An exact algorithm to minimize the makespan in project scheduling with scarce resources and generalized precedence relations

In this paper we propose an exact algorithm for the Resource Constrained Project Scheduling Problem (RCPSP) with generalized precedence relationships (GPRs) and minimum makespan objective. For the RCPSP with GPRs we give a new mathematical formulation and a branch and bound algorithm exploiting such a formulation. The exact algorithm takes advantage also of a lower bound based on a Lagrangian relaxation of the same mathematical formulation. We provide an extensive experimentation and a comparison with known lower bounds and competing exact algorithms drawn from the state of the art.     

Single machine scheduling subject to precedence delays

A single-machine scheduling problem with precedence delays is analyzed. A set of n tasks is to be scheduled on the machine in such a way that the makespan is minimized. The executions of the tasks are constrained by precedence delays, i.e., a task can start its execution only after any of its predecessors has completed and the delay between the two tasks has elapsed. In the case of unit execution times and integer lengths of delays, the problem is shown to be NP-hard in the strong sense. In the case of integer execution times and unit length of delays, the problem is polynomial, and an O(n²) optimal algorithm is provided. Both preemptive and non-preemptive cases are considered.     

Multiprocessor scheduling under precedence constraints: Polyhedral results

We consider the problem of scheduling a set of tasks related by precedence constraints to a set of processors, so as to minimize their makespan. Each task has to be assigned to a unique processor and no preemption is allowed. A new integer programming formulation of the problem is given and strong valid inequalities are derived. A subset of the inequalities in this formulation has a strong combinatorial structure, which we use to define the polytope of partitions into linear orders. The facial structure of this polytope is investigated and facet defining inequalities are presented which may be helpful to tighten the integer programming formulation of other variants of multiprocessor scheduling problems. Numerical results on real-life problems are presented.     

Combined vehicle routing and scheduling with temporal precedence and synchronization constraints

We present a mathematical programming model for the combined vehicle routing and scheduling problem with time windows and additional temporal constraints. The temporal constraints allow for imposing pairwise synchronization and pairwise temporal precedence between customer visits, independently of the vehicles. We describe some real world problems where in the literature the temporal constraints are usually remarkably simplified in the solution process, even though these constraints may significantly improve the solution quality and/or usefulness. We also propose an optimization based heuristic to solve real size instances. The results of numerical experiments substantiate the importance of the temporal constraints in the solution approach. We also make a computational study by comparing a direct use of a commercial solver against the proposed heuristic, where the latter approach can find high quality solutions within specific time limits.     

Extending Tilting Modules to One-Point Extensions by Projectives

Let k be an algebraically closed field, B be a finite dimensional k-algebra and A be the one-point extension of B by the projective B-module P ₀. We compare the posets 𝒯 _A and 𝒯 _B of tilting A-modules and B-modules, respectively. We prove that the restriction and the extension functors define morphisms of posets r: 𝒯 _A  → 𝒯 _B and e: 𝒯 _B  → 𝒯 _A such that re = id. Moreover, e induces a full embedding of the quiver of 𝒯 _B into that of 𝒯 _A , whose image is closed under successors, and mapping distinct connected components of the first into distinct connected components of the second.     

Time and work generalised precedence relationships in project scheduling with pre-emption: An application to the management of Service Centres

In this paper we present an application of project scheduling concepts and solution procedures for the solution of a complex problem that comes up in the daily management of many company Service Centres. The real problem has been modelled as a multi-mode resource-constrained project scheduling problem with pre-emption, time and work generalised precedence relationships with minimal and maximal time lags between the tasks and due dates. We present a complete study of work GPRs which includes proper definitions, a new notation and all possible conversions amongst them. Computational results that show the efficiency of the proposed hybrid genetic algorithm and the advantages of allowing pre-emption are also presented.     

On the single machine serial batching scheduling problem to minimize total completion time with precedence constraints, release dates and identical processing times

We consider the single machine, serial batching, total completion time scheduling problem with precedence constraints, release dates and identical processing times in this paper. The complexity of this problem is reported as open in the literature. We provide an O(n⁵) time algorithm to solve this problem.     

