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 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.     

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.     

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.     

An exact algorithm for minimizing resource availability costs in project scheduling

A new exact algorithm that solves the Resource Availability Cost Problem (RACP) in project scheduling is shown to yield a significant improvement over the existing algorithm in the literature. The new algorithm consists of a hybrid method where an initial feasible solution is found heuristically. The branching scheme solves a Resource-Constrained Project Scheduling Problem (RCPSP) at each node where the resources of the RACP are fixed. The knowledge of previously solved RCPSPs is used to produce cuts in the search tree. A worst-case-performance theorem is established for this new algorithm. Experiments are performed on instances adapted from the PSPLIB database. The new algorithm can be used to minimize any resource availability cost problem once a procedure for the underlying resource-constrained problem is available.     

A breakpoint search approach for convex resource allocation problems with bounded variables

We present an efficient approach to solve resource allocation problems with a single resource, a convex separable objective function, a convex separable resource-usage constraint, and variables that are bounded below and above. Through a combination of function evaluations and median searches, information on whether or not the upper- and lowerbounds are binding is obtained. Once this information is available for all upper and lower bounds, it remains to determine the optimum of a smaller problem with unbounded variables. This can be done through a multiplier search procedure. The information gathered allows for alternative approaches for the multiplier search which can reduce the complexity of this procedure.     

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.     

Job scheduling with forbidden setups and two objectives using genetic algorithms and penalties

One of the most important tasks in service and manufacturing systems is how to schedule arriving jobs such that some criteria will be satisfied. Up to now there have been defined a great variety of scheduling problems as well as corresponding models and solution approaches. Most models suffer from such more or less restrictive assumptions like single machine, unique processing times, zero set-up times or a single criterion. On the other hand some classical approaches like linear or dynamic programming are practicable for small-size problems only. Therefore over the past years we can observe an increasing application of heuristic search methods. But scheduling problems with multiple machines, forbidden setups and multiple objectives are scarcely considered. In our paper we apply a Genetic Algorithm to such a problem which was found at a continuous casting plant. Because of the forbidden setups the probability for a random generated schedule to be feasible is nearly zero. To resolve this problem we use three kinds of penalties, a global, a local and a combined approach. For performance investigations of these penalty types we applied our approaches to a real world test instance with 96 jobs, three machines and two objectives. We tested five different penalty levels with 51 independent runs to evaluate the impact of the penalties.     

On the two-phase method for preemptive scheduling

Preemptive scheduling problems on parallel processors may in some cases be solved by a two-phase method: forst solve a LP problem which will give the minimum total completion time T and the processing times of the jobs on the various processors; second construct a feasible schedule using T time units. Some extensions of this procedure are discussed. A general model for preemptive scheduling is described with a two-phase method for solving problems of this type.     

Tabu search for a class of scheduling problems

Scheduling problems are often modeled as resourceconstrained problems in which critical resource assignments to tasks are known and the best assignment of resource time must be made subject to these constraints. Generalization toresource scheduling, where resource assignments are chosen concurrently with times results is a problem which is much more difficult. A simplified model of the general resource scheduling model is possible, however, in which tasks must be assigned a singleprimary resource, subject to constraints resulting from preassignment ofsecondary, or auxiliary, resources. This paper describes extensions and enhancements of tabu search for the special case of the resource scheduling problem described above. The class of problems is further restricted to those where it is reasonable to enumerate both feasible time and primary resource assignments. Potential applications include shift oriented production and manpower scheduling problems as well as course scheduling where classrooms (instructors) are primary and instructors (rooms) and students are secondary resources. The underlying model is a type of quadratic multiple choice problem which we call multiple choice quadratic vertex packing (MCQVP). Results for strategic oscillation and biased candidate sampling strategies are shown for reasonably sized real and randomly generated, synthetic, problem instances. The strategies are compared with other variations using consistent measures of solution time and quality developed for this study.     

A variable neighbourhood search algorithm for the constrained task allocation problem

A variable neighbourhood search algorithm that employs new neighbourhoods is proposed for solving a task allocation problem whose main characteristics are: (i) each task requires a certain amount of resources and each processor has a capacity constraint which limits the total resource of the tasks that are assigned to it; (ii) the cost of solution includes fixed costs when using processors, task assignment costs, and communication costs between tasks assigned to different processors. A computational study shows that the algorithm performs well in terms of time and solution quality relative to other local search procedures that have been proposed.     

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.     

