Simultaneous Job Scheduling and Resource Allocation on Parallel Machines

Most deterministic production scheduling models assume that the processing time of a job on a machine is fixed externally and known in advance of scheduling. However, in most realistic situations, apart from the machines, it requires additional resources to process jobs, and the processing time of a job is determined internally by the amount of the resources allocated. In these situations, both the cost associated with the job schedule and the cost of the resources allocated should be taken into account. Therefore, job scheduling and resource allocation should be carefully coordinated and optimized jointly in order to achieve an overall cost-effective schedule. In this paper, we study a parallel-machine scheduling model involving both job processing and resource allocation. The processing time of a job is non-increasing with the cost of the allocated resources. The objective is to minimize the total cost including the cost measured by a scheduling criterion and the cost of all allocated resources. We consider two particular problems of this model, one with the scheduling criterion being the total weighted completion time, and the other with that being the weighted number of tardy jobs. We develop a column generation based branch and bound method for finding optimal solutions for these NP-hard problems. The method first formulates the problems as set partitioning type formulations, and then solves the resulting formulations exactly by branch and bound. In the branch and bound, linear relaxations of the set partitioning type formulations are decomposed into master problems and single-machine subproblems and solved by a column generation approach. The algorithms developed based on this method are capable of solving the two problems with a medium size to optimality within a reasonable computational time.     

Heuristic approaches to discrete-continuous project scheduling problems to minimize the makespan

In this paper some discrete-continuous project scheduling problems to minimize the makespan are considered. These problems are characterized by the fact that activities of a project simultaneously require for their execution discrete and continuous resources. A class of these problems is considered where the number of discrete resources is arbitrary, and one continuous, renewable, limited resource occurs. A methodology for solving the defined problems is presented. The continuous resource allocation problem is analyzed. An exact, as well as a heuristic approach to the problem is discussed. The idea of the continuous resource discretization is described, and a special case of the problem with identical processing rate functions is analyzed. Some computational experiments for evaluating the efficiency of the proposed heuristic approaches are presented. Conclusions and directions for future research are given.     

Single-machine group scheduling with deteriorating jobs and allotted resource

In this paper, we consider single-machine scheduling problems with deteriorating jobs and resource allocation in a group technology environment. In the proposed model of this paper the actual processing time of a job depend on its starting time and the amount of resource allocated to it, and the actual setup time of a group depend on its starting time and the amount of resource allocated. Deterioration effect and two resource allocation functions are examined for minimizing the weighted sum of makespan and total resource cost. For the linear resource allocation function and the convex resource allocation function, we show that the problem remains polynomially solvable under certain conditions.     

Approximation algorithms for scheduling on multi-core processor with shared speedup resources

We consider a joint resource partition and scheduling problem. We are given $m$ identical cores and discrete resources of total size $k$. We need to partition the resources among these cores. A set of jobs must be processed non-preemptively on these cores after the resource partition. The processing time of a job on a core depends on the size of resources allocated to that corresponding core. The resource allocation scheme is static, i.e., we cannot change the amount of resources that was allocated to a core during the whole scheduling. Hassidim et al. (2013) investigated this problem with a general processing time function, i.e., the processing time of a job is an arbitrary function of the level of resources allocated to that core. They provided an algorithm with approximation ratio of 36. In this paper, we improve the approximation ratio to 8 by presenting a new resource partition scheme. Next, we consider a special model where the core’s speed is proportional to its allocated resource, then we present two algorithms with improved approximation ratios.     

Dynamic transport scheduling under multiple resource constraints

This paper presents a heuristic for the dynamic vehicle scheduling problem with multiple resource capacity constraints. In the envisaged application, an automated transport system using Automated Guided Vehicles, bottleneck resources are (1) vehicles, (2) docks for loading/unloading, (3) vehicle parking places, and (4) load storage space. This problem is hard, because interrelated activities (loading, transportation, unloading) at several geographical locations have to be scheduled under multiple resource constraints, where the bottleneck resource varies over time. Besides, the method should be suitable for real-time planning. We developed a dedicated serial scheduling method and analyzed its dynamic behavior using discrete event simulation. We found that our method is very well able to find good vehicle schedules satisfying all resource constraints. For comparison, we used a simple approach where we left out the resource constraints and extended the processing times by statistically estimated waiting times to account for finite capacities. We found that our newly designed method finds better schedules in terms of service levels.     

