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In this paper, we study and compare several integer and mixed‐integer linear programming formulations for the multiskill resource‐constrained project scheduling problem. This is a problem characterized by a set of activities to be executed that in addition to the usual precedence constraints require several resources for each skill needed for their execution. A set of multiskill resources is assumed. The objective is to minimize the project's makespan. We revisit two existing models for the problem and propose two new ones. Additionally, we perform a theoretical comparison between the lower bounds provided by the linear programming relaxations of the models. Furthermore, a numerical experiment is performed on a set of instances from the literature to evaluate the suitability of an off‐the‐shelf solver to compute optimal solutions and lower bounds to the studied problem, using the aforementioned models.

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... Artigues, Koné, Lopez, and Mongeau (2015) set out possible mixed-integer linear programming formulations for the RCPSP. More recently, Almeida, Correia, and Saldanha-da Gama (2019) proposed multiple linear programming formulations for the MSRCPSP. In this paper, we reiterate the seven different linear programming formulations (see Section 2.2) for the MSRCPSP presented in Almeida et al. (2019), which we will then apply to the six extensions of the MSRCPSP with hierarchical levels of skills in Section 3. It must be specified that we have adjusted several decision variables and/or constraints in order to complement our notations. ...

... More recently, Almeida, Correia, and Saldanha-da Gama (2019) proposed multiple linear programming formulations for the MSRCPSP. In this paper, we reiterate the seven different linear programming formulations (see Section 2.2) for the MSRCPSP presented in Almeida et al. (2019), which we will then apply to the six extensions of the MSRCPSP with hierarchical levels of skills in Section 3. It must be specified that we have adjusted several decision variables and/or constraints in order to complement our notations. The formulations used in this paper will be classified based on the purpose that the decision variables serve and how they work in relation to each other in the scheduling problem, which is usually the case in project scheduling literature. ...

... For this formulation, we can again create a version with disaggregated precedence relations 5, this time also adding constraints (22). Moreover, formulation 6 proposed by Almeida et al. (2019) can be easily derived from formulation 4 by replacing constraints (20) and (21) by constraints (13). Formulation 6 can also be modified with disaggregated precedence relations in formulation 7. ...

In this paper, we present six extensions to the multi-skilled resource-constrained project scheduling problem (MSRCPSP) by introducing hierarchical levels of skills. These hierarchical skills can impact the MSRCPSP in multiple different ways. This paper studies efficiency differences, cost differences, quality differences and more. For each of these problems we propose and analyse seven continuous and time-indexed (mixed-)integer linear programming formulations. A modular artificial dataset is generated that assembles instances of the presented problems as well as combinations of these problems. In the computational experiments, we solve these instances using the proposed mathematical formulations with the CPLEX solver. Finally, we compare the results of the different formulations for the resource-constrained project scheduling problems with hierarchical levels of skills in order to explain their inherent similarities and differences.

... In some industrial environments, such as nuclear research facilities described in Polo-Mejía et al. (2020) that inspired our study, scheduling activities need to take into account a set of skills and clearances required to execute the activities and to match them with those that some technicians master. In this context, where regulations require the presence of a group of technicians having a set of well-defined skills for the execution of an activity, the multi-skill project scheduling problem (MSPSP) is important, see for example Bellenguez-Morineau and Néron (2007); Bellenguez-Morineau (2008); Montoya et al. (2014); Correia and Saldanha-da Gama (2015); Almeida et al. (2016); Young et al. (2017); Almeida (2018); Almeida et al. (2019). The MSPSP, initially proposed to schedule IT development projects (Néron, 2002), turns out to be more challenging than traditional scheduling problems, due to additional decisions to be made: it is needed to decide not only which resources will be allocated to each activity, but also the skills with which they will contribute. ...

... As indicated in Bellenguez-Morineau (2006), the MSPSP is NP-hard, thus solving industrial-sized instances of the MSPSP is time-consuming. Exact methods are proposed, among them branch-and-bound (Bellenguez-Morineau and Néron, 2007), branch-and-price (Montoya et al., 2014), constraint programming (Young et al., 2017) and mixed-integer linear programming (Correia et al., 2012;Almeida et al., 2019). Heuristic methods are then necessary to tackle this type of instances in short computing times. ...

The multi-skill project scheduling problem (MSPSP) has been first addressed in the scheduling community for more than 15 years. This article deals with a new variant of this problem, the MSPSP with partial preemption, where only a subset of resources can be released during the preemption periods. Like the standard problem, this variant is NP-hard, because of which we propose in this article a series of heuristic algorithms to solve instances arising from an industrial application. First, we present a serial greedy algorithm, based on priority rules and a flow problem for resource allocation. To improve the solutions of the greedy algorithm, we then introduce a binary tree based search algorithm and a greedy randomized adaptive search procedure (GRASP). Finally, we propose a large neighborhood search (LNS) algorithm integrating exact and heuristic methods. The best results in terms of solution quality and execution time are obtained by combining the GRASP algorithm and LNS approach. Furthermore, the proposed GRASP algorithm is able to find new best results on 56 of 216 instances on a standard MSPSP instance set, which shows the quality of the approach even on special cases of the considered problem.

