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Showing papers on "Job shop scheduling published in 1987"


Journal ArticleDOI
TL;DR: A new approach to constructing approximation algorithms, which the aim is find superoptimal, but infeasible solutions, and the performance is measured by the degree of infeasibility allowed, which should find wide applicability for any optimization problem where traditional approximation algorithms have been particularly elusive.
Abstract: The problem of scheduling a set of n jobs on m identical machines so as to minimize the makespan time is perhaps the most well-studied problem in the theory of approximation algorithms for NP-hard optimization problems. In this paper the strongest possible type of result for this problem, a polynomial approximation scheme, is presented. More precisely, for each e, an algorithm that runs in time O((n/e)1/e2) and has relative error at most e is given. In addition, more practical algorithms for e = 1/5 + 2-k and e = 1/6 + 2-k, which have running times O(n(k + log n)) and O(n(km4 + log n)) are presented. The techniques of analysis used in proving these results are extremely simple, especially in comparison with the baroque weighting techniques used previously.The scheme is based on a new approach to constructing approximation algorithms, which is called dual approximation algorithms, where the aim is to find superoptimal, but infeasible, solutions, and the performance is measured by the degree of infeasibility allowed. This notion should find wide applicability in its own right and should be considered for any optimization problem where traditional approximation algorithms have been particularly elusive.

766 citations


Book
01 Jan 1987
TL;DR: In this thesis, a system called ISIS is presented, which uses a constraint-directed search paradigm to solve the scheduling problem and provides a knowledge representation language for modeling organizations and their constraints.
Abstract: : This thesis investigates the problem of constraint-directed reasoning in the job-shop scheduling domain. The job-shop scheduling problem is defined as: selecting a sequence of operations whose execution results in the completion of an order, and assigning times (i.e., start and end times) and resources to each operation. The number of possible schedules grows exponentially with the number of orders, alternative production plans, substitutable resources, and possible times to assign resources and perform operations. The acceptability of a particular schedule depends not only on the availability of alternatives, but on other knowledge such as organizational goals, physical limitations of resources, causal restrictions amongst resources and operations, availability of resources, and preferences amongst alternatives. By viewing the scheduling problem from a constraint-directed search perspective, much of this knowledge can be viewed as constraints on the schedule generation and selection process. In this thesis, we present a system called ISIS. ISIS uses a constraint-directed search paradigm to solve the scheduling problem. ISIS provides: a knowledge representation language (SRL) for modeling organizations and their constraints; hierarchical, constraint-directed scheduling of orders, which includes: constraint-directed bounding of the solution space; context-sensitive selection of constraints, and weighted interpretation of constraints; analytic and generative constraint relaxation; and techniques for the diagnosis of poor schedules.

532 citations


Proceedings ArticleDOI
12 Oct 1987
TL;DR: In this paper, a polynomial algorithm was proposed to find a schedule that minimizes the makespan of a linear programming problem with a fixed number of machines and constant number of processing times.
Abstract: We consider the following scheduling problem. There are m parallel machines and n independent jobs. Each job is to be assigned to one of the machines. The processing of job j on machine i requires time pij. The objective is to find a schedule that minimizes the makespan. Our main result is a polynomial algorithm which constructs a schedule that is guaranteed to be no longer than twice the optimum. We also present a polynomial approximation scheme for the case that the number of machines is fixed. Both approximation results are corollaries of a theorem about the relationship of a class of integer programming problems and their linear programming relaxations. In particular, we give a polynomial method to round the fractional extreme points of the linear program to integral points that nearly satisfy the constraints. In contrast to our main result, we prove that no polynomial algorithm can achieve a worst-case ratio less than 3/2 unless P = NP. We finally obtain a complexity classification for all special cases with a fixed number of processing times.

384 citations


Journal ArticleDOI
TL;DR: An algorithm which maximizes the value of jobs completed by k identical machines is described, which runs in time O (n 2 log n) , where n is the number of jobs.

