Tabu search for a class of scheduling problems

Scheduling problems are often modeled as resourceconstrained problems in which critical resource assignments to tasks are known and the best assignment of resource time must be made subject to these constraints. Generalization toresource scheduling, where resource assignments are chosen concurrently with times results is a problem which is much more difficult. A simplified model of the general resource scheduling model is possible, however, in which tasks must be assigned a singleprimary resource, subject to constraints resulting from preassignment ofsecondary, or auxiliary, resources. This paper describes extensions and enhancements of tabu search for the special case of the resource scheduling problem described above. The class of problems is further restricted to those where it is reasonable to enumerate both feasible time and primary resource assignments. Potential applications include shift oriented production and manpower scheduling problems as well as course scheduling where classrooms (instructors) are primary and instructors (rooms) and students are secondary resources. The underlying model is a type of quadratic multiple choice problem which we call multiple choice quadratic vertex packing (MCQVP). Results for strategic oscillation and biased candidate sampling strategies are shown for reasonably sized real and randomly generated, synthetic, problem instances. The strategies are compared with other variations using consistent measures of solution time and quality developed for this study.     

Improved 0/1-interchange scheduling

Consideration is given to the problem of nonpreemptively scheduling a set ofN independent tasks to a system ofM identical processors, with the objective to minimize the overall finish time. It is proved that the 0/1-INTERCHANGE scheduling heuristic can be modified, without increasing its time complexity fromO(N logM), so that its worst-case performance bound is reduced from 2 to 4/3 times optimal.     

Note on scheduling with general learning curves to minimize the number of tardy jobs

Several research studies have confirmed that people and organizations become better at their tasks as the tasks are repeated. The effect of this learning phenomenon on classical scheduling problems has been studied recently. One of the single-machine scheduling problems which seems to become nontrivial when learning effects are introduced is that of minimizing the number of tardy jobs. In this note, we study the special case where all jobs share a common due-date. We show that even when the learning process is assumed to be general and job-dependent, the problem remains polynomially solvable.     

An efficient solution approach for real-world driver scheduling problems in urban bus transportation

The driver scheduling problem in public transportation is defined in the following way. There is a set of operational tasks, and the goal is to define the sequence of these tasks as shifts in such a way that every task must be assigned to a shift without overlapping. In real-world situations several additional constraints need to be considered, which makes large practical problems challenging to be solved efficiently. In practice it is also an important request with respect to a public transportation scheduling system to offer several versions of quasi-optimal solutions. In this paper we present an efficient driver scheduling solution methodology which is flexible in the above sense.     

A new approach for scheduling independent tasks with multiple modes

Heuristic algorithms for scheduling tasks with multiple modes and minimizing the schedule length involve in general two distinct phases, task mode assignment and then task scheduling. We propose a novel approach where these two features are managed in an integrated mechanism with mode assignment embedded in scheduling. The problem is first reformulated as a special single-mode task scheduling problem, and then is modeled as a graph interval T-coloring. Finally, a tabu-like metaheuristic is proposed for this latter graph coloring problem, and the performance of our approach is compared to known multi-mode scheduling heuristics.     

Legal coloring of graphs

The following computational problem was initiated by Manber and Tompa (22nd FOCS Conference, 1981): Given a graphG=(V, E) and a real functionf:V→R which is a proposed vertex coloring. Decide whetherf is a proper vertex coloring ofG. The elementary steps are taken to be linear comparisons. Lower bounds on the complexity of this problem are derived using the chromatic polynomial ofG. It is shown how geometric parameters of a space partition associated withG influence the complexity of this problem. Existing methods for analyzing such space partitions are suggested as a powerful tool for establishing lower bounds for a variety of computational problems.     

Scheduling vessels and container-yard operations with conflicting objectives

We consider the problem of coordinating the operations of two supply chain partners: a foreign shipping company and a domestic port. The two partners have conflicting business objectives, and the issue is to determine the optimal cycle time, by which the shipping company removes the empty containers from the domestic port, so that the joint profit of the two partners is maximized. The domestic port prefers a shorter cycle time to mitigate its empty container accumulation and land use problems, while the shipping company wishes a longer cycle time to save its expensive vessel capacities. We propose an iterative procedure to search for this optimal cycle time. In each iteration, a candidate cycle time is evaluated by solving a deterministic vessel scheduling problem and a stochastic container-yard capacity optimization problem. We prove the properties of the vessel scheduling problem, derive the optimality condition under which the vessel scheduling problem can be decomposed, and show that the profit function of the domestic port is convex and thus the optimal container-yard capacity can be determined efficiently. Empirical observations on the algorithm computational performance collected from over 300 randomly generated test cases under various problem settings are reported.     

