Discrete-continuous Scheduling Research Articles

A Multisize no Migration Island-Based Differential Evolution Algorithm With Removal of Ineffective Islands

The paper is a continuation of our previous research where a novel concept of multisize island model was proposed. Such multisize approach facilitates the design of island-based algorithms and brings such benefits as: improved fitness dynamics throughout the entire time of operation even without the migration of solutions between the islands. The absence of migration eliminates the need to establish the topology and the policy of migration. It also makes the efficiency of multisize island-based algorithms independent of the particular islands’ size and eliminates the need of tuning the size of islands which is usually done in the case of the canonical island model. All these features indicate the superiority of the multisize island model over the canonical one. In this paper we improved earlier proposed multisize island-based DE algorithm by adding to it the ability to automatically optimize the number of islands in operation. This feature enables the release of most computational units before the algorithm completes its operation in the case of concurrent execution of the algorithm on multiple computational units, or reduction of the algorithm running time in the case of its execution on a single computational unit. The proposed algorithm was tested by solving computationally difficult scheduling problem, which is the discrete-continuous scheduling with continuous resource discretization.

Combined scheduling model to optimize tritium inventory in tritium plant of ITER

The goal of this study was to provide a discrete-continuous scheduling model for optimal operation of the fuel cycle of ITER. The model is designed to minimize the tritium working inventory in the Isotope Separation System (ISS) of the fuel cycle by adopting a two-stage scheduling approach. To consider the effect of pump regeneration schedule on tritium fluctuation in the ISS, the suggested solving algorithm is presented as a two-stage scheduling models: Stage I obtains the optimal pump-regeneration schedule, and Stage II suggests the optimal fuel-transition timing from the ISS to the Storage and Delivery System to minimize tritium working inventory in the ISS. The scheduling model uses discrete-time representation in Stage I and continuous-time representation in Stage II.

An island-based differential evolution algorithm with the multi-size populations

Computational intelligence methods can provide high-quality solutions to a variety of complex optimization problems where exact analytical solutions are impossible to obtain within a reasonable time and other resources used. The article proposes a novel concept of island model with islands of different sizes as well as differential evolution algorithms implementing this concept. Such a multi-size approach facilitates the design of island-based algorithms and brings a variety of benefits. Among them: improved fitness dynamics throughout the entire time of operation even without migration of solutions among the islands. The absence of migration eliminates the need to establish the topology and the policy of migration. It also makes the efficiency of multi-size island-based algorithms independent of the particular islands' size and practically eliminates the need of tuning the size of islands which is usually done in the case of the canonical island model. All these features indicate the superiority of the proposed multi-size island model over the canonical one. The efficiency of the proposed multi-size approach has been tested by solving one of the most difficult scheduling problems which is the discrete-continuous scheduling with continuous resource discretization.

Scheduling shared continuous resources on many-cores

We consider the problem of scheduling a number of jobs on m identical processors sharing a continuously divisible resource. Each job j comes with a resource requirement . The job can be processed at full speed if granted its full resource requirement. If receiving only an x-portion of $$r_j$$ , it is processed at an x-fraction of the full speed. Our goal is to find a resource assignment that minimizes the makespan (i.e., the latest completion time). Variants of such problems, relating the resource assignment of jobs to their processing speeds, have been studied under the term discrete–continuous scheduling. Known results are either very pessimistic or heuristic in nature. In this article, we suggest and analyze a slightly simplified model. It focuses on the assignment of shared continuous resources to the processors. The job assignment to processors and the ordering of the jobs have already been fixed. It is shown that, even for unit size jobs, finding an optimal solution is NP-hard if the number of processors is part of the input. Positive results for unit size jobs include a polynomial-time algorithm for any constant number of processors. Since the running time is infeasible for practical purposes, we also provide more efficient algorithm variants: an optimal algorithm for two processors and a -approximation algorithm for m processors.