Machine scheduling with resource dependent processing times

We consider machine scheduling on unrelated parallel machines with the objective to minimize the schedule makespan. We assume that, in addition to its machine dependence, the processing time of any job is dependent on the usage of a discrete renewable resource, e.g. workers. A given amount of that resource can be distributed over the jobs in process at any time, and the more of that resource is allocated to a job, the smaller is its processing time. This model generalizes the classical unrelated parallel machine scheduling problem by adding a time-resource tradeoff. It is also a natural variant of a generalized assignment problem studied previously by Shmoys and Tardos. On the basis of an integer linear programming formulation for a relaxation of the problem, we use LP rounding techniques to allocate resources to jobs, and to assign jobs to machines. Combined with Graham’s list scheduling, we show how to derive a 4-approximation algorithm. We also show how to tune our approach to yield a 3.75-approximation algorithm. This is achieved by applying the same rounding technique to a slightly modified linear programming relaxation, and by using a more sophisticated scheduling algorithm that is inspired by the harmonic algorithm for bin packing. We finally derive inapproximability results for two special cases, and discuss tightness of the integer linear programming relaxations.     

Continuous management of airlift and tanker resources: A constraint-based approach

Efficient allocation of aircraft and aircrews to transportation missions is an important priority at the USAF Air Mobility Command (AMC), where airlift demand must increasingly be met with less capacity and at lower cost. In addition to presenting a formidable optimization problem, the AMC resource management problem is complicated by the fact that it is situated in a continuously executing environment. Mission requests are received (and must be acted upon) incrementally, and, once allocation decisions have been communicated to the executing agents, subsequent opportunities for optimizing resource usage must be balanced against the cost of solution change. In this paper, we describe the technical approach taken to this problem in the AMC barrel allocator, a scheduling tool developed to address this problem and provide support for day-to-day allocation and management of AMC resources. The system utilizes incremental and configurable constraint-based search procedures to provide a range of automated and semi-automated scheduling capabilities. Most basically, the system provides an efficient solution to the fleet scheduling problem. More importantly to continuous operations, it also provides techniques for selectively reoptimizing to accommodate higher priority missions while minimizing disruption to most previously scheduled missions, and for selectively “merging” previously planned missions to minimize nonproductive flying time. In situations where all mission requirements cannot be met, the system can generate and compare alternative constraint relaxation options. The barrel allocator technology is currently transitioning into operational use within AMC's Tanker/Airlift Control Center (TACC). A version of the barrel allocator supporting airlift allocation was first incorporated as an experimental module of the AMC's Consolidated Air Mobility Planning System (CAMPS) in September 2000. In May 2003, a new tanker allocation module is scheduled for initial operational release to users as part of CAMPS Release 5.4.     

Fixed cost and resource allocation based on DEA cross-efficiency

In many managerial applications, situations frequently occur when a fixed cost is used in constructing the common platform of an organization, and needs to be shared by all related entities, or decision making units (DMUs). It is of vital importance to allocate such a cost across DMUs where there is competition for resources. Data envelopment analysis (DEA) has been successfully used in cost and resource allocation problems. Whether it is a cost or resource allocation issue, one needs to consider both the competitive and cooperative situation existing among DMUs in addition to maintaining or improving efficiency. The current paper uses the cross-efficiency concept in DEA to approach cost and resource allocation problems. Because DEA cross-efficiency uses the concept of peer appraisal, it is a very reasonable and appropriate mechanism for allocating a shared resource/cost. It is shown that our proposed iterative approach is always feasible, and ensures that all DMUs become efficient after the fixed cost is allocated as an additional input measure. The cross-efficiency DEA-based iterative method is further extended into a resource-allocation setting to achieve maximization in the aggregated output change by distributing available resources. Such allocations for fixed costs and resources are more acceptable to the players involved, because the allocation results are jointly determined by all DMUs rather than a specific one. The proposed approaches are demonstrated using an existing data set that has been applied in similar studies.     

Multi-mode resource constrained multi-project scheduling and resource portfolio problem

This paper introduces a multi-project problem environment which involves multiple projects with assigned due dates; activities that have alternative resource usage modes; a resource dedication policy that does not allow sharing of resources among projects throughout the planning horizon; and a total budget. Three issues arise when investigating this multi-project environment. First, the total budget should be distributed among different resource types to determine the general resource capacities, which correspond to the total amount for each renewable resource to be dedicated to the projects. With the general resource capacities at hand, the next issue is to determine the amounts of resources to be dedicated to the individual projects. The dedication of resources reduces the scheduling of the projects’ activities to a multi-mode resource constrained project scheduling problem (MRCPSP) for each individual project. Finally, the last issue is the efficient solution of the resulting MRCPSPs. In this paper, this multi-project environment is modeled in an integrated fashion and designated as the resource portfolio problem. A two-phase and a monolithic genetic algorithm are proposed as two solution approaches, each of which employs a new improvement move designated as the combinatorial auction for resource portfolio and the combinatorial auction for resource dedication. A computational study using test problems demonstrated the effectiveness of the solution approach proposed.     

