A survey of variants and extensions of the resource-constrained project scheduling problem <s> Resource Requests 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 Requests Varying with Time <s> In this paper we consider a labor constrained scheduling problem (LCSP) which is a simplification of a practical problem arising in industry. Jobs are subject to precedence constraints and have specified processing times. Moreover, for each job the labor requirement varies as the job is processed. Given the amount of labor available in each period, the problem is to finish all the jobs as soon as possible, that is, to minimize makespan, subject to the precedence and labor constraints. Several Integer Programming (IP) formulations for this problem are discussed and valid inequalities for these different models are introduced. It turns out that a major drawback in using the IP approach is the weakness of the lower bound relaxations. However, we report computational experiments showing how the solution of the linear relaxation of the IP models can be used to provide good schedules. Solutions arising from these LP-based heuristics are considerably improved by local search procedures. We further exploit the capabilities of local search for LCSP by designing a Tabu Search algorithm. The computational experiments on a benchmark data set show that the Tabu algorithm generates the best known upper bounds for almost all these instances. We also show how IP can be used to provide reasonably good lower bounds for LCSP when the makespan is replaced by suitably modified objective functions. Finally some directions for further investigations which may turn IP techniques into a more interesting tool for solving such a problem are suggested. <s> BIB002 </s> A survey of variants and extensions of the resource-constrained project scheduling problem <s> Resource Requests Varying with Time <s> We consider in this paper a project scheduling problem where resources are employees and activity requirements are time-dependent. Furthermore, employees have different skills and legal constraints dictated by the French workforce legislation have to be respected. The project has to be scheduled and employees have to be assigned to activities, so that most of the constraints are respected and the maximal lateness is minimized. We first give a linear programming formulation of the problem. Then, greedy algorithms are proposed to give an approximate solution to the problem. These methods give an initial solution to a Tabu search algorithm. Finally, computational experiments are conducted and results are discussed. <s> BIB003
The activities in the standard RCPSP require constant amounts of renewable resources, that is, the per-period request for a resource does not change during the duration of an activity. This can be generalized by resource requests varying with time. To incorporate this formally, we denote with r jkt the request of activity j for renewable resource k in the t-th period of its processing time. Hartmann  describes a real-world medical research project with time-dependent resource requests. There, each activity requires certain laboratory equipment, but only in the last period of its duration. Cavalcante et al. BIB002 follow a similar problem setting. They define activities with timedependent resource requests for one renewable resource which is interpreted as labor. Drezet and Billaut BIB003 deal with time-dependent requests for resources which represent software developers. A minimum and a maximum resource request per period is given. Additional constraints reflect legal requirements such as the maximal working time per day. Bartusch et al. BIB001 point out that resource requests varying with time can be transformed into constant requests if maximal time lags (see Section 3.2) are available. An activity is split into parts with constant requests. Between two sub-activities i and i + 1 a precedence relation is introduced as well as a maximal time lag to avoid preemption of the original activity.