Counting and enumeration complexity with application to multicriteria scheduling

In this paper we tackle an important point of combinatorial optimisation: that of complexity theory when dealing with the counting or enumeration of optimal solutions. Complexity theory has been initially designed for decision problems and evolved over the years, for instance, to tackle particular features in optimisation problems. It has also evolved, more or less recently, towards the complexity of counting and enumeration problems and several complexity classes, which we review in this paper, have emerged in the literature. This kind of problems makes sense, notably, in the case of multicriteria optimisation where the aim is often to enumerate the set of the so-called Pareto optima. In the second part of this paper we review the complexity of multicriteria scheduling problems in the light of the previous complexity results. This paper appeared in 4OR 3(1), 1–21, 2005.     

Personnel scheduling in the Call Center industry

We discuss the main results of the author’s PhD thesis (Canon 2005) presented in December 2005 at the Université François-Rabelais de Tours and supervised by J.-C. Billaut and J.-L. Bouquard. This thesis is written in French and is available from the author upon request. It deals with three different problems arising in the call center industry: the problem of dimensioning a call center, the annualisation problem and finally the Shift Design Problem.     

On the complexity of project scheduling to minimize exposed time

We consider project scheduling where the project manager’s objective is to minimize the time from when an adversary discovers the project until the completion of the project. We analyze the complexity of the problem identifying both polynomially solvable and NP-hard versions of the problem. The complexity of the problem is seen to be dependent on the nature of renewable resource constraints, precedence constraints, and the ability to crash activities in the project.     

Fixed interval scheduling: Models,applications, computational complexity and algorithms

The defining characteristic of fixed interval scheduling problems is that each job has a finite number of fixed processing intervals. A job can be processed only in one of its intervals on one of the available machines, or is not processed at all. A decision has to be made about a subset of the jobs to be processed and their assignment to the processing intervals such that the intervals on the same machine do not intersect. These problems arise naturally in different real-life operations planning situations, including the assignment of transports to loading/unloading terminals, work planning for personnel, computer wiring, bandwidth allocation of communication channels, printed circuit board manufacturing, gene identification and examining computer memory structures. We present a general formulation of the interval scheduling problem, show its relations to cognate problems in graph theory, and survey existing models, results on computational complexity and solution algorithms.     

New complexity results on scheduling with small communication delays

Although most of the scheduling problems with interprocessor communication delays have been shown to be NP-complete, some important special cases were still unsolved. This paper deals with the problem where communication times are smaller than processing times and task duplication is not allowed. We prove that this problem is NP-complete and we give an efficient approximate algorithm with performance guarantee.     

Minimizing shifts for personnel task scheduling problems: A three-phase algorithm

The personnel task scheduling problem is a subject of commercial interest which has been investigated since the 1950s. This paper proposes an effective and efficient three-phase algorithm for solving the shift minimization personnel task scheduling problem (SMPTSP). To illustrate the increased efficacy of the proposed algorithm over an existing algorithm, computational experiments are performed on a test problem set with characteristics motivated by employee scheduling applications. Experimental results show that the proposed algorithm outperforms the existing algorithm in terms of providing optimal solutions, improving upon most of the best-known solutions and revealing high-quality feasible solutions for those unsolved test instances in the literature.     

Workforce scheduling with multiple objectives

In workforce scheduling, the optimal schedule has traditionally been determined by minimizing the cost of labor subject to an acceptable service level, which is defined as the percentage of customers served within a predetermined time interval. We propose an alternative multidimensional paradigm, where cost minimization and service level maximization are considered simultaneously, together with other, complementary criteria. The ultimate goal of the proposed approach is to open a broader workforce scheduling paradigm that incorporates service quality into the analysis and provides the possibility to study the interaction between cost and service quality. Furthermore, the approach enables us to avoid strong assumptions. An example with real-world, empirical demand data is provided.     

The just-in-time scheduling problem in a flow-shop scheduling system

We study the problem of maximizing the weighted number of just-in-time (JIT) jobs in a flow-shop scheduling system under four different scenarios. The first scenario is where the flow-shop includes only two machines and all the jobs have the same gain for being completed JIT. For this scenario, we provide an O(n³) time optimization algorithm which is faster than the best known algorithm in the literature. The second scenario is where the job processing times are machine-independent. For this scenario, the scheduling system is commonly referred to as a proportionate flow-shop. We show that in this case, the problem of maximizing the weighted number of JIT jobs is NP-hard in the ordinary sense for any arbitrary number of machines. Moreover, we provide a fully polynomial time approximation scheme (FPTAS) for its solution and a polynomial time algorithm to solve the special case for which all the jobs have the same gain for being completed JIT. The third scenario is where a set of identical jobs is to be produced for different customers. For this scenario, we provide an O(n³) time optimization algorithm which is independent of the number of machines. We also show that the time complexity can be reduced to O(n log n) if all the jobs have the same gain for being completed JIT. In the last scenario, we study the JIT scheduling problem on m machines with a no-wait restriction and provide an O(mn²) time optimization algorithm.     

