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In this article, we study compact Mixed-Integer Programming (MIP) models for the Resource-Constrained Project Scheduling Problem (RCPSP). Compared to the classical time-indexed formulation, the size of compact models is strongly polynomial in the number of jobs. In addition to two compact models from the literature, we propose a new compact model. We can show that all three compact models are equivalent by successive linear transformations. For their LP-relaxations, however, we state a full inclusion hierarchy where our new model dominates the previous models in terms of polyhedral strength. Moreover, we reveal a polyhedral relationship to the common time-indexed model. Furthermore, a general class of valid cutting planes for the compact models is introduced and finally all models are evaluated by computational experiments.
In this paper, we address the Energetic Reasoning propagation rule for the Cumulative Scheduling Problem (CuSP). An energetic reasoning propagation algorithm is called exact, if it computes the maximum possible energetic reasoning propagation for all the jobs. The currently best known exact energetic reasoning algorithm has complexity O(n^3). In this paper, we present a new exact energetic reasoning propagation algorithm with improved complexity of O(n^2 \log^2 n).
In the Resource-Constrained Project Scheduling Problem (RCPSP) a set of jobs is planned subject to resource- and precedence constraints. The objective is to minimize the makespan, that is the time when all jobs have been completed. There exist several Mixed-Integer-Programming (MIP) models in order to solve the problem. Most common models are based on time-discretization. In this case, the scheduling horizon is split into unit size intervals and each job gets assigned a unique starting interval. The drawback of time-discrete models is the computational intractability for large scheduling horizons or fine discretizations. In this connection, this thesis deals with compact MIP models where the model size is independent of the scheduling horizon.
In addition to two compact models from the literature, we present two new compact models. We investigate their induced polyhedra and deduce an inclusion hierarchy via linear transformations. Moreover, we give a combinatorial interpretation of these transformations. Furthermore, we study a class of valid cutting planes for the compact models, which are known as cover inequalities. In order to strengthen these cutting planes we introduce a lifting algorithm that is independent of the model size.
Subsequently, we examine lower bounds for the RCPSP from linear programming. Based on a linear transformation, we reveal a connection between two approaches from the literature. For one model we generate strong cutting planes that are obtained from a primal-dual relation between the models. Two cutting plane algorithms are derived. Likewise, we show that similar cutting planes can be transferred to the compact MIP models.
Our models have been implemented, tested and evaluated on the instances of the PSPLIB problem library.
Scientific conferences play an important role in almost all areas of today’s science and
research. They offer a wide platform for international researchers and developers for
communication and scientific exchange of current problems, experiences and knowledge
in nearly all fields of research. Such conferences usually consist of talks about different
topics that are given by authorized speakers. The planning process of a conference
includes the creation of a scientific program schedule, that means a local and temporal
allocation of all available talks.
As the number of talks can grow into thousands for large-scale conferences, the
conference scheduling becomes a complex challenge. Furthermore there exist a great
bandwidth on various and individual requirements on the allocation of the talks. In this
thesis we discuss the major problems that occur during the development of a conference
schedule and provide a framework how proper conference schedules can be generated
computationally. For that, we use Mixed-Integer-Programming techniques to find sched-
ules that respect the difficult range of additional constraints.
The basis of this thesis was the development of an optimized conference schedule
of the International Symposium on Mathematical Programming 2012 (ISMP 2012) in
Berlin, Germany. The ISMP is one of the largest conferences in the field of applied
mathematics and mathematical optimization and constitutes a great attraction for in-
ternational mathematicians and researchers. We will present our results based on the
problem instance that we used for the ISMP 2012.
