A survey of variants and extensions of the resource-constrained project scheduling problem <s> Resource Capacities Varying with Time <s> Project networks with time windows are generalizations of the well-known CPM and MPM networks that allow for the introduction of arbitrary minimal and maximal time lags between the starting and completion times of any pair of activities. <s> BIB001 </s> A survey of variants and extensions of the resource-constrained project scheduling problem <s> Resource Capacities Varying with Time <s> In this paper we present three new models for the resource constrained project scheduling problem, using two objective functions: minimizing the makespan and maximizing the net present value of the cash flows of the activities. In these models, compared to the models in the literature, we relax one of the most constraining assumptions about the integrality of the activity durations and the project time line. Furthermore resources are assumed to be partially-renewable. These models have multiple uses in Operations Management in that they can be used to solve resource constrained MRP problems. The models are found to be amenable to solution using the Optimization Subroutine Library (OSL). Extensive computational experiments with OSL on a total of 1400 problems with different parameter settings are reported. Detailed analysis is done to determine the factors that affect the computational efficiency of the code and the models. In general, results demonstrate that OSL is a powerful tool for solving the models with either objective function. Optimal solutions are obtained with reasonable computational effort. A discussion of the application of several special features of OSL is also given. <s> BIB002 </s> A survey of variants and extensions of the resource-constrained project scheduling problem <s> Resource Capacities Varying with Time <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> BIB003 </s> A survey of variants and extensions of the resource-constrained project scheduling problem <s> Resource Capacities Varying with Time <s> The paper deals with batch scheduling problems in process industries where final products arise from several successive chemical or physical transformations of raw materials using multi–purpose equipment. In batch production mode, the total requirements of intermediate and final products are partitioned into batches. The production start of a batch at a given level requires the availability of all input products. We consider the problem of scheduling the production of given batches such that the makespan is minimized. Constraints like minimum and maximum time lags between successive production levels, sequence–dependent facility setup times, finite intermediate storages, production breaks, and time–varying manpower contribute to the complexity of this problem. We propose a new solution approach using models and methods of resource–constrained project scheduling, which (approximately) solves problems of industrial size within a reasonable amount of time. <s> BIB004 </s> A survey of variants and extensions of the resource-constrained project scheduling problem <s> Resource Capacities Varying with Time <s> Crew or staff scheduling is a complex combinatorial problem. It consists of assigning tasks to persons which have to perform these task and to schedule the performance of the tasks. Railway and airline crew scheduling problems, audit scheduling problems, school and university course scheduling and staff scheduling in hospitals are typical examples of such combinatorial problems. <s> BIB005
Up to this point, the availability of the renewable resources has been assumed to be constant over time, i.e., the capacity is the same in each period. This assumption might be too strict in some cases such as changing availability of workers due to vacation or varying equipment capacities due to maintenance. To capture resource availabilities varying with time, we denote the capacity of renewable resource k in period t as R kt . Time-dependent resource capacities have been discussed by, e.g., Bomsdorf BIB004 . Hartmann  uses time-dependent capacities to capture the availability of researchers and laboratory equipment in a medical research project. Resource capacities varying with time are sometimes discussed together with preemptive scheduling, see Section 2.1 and the references cited therein. In the notation of Brucker et al. BIB003 , time-varying capacities of renewable resources can be captured using α = PS; Rm, ·, · which indicates that we have m renewable resources with a time-dependent capacity and an arbitrary request. Brucker and Knust BIB005 consider so-called disjunctive resources which have time-dependent capacities of up to 1, that is, R kt ∈ {0, 1}. Observe that these resources are a time-dependent version of the dedicated resources described in Section 4.5. Icmeli and Rom BIB002 deal with time-dependent capacities in a problem with a continuous timeline where changes in the resource availability occur at certain points in time called milestones. Bartusch et al. BIB001 show that resource capacities varying with time can be transformed into constant capacities if minimal and maximal time lags are available. The constant capacity would be defined as the maximum of the time-dependent capacity over time, and for each time interval with a smaller capacity, an artificial activity is defined to reduce the capacity appropriately. Each artificial activity is fixed to the desired time interval using a minimal and a maximal time lag. Also note that the RCPSP with time-varying resource capacities is a special case of the RCPSP with partially renewable resources, since we can define a period subset for each period with an individual capacity.