306 citations


Journal ArticleDOI
01 Dec 1987
TL;DR: In this article, the authors present mathematical formulations to the problem and survey recent advances in optimization-based solution methods for the problem of short-term unit commitment, shortterm hydro scheduling, and hydro-thermal coordination.
Abstract: The operations scheduling problem in electric power systems is to determine which generating units should be on-line and available for generation at each hour and the associated nominal generation or dispatch. This paper describes the scope of the operations scheduling problem, presents mathematical formulations to the problem, and surveys recent advances in optimization-based solution methods. It is mainly concerned with the shorter term aspects of the problem where the time horizon is of the order of a week. The unit commitment, short-term hydro scheduling, and hydro-thermal coordination problems as well as solution methods are emphasized.

193 citations


Journal ArticleDOI
TL;DR: It is proved that the deviation of Jackson's schedule from the optimum is smaller than twice the largest processing time, and a new branching scheme is proposed by associating with each job an interval of time during which it has to be processed.

132 citations


Journal ArticleDOI
TL;DR: This work considers scheduling a set of jobs on parallel processors, when all jobs have a common due date and earliness and lateness are penalized at different cost rates.
Abstract: We consider scheduling a set of jobs on parallel processors, when all jobs have a common due date and earliness and lateness are penalized at different cost rates. For identical processors, the secondary criteria of minimizing makespan and machine occupancy are addressed. The extension to different, uniform processors is also solved.

123 citations


Journal ArticleDOI
TL;DR: In this article, the authors consider the problem of scheduling a set of n preemptable tasks in a system having r resources, where each task has an arbitrary, but known, worst case processing time and a deadline, and may request simultaneous use of a number of resources.
Abstract: We consider the problem of scheduling a set of n preemptable tasks in a system having r resources. Each task has an arbitrary, but known, worst case processing time and a deadline, and may request simultaneous use of a number of resources. A resource can be used either in shared mode or exclusive mode. In this paper, we develop and evaluate algorithms for determining whether or not a set of preemptive tasks is schedulable in such a real-time system, and if so, determining a schedule for it. This scheduling problem is known to be computationally intensive. In many real-time application environments, tasks are scheduled dynamically, and hence the scheduling algorithms used must have low run-time costs. To keep run-time costs low, we propose the use of suboptimal but practical algorithms that employ computationally simple heuristics. The computational complexity of our algorithms for scheduling n tasks in a system having r resources is O(rn2), which is very much lower than that of known optimal algorithms. We report on the results of simulation studies performed on such heuristic preemptive scheduling algorithms and the sensitivity of the performance of the algorithms with respect to various scheduling parameters. These studies show that due to the complexity of the problem, straightforward heuristics do not perform satisfactorily. However, an algorithm that uses combinations of such heuristics in conjunction with limited backtracks works very well.

122 citations


Journal ArticleDOI
TL;DR: This work examines the nonpreemptive assignment of independent tasks to a system of uniform processors with the objective of minimizing the makespan.
Abstract: We examine the nonpreemptive assignment of independent tasks to a system of uniform processors with the objective of minimizing the makespan The worst case bound for the largest processing time first algorithm for this problem is tightened to be in the interval (152, 167)

106 citations


Journal ArticleDOI
TL;DR: In this paper, the authors consider a single-machine scheduling problem in which all jobs have the same due date and penalties are assessed for both early and late completion of jobs, but earliness and tardiness are penalized at different rates.
Abstract: We consider a single-machine scheduling problem in which all jobs have the same due date and penalties are assessed for both early and late completion of jobs. However, earliness and tardiness are penalized at different rates. The scheduling objective is to minimize either the weighted sum of absolute deviations (WSAD) or the weighted sum of squared deviations (WSSD). For each objective we consider two versions of the problem. In the unconstrained version an increase in the due date does not yield any further decrease in the objective function. We present a constructive algorithm for the unconstrained WSAD problem and show that this problem is equivalent to the two-parallel, nonidentical machine, mean flow-time problem. For the unconstrained WSSD and the constrained WSAD and WSSD problems we propose implicit enumeration procedures based on several dominance conditions. We also report on our computational experience with the enumeration procedures.

104 citations



Journal ArticleDOI
TL;DR: Algorithms used for determining whether or not a set of tasks is schedulable in a system having r resources are studied, and if so, determining a schedule for it.

