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 branch-and-bound procedure for the resource-constrained project scheduling problem with generalized precedence relations

We present an optimal solution procedure for the resource-constrained project scheduling problem (RCPSP) with generalized precedence relations (RCPSP-GPR) with the objective of minimizing the project makespan. The RCPSPGPR extends the RCPSP to arbitrary minimal and maximal time lags between the starting and completion times of activities. The proposed procedure is suited for solving a general class of project scheduling problems and allows for arbitrary precedence constraints, activity ready times and deadlines, multiple renewable resource constraints with time-varying resource requirements and availabilities, several types of permissible and mandatory activity overlaps and multiple projects. It can be extended to other regular and non-regular measures of performance. Essentially, the procedure is a depth-first branch-and-bound algorithm in which the nodes in the search tree represent the original project network extended with extra precedence relations to resolve a number of resource conflicts. These conflicts are resolved using the concept of minimal delaying modes, which is an extension of the notion of minimal delaying alternatives for the RCPSP. Several bounds and dominance rules are used to fathom large portions of the search tree. Extensive computational experience is reported.     

A simulated annealing algorithm for resource constrained project scheduling problems

This paper presents a simulated annealing algorithm for resource constrained project scheduling problems with the objective of minimising makespan. In the search algorithm, a solution is represented with a priority list, a vector of numbers each of which denotes the priority of each activity. In the algorithm, a priority scheduling method is used for making a complete schedule from a given priority list (and hence a project schedule is defined by a priority list). The search algorithm is applied to find a priority list which corresponds to a good project schedule. Unlike most of priority scheduling methods, in the suggested algorithm some activities are delayed on purpose so as to extend search space. Solutions can be further improved by delaying certain activities, since non-delay schedules are not dominant in the problem (the set of non-delay schedules does not always include an optimal solution). The suggested algorithm is flexible in that it can be easily applied to problems with an objective function of a general form and/or complex constraints. The performance of the simulated annealing algorithm is compared with existing heuristics on problems prepared by Patterson and randomly generated test problems. Computational results showed that the suggested algorithm outperformed existing ones.     

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.     

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.     

A branch-and-bound algorithm for the resource-constrained project scheduling problem

We describe a time-oriented branch-and-bound algorithm for the resource-constrained project scheduling problem which explores the set of active schedules by enumerating possible activity start times. The algorithm uses constraint-propagation techniques that exploit the temporal and resource constraints of the problem in order to reduce the search space. Computational experiments with large, systematically generated benchmark test sets, ranging in size from thirty to one hundred and twenty activities per problem instance, show that the algorithm scales well and is competitive with other exact solution approaches. The computational results show that the most difficult problems occur when scarce resource supply and the structure of the resource demand cause a problem to be highly disjunctive.     

一类具有资源限制的调度问题

研究了具有线性退化及学习效应作用下的单台机器调度问题,对于工件的到达时间是其资源消耗量的正的严格单调递减函数时,考虑了总资源消耗量限定情形下求最大完工时间最小化问题给出了最优算法.     

Heuristics for multimode scheduling problems with dedicated resources

We study scheduling problems with multiple modes and dedicated resources arising in production and project management, which constitute a special class of the general multimode resource-constrained project scheduling problem. A task may require simultaneously a set of discrete, renewable resources to be processed and the processing can be performed in different modes, that is with different resource sets, processing times, or costs. Precedence constraints can exist among tasks. The total budget that can be allocated to the project can be limited. The problem consists of identifying a mode for each task and a starting time for its processing respecting precedence, resource, and budget constraints. A graph model and an iterative solution scheme are presented. Specific heuristic algorithms for the cases with and without budget constraints are given and computational results are discussed.     

带有资源转移时间的RCPSP资源流模型及算法

本文在传统资源受限项目调度问题(resource-constrained project scheduling problem, RCPSP)中引入资源转移时间,为有效获得问题的最优解,采用资源流编码方式表示可行解,建立了带有资源转移时间的RCPSP资源流优化模型,目标为最小化项目工期。根据问题特征设计了改进的资源流重构邻域算子,分别设计了改进的禁忌搜索算法和贪心随机自适应禁忌搜索算法求解模型。数据实验结果表明,相较于现有文献中的方法,所提两种算法均可针对更多的项目实例求得最优解,并且得到最优解的时间更短,求解效率更高。此外,分析了算法在求解具有不同特征的项目实例时的性能,所得结果为项目经理结合项目特征评价算法适用性提供了指导。     

Multiple-Project Scheduling with Controllable Project Duration and Hard Resource Constraint: Some Solvable Cases

In many large-scale project scheduling problems, multiple projects are either taking place at the same time or scheduled into a tight sequence in order to efficiently share a common resource. One example of this is the computing resource allocation at an Application Service Provider (ASP) which provides data processing services for multiple paying customers. Typical services provided by ASPs are data mining, payroll processing, internet-based storage backup services and Customer Relation Management (CRM) services. The processing mode of an ASP can be either batch or concurrent, depending on the type service rendered. For example, for CPU intensive or long processing time required services, it would be more economical to processes one customer request at a time in order to minimize the context switching overhead. While the data transaction processes within a service request are subject to certain precedence relationships, the requests from different customers to an ASP are independent of each other, and the total time required to process a service request depends on the computing resource allocated to that request. The related issue of achieving an optimal use of resources at ASPs leads to problem of project scheduling with controllable project duration.In this paper, we present efficient algorithms for solving several special cases of such multi-project scheduling problems with controllable project duration and hard resource constraints. Two types of problems are considered. In type I, the duration of each project includes a constant and a term that is inversely proportional to the amount of resource allocated. In type II, the duration of each individual project is a continuous decreasing function of the amount of resource allocated.     

