A linear time approximation algorithm for multiprocessor scheduling

Givenn jobs andm identical processors anO(n) approximation algorithm is presented which tries to determine a nonpreemptive schedule with minimum finish time. Ifr is the number of jobs placed onto the processor with maximum finish time, then the worst case ratio of the new algorithm's finish time to the optimal solution is shown to be less thanrm/(rm–m+1). Extensive empirical results show that the new algorithm is competitive with the LPT algorithm in terms of quality of solution and faster in terms of computing time.     

Polynomial time approximation schemes and parameterized complexity

In this paper, we study the relationship between the approximability and the parameterized complexity of NP optimization problems. We introduce a notion of polynomial fixed-parameter tractability and prove that, under a very general constraint, an NP optimization problem has a fully polynomial time approximation scheme if and only if the problem is polynomial fixed-parameter tractable. By enforcing a constraint of planarity on the W-hierarchy studied in parameterized complexity theory, we obtain a class of NP optimization problems, the planar W-hierarchy, and prove that all problems in this class have efficient polynomial time approximation schemes (EPTAS). The planar W-hierarchy seems to contain most of the known EPTAS problems, and is significantly different from the class introduced by Khanna and Motwani in their efforts in characterizing optimization problems with polynomial time approximation schemes.     

Approximation schemes for job shop scheduling problems with controllable processing times

In this paper we study the job shop scheduling problem under the assumption that the jobs have controllable processing times. The fact that the jobs have controllable processing times means that it is possible to reduce the processing time of the jobs by paying a certain cost. We consider two models of controllable processing times: continuous and discrete. For both models we present polynomial time approximation schemes when the number of machines and the number of operations per job are fixed.     

Polynomial time algorithms for two special classes of the proportionate multiprocessor open shop

This paper describes a complex scheduling problem taken from a hospital diagnostic testing center that schedules hundreds of patients in an open shop environment consisting of multiple facilities and multiple processors. This scheduling problem, known as the multiprocessor open shop (MPOS) problem, is strongly NP-hard with few published results. Realizing that in many MPOS environments processing times are stage-dependent, not both job and stage-dependent, this paper examines a new class of problems for the MPOS—proportionate ones. This paper exploits the structural nature of the proportionate MPOS and defines new terms. Despite the enormous complexity of the MPOS problem, this work demonstrates that polynomial time algorithms exist for two special cases. Since other applications of this problem exist in service and manufacturing environments, solving the proportionate MPOS problem is not only significant in the theory of optimization, but also in many real-world applications.     

Polynomial time approximation schemes for dense instances of minimum constraint satisfaction

It is known that large fragments of the class of dense Minimum Constraint Satisfaction (MIN‐CSP) problems do not have polynomial time approximation schemes (PTASs) contrary to their Maximum Constraint Satisfaction analogs. In this paper we prove, somewhat surprisingly, that the minimum satisfaction of dense instances of kSAT ‐formulas, and linear equations mod 2, Ek‐LIN2, do have PTASs for any k. The MIN‐Ek‐LIN2 problems are equivalent to the k‐ary versions of the Nearest Codeword problem, the problem which is known to be exceedingly hard to approximate on general instances. The method of solution of the above problems depends on the development of a new density sampling technique for k‐uniform hypergraphs which could be of independent interest. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 73–91, 2003     

High-multiplicity cyclic job shop scheduling

We consider the High-Multiplicity Cyclic Job Shop Scheduling Problem. There are two objectives of interest: the cycle time and the flow time. We give several approximation algorithms after showing that a very restricted case is APX-hard.     

Inapproximability results for no-wait job shop scheduling

We investigate the approximability of the no-wait job shop scheduling problem under the makespan criterion. We show that this problem is -hard (i) for the case of two machines with at most five operations per job, and (ii) for the case of three machines with at most three operations per job. Hence, these problems do not possess a polynomial time approximation scheme, unless .     

Approximative procedures for no-wait job shop scheduling

In this article we consider the no-wait job shop problem with makespan objective. Based on a decomposition of the problem into a sequencing and a timetabling problem, we propose two local search algorithms. Extensive computational tests in which the algorithms compare favorably to the best existing strategies are reported. Although not specifically designed for that purpose, our algorithms also outperform one of the best no-wait flow shop algorithms in literature.     

Surrogate duality relaxation for job shop scheduling

Surrogate duality bounds for the job shop scheduling problem are obtained by replacing certain constraints by their weighted sum and strengthening the aggregate constraint by iterating over all possible weights. The constraints successively considered for this purpose are the capacity constraints on the machines and the precedence constraints determining the machine order for each job. The resulting relaxations are investigated from a theoretical and a computational point of view.     

A tabu search approach for proportionate multiprocessor open shop scheduling

In the multiprocessor open shop scheduling problem, jobs are to be processed on a set of processing centers—each having one or more parallel identical machines, while jobs do not have a pre-specified obligatory route. A special case is the proportionate multiprocessor open shop scheduling problem (PMOSP) in which the processing time on a given center is not job-dependent. Applications of the PMOSP are evident in health care systems, maintenance and repair shops, and quality auditing and final inspection operations in industry. In this paper, a tabu search (TS) approach is presented for solving the PMOSP with the objective of minimizing the makespan. The TS approach utilizes a neighborhood search function that is defined over a network representation of feasible solutions. A set of 100 benchmark problems from the literature is used to evaluate the performance of the developed approach. Experimentations show that the developed approach outperforms a previously developed genetic algorithm as it produces solutions with an average of less than 5 % deviation from a lower bound, and 40 % of its solutions are provably optimal.     