... In the case of MRCPSPs, when the process allows the choice of modes, further complexity is added that enlarges the search space for finding the optimize/appropriate resources for activities (Kyriakidis et al., 2012). There has been a significant number of exact methods, and heuristic, metaheuristic, and hyper-heuristic approaches for solving different RCPSPs and MRCPSPs (Elsayed et al., 2017;Almeida et al., 2019). Among them, the exact methods for MRCPSP include the "branch and cut" algorithm (Zhu et al., 2006), the "tree-based branch and bound" algorithm (Hartmann and Drexl, 1998;Cheng et al., 2015), and the "linear programming-based" algorithm (Kopanos et al., 2014). ...

This paper aims at providing a fast near‐optimum solution to the multi‐mode resource‐constrained project scheduling problems (MRCPSPs), for projects with activities that have known deterministic renewable and nonrenewable resource requirements. The MRCPSP is known to be nondeterministic polynomial‐time hard and has been solved using various exact, heuristic, and meta‐heuristic procedures. In this paper, a modified variable neighborhood search heuristic algorithm is used as an advanced optimization technique that suits scheduling problems. For the experimental study, we have considered a standard set of 3929 multi‐mode benchmark instances from the project scheduling library with a range of projects comprising 10–30 activities. Moreover, for a better comparison, this research also considers a standard set of 4320 newly developed multi‐mode instances from MMLIB50, MMLIB100, and MMLIB+ datasets. With the limit of 50,000 schedules on these datasets, our proposed algorithm provides better makespan for 106, 34, and 1601 instances, respectively, which justifies the efficiency of the proposed algorithm, particularly for projects with a larger number of activities. The results reported in this paper can be used as a benchmark for other researchers to compare and improve.

... They found that the key ingredients for a successful solving algorithm was the combination of problem tailored search with the use of redundant resource constraints. Almeida et al. (2018) compared several sequence-based and time-indexed MIPs for the MS-RCPSP. They also compared the lower bounds provided by the LP-relaxations of the models using the 216 instances introduced in literature. ...

Multi-skilling provides an organization with the ability to arrange its workers to meet the certain needs for several skills. This paper reviews the literature on scheduling problems under the multi-skilled and flexible resources which is the case for a wide range of disciplines including construction industry, IT projects, healthcare, process systems, and so on. The main purpose of this review is that it helps researchers and scholars entering the multi-skilling experience an all-encompassing overview of existing models and methods and to identify new research directions. To structure the emerging literature in this area, we review and classify 160 articles published from 2000 to middle 2020 based on characteristics of the objective functions, the mathematical formulations, the solving methodologies, and the potential applications. This review outsets with a general framework for multi-skilling and accomplishes a comprehensive taxonomy for the literature of multi-skilling in scheduling problems. The results show that the main focus of the existing research in this field has been devoted to project scheduling problems (53.12%), mixed integer programming models (54.2%) and metaheuristics (28.7%) as solving method, cost (39.4%) as single objective function (68.6%), and deterministic environment for parameters (85.5%) on the top. It also turned out that 68.8% of research considered a single objective, whereas 13.8% and 17.6% of the research papers have developed models with two and more objectives, respectively. With the goal of providing a vivid roadmap for researchers, through meta-narrative analysis of the collected papers and a rigorous analysis, promising future lines of research are outlined.

... Multimode Resource Constrained Project Scheduling Problem (MRCPSP), as an extension version of RCPSP, has been attended very much by researchers in recent years. Under the resource and precedence constraints, the aim of MRCPSP is to assign an execution mode and a feasible start time for each activity, such that makespan of the project is minimized (Hartmann and Briskorn 2010;Chaleshtari and Shadokh 2014;Qi et al. 2014;Ahmed et al. 2018;Chakrabortty et al. 2018;Muritiba et al. 2018;Almeida et al. 2019). ...

The existence of limited resources and precedence relations between activities makes project scheduling a difficult task. In this paper, the multimode resource-constrained project scheduling problem (MRCPSP) as one of the important project scheduling problems is focused. In this regard, a genetic algorithm (GA) is proposed for solving the multimode resource-constrained project scheduling problem (MRCPSP) improved by a new local search method. The Activity Mode List (AML) is used as encoding and the multimode serial schedule generation scheme (MSSGS) is employed as the decoding procedure. The aim of the suggested local search is maximum use of existed nonrenewable resources of the project such that its makespan is decreased. For this purpose, the local search procedure allocates the unused nonrenewable resources of each feasible solution to the activities with lower total float. Comparing results of the proposed method with other ones using the set J20 in Project Scheduling Problem Library (PSPLIB) validates the effectiveness of the proposed algorithm to solve the MRCPSP.