PROGRESS: Optimally solving the generalized resource-constrained project scheduling problem

This paper deals with the generalized resource-constrained project scheduling problem (GRCPSP) which extends the well-known resource-constrained project scheduling problem (RCPSP) by considering job specific release and due dates, non-negative minimum start-to-start time lags as well as time-varying resource availabilities. The structure of the project is represented by an acyclic network diagram. Though the extensions are of high practical importance, only a few exact solution procedures have been presented in the literature so far. Therefore, a new exact procedure PROGRESS is developed which includes new dominance rules as well as enhancements of existing ones. For evaluating the efficiency experimentally, new GRCPSP instances with 30 and 60 jobs are considered which extend the standard benchmark sets for the RCPSP generated by ProGen. PROGRESS shows superior performance when applied to the GRCPSP and is also very competitive in comparison to approaches proposed for the RCPSP.     

Optimal scheduling in film production to minimize talent hold cost

The problem of minimizing the cost due to talent hold days in the production of a feature film is considered. A combinatorial model is developed for the sequencing of shooting days in a film shoot. The problem is shown to be strongly NP-hard. A branch-and-bound solution algorithm and a heuristic solution method for large instances of the problem (15 shooting days or more) are developed and implemented on a computer. A number of randomly generated problem instances are solved to study the power and speed of the algorithms. The computational results reveal that the heuristic solution method is effective and efficient in obtaining near-optimal solutions.This research was supported in part by the Natural Sciences and Engineering Research Council of Canada under Grant OPG-0036424. The authors are thankful to two anonymous referees for their helpful comments on an earlier version of this paper.     

Decomposition approaches in permutation scheduling problems with application to the M-machine flow shop scheduling problems

In this article, we consider three decomposition techniques for permutation scheduling problems. We introduce a general iterative decomposition algorithm for permutation scheduling problems and apply it to the permutation flow shop scheduling problem. We also develop bounds needed for this iterative decomposition approach and compare its computational requirements to that of the traditional branch and bound algorithms. Two heuristic algorithms based on the iterative decomposition approach are also developed. extensive numerical study indicates that the heuristic algorithms are practical alternatives to very costly exact algorithms for large flow shop scheduling problems.     

Stochastic scheduling in an in-forest

Consider a problem of scheduling activities of a research and development project, where precedence relations of the activities constituting the project are represented by edges of an in-forest. Each activity is characterized by two parameters: a cost for attempting that activity and a probability that attempting the activity will lead to successful completion. The problem is to find a policy for attempting activities that minimizes the expected cost incurred until termination (successful completion of the project or the first activity failure). The main result of the paper is the design of an efficient algorithm to determine an optimal sequence in which to attempt the activities; a result which is established for linear and exponential utility functions. It is also shown that, unlike the related problem with out-forest precedence relations, there need not exist an optimal policy that is based on an index-rule for determining priority of edges by evaluating their successors.     

Application of an optimization problem in Max-Plus algebra to scheduling problems

The problem tackled in this paper deals with products of a finite number of triangular matrices in Max-Plus algebra, and more precisely with an optimization problem related to the product order. We propose a polynomial time optimization algorithm for 2×2 matrices products. We show that the problem under consideration generalizes numerous scheduling problems, like single machine problems or two-machine flow shop problems. Then, we show that for 3×3 matrices, the problem is NP-hard and we propose a branch-and-bound algorithm, lower bounds and upper bounds to solve it. We show that an important number of results in the literature can be obtained by solving the presented problem, which is a generalization of single machine problems, two- and three-machine flow shop scheduling problems. The branch-and-bound algorithm is tested in the general case and for a particular case and some computational experiments are presented and discussed.     

Periodic real-time scheduling: from deadline-based model to latency-based model

This paper presents a connection between two real-time models: a deadline-based model and a latency-based model. The importance of the latency-based model is proved through a result showing that two deadlines, instead of a latency constraint, over-constrain the real-time applications. Moreover, we give a deadline-marking algorithm based on the relation between deadlines and latency constraints. This algorithm provides non-preemptive feasible schedules for systems with precedence constraints and deadlines, or more complex systems with deadlines and latencies. This is the first step toward non-preemptive schedulability for distributed architectures (without over-constraining the system) like, for example, the automotive applications using protocols such as Controller Area Network (CAN). These results were obtained while the first author was at INRIA Rocquencourt.     