In constraint programming, energetic reasoning constitutes a powerful start time propagation rule for cumulative scheduling problems (CuSP). In this paper, we first present an improved time interval checking algorithm that is derived from a polyhedral model. In a second step, we extend this algorithm to an energetic reasoning propagation algorithm with complexity O(n^2 log n) where n denotes the number of jobs. The key idea is based on a new sweep line subroutine that efficiently evaluates the relevant time intervals for all jobs. In particular, our algorithm yields at least one possible energetic reasoning propagation for each job. Finally, we show that on the vast number of relevant time intervals our approach yields the maximum possible propagation according to the energetic reasoning rule.
In this article, we study compact Mixed-Integer Programming (MIP) models for the Resource-Constrained Project Scheduling Problem (RCPSP). Compared to the classical time-indexed formulation, the size of compact models is strongly polynomial in the number of jobs. In addition to two compact models from the literature, we propose a new compact model. We can show that all three compact models are equivalent by successive linear transformations. For their LP-relaxations, however, we state a full inclusion hierarchy where our new model dominates the previous models in terms of polyhedral strength. Moreover, we reveal a polyhedral relationship to the common time-indexed model. Furthermore, a general class of valid cutting planes for the compact models is introduced and finally all models are evaluated by computational experiments.
In constraint programming, energetic reasoning constitutes a powerful start time propagation rule for cumulative scheduling problems (CuSP). In this paper, we first present an improved time interval checking algorithm that is derived from a polyhedral model. In a second step, we extend this algorithm to an energetic reasoning propagation algorithm with complexity O(n^2 log n) where n denotes the number of jobs. The key idea is based on a new sweep line subroutine that efficiently evaluates the relevant time intervals for all jobs. In particular, our algorithm yields at least one possible energetic reasoning propagation for each job. Finally, we show that on the vast number of relevant time intervals our approach yields the maximum possible propagation according to the energetic reasoning rule.
A Polyhedral Study of Event-Based Models for the Resource-Constrained Project Scheduling Problem
(2020)
We consider event-based Mixed-Integer Programming (MIP) models for the Resource-Constrained Project Scheduling Problem (RCPSP) that represent an alternative to the common time-indexed model (DDT) of Pritsker et al. (1969) for the case where the underlying time horizon is large or job processing times are subject to huge variations. In contrast to the time-indexed model, the size of event-based models does not depend on the time horizon. For two event-based formulations OOE and SEE of Koné et al. (2011) we present new valid inequalities that dominate the original formulation. Additionally, we introduce a new event-based model: the Interval Event-Based Model (IEE). We deduce linear transformations between all three models that yield the strict domination order IEE > SEE > OOE for their linear programming (LP) relaxations, meaning that IEE has the strongest linear relaxation among the event-based models. We further show that the popular DDT formulation can be retrieved from IEE by certain polyhedral operations, thus giving a unifying view on a complete branch of MIP formulations for the RCPSP. In addition, we analyze the computational performance of all presented models on test instances of the PSPLIB (Kolisch and Sprecher 1997).
A Polyhedral Study of Event-Based Models for the Resource-Constrained Project Scheduling Problem
(2018)
We consider event-based Mixed-Integer Programming (MIP) models for the Resource-Constrained Project Scheduling Problem (RCPSP) that represent an alternative to the common time-indexed model (DDT) of Pritsker et al. (1969) for the case where the underlying time horizon is large or job processing times are subject to huge variations. In contrast to the time-indexed model, the size of event-based models does not depend on the time horizon. For two event-based formulations OOE and SEE of Koné et al. (2011) we present new valid inequalities that dominate the original formulation. Additionally, we introduce a new event-based model: the Interval Event-Based Model (IEE). We deduce linear transformations between all three models that yield the strict domination order IEE > SEE > OOE for their linear programming (LP) relaxations, meaning that IEE has the strongest linear relaxation among the event-based models. We further show that the popular DDT formulation can be retrieved from IEE by certain polyhedral operations, thus giving a unifying view on a complete branch of MIP formulations for the RCPSP. In addition, we analyze the computational performance of all presented models on test instances of the PSPLIB (Kolisch and Sprecher 1997).