A bi-objective hierarchical program scheduling problem and its solution based on NSGA-III

In enterprise project management systems, a program at the tactical level coordinates and manages multiple projects at the operational level. There are close relationships between multiple projects in a program, which are typically manifested as shared resources and precedence relationships. Most research efforts have concentrated on the resource sharing by projects, while the precedence relationships between projects have yet to be comprehensively investigated. In this paper, a bi-objective hierarchical resource-constrained program scheduling problem proposed, where both resource sharing and precedence relationships between projects are considered in a distributed environment. The problem contains two different sub-problems at the operational level and the tactical level, and they are modeled in the same way as two bi-objective multi-mode scheduling problems. Shared resources are allocated from the tactical level to the operational level, and once they are allocated to a project, they can only be re-allocated to other projects once the current project is finished. Subsequently, a two-phase algorithm based on NSGA-III is developed. The algorithm runs at the operational level and the tactical level in turn. According to the Pareto fronts of projects that are submitted from the operational level, the bi-objective program planning at the tactical level is conducted under the constraints of precedence relationships and shared resources. The results of computational simulations demonstrate the satisfactory performance of the improved algorithm. By coordinating the local optimization of projects and the global optimization of the program in a hierarchical framework, the method proposed in this paper provides an effective integrated scheduling method for decision-makers at various levels of a program.

     

A heuristic solution framework for the resource constrained (multi-)project scheduling problem with sequence-dependent transfer times

We consider the problem of scheduling multiple projects subject to joint resource constraints. Most approaches proposed in the literature so far are based on the unrealistic assumption that resources can be transferred from one project to the other without any expense in time or cost. In order to contribute to closing this gap to reality, we generalise the multi-project scheduling problem by additionally including sequence- and resource-dependent transfer times, which represent setup activities necessary when a resource is removed from one project and reassigned to another (or from one job to another within the same project). In this paper, we define the modified resource constrained multi-project scheduling problem with transfer times (called RCMPSPTT), which aims at minimising the multi-project duration for the single-project approach or the mean project duration for the multi-project approach. We formulate both perspectives as an integer linear program, propose priority rule based solution procedures and present results of comprehensive computational experiments. Provided that the combination of scheduling scheme and priority rules is chosen appropriately, the procedures obtain good results. In particular, resource oriented priority rules are identified to be successful.     

A decomposed branch-and-price procedure for integrating demand planning in personnel staffing problems

The personnel staffing problem calculates the required workforce size and is determined by constructing a baseline personnel roster that assigns personnel members to duties in order to cover certain staffing requirements. In this research, we incorporate the planning of the duty demand in the staff scheduling problem in order to lower the staffing costs. More specifically, the demand originates from a project scheduling problem with discrete time/resource trade-offs, which embodies additional flexibility as activities can be executed in different modes. In order to tackle this integrated problem, we propose a decomposed branch-and-price procedure. A tight lower and upper bound are calculated using a problem formulation that models the project scheduling constraints and the time-related resource scheduling constraints implicitly in the decision variables. Based upon these bounds, the strategic problem is decomposed into multiple tactical subproblems with a fixed workforce size and an optimal solution is searched for each subproblem via branch-and-price. Fixing the workforce size in a subproblem facilitates the definition of resource capacity cuts, which limit the set of eligible project schedules, decreasing the size of the branching tree. In addition, in order to find the optimal integer solution, we propose a specific search strategy based upon the lower bound and dedicated rules to branch upon the workload generated by a project schedule. The computational results show that applying the proposed search space decomposition and the inclusion of resource capacity cuts lead to a well-performing procedure outperforming different other heuristic and exact methodologies.     

Multi-mode resource-constrained project scheduling using RCPSP and SAT solvers

This paper reports on a new solution approach for the well-known multi-mode resource-constrained project scheduling problem (MRCPSP). This problem type aims at the selection of a single activity mode from a set of available modes in order to construct a precedence and a (renewable and non-renewable) resource feasible project schedule with a minimal makespan. The problem type is known to be NP-hard and has been solved using various exact as well as (meta-)heuristic procedures.The new algorithm splits the problem type into a mode assignment and a single mode project scheduling step. The mode assignment step is solved by a satisfiability (SAT) problem solver and returns a feasible mode selection to the project scheduling step. The project scheduling step is solved using an efficient meta-heuristic procedure from literature to solve the resource-constrained project scheduling problem (RCPSP). However, unlike many traditional meta-heuristic methods in literature to solve the MRCPSP, the new approach executes these two steps in one run, relying on a single priority list. Straightforward adaptations to the pure SAT solver by using pseudo boolean non-renewable resource constraints has led to a high quality solution approach in a reasonable computational time. Computational results show that the procedure can report similar or sometimes even better solutions than found by other procedures in literature, although it often requires a higher CPU time.     