A subproblem-centric model and approach to the nurse scheduling problem

The number of hospitals in Japan exceeds 10,000, and every month nurses are scheduled to shifts in about 30,000 units in total. There is serious demand for automating this scheduling task. In this paper, we introduce a mathematical programming formulation of the nurse scheduling problem in Japan, and develop a meta-heuristic approach to solve the problem. This scheduling problem is a hard combinatorial problem due to tight constraints involving such factors as the skill level of a team, the need to balance workload among nurses, and the consideration of nurses' preferences, even though the number of the nurses to be scheduled is not large, at between 20 and 40. The performance of our approach is demonstrated by the successful solution of data taken from actual scheduling problems. The proposed model and approach can be adapted for the majority of hospitals in Japan, as well as for some hospitals in other countries, and is likely applicable to many other scheduling problems in the fields of business and logistics. Key words.nurse scheduling – block-angular problem – subproblem – integer programming – relaxation – tabu search – branch-and-boundMathematics Subject Classification (1991):20E28, 20G40, 20C20     

An Analysis of Partially Clairvoyant Scheduling

Real-time scheduling problems confront two issues not addressed by traditional scheduling models, viz., parameter variability and the existence of complex relationships constraining the executions of jobs. Accordingly, modeling becomes crucial in the specification of scheduling problems in such systems. In this paper, we analyze scheduling algorithms in Partially Clairvoyant Real-time scheduling systems and present a new dual-based algorithm for the feasibility problem in the case of strict relative constraints. We also study the problem of online dispatching in Partially Clairvoyant systems and show that the complexity of dispatching is logarithmically related to the complexity of the schedulability problem.     

Preemptive scheduling of interval orders is polynomial

In 1979, Papadimitriou and Yannakakis gave a polynomial time algorithm for the scheduling of jobs requiring unit completion times when the precedence constraints form an interval order. The authors solve here the corresponding problem, for preemptive scheduling (a job can be interrupted to work on more important tasks, and completed at a later time, subject to the usual scheduling constraints.) The m-machine preemptive scheduling problem is shown to have a polynomial algorithm, for both unit time and variable execution times as well, when the precedence constraints are given by an interval order.     

On the geometry,preemptions and complexity of multiprocessor and shop scheduling

In this paper we study multiprocessor and open shop scheduling problems from several points of view. We explore a tight dependence of the polynomial solvability/intractability on the number of allowed preemptions. For an exhaustive interrelation, we address the geometry of problems by means of a novel graphical representation. We use the so-called preemption and machine-dependency graphs for preemptive multiprocessor and shop scheduling problems, respectively. In a natural manner, we call a scheduling problem acyclic if the corresponding graph is acyclic. There is a substantial interrelation between the structure of these graphs and the complexity of the problems. Acyclic scheduling problems are quite restrictive; at the same time, many of them still remain NP-hard. We believe that an exhaustive study of acyclic scheduling problems can lead to a better understanding and give a better insight of general scheduling problems. We show that not only acyclic but also a special non-acyclic version of periodic job-shop scheduling can be solved in polynomial (linear) time. In that version, the corresponding machine dependency graph is allowed to have a special type of the so-called parti-colored cycles. We show that trivial extensions of this problem become NP-hard. Then we suggest a linear-time algorithm for the acyclic open-shop problem in which at most m−2 preemptions are allowed, where m is the number of machines. This result is also tight, as we show that if we allow one less preemption, then this strongly restricted version of the classical open-shop scheduling problem becomes NP-hard. In general, we show that very simple acyclic shop scheduling problems are NP-hard. As an example, any flow-shop problem with a single job with three operations and the rest of the jobs with a single non-zero length operation is NP-hard. We suggest linear-time approximation algorithm with the worst-case performance of ( , respectively) for acyclic job-shop (open-shop, respectively), where (‖ℳ‖, respectively) is the maximal job length (machine load, respectively). We show that no algorithm for scheduling acyclic job-shop can guarantee a better worst-case performance than . We consider two special cases of the acyclic job-shop with the so-called short jobs and short operations (restricting the maximal job and operation length) and solve them optimally in linear time. We show that scheduling m identical processors with at most m−2 preemptions is NP-hard, whereas a venerable early linear-time algorithm by McNaughton yields m−1 preemptions. Another multiprocessor scheduling problem we consider is that of scheduling m unrelated processors with an additional restriction that the processing time of any job on any machine is no more than the optimal schedule makespan C _max^*. We show that the (2m−3)-preemptive version of this problem is polynomially solvable, whereas the (2m−4)-preemptive version becomes NP-hard. For general unrelated processors, we guarantee near-optimal (2m−3)-preemptive schedules. The makespan of such a schedule is no more than either the corresponding non-preemptive schedule makespan or max {C _max^*,p _max}, where C _max^* is the optimal (preemptive) schedule makespan and p _max is the maximal job processing time. E.V. Shchepin was partially supported by the program “Algebraical and combinatorial methods of mathematical cybernetics” of the Russian Academy of Sciences. N. Vakhania was partially supported by CONACyT grant No. 48433.     

