase an examination timetable is sought where no student is required to write more than one examination paper during any timeslot. After the first phase, the number of examination papers scheduled during each timeslot is balanced in the second phase of the algorithm. This is to ensure that amongst others, enough lecture halls are available during each timeslot to accommodate all students. After a feasible initial solution is obtained, hill climbing and the great deluge algorithm (GDA) are used to improve upon the equally spread of students’ examination papers as much as possible over the entire examination period. Three moves are defined in this project to move from solution to solution in the solution space. The first move moves one module in the timetable to another timeslots, the second move swaps all of the modules in two timeslots and the third move is to swap two modules that are in different timeslots. To evaluate how well students’ examination papers are spread over the entire examination period for each timetable, a newly derived cost function is used. The cost function strives to be fair towards all students. Parameter calibration is done on the parameters used in the cost function and the search algorithms. The resulting timetables when using hill climbing and the GDA are compared, and it is found that the GDA outperforms hill climbing. Furthermore, the cost function used in this project is compared to the cost function of the 2nd International Timetabling Competition (ITC). Using Stellenbosch University’s variant of the ETP, it is found that the cost function of this project outperforms the cost function used in the ITC.AFRIKAANSE OPSOMMING : In die eksamenrooster–probleem (ERP) moet eksamenroosters sAs ingedeel word dat geen student meer as een eksamenvraestel gelyktydig moet skryf nie. Hierdie probleem is NP–volledig en is al deeglik deur navorsers ondersoek. Daar is baie variante van die ERP, en in hierdie projek word daar gefokus op die variant wat op die Universiteit van Stellenbosch (US) van toepassing is. Die doel van hierdie projek is om eksamenroosters op te stel wat studente se eksamenvraestelle so ver as moonlik eweredig oor die hele eskamenperiode versprei. Die US word as gevallestudie gebruik. Die primêre doelwit van enige algoritme is om aan die US se vaste vereistes vir ’n eksamenrooster te voldoen. Een van hierdie vereiste is, byvoorbeeld, dat sekere eksamenvraestelle op spesifieke tydgleuwe geskeduleer moet word. Grafiekkleuring is gebruik in ’n heuristiek wat van twee fases gebruik maak om ’n aanvanklike eksamenrooster wat aan die US se vereistes voldoen, te kry. In die eerste fase word ’n eksamenrooster gesoek waarin daar van geen student verwag word om twee of meer eksamenvraestelle gelyktydig te skryf nie, en in die tweede fase word die hoeveelheid eksamenvraestelle per tydgleuf gebalanseer. Die tweede fase word byvoorbeeld benodig sodat daar ten alle tye genoeg eksamenlokale beskikbaar sal wees om alle studente te akkommodeer. Nadat ’n aanvanklike toelaatbare oplossing verkry is, word hill climbing en the great deluge algoritme (GDA) gebruik om studente se opeenvolgende eksamenvraestelle so ver as moontlik uitmekaar te versprei. Drie eenvoudige skuiwe word in die algoritmes gebruik om sodoende deur die oplossingsruimte te beweeg. Die eerste skuif is om ’n eksamenvraestel te skuif na ’n ander tydgleuf, die tweede skuif is om al die eksamenvraestelle van twee tydgleuwe met mekaar om te ruil en die laatse skuif is om twee eksamenvraestelle wat in twee verskillende tydgleuwe is met mekaar om te ruil. Om ’n aanduiding te kry van hoe goed studente se eksamenrooster ingedeel is, is ’n koste–funksie bepaal. Die doelfunksie handhaaf regverdigheid ten opsigte van die hoeveelheid eksamenvraestelle wat per student geskryf moet word. Die parameters van die doelfunksie en die algoritmes word gekalibreer om sodoende goeie parameterwaardes te bepaal. Die twee algoritmes word met mekaar vergelyk, en daar is bevind dat GDA beter vaar as die hill climbing algoritme. Verder word die koste–funksie van hierdie projek vergelyk met die koste–funksie van die 2de Internationale Roosterindelingskompetisie (ITC). Daar is bevind dat die koste–funksie van hierdie projek beter vertoon om die US se eksamenroosters in te deel as die koste–funksie van die ITC

A Polyhedral Study of Mixed 0-1 Set

We consider a variant of the well-known single node fixed charge network flow set with constant capacities. This set arises from the relaxation of more general mixed integer sets such as lot-sizing problems with multiple suppliers. We provide a complete polyhedral characterization of the convex hull of the given set

Working Notes from the 1992 AAAI Spring Symposium on Practical Approaches to Scheduling and Planning

