A survey of variants and extensions of the resource-constrained project scheduling problem <s> Cumulative 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> Cumulative Resources <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> BIB002 </s> A survey of variants and extensions of the resource-constrained project scheduling problem <s> Cumulative Resources <s> Inventory constraints refer to so-called cumulative resources, which can store a single or several different products and have a prescribed minimum and maximum inventory, where the inventory is depleted and replenished over time. Some additional applications of cumulative resources, e.g. to investment projects, are also discussed in this paper. <s> BIB003 </s> A survey of variants and extensions of the resource-constrained project scheduling problem <s> Cumulative Resources <s> This paper deals with scheduling batch (i.e., discontinuous), continuous, and semicontinuous production in process industries (e.g., chemical, pharmaceutical, or metal casting industries) where intermediate storage facilities and renewable resources (processing units and manpower) of limited capacity have to be observed. First, different storage configurations typical of process industries are discussed. Second, a basic scheduling problem covering the three above production modes is presented. Third, (exact and truncated) branch-and-bound methods for the basic scheduling problem and the special case of batch scheduling are proposed and subjected to an experimental performance analysis. The solution approach presented is flexible and in principle simple, and it can (approximately) solve relatively large problem instances with sufficient accuracy. <s> BIB004
Cumulative resources have been introduced by Neumann and Schwindt BIB003 who incorporate them into an RCPSP with minimal and maximal time lags. If a project represents a production process, then an activity may require some intermediate product which is withdrawn from the storage area, or it may manufacture a product which is put into the storage area. A cumulative resource k is given by the capacity R k and the minimum inventory level (or safety stock) R k of the storage area. An activity j may produce r jk items (then we have r jk > 0) or withdraw r jk items (then we have r jk < 0); otherwise we have r jk = 0 (note that events rather than activities are used to describe production and withdrawal formally). Neumann and Schwindt BIB003 point out that cumulative resources subsume both renewable and nonrenewable resources. We propose to use α = PS;Cu to specify the RCPSP with cumulative resources in the notation of Brucker et al. BIB001 . Bartels and Zimmermann  employ cumulative resources within a multi-mode RCPSP with minimal and maximal time lags. They consider a project of engineering and testing activities in the automotive industry. A test vehicle is modeled as a cumulative resource since it can be built, used (which temporarily withdraws its capacity) and destroyed in a crash test (which withdraws its capacity until the project end). Neumann et al. BIB004 use cumulative resources in an RCPSP incorporating minimal and maximal time lags to capture a scheduling problem related to batch production in the process industry. Schwindt and Trautmann BIB002 further extend this setting.