01 Jun 1987
TL;DR: Constraint propagation techniques used in the OPIS scheduling system to update schedule descriptions and detect introduced inconsistencies are presented.
Abstract: : This paper, presents constraint propagation techniques used in the OPIS scheduling system to update schedule descriptions and detect introduced inconsistencies. This approach is summarized as follows: 1) a hierarchical model is used to represent resources and operations to be performed. Schedules are developed and maintained at different levels of precision which are explicitly associated with resources and operations; constraint propagation is correspondingly performed at different levels, 2) Various scheduling constraints are attached to resources and operations, and combined to derive time bound constraints. A description of the original constraints that collectively impose a bound is explicitly recorded, 3) Time bound constraints are maintained by an object-oriented propagation process: through messages, resources and operations communicate constraints and cooperate to compute time bounds, and 4) When time bounds are inconsistent, their origins provide the information required to construct an appropriate description of the conflicting situation. This description provides OPIS with information needed to make reactive decisions.

Journal ArticleDOI
TL;DR: How the underlying decision problem was analyzed was analyzed using both a network flow model and a mixed integer programming model, and the components of the decision support system developed to generate schedules are described.

Journal ArticleDOI
TL;DR: A solution routine, based on iterative improvement of an initial solution, has been developed for vehicle-scheduling problems and tries to reduce the overall distance travelled by moving customers in a systematic way.
Abstract: A solution routine, based on iterative improvement of an initial solution, has been developed for vehicle-scheduling problems. The improving routine combines a number of very simple procedures, each of which tries to reduce the overall distance travelled by moving customers in a systematic way. Tests were run using standard problems, and results obtained were of good quality. An application of the procedure is described briefly.

Journal ArticleDOI
TL;DR: Under mild conditions on the probability distribution, strong asymptotic optimality results are obtained for the LPT Longest Processing Time rule, in which the jobs are assigned to the machines in order of nonincreasing processing requirements.
Abstract: For the problem of minimizing makespan on parallel machines of different speed, the behaviour of list scheduling rules is subjected to a probabilistic analysis under the assumption that the processing requirements of the jobs are independent, identically distributed nonnegative random variables. Under mild conditions on the probability distribution, we obtain strong asymptotic optimality results for the LPT Longest Processing Time rule, in which the jobs are assigned to the machines in order of nonincreasing processing requirements.

Journal ArticleDOI
01 Jan 1987-Networks
TL;DR: A new and much faster algorithm is presented for the problem of assigning buses to a large number of short trips in an urban area, based on the hungarian method and making full use of the sparsity of the assignment matrix for the bus scheduling problem.
Abstract: A new and much faster algorithm is presented for the problem of assigning buses to a large number of short trips in an urban area. The trips are grouped into chains, beginning and ending at the same bus depot, and a vehicle is assigned to each one of them. Fleet size costs and dead heading time are to be minimized. This problem has been already formulated as a transportation problem and more recently, as an assignment model. However, some difficulties, such as the zero pivot phenomenon, rising in many practical cases drastically affected computing times required to obtain the optimal solution. This is overcome by an algorithm based on the hungarian method and making full use of the sparsity of the assignment matrix for the bus scheduling problem. Computational results comparing the different methods are given in the last section of the paper. Significant reductions in computing time are obtained for either real case applications or random generated test problems.

Journal ArticleDOI
TL;DR: The interface between lot sizing and sequencing is introduced and the effect of lot sizes on makespan and flow times is qualitatively analyzed.
Abstract: In production scheduling, time performance is usually taken to be the province of sequencing models which take task processing times to be given. However, in practice, processing times can often be controlled by the choice of lot sizes which thus have a major impact on makespan, waiting times, flow times and other measures. Here the interface between lot sizing and sequencing is introduced and the effect of lot sizes on makespan and flow times is qualitatively analyzed.

Journal ArticleDOI
TL;DR: It is shown that minimizing makespan in an ordered three-machine open shop is NP-complete, which implies many other open-shop problems are also NP- complete, and that slightly different assumptions can change a “hard” problem to an “easy” one.