... In its standard state, this is the problem of defining practicable scheduling for activities considering precedence relations which are only the multiskilled human resource type. Almeida et al. (2019) compared several integer and mixed integer linear programming formulations for the multi-skill resource-constrained project scheduling problem. Maghsoudlou et al. (2017) investigated a version of the multiskilled resource-constrained project scheduling problem by a bi-objective optimization model to minimize total costs of processing and minimize reworking risks of activities. ...

In this study, we aim to present a new model for the resource-constrained project scheduling problem (RCPSP) considering a working calendar for project members and determined the skill factor of any member using the efficiency concept. For this purpose, the recyclable resources are staff resources where any person with multiple skills can meet the required skills of activities in a given time. Then, considering uncertainty condition for parameters, it provided a fuzzy scheduling model and validated models by solving different examples. The proposed mathematical programming model optimizes the allocation of limited resources to project activities for scheduling purposes in an essential activity in the real condition of scheduling problems. Moreover, the proposed model can decrease the risk of deviation from scheduling by allocating members with higher skill factors to critical activities. Then, considering uncertainty condition for parameters, it provided a fuzzy scheduling model and validated models by solving different examples. Considering fuzzy conditions for the calendar of the project and multi-skill operators are firstly considered in this paper. Also, the recyclable resources are staff resources which are being considered for the model concurrently in response to the risks of availability to resources and delay in completing the project under uncertainty. The results derived from the model solved by CPLEX indicated a decreased need for employment and shortened project completion duration. Assuming the uncertainty of available resource capacity at any time, the results obtained from the fuzzy model for the value of objective function were evaluated under the influence of the resource calendar and showed the benefits. Effect of the multi-skill members with calendar constraints on the model is tested, and the advantages are determined.

... Several classical activity priority rules for RCPSP were adapted and two new concepts -activity group and resource weightwere designed for solving MS-RCPSP. More recently, Almeida et al. (2019) analyzed different integer and mixed-integer formulations for the MS-RCPSP, and presented a general modeling framework for project scheduling and staffing problems. ...

This paper proposes a new extension of the multi-skill resource-constrained project scheduling problem: MS-RCPSP with skill switches. In real-world environment, the switch between skills for a resource often incurs considerable extra operation time and cost to meet the requirement of task. A mixed-integer programming model, aiming to minimize project completion time and total cost, is developed. Then, to solve this new problem effectively, we investigate the related existing solution representations and schedule builders from previous literature. Based on the investigation, we propose a new flexible solution representation scheme with reduced search space and a novel greedy-like schedule builder scheme that reorders tasks to reduce skill switches. Besides, we select two efficient mutation operators, i.e., a swap operator and a resource-leveling operator. The swap operator adjusts the task assignment sequence while the resource-leveling operator reassigns the tasks to other resources based on the resource loads of the current schedule. This operator improves solutions by balancing resource loads. We embed all the proposed new components into a multi-objective evolution strategy (MOES) framework. We analyze different configurations of MOES in terms of representation, schedule builder, and mutation operators, and identify the best configuration. We compare the result of MOES with the state-of-the-art algorithms on a wide range of test instances, and the results show that our proposed representation scheme and schedule builder can improve the convergence of the Pareto Front, while the resource-leveling operator can greatly improve the spread and diversity of the front.

... Also note that a partially renewable resource is very general because it includes renewable and nonrenewable resources as well as time-dependent resource capacities as special cases. The second category is resources with multiple skills (Montoya et al., 2014;Almeida et al., 2019). A set of skills is given, and for each resource it is known which of these skills it masters. ...

Determining high-quality schedules is an important task in project management. This paper gives an overview of research in project scheduling. While other recent survey papers from the literature focus either only on models or discuss solution methods for a specific type of model, this contribution provides a broader survey that covers the most important models as well as related solution approaches. The paper sketches out and classifies the most important problem settings, with a focus on settings that are relevant for construction projects. This includes the basic resource-constrained project scheduling problem (RCPSP) as well as its major extensions such as multiple modes, generalized precedence relations, different resource categories, various objectives, and stochastic aspects. Moreover, related algorithms are outlined, ranging from exact to heuristic (and in particular metaheuristic) algorithms. Finally, current research directions in project scheduling are discussed.

... In the MMRCPSP project interactions that result from the utilization of shared resources must be taken into consideration (Zapata, Hodge, & Reklaitis, 2008). A significant number of exact, heuristic, and metaheuristic approaches have been proposed for solving different MMRCPSPs (Almeida, Correia, & Saldanha-da-Gama, 2019). The MMRCPSP is distinguishably more complex than the SMRCPSP, which is itself NP-hard (Elloumi & Fortemps, 2010). ...