Algorithms for large scale Shift Minimisation Personnel Task Scheduling Problems

In this paper we introduce the Personnel Task Scheduling Problem (PTSP) and provide solution algorithms for a variant of this problem known as the Shift Minimisation Personnel Task Scheduling Problem (SMPTSP). The PTSP is a problem in which a set of tasks with fixed start and finish times have to be allocated to a heterogenous workforce. Personnel work in shifts with fixed start and end times and have skills that enable them to perform some, but not all tasks. In other words, some personnel are qualified to only perform a subset of all tasks. The objective is to minimise the overall cost of personnel required to perform the given set of tasks. In this paper we introduce a special case in which the only cost incurred is due to the number of personnel (shifts) that are used. This variant of the PTSP is referred to as the Shift Minimisation Personnel Task Scheduling Problem (SMPTSP). While our motivation is a real-life Personnel Task Scheduling Problem, the formulation may also be applied to machine shop scheduling. We review the existing literature, provide mathematical formulations, and develop a heuristic approach for the SMPTSP.     

A polynomial algorithm for 2-cyclic robotic scheduling: A non-Euclidean case

In this paper we consider the problem of no-wait cyclic scheduling of identical parts in an m-machine production line in which a robot is responsible for moving each part from a machine to another. The aim is to find the minimum cycle time for the so-called 2-cyclic schedules, in which exactly two parts enter and two parts leave the production line during each cycle. The earlier known polynomial-time algorithms for this problem are applicable only under the additional assumption that the robot travel times satisfy the triangle inequalities. We lift this assumption on robot travel times and present a polynomial-time algorithm with the same time complexity as in the metric case, O(m⁵logm).     

Counting and enumeration complexity with application to multicriteria scheduling

In this paper we tackle an important point of combinatorial optimisation: that of complexity theory when dealing with the counting or enumeration of optimal solutions. Complexity theory has been initially designed for decision problems and evolved over the years, for instance, to tackle particular features in optimisation problems. It has also evolved, more or less recently, towards the complexity of counting and enumeration problems and several complexity classes, which we review in this paper, have emerged in the literature. This kind of problems makes sense, notably, in the case of multicriteria optimisation where the aim is often to enumerate the set of the so-called Pareto optima. In the second part of this paper we review the complexity of multicriteria scheduling problems in the light of the previous complexity results.Received: November 2004 / Received version: March 2005MSC classification: 90B40, 90C29, 68Q15     

A note on the complexity of flow shop scheduling with transportation constraints

This note investigates two-machine flow shop scheduling with transportation constraints to minimize makespan. Recently, Soukhal et al. [A. Soukhal, A. Oulamara, P. Martineau, Complexity of flow shop scheduling problems with transportation constraints, European Journal of Operational Research 161 (2005) 32–41] proved that this problem is strongly NP-hard when the capacity of the truck is limited to two or three parts. The considered problem with blocking constraints is also proved to be strongly NP-hard by Soukhal et al. Unfortunately, their proofs contain mistakes. We point out their proofs’ invalidity and then show that, when the capacity of the truck is limited to two parts, the problem is binary NP-hard, and when the capacity of the truck is limited to three parts the problem is strongly NP-hard even if the jobs have a common processing time on machine one and all jobs have the same transportation time. We show also that the last result can be generalized to any fixed c (c ? 3) parts.     

The complexity of machine scheduling for stability with a single disrupted job

A stable schedule is a robust schedule that will change little when uncertain events occur. The purpose of this paper is to investigate the complexity status of a number of machine scheduling problems with stability objective, when the duration of a single job is anticipated to be disrupted.     

A survey of scheduling with controllable processing times

In classical deterministic scheduling problems, the job processing times are assumed to be constant parameters. In many practical cases, however, processing times are controllable by allocating a resource (that may be continuous or discrete) to the job operations. In such cases, each processing time is a decision variable to be determined by the scheduler, who can take advantage of this flexibility to improve system performance. Since scheduling problems with controllable processing times are very interesting both from the practical and theoretical point of view, they have received a lot of attention from researchers over the last 25 years. This paper aims to give a unified framework for scheduling with controllable processing times by providing an up-to-date survey of the results in the field.     

“Product Partition” and related problems of scheduling and systems reliability: Computational complexity and approximation

Problem Product Partition differs from the NP-complete problem Partition in that the addition operation is replaced by the multiplication operation. Furthermore it differs from the NP-complete problem Subset Product in that it does not contain the product value B in its input. We prove that problem Product Partition and several of its modifications are NP-complete in the strong sense. Our results imply the strong NP-hardness of a number of scheduling problems with start-time-dependent job processing times and a problem of designing a reliable system with a series–parallel structure. It should be noticed that the strong NP-hardness of the considered optimization problems does not preclude the existence of a fully polynomial time approximation scheme (FPTAS) for them. We present a simple FPTAS for one of these problems.     

Space complexity of Abelian groups

We develop a theory of LOGSPACE structures and apply it to construct a number of examples of Abelian Groups which have LOGSPACE presentations. We show that all computable torsion Abelian groups have LOGSPACE presentations and we show that the groups , and the additive group of the rationals have LOGSPACE presentations over a standard universe such as the tally representation and the binary representation of the natural numbers. We also study the effective categoricity of such groups. For example, we give conditions are given under which two isomorphic LOGSPACE structures will have a linear space isomorphism.