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.     

The multi-mode resource-constrained project scheduling problem with generalized precedence relations

In this paper, we tackle the challenging problem of scheduling activities to minimize the project duration, in which the activities (a) are subject to generalized precedence relations, (b) require units of multiple renewable, non-renewable and doubly constrained resources for which a limited availability is imposed, and (c) can be performed in one of several different ways, reflected in multiple activity scenarios or modes. These multiple modes give rise to several kinds of trade-offs (time/resource, time/cost and resource/resource trade-offs) which allow for a more efficient allocation and use of resources. We present a local search-based solution methodology which is able to handle many real-life project scheduling characteristics such as time-varying resource requirements and availabilities, activity ready times, due dates and deadlines, activity overlaps, activity start time constraints and other types of temporal constraints.     

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。     

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.     

Using resource scarceness characteristics to solve the multi-mode resource-constrained project scheduling problem

In the past decades, resource parameters have been introduced in project scheduling literature to measure the scarceness of resources of a project instance. In this paper, we incorporate these resource scarceness parameters in the search process to solve the multi-mode resource constrained project scheduling problem, in which multiple execution modes are available for each activity in the project. Therefore, we propose a scatter search algorithm, which is executed with different improvement methods, each tailored to the specific characteristics of different renewable and nonrenewable resource scarceness values. Computational results prove the effectiveness of the improvement methods and reveal that the procedure is among the best performing competitive algorithms in the open literature.     

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.     

A hybrid rank-based evolutionary algorithm applied to multi-mode resource-constrained project scheduling problem

We consider the multi-mode resource-constrained project scheduling problem (MRCPSP), where a task has different execution modes characterized by different resource requirements. Due to the nonrenewable resources and the multiple modes, this problem is NP-hard; therefore, we implement an evolutionary algorithm looking for a feasible solution minimizing the makespan.     

New computational results on the discrete time/cost trade-off problem in project networks

We describe a new exact procedure for the discrete time/cost trade-off problem in deterministic activity-on-the-arc networks of the CPM type, where the duration of each activity is a discrete, nonincreasing function of the amount of a single resource (money) committed to it. The objective is to construct the complete and efficient time/cost profile over the set of feasible project durations. The procedure uses a horizon-varying approach based on the iterative optimal solution of the problem of minimising the sum of the resource use over all activities subject to the activity precedence constraints and a project deadline. This optimal solution is derived using a branch-and-bound procedure which computes lower bounds by making convex piecewise linear underestimations of the discrete time/cost trade-off curves of the activities to be used as input for an adapted version of the Fulkerson labelling algorithm for the linear time/cost trade-off problem. Branching involves the selection of an activity in order to partition its set of execution modes into two subsets which are used to derive improved convex piecewise linear underestimations. The procedure has been programmed in Visual C ++ under Windows NT and has been validated using a factorial experiment on a large set of randomly generated problem instances.     

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.     

The resource constrained project scheduling problem with multiple crashable modes: A heuristic procedure

We present a heuristic procedure for a nonpreemptive resource constrained project scheduling problem in which the duration/cost of an activity is determined by the mode selection and the duration reduction (crashing) applied within the selected mode. This problem is a natural combination of the time/cost trade-off problem and the resource constrained project scheduling problem. The objective is to determine each activity's start (finish) time, mode and duration so that the total project cost is minimized. Total project cost is the sum of all activity costs and the penalty cost for completing the project beyond its due date. We introduce a multi-pass algorithm. We report computational results with a set of 100 test problems and demonstrate the efficacy of the proposed heuristic procedure.     

Branch and bound based solution algorithms for the budget constrained discrete time/cost trade-off problem

The time/cost trade-off models in project management aim to reduce the project completion time by putting extra resources on activity durations. The budget problem in discrete time/cost trade-off scheduling selects a time/cost mode for each activity so as to minimize the project completion time without exceeding the available budget. There may be alternative modes that solve the budget problem optimally and each solution may have a different total cost value. In this study we consider the budget problem and aim to find the minimum cost solution among the minimum project completion time solutions. We analyse the structure of the problem together with its linear programming relaxation and derive some mechanisms for reducing the problem size. We solve the reduced problem by branch and bound based optimization and heuristic algorithms. We find that our branch and bound algorithm finds optimal solutions for medium-sized problem instances in reasonable times and the heuristic algorithms produce high quality solutions very quickly.     