Local search algorithms for the multiprocessor flow shop scheduling problem

The multiprocessor flow shop scheduling problem is a generalization of the ordinary flow shop scheduling problem. The problem consists of both assigning operations to machines and scheduling the operations assigned to the same machine. We review the literature on local search methods for flow shop and job shop scheduling and adapt them to the multiprocessor flow shop scheduling problem. Other local search approaches we consider are variable-depth search and simulated annealing. We show that tabu search and variable-depth search with a neighborhood originated by Nowicki and Smutnicki outperform the other algorithms.     

Fuzzy job shop scheduling with lot-sizing

This paper deals with a problem of determining lot-sizes of jobs in a real-world job shop-scheduling in the presence of uncertainty. The main issue discussed in this paper is lot-sizing of jobs. A fuzzy rule-based system is developed which determines the size of lots using the following premise variables: size of the job, the static slack of the job, workload on the shop floor, and the priority of the job. Both premise and conclusion variables are modelled as linguistic variables represented by using fuzzy sets (apart from the priority of the job which is a crisp value). The determined lots’ sizes are input to a fuzzy multi-objective genetic algorithm for job shop scheduling. Imprecise jobs’ processing times and due dates are modelled by using fuzzy sets. The objectives that are used to measure the quality of the generated schedules are average weighted tardiness of jobs, the number of tardy jobs, the total setup time, the total idle time of machines and the total flow time of jobs. The developed algorithm is analysed on real-world data obtained from a printing company.     

Depth-first heuristic search for the job shop scheduling problem

We evaluate two variants of depth-first search algorithms and consider the classic job shop scheduling problem as a test bed. The first one is the well-known branch-and-bound algorithm proposed by P. Brucker et al. which uses a single chronological backtracking strategy. The second is a variant that uses partially informed depth-first search strategy instead. Both algorithms use the same heuristic estimation; in the first case, it is only used for pruning states that cannot improve the incumbent solution, whereas in the second it is also used to sort the successors of an expanded state. We also propose and analyze a new heuristic estimation which is more informed and more time consuming than that used by Brucker’s algorithm. We conducted an experimental study over well-known instances showing that the proposed partially informed depth-first search algorithm outperforms the original Brucker’s algorithm.     

Cyclic job shop scheduling problems with blocking

A tabu search algorithm for a cyclic job shop problem with blocking is presented. Operations are blocking if they must stay on a machine after finishing when the next machine is occupied by another job. During this stay the machine is blocked for other jobs. For this problem traditional tabu search moves often lead to infeasible solutions. Recovering procedures are developed which construct nearby feasible solutions. Computational results are presented for the approach.     

Permutation-induced acyclic networks for the job shop scheduling problem

In the literature of the combinatorial optimization problems, it is a commonplace to find more than one mathematical model for the same problem. The significance of a model may be measured in terms of the efficiency of the solution algorithms that can be built upon it. The purpose of this article is to present a new network model for the well known combinatorial optimization problem – the job shop scheduling problem. The new network model has similar structure as the disjunctive graph model except that it uses permutations of jobs as decision variables instead of the binary decision variables associated with the disjunctive arcs. To assess the significance of the new model, the performances of exact branch-and-bound algorithmic implementations that are based on both the new model and the disjunctive graph model are compared.     

Surgical case scheduling as a generalized job shop scheduling problem

Surgical case scheduling allocates hospital resources to individual surgical cases and decides on the time to perform the surgeries. This task plays a decisive role in utilizing hospital resources efficiently while ensuring quality of care for patients. This paper proposes a new surgical case scheduling approach which uses a novel extension of the Job Shop scheduling problem called multi-mode blocking job shop (MMBJS). It formulates the MMBJS as a mixed integer linear programming (MILP) problem and discusses the use of the MMBJS model for scheduling elective and add-on cases. The model is illustrated by a detailed example, and preliminary computational experiments with the CPLEX solver on practical-sized instances are reported.     

A projective algorithm for preemptive open shop scheduling with two multiprocessor groups

We study a multiprocessor extension of the preemptive open shop scheduling problem, where the set of processors is partitioned into processor groups. We show that the makespan minimization problem is polynomially solvable for two multiprocessor groups even if preemptions are restricted to integral times.     

Preemptive scheduling in a two-stage multiprocessor flow shop is NP-hard

In 1954, Johnson gave an efficient algorithm for minimizing makespan in a two-machine flow shop; there is no advantage to preemption in this case. McNaughton's wrap-around rule of 1959 finds a shortest preemptive schedule on identical parallel machines in linear time. A similarly efficient algorithm is unlikely to exist for the simplest common generalization of these problems. We show that preemptive scheduling in a two-stage flow shop with at least two identical parallel machines in one of the stages so as to minimize makespan is NP-hard in the strong sense.