Scheduling preemptable jobs on identical processors under varying availability of an additional continuous resource

Abstract In this work we consider a problem of scheduling preemptable, independent jobs, characterized by the fact that their processing speeds depend on the amounts of a continuous, renewable resource allocated to jobs at a time. Jobs are scheduled on parallel, identical machines, with the criterion of minimization of the schedule length. Since two categories of resources occur in the problem: discrete (set of machines) and continuous, it is generally called a discrete-continuous scheduling problem. The model studied in this paper allows the total available amount of the continuous resource to vary over time, which is a practically important generalization that has not been considered yet for discrete-continuous scheduling problems. For this model we give some properties of optimal schedules on a basis of which we propose a general methodology for solving the considered class of problems. The methodology uses a two-phase approach in which, firstly, an assignment of machines to jobs is defined and, secondly, for this assignment an optimal continuous resource allocation is found by solving an appropriate mathematical programming problem. In the approach various cases are considered, following from assumptions made on the form of the processing speed functions of jobs. For each case an iterative algorithm is designed, leading to an optimal solution in a finite number of steps.

Island-based Differential Evolution Algorithm for the Discrete-continuous Scheduling with Continuous Resource Discretisation

In the paper, we propose an island-based differential evolution algorithm (IBDEA) for solving the discrete-continuous scheduling problem (DCSP) with continuous resource discretisation - ΘZ. The considered problem originates from DCSP, in which nonpreemtable tasks should be scheduled on parallel identical machines under constraint on discrete resource and requiring, additionally, a renewable continuous resource to minimize the schedule length. The continuous resource in DCSP is divisible continuously and is allocated to tasks from a given interval in amounts unknown in advance. Task processing rate depends on the allocated amount of the continuous resource. To eliminate time consuming optimal continuous resource allocation, an NP-hard problem ΘZ with continuous resource discretisation is introduced and sub-optimally solved by IBDEA. Experimental results show that IBDEA is able to find better solutions than an algorithm realizing only the differential evolution method and was able to improve best-known solutions to the considered problem.

Neighbourhood Properties in Some Single Processor Scheduling Problem with Variable Efficiency and Additional Resources

In the paper, we consider a problem of scheduling a set of tasks on a single processor. Each task must be preprocessed before it can be started on a processor. The efficiency of preprocessing is variable, i.e., the rate of the task preprocessing depends on the amount of continuously divisible resource allotted to this task. This dependency is given by concave, continuous, non-negative and strictly increasing function of the resource amount. The total consumption of resource at each moment is upper bounded. The objective is to minimize the maximum task completion time. The considered problem is NP-hard. Such a problem appears, e.g., in steel mill systems, where ingots (before hot rolling on the blooming mill) have to achieve the required temperature in the preheating process in soaking pits. Some new properties of the problem are proved. These properties are used to construct the procedure for evaluation of the neighbourhood. The procedure is proposed to improve the efficiency of algorithms based on the neighbourhood concept, such as metaheuristics. The computational experiment is conducted to examine the efficiency of the proposed procedure. The described approach can be easily used in the other discrete-continuous scheduling problems.

Resource level minimization in the discrete–continuous scheduling

A discrete–continuous problem of non-preemptive task scheduling on identical parallel processors is considered. Tasks are described by means of a dynamic model, in which the speed of the task performance depends on the amount of a single continuously divisible renewable resource allotted to this task over time. An upper bound on the completion time of all the tasks is given. The criterion is to minimize the maximum resource consumption at each time instant, i.e., the resource level. This problem has been observed in many industrial applications, where a continuously divisible resource such as gas, fuel, electric, hydraulic or pneumatic power, etc., has to be distributed among the processing units over time, and it affects their productivity. The problem consists of two interrelated subproblems: task sequencing on processors (discrete subproblem) and resource allocation among the tasks (continuous subproblem). An optimal resource allocation algorithm for a given sequence of tasks is presented and computationally tested. Furthermore, approximation algorithms are proposed, and their theoretical and experimental worst-case performances are analyzed. Computer experiments confirmed the efficiency of all the algorithms.

A cross-entropy-based population-learning algorithm for discrete-continuous scheduling with continuous resource discretisation

The problem of scheduling nonpreemtable tasks on parallel identical machines under constraint on discrete resource and requiring, additionally, renewable continuous resource to minimize the schedule length is considered in the paper. A continuous resource is divisible continuously and is allocated to tasks from given intervals in amounts unknown in advance. Task processing rate depends on the allocated amount of the continuous resource. To eliminate time-consuming optimal continuous resource allocation, a problem Θ Z with continuous resource discretisation is introduced. Because Θ Z is NP-hard a population-learning algorithm (PLA2) is proposed to tackle the problem. PLA2 is a population-based approach which takes advantage of the features common to the social education system rather than to the evolutionary processes. The proposed approach is based on the idea of constructing the hybrid algorithm integrating different optimization techniques complementing each other and producing a synergetic effect. Experimental results proved that PLA2 excels known algorithms for solving the considered problem.