Local search algorithms for the rectangle packing problem with general spatial costs

We propose local search algorithms for the rectangle packing problem to minimize a general spatial cost associated with the locations of rectangles. The problem is to pack given rectangles without overlap in the plane so that the maximum cost of the rectangles is minimized. Each rectangle has a set of modes, where each mode specifies the width and height of the rectangle and its spatial cost function. The spatial costs are general piecewise linear functions which can be non-convex and discontinuous. Various types of packing problems and scheduling problems can be formulated in this form. To represent a solution of this problem, a pair of permutations of n rectangles is used to determine their horizontal and vertical partial orders, respectively. We show that, under the constraint specified by such a pair of permutations, optimal locations of the rectangles can be efficiently determined by using dynamic programming. The search for finding good pairs of permutations is conducted by local search and metaheuristic algorithms. We report computational results on various implementations using different neighborhoods, and compare their performance. We also compare our algorithms with other existing heuristic algorithms for the rectangle packing problem and scheduling problem. These computational results exhibit good prospects of the proposed algorithms. Key words.rectangle packing – sequence pair – general spatial cost – dynamic programming – metaheuristicsMathematics Subject Classification (1991):20E28, 20G40, 20C20     

Completion problems forj pq-inner functions. I.

This paper studies various completion problems for a subclass ofj _pq-inner functions. Special attention is drawn to so-calledA-normalizedj _pq-elementary factors of full-rank, which are closely related to the matricial Schur problem. Finally, as an application an inverse problem for Carathéodory sequences is answered.     

Dynamic Generalized Assignment Problems with Stochastic Demands and Multiple Agent--Task Relationships

The assignment problem is a well-known operations research model. Its various extensions have been applied to the design of distributed computer systems, job assignment in telecommunication networks, and solving diverse location, truck routing and job shop scheduling problems.This paper focuses on a dynamic generalization of the assignment problem where each task consists of a number of units to be performed by an agent or by a limited number of agents at a time. Demands for the task units are stochastic. Costs are incurred when an agent performs a task or a group of tasks and when there is a surplus or shortage of the task units with respect to their demands. We prove that this stochastic, continuous-time generalized assignment problem is strongly NP-hard, and reduce it to a number of classical, deterministic assignment problems stated at discrete time points. On this basis, a pseudo-polynomial time combinatorial algorithm is derived to approximate the solution, which converges to the global optimum as the distance between the consecutive time points decreases. Lower bound and complexity estimates for solving the problem and its special cases are found.     