This study introduces two optimization problems related to university course planning. In the student course planning problem (SCPP), a student needs to design a course plan that allows him/her to graduate in a timely manner. In the department course planning problem (DCPP), an academic department needs to decide which courses to offer during which semester to facilitate students’ timely graduation. Mathematical models of these problems are developed, coded in C++, and solved with IBM ILOG CPLEX. Experiments on small, medium, and large real-world and fictional instances show promising results.

... The term "scheduling" generally in computing, project management and operations research have been used severally with varying goals and context in mind applying it to different instances including manufacturing, operations research/decision theory, modeling or finding solutions to project scheduling problems, machine job processing [10], [11], [13], [14], [15], and [16]. ...

A schedule in project management is a listing of a project's milestones and deliverables, typically with supposed begin and end dates. How do we effectively and automatically schedule projects in order to maximize time and profit? This paper presents the development of a profit-based project scheduling management system which was designed to meet the need of software developers who often have issues with effectively prioritizing projects. Unified Modelling Language (UML) was used in visualizing the design of the system. Also, a weighted job scheduling algorithm was adopted to determine the most viable project based on time management and profit return. The system was implemented following a sequential development approach and client-server architecture. The prototype developed is an interactive web-based one that can be used to manage, schedule and prioritize software developers' projects based on profits majorly considering the project cost, profits, duration of project and estimated time of completion.

... The goal is to determine a start time as well as appropriate resources for each activity with the objective to minimize the makespan. This standard MS-RCPSP has been tackled by Correia et al. (2012), Correia and Saldanha-da Gama (2015), Almeida et al. (2016Almeida et al. ( , 2018Almeida et al. ( , 2019, and Montoya et al. (2014Montoya et al. ( , 2015. ...

The resource-constrained project scheduling problem is to schedule activities subject to precedence and resource constraints such that the makespan is minimized. It has become a standard problem in the context of project scheduling which has attracted numerous researchers who developed both exact and heuristic scheduling procedures. However, it is a rather stylized model with assumptions that are too narrow to capture many real world requirements. Consequently, various extensions of the basic resource-constrained project scheduling problem have been developed. This paper builds on an overview which has been published 10 years ago. Due to the unabated interest in the scientific community since it has been published the overview at hand delivers an update focussing on the last decade. The problem extensions are classified according to the structure of the resource-constrained project scheduling problem. We summarize generalizations of the activity concept, of the precedence relations, and of the resource constraints. Alternative objectives and approaches for scheduling multiple projects are discussed as well.

... We extend the model of Almeida et al. (2019). The mathematical formulation of the reconnaissance mission scheduling problem can be formulated as follows: ...

With the progress of technology, the multi-agent system is successfully applied in many applications. In this paper, we investigate the problem of multi-agent system reconnaissance mission scheduling, which is the core of the reconnaissance decision support system and can be modeled as an extension of Multi-Mode Multi-Skill Resource-Constrained Project Scheduling Problem. Three objectives are considered in this paper: (1) minimizing the reconnaissance mission’s makespan, (2) minimizing the total cost of allocating reconnaissance agents, and (3) maximizing the total quality of all reconnaissance tasks. An effective problem-specific multi-objective invasive weed optimization algorithm (PS-MOIWO) is proposed for solving the problem. Firstly, a new chromosome structure guaranteeing the feasibility of solutions and an initialization method are proposed. Secondly, we propose a self-adaptive penalty-based constraint handling technique to describe the fitness of each individual and adopt a novel non-dominated sorting method to rank the population. Thirdly, by using the problem-specific knowledge, a local search procedure is developed and incorporated into the PS-MOIWO framework to enhance the exploitation ability. Based on the Taguchi method, algorithm’s suitable parameter combinations are determined. Simulation results based on a set of newly generated reconnaissance instances and the comparisons with some existing algorithms demonstrate the proposed algorithm’s effectiveness.

... Recently, papers have been published that focus on solving the MSRCPSP. Some of them encapsulate the MSRCPSP in a linear programming model ( Li and Womer (2009) ;Correia, Lampreia-Lourenço, and Saldanha-da Gama (2012) ; Almeida, Correia, and Saldanha-da Gama (2018b) ). Moreover, various heuristic and exact solutions have been proposed for this problem. ...

This paper addresses a multi-skilled extension of the resource-constrained project scheduling problem (RCPSP). Although a handful of papers dealt with the multi-skilled RCPSP (MSRCPSP), little to no attention is given to the ideal levels of skills for multi-skilled resources. In this paper, skills are measured along two dimensions known as breadth and depth. In a project environment, the breadth of a resource is perceived as the amount of skills an employee masters. The depth of a skill is the efficiency level at which work can be performed by a resource that masters that skill. The MSRCPSP with breadth and depth consists of scheduling activities with skill requirements and assigning multi-skilled resources to those activities. To be able to efficiently solve the MSRCPSP, a genetic algorithm is developed. Using the created activity schedules and resources assignments, the best workforce characteristics are analysed. Key aspects in this analysis are the breadth and depth. The problem-specific procedure combines a new representation, a new crossover and tailor-made local searches. Computational experiments measure the impact of different multi-skilled resources and their efficiency levels on the makespan of the project.

