A survey of variants and extensions of the resource-constrained project scheduling problem <s> Continuous Resources <s> Abstract 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. <s> BIB001 </s> A survey of variants and extensions of the resource-constrained project scheduling problem <s> Continuous Resources <s> Abstract. 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. <s> BIB002 </s> A survey of variants and extensions of the resource-constrained project scheduling problem <s> Continuous Resources <s> In this paper we study a resource constrained project scheduling problem in which the resource usage of each activity may vary over time proportionally to its varying intensity. We formalize the problem by means of a mixed integer-linear program, prove that feasible solution existence is NP-complete in the strong sense and propose a branch-and-cut algorithm for finding optimal solutions. To this end, we provide a complete description of the polytope of feasible intensity assignments to two variable-intensity activities connected by a precedence constraint along with a fast separation algorithm. A computational evaluation confirms the effectiveness of our method on various benchmark instances. <s> BIB003 </s> A survey of variants and extensions of the resource-constrained project scheduling problem <s> Continuous Resources <s> This paper presents a branch-and-cut based exact solution algorithm for scheduling of projects with variable intensity activities connected by feeding precedence constraints the objective being to minimize the violation of resource constraints. Feeding precedence constraints allow some overlap in the execution of the connected activities and capture the flow of material or information between them. New polyhedral results are obtained and computational results are summarized. <s> BIB004 </s> A survey of variants and extensions of the resource-constrained project scheduling problem <s> Continuous Resources <s> In this paper discrete-continuous project scheduling problems with discounted cash flows are considered. In discrete-continuous project scheduling activities require for their processing discrete and continuous resources. The processing rate of an activity depends on the amount of the continuous resource allotted to this activity at a time. A positive cash flow is associated with each activity. Two common payment models-lump-sum payment and payments at activities' completion times-are considered. The objective is the maximization of the net present value of all cash flows of the project. Some properties of optimal schedules are discussed, and the formulation of a mathematical programming problem for an optimal continuous resource allocation is presented. Applications of a local search metaheuristic-tabu search, as well as simple search methods-multi-start iterative improvement and random sampling are described. The algorithms are compared on the basis of a computational experiment, the results are analyzed and discussed. Some conclusions as well as directions for further research are given. <s> BIB005
In the problem settings discussed up to now, the considered resources were available in discrete quantities only, as in the cases of, e.g., manpower and machines. Wȩglarz et al.  generalized the concept of renewable resources by allowing continuously divisible resources. This is useful if the resources correspond to, e.g., energy or raw material like liquids. An overview is provided in Blazewicz et al. [22] . Considering the notation of Brucker et al. BIB001 , we suggest to use α = PS;Co to reflect the RCPSP with continuous resources. Further problems with continuous resources have been considered by Józefowska et al. BIB002 , Kis BIB003 BIB004 and Waligóra BIB005 . Wȩglarz  presents an extension which is essentially a continuous variant of the doubly constrained resource concept.