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.     

Proactive policies for the stochastic resource-constrained project scheduling problem

The resource-constrained project scheduling problem involves the determination of a schedule of the project activities, satisfying the precedence and resource constraints while minimizing the project duration. In practice, activity durations may be subject to variability. We propose a stochastic methodology for the determination of a project execution policy and a vector of predictive activity starting times with the objective of minimizing a cost function that consists of the weighted expected activity starting time deviations and the penalties or bonuses associated with late or early project completion. In a computational experiment, we show that our procedure greatly outperforms existing algorithms described in the literature.     

A hybrid scatter search for the discrete time/resource trade-off problem in project scheduling

We develop a heuristic procedure for solving the discrete time/resource trade-off problem in the field of project scheduling. In this problem, a project contains activities interrelated by finish-start-type precedence constraints with a time lag of zero, which require one or more constrained renewable resources. Each activity has a specified work content and can be performed in different modes, i.e. with different durations and resource requirements, as long as the required work content is met. The objective is to schedule each activity in one of its modes in order to minimize the project makespan. We use a scatter search algorithm to tackle this problem, using path relinking methodology as a solution combination method. Computational results on randomly generated problem sets are compared with the best available results indicating the efficiency of the proposed algorithm.     

Two Approaches to Problems of Resource Allocation Among Project Activities — A Comparative Study

The paper deals with a class of problems of resource allocation among project activities, where the resource requirements of each activity concern numbers of resource units from given finite sets for particular resource types. Three categories of constrained resources are considered in a general model: renewable (only total usage at every moment is constrained), non-renewable (only total consumption over the period of project duration is constrained) and doubly-constrained (both usage and consumption are constrained). For every feasible combination of resource amounts, the performing time of each activity is known. Time and cost criteria are considered for project performance evaluation. For solving these problems, two general approaches using linear programming in specific ways are described. These approaches are different in nature, the difference being reflected in the range of problems solved by them and in their computational properties. This is shown by an extensive comparison of both approaches. This comparison also characterizes the state-of-the-art across the problems and points out desirable directions for further research.     

A linear Bayesian stochastic approximation to update project duration estimates

By relaxing the unrealistic assumption of probabilistic independence on activity durations in a project, this paper develops a hierarchical linear Bayesian estimation model. Statistical dependence is established between activity duration and the amount of resource, as well as between the amount of resource and the risk factor. Upon observation or assessment of the amount of resource required for an activity in near completion, the posterior expectation and variance of the risk factor can be directly obtained in the Bayesian scheme. Then, the expected amount of resources required for and the expected duration of upcoming activities can be predicted. We simulate an application project in which the proposed model tracks the varying critical path activities on a real time basis, and updates the expected project duration throughout the entire project. In the analysis, the proposed model improves the prediction accuracy by 38.36% compared to the basic PERT approach.     

A new heuristic for resource-constrained project scheduling in stochastic networks using critical chain concept

This paper presents a newly developed resource-constrained project scheduling method in stochastic networks by merging the new and traditional resource management methods. In each project, the activities consume various types of resources with fixed capacities. The duration of each activity is a random variable with a given density function. Since the backward pass method is implemented for feeding-in resources. The problem is to determine the finish time of each activity instead of its start time. The objective of the presented model is defined as minimizing the multiplication of expected project duration and its variance. The values of activities finish times are determined at decision points when at least one activity is ready to be operated and there are available resources. If at a certain point of time, more than one activity is ready to be operated but available resources are lacking, a competition among ready activities is carried out in order to select the activities which must be operated first. This paper suggests a competition routine by implementing a policy to maximize the total contribution of selected activities in reducing the expected project duration and its variance. In this respect, a heuristic algorithm is developed and compared with the other existing methods.     