This paper presents the results of a research project aiming to optimise the scheduling of activities within a research laboratory of the ‘Commissariat à l'Energie Atomique et aux Energies Alternatives (CEA)’. To tackle this problem, we decompose every activity into a set of elementary tasks to apply standard scheduling methods. We model the problem as an extended version of the Multi-Skill Project Scheduling Problem (MSPSP). As a first approach, we propose a Multi-Skill Project Scheduling Problem with penalty for preemption, along with its mixed-integer/linear programming (MILP) formulation, where the preemption is allowed applying a penalty every time an activity is interrupted. However, the previous approach does not take into account all safety constraints at the facility, and a more accurate variant of the problem is needed. We propose then to integrate the concept of partial preemption to the MSPSP. This concept, that has not been yet studied in the scientific literature, implies that only a subset of resources is released during preemption periods. The resulting MSPSP with partial preemption (MSPSP-PP) is modelled using two methodologies: MILP and constraint programming. Regarding the industrial need of having good solutions in a short time, we also present a greedy algorithm for the MSPSP-PP.

The authors consider the Multi-Skilled Project Scheduling Problem (MSPSP) as a version of the Multi-mode Resource-Constrained Project Scheduling Problem (MRCPSP), which extends the well-known NP-hard Resource-Constrained Project Scheduling Problem (RCPSP). In the MSPSP, it is made up of activities, which require specific skills to be done. Moreover, resources are staff members who master fixed skill(s). Thus, each resource requirement of an activity corresponds to the number of persons doing the corresponding skill that must be assigned to the activity during its whole processing time. The authors focus on taking into account the setup times when performing work, depending on the appointment of a worker with the individual skill. The authors propose an integer linear programming model for solving the problem, in which minimization of the project makespan according to the setup time is considered. Genetic(GA) and Simulated Annealing(SA) metaheuristic algorithms were developed for solving that problem. For verification of the model, CPLEX software was used. The calculation results showed that with regard to the selected parameters for each algorithm, both of the proposed algorithms had relatively similar performance.

This study addresses a resource-constrained multi-project scheduling and multi-skilled workforce assignment problem for the large-scale equipment manufacturing industry. The goal is to help the production manager perform effective planning and decrease the production cost under delivery constraints. The deterministic duration is used in most past investigations in resource-constrained multi-project scheduling problems. Uncertainty in project execution is neglected. Additional related factors are included in the current study to be close to practical circumstances. The required processing duration and material arrival time are modeled as stochastic variables. The heterogeneous skill efficiency of the internal workforce is considered with the learning effect. Skill efficiency grows with the accumulation of experience. Actual processing time is calculated on the basis of skill efficiency and stochastic process duration. The external workforce is hired when the internal workforce fails to satisfy processing demand. The objective is to minimize the expected integrated cost, which is the sum of tardiness penalty and external workforce cost. A genetic algorithm with a heuristic workforce assignment method is also developed, and a case study is illustrated to explain the scheduling result. The proposed approach with uniform crossover and local search mechanism outperforms other comparative methods by approximately 10.34% considering total costs. Furthermore, the Taguchi method of design of the experiment is conducted to find the optimal parameter setting efficiently. Afterward, the sensitivity analysis of production-related parameters is discussed.

This paper studies and analyses the multi-skilled resource-constrained project scheduling problem (MSRCPSP). We present a new classification scheme based on an existing classification scheme for project scheduling problems. This allows researchers to classify all multi-skilled project scheduling problems and its extensions. Furthermore, we propose a new data generation procedure for the MSRCPSP and introduce multiple artificial datasets for varying research purposes. The new datasets are generated based on new multi-skilled resource parameters and are compared to existing benchmark datasets in the literature. A set of 7 empirical multi-skilled project instances from software and railway construction companies are collected in order to validate the quality of the artificial datasets. Solutions are obtained through a genetic algorithm and by solving a mixed-integer linear programming formulation with CPLEX 12.6. The hardness of the multi-skilled project instances is investigated in the computational experiments. An experimental analysis studies the impact of skill availability, workforce size and multi-skilling on the makespan of the project.