Journal ArticleDOI
TL;DR: In this paper, a heuristic procedure for generating feasible solutions for the single-machine, multi-product, infinite-horizon, lot scheduling problem was developed. But the problem of scheduling independent products is reduced to that of scheduling G groups of products.
Abstract: This paper develops a heuristic procedure for generating feasible solutions for the single-machine, multi-product, infinite-horizon, lot scheduling problem. This problem occurs in several practical situations, for example, in metal stamping factories, on appliance assembly lines, in the beverage blending and bottling industries, in paint production and on motor car assembly lines. The proposed heuristic divides the N products into G groups and the products belonging to the same group are produced in the same cyclical pattern. Thus the problem of scheduling N independent products is reduced to that of scheduling G groups of products. Since G is much less than N, the problem is made simpler. The proposed heuristics has two main advantages: implementation facility and effectiveness. Computer codes are available for several mini and micro computers. The effectiveness is demonstrated by two tests. First, we solved the six problems originally solved by Elmaghraby (1978 a). Obviously, the results based ...

Journal ArticleDOI
TL;DR: A general bound on problem size is obtained that permits Bellman equations to be used to construct an optimal scheduling rule for any given set of m rate parameters, with the memory required to represent that scheduling rule being independent of the number of remaining jobs.
Abstract: We study the problem of scheduling n given jobs on m uniform processors to minimize expected makespan (maximum finishing time). Job execution times are not known in advance, but are known to be exponentially distributed, with identical rate parameters depending solely on the executing processor. For m = 2 and 3, we show that there exist optimal scheduling rules of a certain threshold type, and we show how the required thresholds can be easily determined. We conjecture that similar threshold rules suffice for m > 3 but are unable to prove this. However, for m > 3 we do obtain a general bound on problem size that permits Bellman equations to be used to construct an optimal scheduling rule for any given set of m rate parameters, with the memory required to represent that scheduling rule being independent of the number of remaining jobs.

Journal ArticleDOI
TL;DR: In this paper, the authors considered the problem of job shop problems with allocation of continuously-divisible non-renewable resource and proposed an algorithm based on disjunctive graphs theory and branch and bound technique.

Journal ArticleDOI
TL;DR: It is shown that for some very simple job length distributions, LVF minimizes both the expected flowtime and the expected makespan.
Abstract: We consider a situation in which n jobs, requiring random amounts of processing, all with the same mean, are to be scheduled on m parallel machines with respect to one of two objectives: expected flowtime and expected makespan. We discuss optimality of the rule that says to schedule the jobs with the largest variance first (LVF). We show that for some very simple job length distributions, LVF minimizes both the expected flowtime and the expected makespan.

Journal ArticleDOI
TL;DR: In this paper, an existing small scale manufacturing system called a Flexible Forging Module (FFM) is considered and a network model is developed which easily incorporates both the sequence-dependent changeover costs and the precedence constraints.
Abstract: In this paper, an existing small scale manufacturing system called a Flexible Forging Module (FFM) is considered. For this system the sequence-dependent changeover costs have been identified. In order to obtain an optimal schedule, a network model is developed which easily incorporates both the sequence-dependent changeover costs and the precedence constraints. A branch and bound algorithm is outlined and numerical results are reported.

Journal ArticleDOI
TL;DR: In this article, an heuristic simulation based approach to the production of schedules is proposed to ensure that the schedules are feasible and take account of the constraints of materials handling, tooling and fixtures while the proposed look ahead algorithm does not ensure optimal schedules.

Journal ArticleDOI
TL;DR: This paper treats the class of n job - m machine scheduling problems with job processing times dependent on the number of jobs being simultaneously processed in the system at any point in time, and finds the makespan is found to be independent of the job-machine assignment.
Abstract: We treat the class of n job − m machine scheduling problems with job processing times dependent on the number of jobs being simultaneously processed in the system at any point in time. Such systems occur when jobs are assigned to multiple parallel processors driven by a common power source. In situations typical of hydraulic and pneumatic power sources the level of power delivered to each processor is inversely proportional to the number of processors simultaneously at work. Aside from the variable processing rate assumptions, the remaining assumptions on the structure of the system conform to those of the standard identical parallel processor problem without job preemption. Although the standard n job − m machine problem is NP-hard with respect to a makespan measure, this is not the case when our seemingly complicating variable processing rate function, typical of hydraulic power sources, is included. In this case our main results are exceedingly simple. The makespan is found to be independent of the job...