基于MAS协作机制的动态计算资源调度模型

针对网格环境的自治性、动态性、分布性和异构性等特征.提出基于多智能体系统(Mutil Agent System,MAS)博弈协作的资源动态分配和任务调度模型,建立了能够反映供求关系的网格资源调度模型和任务求解算法,证明了资源分配博弈中Nash均衡点的存在性、唯一性和Nash均衡解,该方法能够利用消费者agent的学习和协商能力,考虑和引入消费者的心理行为,使得消费者的资源申请和任务调度具有较高的合理性和有效性.实验结果表明,资源调度算法不但可以有效减少不必要的延迟,而且在响应时间的平滑性、吞吐率及资源利用率方面比传统算法要好,从而使得整个资源的供需合理、负载均衡.     

Solving the discrete-continuous project scheduling problem via its discretization

In this paper a discrete-continuous project scheduling problem is considered. In this problem activities simultaneously require discrete and continuous resources. The processing rate of each activity depends on the amount of the continuous resource allotted to this activity at a time. All the resources are renewable ones. The activities are nonpreemtable and the objective is to minimize the makespan. Discretization of this problem leading to a classical (i.e. discrete) project scheduling problem in the multi-mode version is presented. A simulated annealing (SA) approach to solving this problem is described and tested computationally in two versions: with and without finding an optimal continuous resource allocation for the final schedule. In the former case a nonlinear solver is used for solving a corresponding convex programming problem. The results are compared with the results obtained using SA for the discrete-continuous project scheduling problem where the nonlinear solver is used for exact solving the continuous part in each iteration. The results of a computational experiment are analyzed and some conclusions are included.     

Large Deviations Analysis for Distributed Algorithms in an Ergodic Markovian Environment

We provide a large deviations analysis of deadlock phenomena occurring in distributed systems sharing common resources. In our model transition probabilities of resource allocation and deallocation are time and space dependent. The process is driven by an ergodic Markov chain and is reflected on the boundary of the d-dimensional cube. In the large resource limit, we prove Freidlin-Wentzell estimates, we study the asymptotic of the deadlock time and we show that the quasi-potential is a viscosity solution of a Hamilton-Jacobi equation with a Neumann boundary condition. We give a complete analysis of the colliding 2-stacks problem and show an example where the system has a stable attractor which is a limit cycle.     

Multi-objectives Tabu Search based algorithm for progressive resource allocation

Military course of action planning involves time and space synchronization as well as resource and asset allocation. A mission could be seen as a defined set of logical ordered tasks with time and space constraints. The resources to task rules require that available assets should be allocated to each task. A combination of assets might be required to execute a given task. The couple (task, resources) constitutes an action. This problem is formulated as a multi-objectives scheduling and resource allocation problem. Any solution is assessed based on a number of conflicting and heterogeneous objectives. In fact, military planning officers should keep perfecting the plan based on the Commander’s criteria for success. The scheduling problem and resource allocation problem are considered as NP-Hard Problems [A. Guitouni, B. Urli, J.-M. Martel, Course of action planning: A project based modelling, Working Paper, Faculté des sciences de l’ administration, Université Laval, Québec, 2005]. This paper is concerned with the multi-objectives resource allocation problem. Our objective is to find adequate resource allocation for given courses of action schedule. To optimize this problem, this paper investigates non-exact solution methods, like meta-heuristic methods for finding potential efficient solutions. A progressive resource allocation methodology is proposed based on Tabu Search and multi-objectives concepts. This technique explores the search space so as to find a set of potential efficient solutions without aggregating the objectives into a single objective function. It is guided by the principle of maximizing the usage of any resource before considering a replacement resource. Thus, a given resource is allocated to the maximum number of tasks for a given courses of action schedule. A good allocation is a potential efficient solution. These solutions are retained by applying a combination of a dominance rule and a multi-criteria filtering method. The performance of the proposed Pareto-based approach is compared to two aggregation approaches: weighted-sum and the lexicographic techniques. The result shows that a Pareto-based approach is providing better solutions and allowing more flexibility to the decision-maker.     

Multi-resource allocation in stochastic project scheduling

We propose a resource allocation model for project scheduling. Our model accommodates multiple resources and decision-dependent activity durations inspired by microeconomic theory. First, we elaborate a deterministic problem formulation. In a second stage, we enhance this model to account for uncertain problem parameters. Assuming that the first and second moments of these parameters are known, the stochastic model minimises an approximation of the value-at-risk of the project makespan. As a salient feature, our approach employs a scenario-free formulation which is based on normal approximations of the activity path durations. We extend our model to situations in which the moments of the random parameters are ambiguous and describe an iterative solution procedure. Extensive numerical results are provided.     