This article addresses the Multi-mode Resource-Constrained Project Scheduling Problem (MRCPSP). It is an NP-Hard problem and consists of scheduling all activities of a project, respecting the precedence order of these activities, which, once initiated, can not be interrupted. Jobs have combinations of execution modes, in the form of different combinations of processing time and resource consumption. The objective is to minimize the duration of the project, finding start times for each project activity. This article proposes a Local Branching matheuristic strategy for solving MRCPSP. The influence of the parameters of the based local search method is verified, as well as the characterization of neighborhood structures that define a subproblem during the enumeration of the LB matheuristic. Two characterizations to define the subproblems are proposed, and one of them surpasses the characterization discussed in the literature. Computational tests were performed using sets of instances from the PSPLIB and MMLIB libraries. The best-known results for these instances in relation to the state-of-the-art are presented as a reference to validate the results obtained by the approach proposed in this article.

We develop and test a strong fractional cutting-plane algorithm for the classical non-preemptive precedence- and resource-constrained project scheduling problem. While our basic approach is to formulate the problem as a 0-1 IP and solve it by LP-based branch-and-bound, we enhance the algorithm considerably through (a) an improved IP reformulation, (b) problem preprocessing techniques, and (c) on-the-fly tightening of the LP relaxation by generating strong and valid inequalities that are violated by the current (fractional) LP optimum. We also report results from an exploratory computational implementation of the algorithm.

Project scheduling is concerned with single-item or small batch production where scarce resources have to be allocated to dependent activities over time. Applications can be found in diverse industries such as construction engineering, software development, etc. Also, project scheduling is increasingly important for make-to-order companies where the capacities have been cut down in order to meet lean management concepts. Likewise, project scheduling is very attractive for researchers, because the models in this area are rich and, hence, difficult to solve. For instance, the resource-constrained project scheduling problem contains the job shop scheduling problem as a special case. So far, no classification scheme exists which is compatible with what is commonly accepted in machine scheduling. Also, a variety of symbols are used by project scheduling researchers in order to denote one and the same subject. Hence, there is a gap between machine scheduling on the one hand and project scheduling on the other with respect to both, viz. a common notation and a classification scheme. As a matter of fact, in project scheduling, an ever growing number of papers is going to be published and it becomes more and more difficult for the scientific community to keep track of what is really new and relevant. One purpose of our paper is to close this gap. That is, we provide a classification scheme, i.e. a description of the resource environment, the activity characteristics, and the objective function, respectively, which is compatible with machine scheduling and which allows to classify the most important models dealt with so far. Also, we propose a unifying notation. The second purpose of this paper is to review some of the recent developments. More specifically, we review exact and heuristic algorithms for the single-mode and the multi-mode case, for the time–cost tradeoff problem, for problems with minimum and maximum time lags, for problems with other objectives than makespan minimization and, last but not least, for problems with stochastic activity durations.

Abstract This paper deals with an extension of the classical Resource Constrained Project Scheduling Problem (RCPSP) We present a new type of resource constraints in which staff members are involved On the one hand, we focus on staff members, each of them having several skills, i e, is able to perform more than one kind of activity On the other hand, an activity has specific skill requirements that must be satisfied To solve this problem, we propose two lower bounds The first one uses a linear programming scheme proposed for the RCPSP and the second one is based on energetic reasoning

In this paper we consider the Project Scheduling Problem with resource constraints, where the objective is to minimize the project makespan. We present a new 0-1 linear programming formulation of the problem that requires an exponential number of variables, corresponding to all feasible subsets of activities that can be simultaneously executed without violating resource or precedence constraints. Different relaxations of the above formulation are used to derive new lower bounds, which dominate the value of the longest path on the precedence graph and are tighter than the bound proposed by Stinson et al. (1978).
A tree search algorithm, based on the above formulation, that uses new lower bounds and dominance criteria is also presented. Computational results indicate that the exact algorithm can solve hard instances that cannot be solved by the best algorithms reported in the literature.

This paper proposes a Path-Relinking (PR) algorithm for the well-known and NP-hard Multi-mode Resource-Constrained Project Scheduling Problem (MRCPSP). This problem generalizes the Resource-Constrained Project Scheduling Problem (RCPSP) where the project activities have a set of execution modes. For each execution mode, the processing time, the renewable and nonrenewable resource demands are given. The MRCPSP goal is to minimize the total makespan of the project. The PR algorithm works by travelling through the solution space between two solutions, it performs local search around the intermediate solutions. This work also presents computational tests using benchmark instances to compare our implementation with the most competitive methods from the literature. The PR’s computational results improve the earlier results reported for the benchmark instance sets.

In this paper we investigate one of the most recent extensions of the Resource Constrained Project Scheduling Problem (RCPSP): the Multi-Skill Resource Constrained Project Scheduling Problem (MSRCPSP). For this complex problem we propose the use of a parallel scheduling scheme. Such scheme has been successfully applied to the RCPSP. Nevertheless, in order to apply it to the MSRCPSP two new concepts are developed: resource weight and activity grouping. We discuss such concepts and use them for the new heuristic framework proposed. A series of computational tests performed using a large number of instances and reported in this paper shows that the new heuristic is very effective in finding high quality solutions within very small CPU times.