Determination of critical chain project buffer based on information flow interactions

It has been well accepted in the literature that co-dependency between project activity durations is caused by resource tightness and network complexity. However, we show that information flow interaction between activities is the key factor for it. Based on whether there exist spliced relationships between activities, we introduce the concept of rework safety time. We propose a method to compute the rework safety time using the information output and input time factors, rework probability matrix, and rework impact matrix. We achieve the optimization of the critical chain sequencing via the design structure matrix so that the dependency between activities is reduced. The project buffer is then determined by the tail concentration method based on the optimized chain. The empirical results show that, as opposed to the traditional RSEM method, our approach improves the project buffer consumption rate, shortens project duration, reduces project cost, and increases project on-time completion rate.     

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.     

LOB中基于工期决定因素分析的资源均衡算法研究

资源均衡是重复性项目中的经典调度问题,本文提出一种新的基于平衡线法(line of balance,LOB)的资源均衡方法。首先,本文提出LOB中关键路线的确定方法,确定关键路线及关键工序类型。而后,本文分析项目总工期的决定因素,对不同类型关键工序的特性及其与总工期、资源调整之间的关系进行了研究,论证了在LOB的资源均衡问题中,由于逆关键工序、点关键工序这些特殊工序的存在,可以在保证项目总工期不变的前提下,通过同时调整关键工序和非关键工序实现资源优化。按照这一思路,论文设计了LOB中资源均衡的遗传算法。算例分析表明该资源均衡算法的优化性能。本文提出的资源均衡思路和算法能帮助项目计划人员拓展资源优化空间,达到更好的资源均衡效果。     

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.     

Discrete-continuous project scheduling with discounted cash inflows and various payment models—a review of recent results

In this work discrete-continuous project scheduling problems with discounted cash flows are considered. These problems are characterized by the fact that activities of a project simultaneously require discrete and continuous resources for their execution. A class of these problems is considered, where the number of discrete resources is arbitrary, and there is one continuous, renewable resource, whose total amount available at a time is limited. Activities are non-preemptable, and the processing rate of an activity is a continuous, increasing function of the amount of the continuous resource allotted to the activity at a time. A positive cash flow (cash inflow) is associated with each activity, and the objective is to maximize the net present value (NPV). The discrete-continuous resource-constrained project scheduling problem with discounted cash flows (DCRCPSPDCF) is defined. Four payment models are considered: lump-sum payment at the completion of the project, payments at activity completion times, payments at equal time intervals, and progress payments. Some properties of optimal schedules are proved for two important classes of processing rate functions: all functions not greater than a linear function (including linear and convex functions), and concave processing rate functions.     

Project Scheduling under Time Dependent Costs – A Branch and Bound Algorithm

In a given project network, execution of each activity in normal duration requires utilization of certain resources. If faster execution of an activity is desired then additional resources at extra cost would be required. Given a project network, the cost structure for each activity and a planning horizon, the project compression problem is concerned with the determination of optimal schedule (duration) of performing each activity while satisfying given restrictions and minimizing the total cost of project execution. This paper considers the project compression problem with time dependent cost structure for each activity. The planning horizon is divided into several regular time intervals over which the cost structure of an activity may vary. But the cost structure of the activities remains the same (constant) within a time interval. Key events of the project attract penalty for finishing earlier or later than the corresponding target times. The objective is to find an optimal project schedule minimizing the total project cost. We present a mathematical model for this problem, develop some heuristics and an exact branch and bound algorithm. Using simulated problems we provide an insight into the computational performances of heuristics and the branch and bound algorithm.     

Properties of multi-mode resource-constrained project scheduling problems with resource vacations and activity splitting

This paper presents results from an extensive computational study of the multi-mode resource-constrained project scheduling problem when activities can be split during scheduling under situations where resources may be temporarily not available. All resources considered are renewable and each resource unit may not be available at all times due to resource vacations, which are known in advance, and assignment to other finite duration activities. A designed experiment is conducted that investigates project makespan improvement when activity splitting is permitted in various project scenarios, where different project scenarios are defined by parameters that have been used in the research literature. A branch-and-bound procedure is applied to solve a number of small project scheduling problems with and without activity splitting. The results show that, in the presence of resource vacations and temporary resource unavailability, activity splitting can significantly improve the optimal project makespan in many scenarios, and that the makespan improvement is primarily dependent on those parameters that impact resource utilization.     