Single-machine scheduling with convex resource dependent processing times and deteriorating jobs

In this study, we consider scheduling problems with convex resource dependent processing times and deteriorating jobs, in which the processing time of a job is a function of its starting time and its convex resource allocation. The objective is to find the optimal sequence of jobs and the optimal convex resource allocation separately. This paper focus on the single-machine problems with objectives of minimizing a cost function containing makespan, total completion time, total absolute differences in completion times and total resource cost, and a cost function containing makespan, total waiting time, total absolute differences in waiting times and total resource cost. It shows that the problems remain polynomially solvable under the proposed model.     

Applications of resource assignment and scheduling with Petri Nets and heuristic search

This paper introduces a new Petri Net based approach for resource allocation and scheduling. The goals are (i) minimize the number of required resources given a set of jobs, (ii) find both an assignment for all jobs in the span of a predefined shift and (iii) the sequence in which such jobs are executed. The studied problem was inspired from a complex real life manufacturing shop as described in this document. The modeling of the processes and jobs is carried out with Petri Nets due to their capability of representing dynamic, concurrent discrete-event dynamic systems. The resource assignment starts with an initial feasible solution (initial number of resources) and then follows with a re-optimization process aimed to further reduce the resource requirements. The algorithm is based on a modified Heuristic Search method previously presented. The algorithm was tested first on a number of instances from the literature and then on the aforementioned system (a car seat cover manufacturer). The proposed approach shows not only good results in terms of performance but also shows the potential of Petri Nets for modeling and optimizing real-life systems. An implementation phase at the first stages of the process is underway at the time of writing.     

项目鲁棒调度的资源分配启发式算法研究

合理的资源配置是提高项目调度鲁棒性一种有效的方法。本文针对项目鲁棒调度问题,提出了Max-PRUA资源分配启发式算法,以期通过生成鲁棒性高的资源分配方案来提高调度计划的鲁棒性。本算法设计了最大化利用优先关系和不可避免弧传递资源的资源分配两项策略来传递最大资源量,以减少由额外约束传递的资源量,降低对项目调度鲁棒性的影响。为寻优最优资源分配方案,配合局部搜索算法,本算法构建了动态活动组GRA,通过对组内活动顺序重排以生成多种资源分配方案,以利于从解空间中寻优出最佳的鲁棒性方案。最后通过大量的仿真实验验证和与其它算法进行比较,结果表明本算法对于不同规模和不同因素影响的项目均有较好的适应性,生成的资源分配方案对调度计划鲁棒性影响较小,是一种有效的算法。     

A Robust Genetic Algorithm for Resource Allocation in Project Scheduling

Genetic algorithms have been applied to many different optimization problems and they are one of the most promising metaheuristics. However, there are few published studies concerning the design of efficient genetic algorithms for resource allocation in project scheduling. In this work we present a robust genetic algorithm for the single-mode resource constrained project scheduling problem. We propose a new representation for the solutions, based on the standard activity list representation and develop new crossover techniques with good performance in a wide sample of projects. Through an extensive computational experiment, using standard sets of project instances, we evaluate our genetic algorithm and demonstrate that our approach outperforms the best algorithms appearing in the literature.     

汽车4S店维修车间的优化调度分析

调度研究的问题是将稀缺资源分配给在一定时间内的不同任务,它是一个决策过程,其目的是优化一个或多个目标。对实际问题的优化调度可以帮助企业提高资源利用率,减少客户等待时间,提升竞争力,对汽车4S维修服务站的优化调度问题进行研究,剖析这一实际应用问题的调度目标、机器环境、加工特征和约束等细节,提出了优化调度模型,设计了调度算法。然后,通过实例,简要分析了模型及算法的可行性.     

Project selection,scheduling and resource allocation with time dependent returns

In this paper we formulate and analyze the joint problem of project selection and task scheduling. We study the situation where a manager has many alternative projects to pursue such as developing new product platforms or technologies, incremental product upgrades, or continuing education of human resources. Project return is assumed to be a known function of project completion time. Resources are limited and renewable. The objective is to maximize present worth of profit. A general mathematical formulation that can address several versions of the problem is presented. An implicit enumeration procedure is then developed and tested to provide good solutions based on project ordering and a prioritization rule for resource allocation. The algorithm uses an imbedded module for solving the resource-constrained project scheduling problem at each stage. The importance of integrating the impact of resource constraints into the selection of projects is demonstrated.