Journal ArticleDOI
TL;DR: In this paper, a 32nd order boiler-turbine model is developed to solve the optimal set-point scheduling problem for main and hot reheat steam conditions in a 235 MW gas fired electric generating plant.
Abstract: A 32nd order boiler-turbine model is developed to solve the optimal set-point scheduling problem for main and hot reheat steam conditions in a 235 MW gas fired electric generating plant. The model has 7 states in the process and 25 in the control system. The optimization strategy uses control vector parameterization in which the boiler controller set-point function generators are modified to produce suboptimum signals. The performance index consists of the total energy input to the system from the fuel and boiler feedpump. Constraints in the process variables are monitored by critical alarm limits, and hoop stresses in the heat exchanger tubes are estimated as well. The optimization problem is solved for steady state and dynamic conditions in the 75 to 235 MW load range. Savings in fuel between 0.7 to 4.0% are predicted by computer simulations in the normal operating load range of the unit. Preliminary plant tests indicate these estimated savings may be realistic. The boiler-turbine model and optimization strategy illustrate the potential use of mathematical models for power plants.

Journal ArticleDOI
TL;DR: An efficient method of compilating the timetable by the proposed algorithm is presented, based on the search for the possible orders of the trains, together with the construction example of the timetable for the actual railway.
Abstract: The trial-and-error efforts by experts have been considered as indispensable in compiling railroad timetables. There is a strong requirement to computerize the procedure, but it has been considered difficult. It is proposed to merge the following two concepts as a means to obtain an efficient trial-and-error procedure, which is the basis of the compilation of railroad timetables: (1) the concept of (time) belt, where a number of alternatives for arrival and departure is handled as one alternative, considering only the order of arrivals and departures; (2) the concept of the propagation of constraints, which is well known in the artificial intelligence where alternatives in conflict to the order of arrivals and departures are eliminated. This paper discusses the scheduling problem for the partially ordered set of events. The requirement for the time difference between events is considered as a constraint. An algorithm is proposed which calculates the permissible range of time for the event generation, as a closed interval on the time axis. It is shown that the constructed algorithm can be applied as the basic algorithm for the railroad timetable compilation system, where the timetable is considered as a specification for the order of arrivals and departures of the trains, and the arrival-departure time is considered as a belt. An efficient method of compilating the timetable by the proposed algorithm is presented, based on the search for the possible orders of the trains, together with the construction example of the timetable for the actual railway.

Journal ArticleDOI
TL;DR: A heuristic that iterates between solving the scheduling problem with constant setup costs and solving a "traveling salesman" formulation with sequence dependent setup costs is presented in this article, which always provides a feasible solution if one exists.
Abstract: The problem of scheduling products, with constant demand rates, on a single facility is difficult. This difficulty is compounded if the setup costs are not constant for each product but depend on the sequence in which the products are made. A heuristic that iterates between solving the scheduling problem with constant setup costs and solving a “traveling salesman” formulation with sequence dependent setup costs is presented. The heuristic works well in practice and always provides a feasible solution if one exists. Some computational experience is also given.

Journal ArticleDOI
TL;DR: The use of artificial intelligence techniques has been investigated as a contribution to a solution to the following problems: daily estimated scheduling in a FMS, and real time control of production disturbances.
Abstract: The use of artificial intelligence techniques has been investigated as a contribution to a solution to the following problems: daily estimated scheduling in a FMS, and real time control of production disturbances. We present our investigation in object oriented languages to represent the relevant information of a scheduling problem, particularly knowledge about constraints and flexibility factors in a shop. We detail the off-line daily scheduling prototype SOJA and the reactive capabilities we are incorporating in it to repair previous schedules with respect to real-time situations. The integration of automatic generation and modification of schedule plans leads us to introduce a coordination level which generates an admissible solution and a local, cooperation level which deals with local modifications of the plan.