This title presents a large variety of models and algorithms dedicated to the resource-constrained project scheduling problem (RCPSP), which aims at scheduling at minimal duration a set of activities subject to precedence constraints and limited resource availabilities. In the first part, the standard variant of RCPSP is presented and analyzed as a combinatorial optimization problem. Constraint programming and integer linear programming formulations are given. Relaxations based on these formulations and also on related scheduling problems are presented. Exact methods and heuristics are surveyed. Computational experiments, aiming at providing an empirical insight on the difficulty of the problem, are provided. The second part of the book focuses on several other variants of the RCPSP and on their solution methods. Each variant takes account of real-life characteristics which are not considered in the standard version, such as possible interruptions of activities, production and consumption of resources, cost-based approaches and uncertainty considerations. The last part presents industrial case studies where the RCPSP plays a central part. Applications are presented in various domains such as assembly shop and rolling ingots production scheduling, project management in information technology companies and instruction scheduling for VLIW processor architectures.

In a recent paper published in Optimization Letters, Montoya et al. (Optim Lett 8:1721–1734, 2014) proposed a branch-and-price approach for a multi-skill project scheduling problem. In that paper, an integer linear programming formulation was first introduced which, unfortunately, has a number of inconsistences. At the interest of mathematical rigor, in this note, we refine such formulation.

For non-preemptive scheduling, time-indexed zero-one linear programming formulations have been deeply analyzed. This note clarifies the current knowledge about the strength of these formulations and shows that some formulations that have been proposed “new” in the literature are in fact weaker or equivalent to those already known. Much of the arguments used follow from a PhD thesis by Sousa, which has been largely overlooked in the literature.

This chapter addresses modeling issues associated with project staffing and scheduling problems. Emphasis is given to mixed-integer linear programming formulations. The need for such formulations is motivated and a general modeling framework is introduced, which captures many features that have been considered in the literature on project staffing and scheduling problems. The use of the general framework is then exemplified using two problems that have been addressed in the literature. Several model enhancements and preprocessing procedures are discussed.

This paper describes a branch and bound algorithm for project scheduling with resource constraints. The algorithmis based on the idea of using disjunctive arcs for resolving conflicts that are created whenever sets of activities have to be scheduled whose total resource requirements exceed the resource availabilities in some periods. Four lower bounds are examined. The first is a simple lower bound based on longest path computations. The second and third bounds are derived from a relaxed integer programming formulation of the problem. The second bound is based on the Linear Programming relaxation with the addition of cutting planes, and the third bound is based on a Lagrangean relaxation of the formulation. This last relaxation involves a problem which is a generalization of the longest path computation and for which an efficient, though not polynomial, algorithm is given. The fourth bound is based on the disjunctive arcs used to model the problem as a graph. We report computational results on the performance of each bound on randomly generated problems involving up to 25 activities and 3 resources.

This work introduces a procedure to solve the multi-skill project scheduling problem (MSPSP) (Néron and Baptista, International symposium on combinatorial, optimization (CO’2002), 2002). The MSPSP mixes both the classical resource constrained project scheduling problem and the multi-purpose machine model. The aim is to find a schedule that minimizes the completion time (makespan) of a project, composed of a set of activities. In addition, precedence relations and resources constraints are considered. In this problem, resources are staff members that master several skills. Thus, a given number of workers must be assigned to perform each skill required by an activity. Practical applications include the construction of buildings, as well as production and software development planning. We present a column generation approach embedded within a branch-and-price (B&P) procedure that considers a given activity and time-based decomposition approach. Obtained results show that the proposed B&P procedure is able to reach optimal solutions for several small and medium sized instances in an acceptable computational time. Furthermore, some previously open instances were optimally solved.

In this paper, we address a cost-oriented multi-skill project scheduling problem. The project consists on a set of activities such that, for some pairs, a start-to-start time dependency exists. The execution of each activity requires several skills. More than one resource of each skill may be required for processing an activity. A pull of multi-skilled resources is assumed. Costs are associated with resource usage and include fixed and variable costs. The former are incurred simply by using the resources; the latter depend on the final makespan of the project. For this problem, a mathematical programming modeling framework is proposed. The ‘natural’ model contains a non-linear objective function which, nonetheless, can be linearized at the expense of one additional set of continuous variables. The linearized model is enhanced using several sets of additional inequalities. The results of an extensive set of computational tests performed with the final model are reported. One major goal is to evaluate the possibility of using an off-the-shelf solver for tackling the problem. Another relevant goal is to understand the extent to which a cost-oriented objective influences the solutions obtained. Accordingly, we compare the solutions obtained using such objective with the solutions obtained using the traditional makespan minimization objective, often considered in project scheduling problems.