Tabu search for discrete–continuous scheduling problems with heuristic continuous resource allocation

Problems of scheduling non-preemptable, independent jobs on parallel identical machines under an additional continuous renewable resource to minimize the makespan are considered. Each job simultaneously requires for its processing a machine and an amount (unknown in advance) of the continuous resource. The processing rate of a job depends on the amount of the resource allotted to this job at a time. The problem is to find a sequence of jobs on machines and, simultaneously, a continuous resource allocation that minimize the makespan. A heuristic procedure for allocating the continuous resource is used. The tabu search metaheuristic to solve the considered problem is presented. The results produced by tabu search are compared with optimal solutions for small instances, as well as with the results generated by simple search methods – multi-start iterative improvement and random sampling for larger instances.

The discrete part of the discrete-continuous scheduling problems - new properties

Abstract The problems of scheduling of tasks described with dynamic models appear in the real-world situations, where management of the processes described with differential equations is needed. Possible applications contains e.g: refuelling of the feet of the boats in the given critical time, scheduling of tasks in the multiple computer systems and the forging process in the steel plants. The solution for such problems consists of two parts: continuous one (the allocation of the continuously divisible resource) and the discrete one (sequence of task subsets). The research has been done mostly for the former part so far, where the latter one was neglected. In the paper we recollect properties of the discrete part of the solution space and we prove some new properties. These new properties can be used to construct more efficient algorithms for the scheduling problems with the dynamic models of tasks.

Multicriteria approach to rank scheduling strategies

This paper deals with multicriteria decision-making applied to discrete-continuous scheduling problems. The need for reusability and modularity leads us to build a “generic” optimization and simulation framework, while the need to quickly generate good compromises between conflicting objectives requires the implementation of multicriteria scheduling models. This paper describes the practical possibilities of two hybrid models within this framework, the first one uses a lexicographical sort and the second one a multicriteria method to rank scheduling strategies. The two hybrids are applied to a real-life highly constrained industrial problem.

An Island-Based Evolution Algorithm for Discrete-Continuous Scheduling with Continuous Resource Discretisation

A problem of scheduling nonpreemtable tasks on parallel identical machines under an additional continuous resource to minimize the schedule length is considered in the paper. A continuous resource is divisible continuously and is allocated to tasks from given intervals in amounts unknown in advance. Task processing rate depends on the allocated amount of the continuous resource. The considered problem can be solved in two steps. The first step involves generating all possible task schedules and second - finding an optimal schedule among all schedules with optimal continuous resource allocation. The second step can be formulated as a convex programming problem with linear constraints that can be solved by specialized solvers. To eliminate time consuming optimal continuous resource allocation a problem ΘZ with continuous resource discretisation is introduced. Because ΘZ is NP-hard an island-based evolution algorithm (IBEA) is proposed to tackle the problem. Experimental results proved IBEA to be competitive with known algorithms for solving the considered problem.

Integrating simulation and optimization of manufacturing systems

This paper presents the ongoing development of a modeling methodology and a tool (the so-called simulation integrated system with modeling and optimization (SISMO) solver) that permits manufacturing systems to be both simulated and optimized according to several improvement strategies. We point out that the different steps of modeling, simulating, and optimizing uses the same integrated formalism and environment. A major point of this methodology and tool is the original decision-making mechanism over a hierarchy of complex discrete systems that model the real world. For the SISMO platform to be validated, we applied it to an actual highly constrained discrete-continuous scheduling problem. This study on a real-life problem has systematically led to improvements.

Tabu list management methods for a discrete–continuous scheduling problem

A problem of scheduling jobs on parallel, identical machines under an additional continuous resource to minimize the makespan is considered. Jobs are non-preemtable and independent and all are available at the start of the process. The total amount of the continuous resource available at a time is limited and the resource is a renewable one. Each job simultaneously requires for its processing a machine and an amount (unknown in advance) of the continuous resource. Processing rate of a job depends on the amount of the resource allotted to this job at a time. The problem is to find a sequence of jobs on machines and, simultaneously, a continuous resource allocation that minimizes the makespan. The tabu search (TS) metaheuristic is presented to attack the problem. Three different tabu list management methods: the tabu navigation method (TNM), the cancellation sequence method (CSM) and the reverse elimination method (REM) are discussed and examined. A computational experiment is described and the results obtained for the methods tested are compared to optimal solutions. A few conclusions and final remarks are presented.