Solving resource-constrained project scheduling problem with evolutionary programming

This paper presents an evolutionary programming (EP)-based approach to solving the resource-constrained project scheduling problem (RCPSP), a well-known NP-hard problem in scheduling, with minimization of project duration as the objective subject to precedence and resource constraints. The individual representation of EP for the problem is based on random keys. The serial generation scheme is used in the decoding scheme to generate the project plan. Experimental analyses are presented to investigate the performance of the proposed EP-based methodology, including comparison of the four variants of EP, namely, CEP, FEP, MCEP and IMCEP, with each other and GA to find the best variant of EP for the RCPSP, and comparison of this best variant of EP (MCEP) with other approaches using the J30 standard instances set in PSPLIB. The computational results validate the effectiveness of the proposed algorithm.     

An approach for solving a class of transportation scheduling problems

An algorithm is developed for solving a class of transportation scheduling problems. It applies for a variety of problems such as: the Combining Truck Trip problem, the Delivery problem, the School Bus problem, the Assignment of Buses to Schedules, and the Travelling Salesman problem. The objective functions of the above problems differ from each other. Yet, by using the “savings method” proposed by Clarke and Wright, and extended by Gaskell, we are able to define each one of the above problems as a series of assignment problems. The cost matrix entries of each one of the assignment problems are a function of the constraints of the particular routing or scheduling problem. The solution to the assignment problem determines an upper bound of the optimal solution to the original problem. By combining the above procedure with a Branch and Bound procedure, it is possible to obtain the optimal solution in a finite number of steps. In some cases the Branch and Bound process can be eliminated due to the nature of the problem and in those cases the algorithm is efficient.     

Nodal aggregation of resource constraints in a shortest path problem

The shortest path problem with resource constraints consists of finding the minimum cost path between two specified points while respecting constraints on resource consumption. Its solving by a dynamic programming algorithm requires a computation time increasing with the number of resources. With the aim of producing rapidly a good heuristic solution we propose to reduce the state space by aggregating resources. Our approach consists of projecting the resources on a vector of smaller dimension and then to dynamically adjust the projection matrix to get a better approximation of the optimal solution. We propose an adjustment based on Lagrangian and surrogate relaxations in a column generation framework, in which the sub-problems are shortest path problems with resource constraints. We adjust the multipliers only one time at each column generation iteration. This permit to obtain good solutions of the scheduling problem in few time.     

A branch and bound algorithm for scheduling jobs with controllable processing times on a single machine to meet due dates

In most deterministic scheduling problems, job-processing times are regarded as constant and known in advance. However, in many realistic environments, job-processing times can be controlled by the allocation of a common resource to jobs. In this paper, we consider the problem of scheduling jobs with arbitrary release dates and due dates on a single machine, where job-processing times are controllable and are modeled by a non-linear convex resource consumption function. The objective is to determine simultaneously an optimal processing permutation as well as an optimal resource allocation, such that no job is completed later than its due date, and the total resource consumption is minimized. The problem is strongly NP\mathcal{NP}-hard. A branch and bound algorithm is presented to solve the problem. The computational experiments show that the algorithm can provide optimal solution for small-sized problems, and near-optimal solution for medium-sized problems in acceptable computing time.     

A priority-rule method for project scheduling with work-content constraints

The activities of a project are in general characterized by a work content in terms of resource–time units, e.g. person-days. Even though most project scheduling models assume a time-invariant resource usage, normally it is possible to vary the resource usage during the execution of an activity. Typically, a lower and an upper bound on this resource usage and a minimum time lag between consecutive changes of this resource usage are prescribed. The project scheduling problem studied in this paper consists in determining a feasible resource-usage profile for each activity such that the project duration is minimized subject to precedence and resource-capacity constraints. While the known solution methods interpret the prescribed work content as a lower bound, we assume that each activity’s work content must be processed exactly.     

Multi-project scheduling with two-stage decomposition

We consider a non-preemptive, zero time lag multi-project scheduling problem with multiple modes and limited renewable and nonrenewable resources. A two-stage decomposition approach is adopted to formulate the problem as a hierarchy of 0-1 mathematical programming models. In stage one; each project is reduced to a macro-activity with macro-modes. The macro-activities are combined into a single macro-activity network over which the macro-activity scheduling problem (MP) is defined, where the objective is the maximization of the net present value with positive cash flows and the renewable resource requirements are time-dependent. An exact solution procedure and a genetic algorithm (GA) approach are proposed for solving the MP. A GA is also employed to generate an initial solution for the exact solution procedure. The first stage terminates with a post-processing procedure to distribute the remaining resource capacities. Using the start times and the resource profiles obtained in stage one, each project is scheduled in stage two for minimum makespan. Three new test problem sets are generated with 81, 84 and 27 problems each, and three different configurations of solution procedures are tested.