Interactive analysis of multiple-criteria project scheduling problems

The paper considers an interactive search over a non-dominated solution space of a multiple-criteria project scheduling problem. The approach described handles quite a general class of non-preemptive project scheduling problems with renewable, non-renewable and doubly constrained resources, multiple performing modes of activities, precedence constraints in the form of an activity network and multiple project performance criteria of time and cost type. The approach consists of two stages. In the first stage, a large representative sample of approximately non-dominated schedules is generated by the Pareto Simulated Annealing (PSA) metaheuristic method. Then, in the second stage, an interactive search over the sample is organized by the `Light Beam Search' (LBS) procedure in its discrete version.     

Matching parentheses in parallel

Parallel algorithms for evaluating arithmetic expressions generally assume the computation tree form to be at hand. The computation tree form can be generated within the same resource bounds as the parenthesis matching problem can be solved. We provide a new cost optimal parallel algorithm for the latter problem, which runs in time O(log n) using O(n/log n) processors on an . We also prove that the algorithm is the fastest possible independently of the number of processors available.     

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.     

Implementation of parallel branch-and-bound algorithms – experiences with the graph partitioning problem

Parallel processing is one of the essential concepts in the attempts to increase the computational power available for solving continuous and discrete optimization problems. In the case where an optimization algorithm is search-based, crucial issues of parallel distributed implementations are work-load distribution and granularity, i.e. how to distribute the search space among processors and how to control the amount of processing between interprocessor communication. The present paper compares distributed implementations of two branch-and-bound algorithms for the graph partitioning problem: Given an undirected graph with an even number of edges and weights assigned to each edge, partition the vertices into two subsets of equal size such that the sum of the costs of edges connecting vertices in different subsets is as small as possible. The problem is known to be NP-complete. The two branch-and-bound methods compared differ in design strategy: One is based on time-consuming bound calculations leading to tight bounds and thus a narrow search tree with few nodes, whereas the other employs an easy bound calculation scheme leading to a larger search tree. Both have been implemented on an iPSC-hypercube with 32 processors. We investigate the influence of the design strategy on the performance of the algorithms.     

一种求解柔性资源约束前摄性项目调度问题的启发式算法

在不确定环境中,一个具有较高鲁棒性的进度计划可以保证项目的稳定实施。考虑到现实中资源可能具有多种技能,会对制定鲁棒性较高进度计划的过程产生影响,因此本文研究了柔性资源约束下前摄性项目调度优化问题。首先界定研究问题;然后从鲁棒性最大化的视角出发,构建了研究问题的优化模型,在对模型进行分析的基础上将其分解为经典鲁棒优化和资源技能分配两个子模型;随后设计了求解问题的基于削峰算法的启发式算法;最后用一个实际案例验证了算法有效性,并分析了关键参数对进度计划鲁棒性的影响,得到如下结论:项目进度计划鲁棒性随着项目工期的延长、资源可用量的增加或资源柔性的提高而增大。     

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.     

Four Payment Models for the Multi-Mode Resource Constrained Project Scheduling Problem with Discounted Cash Flows

In this paper, the multi-mode resource constrained project scheduling problem with discounted cash flows is considered. The objective is the maximization of the net present value of all cash flows. Time value of money is taken into consideration, and cash in- and out-flows are associated with activities and/or events. The resources can be of renewable, nonrenewable, and doubly constrained resource types. Four payment models are considered: lump sum payment at the terminal event, payments at prespecified event nodes, payments at prespecified time points and progress payments. For finding solutions to problems proposed, a genetic algorithm (GA) approach is employed, which uses a special crossover operator that can exploit the multi-component nature of the problem. The models are investigated at the hand of an example problem. Sensitivity analyses are performed over the mark up and the discount rate. A set of 93 problems from literature are solved under the four different payment models and resource type combinations with the GA approach employed resulting in satisfactory computation times. The GA approach is compared with a domain specific heuristic for the lump sum payment case with renewable resources and is shown to outperform it.     

Efficient algorithms for solving multiconstraint zero-one knapsack problems to optimality

The multiconstraint 0–1 knapsack problem is encountered when one has to decide how to use a knapsack with multiple resource constraints. Even though the single constraint version of this problem has received a lot of attention, the multiconstraint knapsack problem has been seldom addressed. This paper deals with developing an effective solution procedure for the multiconstraint knapsack problem. Various relaxation of the problem are suggested and theoretical relations between these relaxations are pointed out. Detailed computational experiments are carried out to compare bounds produced by these relaxations. New algorithms for obtaining surrogate bounds are developed and tested. Rules for reducing problem size are suggested and shown to be effective through computational tests. Different separation, branching and bounding rules are compared using an experimental branch and bound code. An efficient branch and bound procedure is developed, tested and compared with two previously developed optimal algorithms. Solution times with the new procedure are found to be considerably lower. This procedure can also be used as a heuristic for large problems by early termination of the search tree. This scheme was tested and found to be very effective.