Computing optimal preemptive schedules for parallel tasks: linear programming approaches

 We study the problem of scheduling a set of n independent parallel tasks on m processors, where in addition to the processing time there is a size associated with each task indicating that the task can be processed on any subset of processors of the given size. Based on a linear programming formulation, we propose an algorithm for computing a preemptive schedule with minimum makespan, and show that the running time of the algorithm depends polynomially on m and only linearly on n. Thus for any fixed m, an optimal preemptive schedule can be computed in O(n) time. We also present extensions of this approach to other (more general) scheduling problems with malleable tasks, due dates and maximum lateness minimization. Received: November 1999 / Accepted: November 2002 Publication online: December 19, 2002 RID="⋆" ID="⋆" This work was done while the authors were associated with the research institutes IDSIA Lugano and MPII Saarbrücken and were supported in part by the Swiss Office Fédéral de l'éducation et de la Science project n 97.0315 titled ``Platform' and by EU ESPRIT LTR Project No. 20244 (ALCOM-IT)     

Preemptive scheduling with rejection

 We consider the problem of preemptively scheduling a set of n jobs on m (identical, uniformly related, or unrelated) parallel machines. The scheduler may reject a subset of the jobs and thereby incur job-dependent penalties for each rejected job, and he must construct a schedule for the remaining jobs so as to optimize the preemptive makespan on the m machines plus the sum of the penalties of the jobs rejected. We provide a complete classification of these scheduling problems with respect to complexity and approximability. Our main results are on the variant with an arbitrary number of unrelated machines. This variant is APX-hard, and we design a 1.58-approximation algorithm for it. All other considered variants are weakly -hard, and we provide fully polynomial time approximation schemes for them. Finally, we argue that our results for unrelated machines can be carried over to the corresponding preemptive open shop scheduling problem with rejection. Received: October 30, 2000 / Accepted: September 26, 2001 Published online: September 5, 2002 Key words. scheduling – preemption – approximation algorithm – worst case ratio – computational complexity – in-approximability Supported in part by the EU Thematic Network APPOL, Approximation and Online Algorithms, IST-1999-14084 Supported by the START program Y43-MAT of the Austrian Ministry of Science.     

A constrained matching problem

We show that certain manpower scheduling problems can be modeled as the following constrained matching problem. Given an undirected graphG = (V,E) with edge weights and a digraphD = (V,A). AMaster/Slave-matching (MS-matching) ofG with respect toD is a matching ofG such that for each arc (u, v) A for which the nodeu is matched, the nodev is matched, too. TheMS-Matching Problem is the problem of finding a maximum-weight MS-matching. Letk(D) be the maximum size of a (weakly) connected component ofD. We prove that MS-matching is an NP-hard problem even ifG is bipartite andk(D) 3. Moreover, we show that in the relevant special case wherek(D) 2, the MS-Matching Problem can be transformed to the ordinary Matching Problem.This research was supported by Grant 03-KL7PAS-6 of the German Federal Ministry of Research and Technology.     

Scheduling tasks with communication delays on parallel processors

LetP={v ₁,...,v _n } be a set ofn jobs to be executed on a set ofm identical machines. In many instances of scheduling problems, if a jobv _i has to be executed before the jobv _j and both jobs are to be executed on different machines, some sort of information exchange has to take place between the machines executing them. The time it takes for this exchange of information is called a communication delay.In this paper we give anO(n) algorithm to find an optimal scheduling with communication delays when the number of machines is not limited and the precedence constraints on the jobs form a tree.     

Reducing network and computation complexities in neural based real-time scheduling scheme

Multiprocessor real-time scheduling is an important issue in many applications. A neural network provides a highly effective method to obtain good solutions for real-time scheduling problems. However, multiprocessor real-time scheduling problems include multiple variables; processor, process and time, and the neural networks have to be presented in three dimensions with these variables. Hence, the corresponding neural networks have more neurons, and synaptic weights, and thus associated network and computational complexities increase. Meanwhile, a neural network using the competitive scheme can provide a highly effective method with less network complexity. Therefore, in this study, a simplified two-dimensional Hopfield-type neural network using competitive rule is introduced for solving three-dimensional multiprocessor real-time scheduling problems. Restated, a two-dimensional network is proposed to lower the neural network dimensions and decrease the number of neurons and hence reduce the network complexity; an M-out-of-N competitive scheme is suggested to greatly reduce the computational complexity. Simulation results reveal that the proposed scheme imposed on the derived energy function with respect to process time and deadline constraints is an appropriate approach to solving these class scheduling problems. Moreover, the computational complexity of the proposed scheme is greatly lowered to O(N × T²).