In this paper, we study a variant of the resource-constrained project scheduling problem in which resources are flexible,
i.e., each resource has several skills. Each activity in the project may need several resources for each required skill. We
present a mixed-integer linear programming formulation for this problem. Several sets of additional inequalities are also
proposed. Due to the fact that some of the above-mentioned inequalities require a valid upper bound to the problem, a heuristic
procedure is proposed. Computational experience is reported based on randomly generated data, showing that for instances of
reasonable size the proposed model enlarged with the additional inequalities can be solved efficiently.

A zero-one (0-1) linear programming formulation of multiproject and job-shop scheduling problems is presented that is more general and computationally tractable than other known formulations. It can accommodate a wide range of real-world situations including multiple resource constraints, due dates, job splitting, resource, substitutability, and concurrency and nonconcurrency of job performance requirements. Three possible objective functions are discussed; minimizing total throughput time for all projects: minimizing the time by which all projects are completed (i.e., minimizing makespan); and minimizing total lateness or lateness penalty for all projects.

This paper surveys single-project, single-objective, deterministic project scheduling problems in which activities can be processed using a finite or infinite (and uncountable) number of modes concerning resources of various categories and types. The survey is based on a unified framework of a project scheduling model including resources, activities, objectives, and schedules. Most important models and solution approaches across the class of problems are characterized, and directions for future research are pointed out.

In this paper we make a comparative study of several mixed integer linear programming (MILP) formulations for resource-constrained project scheduling problems (RCPSPs).First, we present three discrete and continuous time MILP formulations issued from the literature.Second, instead of relying on the traditional discretization of the time horizon, we propose MILP formulations for the RCPSP based on the concept of event: the Start/End formulation and the On/Off formulation. These formulations present the advantage of involving fewer variables than the formulations indexed by time. Because the variables of this type of formulations are not function of the time horizon, we have a better capacity to deal with instances of very large scheduling horizon.Finally, we illustrate our contribution with a series of tests on various types of instance with the MILP formulations issued from the literature, together with our new formulations. We also compare our results with a recent RCPSP-specific exact method. We show that, in terms of exact solving, no MILP formulation class dominates the other ones and that a state-of-the art specialized (non-MILP) method for the RCPSP is even outperformed by MILP on a set of hard instances. Furthermore, on another set of hard “highly cumulative” RCPSP instances with a high processing time range, our On/Off formulation outperforms all the others MILP formulations and obtains results close to the ones of the specialized method.

A flow network model is presented for the static resource-constrained project scheduling problem. Static and dynamic scheduling methods, based on a new polynomial insertion algorithm taking advantage on the flow structure, are proposed. The performed computational experiments on some state-of-the-art problem instances show the potential of this approach.

The resource-constrained project scheduling problem (RCPSP) consists of activities that must be scheduled subject to precedence and resource constraints such that the makespan is minimized. It has become a well-known standard problem in the context of project scheduling which has attracted numerous researchers who developed both exact and heuristic scheduling procedures. However, it is a rather basic model with assumptions that are too restrictive for many practical applications. Consequently, various extensions of the basic RCPSP have been developed. This paper gives an overview over these extensions. The extensions are classified according to the structure of the RCPSP. We summarize generalizations of the activity concept, of the precedence relations and of the resource constraints. Alternative objectives and approaches for scheduling multiple projects are discussed as well. In addition to popular variants and extensions such as multiple modes, minimal and maximal time lags, and net present value-based objectives, the paper also provides a survey of many less known concepts.

We study an assignment type resource-con- strained project scheduling problem with resources being multi-skilled personnel to minimize the total staffing costs. We develop a hybrid Benders decomposition (HBD) algo- rithm that combines the complimentary strengths of both mixed-integer linear programming (MILP) and constraint programming (CP) to solve this NP-hard optimization prob- lem. An effective cut-generating scheme based on temporal analysis in project scheduling is devised for resolving re- source conflicts. The computational study shows that our hybrid MILP/CP algorithm is both effective and efficient compared to the pure MILP or CP method alone.

The Project scheduling with resource constraints can be formulated as follows: given a graph G with node set N, a set H of directed arcs corresponding to precedence relations, and a set H′ of disjunctive arcs reflecting the resource incompatibilities, find among the subsets of H′ satisfying the resource constraints the set S that minimizes the longest path in graph (N, H ∪ S). We define the project scheduling polyhedron Qs as the convex hull of the feasible solutions. We investigate several classes of inequalities with respect to their facet-defining properties for the associated polyhedron. The dimension of Qs is calculated and several inequalities are shown to define facets. For the inequalities that do not define facets, we derive some general lifting theorems and apply them to obtain in general stronger inequalities and facets in some special cases.

Multi project scheduling with limited resources: a zero-one programming approach

- A B Pritsker
- L J Watters
- P M Wolfe

Pritsker, A.B., Watters, L.J., Wolfe, P.M., 1969. Multi project scheduling with limited resources: a zero-one programming
approach. Management Science 16, 93-108.