The symposium presented issues involved in the development of scheduling systems that can deal with resource and time limitations. To qualify, a system must be implemented and tested to some degree on non-trivial problems (ideally, on real-world problems). However, a system need not be fully deployed to qualify. Systems that schedule actions in terms of metric time constraints typically represent and reason about an external numeric clock or calendar and can be contrasted with those systems that represent time purely symbolically. The following topics are discussed: integrating planning and scheduling; integrating symbolic goals and numerical utilities; managing uncertainty; incremental rescheduling; managing limited computation time; anytime scheduling and planning algorithms, systems; dependency analysis and schedule reuse; management of schedule and plan execution; and incorporation of discrete event techniques

Link Patterns in Complex Networks

Network theorists define patterns in complex networks in various ways to make them accessible to human beholders. Prominent definitions are thereby based on the partition of the network's nodes into groups such that underlying patterns in the link structure become apparent. Clustering and blockmodeling are two well-known approaches of this kind. In this thesis, we treat pattern search problems as discrete mathematical optimization problems. From this viewpoint, we develop a new mathematical classification of clustering and blockmodeling approaches, which unifies these two fields and replaces several NP-hardness proofs by a single one. We furthermore use this classification to develop integer mathematical programming formulations for pattern search problems and discuss new linearization techniques for polynomial functions therein. We apply these results to a model for a new pattern search problem. Even though it is the most basic problem in combinatorial terms, we can prove its NP-hardness. In fact, we show that it is a generalization of well-known problems including the Traveling Salesman and the Quadratic Assignment Problem. Our derived exact pattern search procedure is up to 10,000 times faster than comparable methods from the literature. To demonstrate its practicability, we finally apply the procedure to the world trade network from the United Nations' database and show that the network deviates by less than 0.14% from the patterns we found

Optimizing the Efficiency of the United States Organ Allocation System through Region Reorganization

Allocating organs for transplantation has been controversial in the United States for decades. Two main allocation approaches developed in the past are (1) to allocate organs to patients with higher priority at the same locale; (2) to allocate organs to patients with the greatest medical need regardless of their locations. To balance these two allocation preferences, the U.S. organ transplantation and allocation network has lately implemented a three-tier hierarchical allocation system, dividing the U.S. into 11 regions, composed of 59 Organ Procurement Organizations (OPOs). At present, an procured organ is offered first at the local level, and then regionally and nationally. The purpose of allocating organs at the regional level is to increase the likelihood that a donor-recipient match exists, compared to the former allocation approach, and to increase the quality of the match, compared to the latter approach. However, the question of which regional configuration is the most efficient remains unanswered. This dissertation develops several integer programming models to find the most efficient set of regions. Unlike previous efforts, our model addresses efficient region design for the entire hierarchical system given the existing allocation policy. To measure allocation efficiency, we use the intra-regional transplant cardinality. Two estimates are developed in this dissertation. One is a population-based estimate; the other is an estimate based on the situation where there is only one waiting list nationwide. The latter estimate is a refinement of the former one in that it captures the effect of national-level allocation and heterogeneity of clinical and demographic characteristics among donors and patients. To model national-level allocation, we apply a modeling technique similar to spill-and-recapture in the airline fleet assignment problem. A clinically based simulation model is used in this dissertation to estimate several necessary parameters in the analytic model and to verify the optimal regional configuration obtained from the analytic model. The resulting optimal region design problem is a large-scale set-partitioning problem in whichthere are too many columns to handle explicitly. Given this challenge, we adapt branch and price in this dissertation. We develop a mixed-integer programming pricing problem that is both theoretically and practically hard to solve. To alleviate this existing computational difficulty, we apply geographic decomposition to solve many smaller-scale pricing problems based on pre-specified subsets of OPOs instead of a big pricing problem. When solving each smaller-scale pricing problem, we also generate multiple ``promising' regions that are not necessarily optimal to the pricing problem. In addition, we attempt to develop more efficient solutions for the pricing problem by studying alternative formulations and developing strong valid inequalities. The computational studies in this dissertation use clinical data and show that (1) regional reorganization is beneficial; (2) our branch-and-price application is effective in solving the optimal region design problem

Mathematical Methods and Operation Research in Logistics, Project Planning, and Scheduling

In the last decade, the Industrial Revolution 4.0 brought flexible supply chains and flexible design projects to the forefront. Nevertheless, the recent pandemic, the accompanying economic problems, and the resulting supply problems have further increased the role of logistics and supply chains. Therefore, planning and scheduling procedures that can respond flexibly to changed circumstances have become more valuable both in logistics and projects. There are already several competing criteria of project and logistic process planning and scheduling that need to be reconciled. At the same time, the COVID-19 pandemic has shown that even more emphasis needs to be placed on taking potential risks into account. Flexibility and resilience are emphasized in all decision-making processes, including the scheduling of logistic processes, activities, and projects