Minimizing total weighted late work in the resource-constrained project scheduling problem

This paper deals with resource-constrained project scheduling problem under the weighted late work criterion. Late work objective functions estimate the quality of a schedule based on durations of late parts of activities, not taking into account the amount of delay for fully late activities. It is assume that a project contains activities interrelated by finish-to-start type precedence relations with time lag of zero, which require one or more constrained renewable resources. The objective is to schedule each activity such that the total weighted late work is minimized. The problem has been formulated using a linear integer programming model and solved by the CPLEX. Also, a set of priority rules have been designed to quickly generate a set of initial solutions. In order to solve the problem optimally, a depth-first branch-and-bound algorithm is applied based on idea of minimal delaying alternatives. The branching order of nodes that belong to the same level of the search tree is determined on the basis of the developed priority rules. This results in generation six different versions of the branch-and-bound algorithm. Computational results on randomly generated problem sets are provided to analyze the efficiency of the priority rules and the branch-and-bound algorithm.     

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。     

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.     

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.     

Time slack-based techniques for robust project scheduling subject to resource uncertainty

The resource-constrained project scheduling problem (RCPSP) has been the subject of a great deal of research during the previous decades. This is not surprising given the high practical relevance of this scheduling problem. Nevertheless, extensions are needed to be able to cope with situations arising in practice such as multiple activity execution modes, activity duration changes and resource breakdowns. In this paper we analytically determine the impact of unexpected resource breakdowns on activity durations. Furthermore, using this information we develop an approach for inserting explicit idle time into the project schedule in order to protect it as well as possible from disruptions caused by resource unavailabilities. This strategy will be compared to a traditional simulation-based procedure and to a heuristic developed for the case of stochastic activity durations.     

考虑返修的建设项目时间-成本-质量问题

当施工过程中质量不达标需要返修以改进项目质量,但相应地会影响项目工期和成本。本文基于经典多模式资源受限项目调度问题构建一种新的非线性规划模型,目标为项目成本最小和工期最短,其中项目成本考虑返修成本以提高项目质量。首先,使用二元非独立正态分布函数描述活动质量,根据活动间的串联或并联关系定义隐蔽工程质量为活动质量的函数。其次,本文提出一种基于NSGA的混合蛙跳算法,采用串行进度产生方案和调整的活动列表编码,其中蛙跳过程结合了遗传算法中的交叉操作和基于置换的局域搜索。最后,整个模型算法应用于框架铁路立交桥施工项目,验证本文算法性能在支配解数量和质量上都优于标准NSGA。     

A new path-based cutting plane approach for the discrete time-cost tradeoff problem

Determining discrete time-cost tradeoffs in project networks allows for the control of the processing time of an activity via the amount of non-renewable resources allocated to it. Larger resource allocations with associated higher costs reduce activities’ durations. Given a set of execution modes (time-cost pairs) for each activity, the discrete time-cost tradeoff problem (DTCTP) involves selecting a mode for each activity so that either: (i) the project completion time is minimized, given a budget, or (ii) the total project cost is minimized, given a deadline, or (iii) the complete and efficient project cost curve is constructed over all feasible project durations. The DTCTP is a problem with great applicability prospects but at the same time a strongly N P{\mathcal N}\,P-hard optimization problem; solving it exactly has been a real challenge. Known optimal solution methodologies are limited to networks with no more than 50 activities and only lower bounds can be computed for larger, realistically sized, project instances. In this paper, we study a path-based approach to the DTCTP, in which a new path-based formulation in activity-on-node project networks is presented. This formulation is subsequently solved using an exact cutting plane algorithm enhanced with speed-up techniques. Extensive computational results reported for almost 5,000 benchmark test problems demonstrate the effectiveness of the proposed algorithm in solving to optimality for the first time some of the hardest and largest instances in the literature. The promising results suggest that the algorithms may be embedded into project management software and, hence, become a useful tool for practitioners in the future.     

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.     

Project Scheduling with Multiple Modes: A Genetic Algorithm

In this paper we consider the resource-constrained project scheduling problem with multiple execution modes for each activity and makespan minimization as objective. We present a new genetic algorithm approach to solve this problem. The genetic encoding is based on a precedence feasible list of activities and a mode assignment. After defining the related crossover, mutation, and selection operators, we describe a local search extension which is employed to improve the schedules found by the basic genetic algorithm. Finally, we present the results of our thorough computational study. We determine the best among several different variants of our genetic algorithm and compare it to four other heuristics that have recently been proposed in the literature. The results that have been obtained using a standard set of instances show that the new genetic algorithm outperforms the other heuristic procedures with regard to a lower average deviation from the optimal makespan.