Time-cost trade-off via optimal control theory in Markov PERT networks

We develop a new analytical model for the time-cost trade-off problem via optimal control theory in Markov PERT networks. It is assumed that the activity durations are independent random variables with generalized Erlang distributions, in which the mean duration of each activity is a non-increasing function of the amount of resource allocated to it. Then, we construct a multi-objective optimal control problem, in which the first objective is the minimization of the total direct costs of the project, in which the direct cost of each activity is a non-decreasing function of the resources allocated to it, the second objective is the minimization of the mean of project completion time and the third objective is the minimization of the variance of project completion time. Finally, two multi-objective decision techniques, viz, goal attainment and goal programming are applied to solve this multi-objective optimal control problem and obtain the optimal resources allocated to the activities or the control vector of the problem     

柔性工期下的资源受限项目调度双目标优化研究

在项目调度过程中,活动工期应根据项目截止工期以及资源供给情况进行合理设置,而在传统的资源受限项目调度问题(RCPSP)中,活动的工期往往是已知且固定的,这在一定程度上限制了项目调度的灵活性。多模式下的项目调度方式虽然弥补了这一缺点,但其提供的工期-资源组合种类固定且有限,并不一定能保证包含最优的工期-资源组合。本文将活动工期作为项目调度问题的决策变量,允许其在一定范围内取值。这种柔性工期调度方式虽然增加了项目调度难度,但提高了项目调度灵活性,同时可以起到压缩项目完工时间的作用。为验证柔性工期调度方式对项目工期和成本的影响,本文建立了工期-成本双目标权衡优化模型,设计了两阶段嵌套算法(NSGAⅡ-RS)对其求解,实验证明,柔性工期调度策略是一种鲁棒性较好的项目完工时间压缩策略。     

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.     

Multiple schedules and measures of resource constrained float

The conventional analysis of a resource constrained network produces a single schedule. A measure of float for each activity can be defined by examining the sequences of activities, incorporating the linkages implied by the sharing of resources in this particular schedule. However, the resultant floats are specific to one schedule and the analysis ignores the flexibility inherent in many resource constrained networks. It is often possible to employ alternative resource allocations resulting in schedules with identical overall durations but different timings for individual activities; in one schedule an activity may be critical but in another it may have significant float. A systematic exploration of alternative schedules reveals the flexibility of the network and the consequent variation in activities' floats. This paper defines new parameters describing these characteristics of activities in resource constrained networks. The practical value of the parameters is demonstrated in an example in which the measures of float guide the project planner to an alternative schedule, avoiding a critical dependency on a problematic activity and reducing the risk to the project at no additional cost.     

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.     

Using Schedule Risk Analysis with resource constraints for project control

Schedule Risk Analysis (SRA) has shown to provide reliable activity sensitivity information for taking corrective actions during project control. More precisely, by selecting a small subset of activities with high sensitivity values for taking corrective actions, the project outcome can be improved. In resource constrained projects, disrupted activities can affect both their successors as well as other activities when resource conflicts are induced. Since SRA focuses solely on the project network to determine the sensitivity of activities, the traditional SRA metrics do not accurately reflect the activity sensitivity for resource constrained projects. In this paper, the traditional SRA metrics are extended for resource constrained projects, and a novel resource-based sensitivity metric is introduced (RC-SRA metrics).A computational experiment is conducted to investigate the ability of the RC-SRA metrics to identify activities with higher sensitivity values. In addition, two activity selection strategies, defined as the normal strategy and sequential strategy, are designed to select activities for taking corrective actions. Further, two types of corrective actions are proposed to reduce the activity duration or resource demand in case of delays, respectively. Finally, the impact of dynamically updating the RC-SRA metrics during project execution is examined.The computational results show that the normal activity selection strategy is recommended for serial projects, while the sequential strategy is preferred for parallel projects. The results also indicate that reducing the activity durations performs better than reducing the resource demand of activities. Finally, it is shown that updating the RC-SRA metrics dynamically during project execution improves the efficiency of the corrective action taking process.     

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.