A heuristic approach to allocating the continuous resource in discrete-continuous scheduling problems to minimize the makespan

A problem of scheduling jobs on parallel, identical machines under an additional continuous resource to minimize the makespan is considered. Jobs are non-preemtable and independent and all are available at the start of the process. The total amount of the continuous resource available at a time is limited and the resource is a renewable one. Each job simultaneously requires for its processing a machine and an amount (unknown in advance) of the continuous resource. The processing rate of a job depends on the amount of the resource allotted to this job at a time. The problem is to find a sequence of jobs on machines and, simultaneously, a continuous resource allocation which minimize the makespan. A heuristic approach to allocating the continuous resource is proposed. A tabu search algorithm to solve the considered problem is presented and the results for the algorithms with exact and heuristic procedures for allocating the continuous resource are compared on the basis of some computational experiments. Copyright © 2002 John Wiley & Sons, Ltd.

Discrete-continuous scheduling to minimize the makespan for power processing rates of jobs

A class of discrete-continuous scheduling problems is considered when each nonpreemptable, independent job simultaneously requires for its processing a machine from a set of m identical, parallel machines and an amount, arbitrary within interval [0,1], of a continuously divisible, renewable resource available in amount 1. Job processing rates are described by power functions of the resource amount allotted at a time. Some properties of optimal schedules are proved which allow to find such schedules analytically in some cases and to construct efficient heuristics in the general case. Results of a computational experiment are described.

On a methodology for discrete–continuous scheduling

Discrete–continuous problems of scheduling nonpreemptable jobs on parallel machines are considered. The problems arise e.g. when jobs are assigned to multiple parallel processors driven by a common electric, hydraulic or pneumatic power source. Existing models have assumed job processing rates as a function of the number of jobs currently being processed, or equivalently the number of machines currently in operation. In this paper a more general model is proposed in which processing rates of a job assigned to a machine depend on the amount of a continuous, i.e. continuously divisible resource (e.g. power) allotted to this job at a time. Thus the problem consists of two interrelated subproblems: (i) to sequence jobs on machines, and (ii) to allocate the continuous resource among jobs already sequenced. We provide a comprehensive analysis of the problem. This includes properties of optimal schedules, efficiently (in particular analytically) solvable cases, formulations of the possibly simplest mathematical programming problems for finding optimal schedules in the general case, heuristics and the worst-case analysis. Although our objective function in this paper is to minimize makespan of a set of independent jobs, the presented methodology can be applied to other criteria, precedence-related jobs, and many resource types (apart from, or instead of machines).

Local search metaheuristics for discrete–continuous scheduling problems

Problems of scheduling nonpreemptable jobs which require simultaneously a machine from a set of parallel, identical machines and a continuous, renewable resource are considered. For each job there are known: its processing speed as a continuous, concave function of a continuous resource allotted at a time and its processing demand. The optimization criterion is the schedule length. The problem can be decomposed into two interrelated subproblems: (i) to sequence jobs on machines, and (ii) to find an optimal (continuous) resource allocation among jobs already sequenced. Problem (ii) can be formulated as a convex programming problem with linear constraints and solved using proper solvers. Thus, the problem remains to generate a set of all feasible sequences of jobs on machines (this guarantees finding an optimal schedule in the general case). However, the cardinality of this set grows exponentially with the number of jobs. Thus, we propose to use heuristic search methods defined on the space of feasible sequences. Three metaheuristics: tabu search (TS), simulated annealing (SA) and genetic algorithm (GA) have been implemented and compared computationally with a random sampling technique. The computational experiment has been carried out on an SGI PowerChallenge XL computer with 12 RISC R8000 processors. Some directions for further research have been pointed out.

Discrete-continuous scheduling problems — Mean completion time results

Problems of scheduling independent, nonpreemptable jobs on m parallel machines with an additional continuously divisible, renewable resource to minimize the mean completion time are considered. A job is described by its processing demand and its processing rate being a continuous, nondecreasing function of the amount of the continuous resource allotted to it at a time. Properties of optimal schedules are proved and two special cases with efficient solution algorithms are presented.