Interval Reductions and Extensions of Orders: Bijections to Chains in Lattices

We discuss bijections that relate families of chains in lattices associated to an order P and families of interval orders defined on the ground set of P. Two bijections of this type have been known:(1) The bijection between maximal chains in the antichain lattice A(P) and the linear extensions of P.(2) The bijection between maximal chains in the lattice of maximal antichains A_M(P) and minimal interval extensions of P.We discuss two approaches to associate interval orders with chains in A(P). This leads to new bijections generalizing Bijections 1 and 2. As a consequence, we characterize the chains corresponding to weak-order extensions and minimal weak-order extensions of P.Seeking for a way of representing interval reductions of P by chains we came upon the separation lattice S(P). Chains in this lattice encode an interesting subclass of interval reductions of P. Let S_M(P) be the lattice of maximal separations in the separation lattice. Restricted to maximal separations, the above bijection specializes to a bijection which nicely complements 1 and 2.(3) A bijection between maximal chains in the lattice of maximal separations S_M(P) and minimal interval reductions of P.     

Application of Benders’ decomposition to power plant preventive maintenance scheduling

Power plant preventive maintenance scheduling consists of knowing which generating units to take off line for regular safety inspection. This problem is extremely important because a failure in a power station may cause a general breakdown in an electric network. The principal danger is that customer electricity demand will not be satisfied in such cases. The problem is approached from the operations research perspective as a question of optimisation. Benders’ decomposition technique is used to solve the resulting model. An example of this application is included. The algorithm is put to use in a real power plant setting. The obtained result is a schedule that allows the efficient organisation of preventive maintenance over the time horizon considered.     

A new AATP model with considering supply chain lead-times and resources and scheduling of the orders in flowshop production systems: A graph-theoretic view

The present study attempts to synchronize the scheduling problem with determining the advanced available-to-promise (AATP) in a flowshop system to enhance supplier profitability and service level. In the proposed model the AATP, scheduling and graph theory concept have been combined to find the optimum resource allocation and enable accurate estimations of machines scheduling, production costs and delivery dates. To find the near optimum solutions for the large size problems a genetic algorithm is developed, first the orders are ranked based on their scores which are estimated then the optimum cost is calculated by balancing profitability and constraints such as the availability of the machines or the available material in each workstation. Some computer simulated experiments are provided to evaluate the performance of the proposed algorithm.     

Stronger Lagrangian bounds by use of slack variables: Applications to machine scheduling problems

Lagrangian relaxation is a powerful bounding technique that has been applied successfully to manyNP-hard combinatorial optimization problems. The basic idea is to see anNP-hard problem as an easy-to-solve problem complicated by a number of nasty side constraints. We show that reformulating nasty inequality constraints as equalities by using slack variables leads to stronger lower bounds. The trick is widely applicable, but we focus on a broad class of machine scheduling problems for which it is particularly useful. We provide promising computational results for three problems belonging to this class for which Lagrangian bounds have appeared in the literature: the single-machine problem of minimizing total weighted completion time subject to precedence constraints, the two-machine flow-shop problem of minimizing total completion time, and the single-machine problem of minimizing total weighted tardiness.     

An evolutionary approach to rehabilitation patient scheduling: A case study

Focusing on real settings, this study aimed to develop an evolutionary approach based on genetic algorithm for solving the problem of rehabilitation patient scheduling to increase service quality by reducing patient waiting time and improve operation efficiency by increasing the therapy equipment utilization. Indeed, due to partial precedence constraints of rehabilitation therapies, the problem can be structured as a hybrid shop scheduling problem that has received little attention to date. In addition, a mixed integer programming model was also constructed as a benchmark to validate the solution quality with small problems. Based on empirical data from a Medical Center in Taiwan, several experiments were conducted to estimate the validity of the proposed algorithm. The results showed that the proposed algorithm can reduce patient waiting time and enhance resource utilization and thus demonstrated the practicality of the proposed algorithm. Indeed, a decision support system embedded with the developed algorithm has been implemented in this medical center.     

Safe scheduling: Setting due dates in single-machine problems

We consider single-machine stochastic scheduling models with due dates as decisions. In addition to showing how to satisfy given service-level requirements, we examine variations of a model in which the tightness of due-dates conflicts with the desire to minimize tardiness. We show that a general form of the trade-off includes the stochastic E/T model and gives rise to a challenging scheduling problem. We present heuristic solution methods based on static and dynamic sorting procedures. Our computational evidence identifies a static heuristic that routinely produces good solutions and a dynamic rule that is nearly always optimal. The dynamic sorting procedure is also asymptotically optimal, meaning that it can be recommended for problems of any size.     

On-line scheduling to minimize max flow time: an optimal preemptive algorithm

We investigate the maximum flow time minimization problem of on-line scheduling jobs on m identical parallel machines. When preemption is allowed, we derive an optimal algorithm with competitive ratio 2-1/m. When preemption is not allowed and m=2, we show that the First In First Out heuristic achieves the best possible competitive ratio.