Applied Soft Computing 12 (2012) 82–90 Contents lists available at SciVerse ScienceDirect Applied Soft Computing j ourna l ho me p age: www.elsevier.com/l ocate/asoc Multi-mode renewable resource-constrained allocation in PERT networks Siamak Baradaran a , S.M.T. Fatemi Ghomi b,∗ , M. Ranjbar c , S.S. Hashemin d a Department of Industrial Engineering, Islamic Azad University, Damavand Branch, Damavand, Iran b Department of Industrial Engineering, Amirkabir University of Technology, 424 Hafez Avenue, 1591634311 Tehran, Iran c Department of Industrial Engineering, Faculty of Engineering, University of Mashad, Mashad, Iran d Department of Industrial Engineering, Islamic Azad University, Ardabil Branch, Ardabil, Iran a r t i c l e i n f o Article history: Received 19 January 2010 Received in revised form 23 February 2011 Accepted 18 September 2011 Available online 24 September 2011 Keywords: PERT networks Multi-mode resource-constrained allocation Renewable resource Scatter search Path relinking Genetic algorithm a b s t r a c t This paper presents a hybrid metaheuristic algorithm (HMA) for Multi-Mode Resource-Constrained Project Scheduling Problem (MRCPSP) in PERT networks. A PERT-type project, where activities require resources of various types with random duration, is considered. Each activity can be accomplished in one of several execution modes and each execution mode represents an alternative combination of resource requirements of the activity and its duration. The problem is to minimize the regular criterion namely project’s makespan by obtaining an optimal schedule and also the amount of different resources assigned to each activity. The resource project scheduling model is strongly NP-hard, therefore a metaheuristic algorithm is suggested namely HMA. In order to validate the performance of new hybrid metaheuristic algorithm, solutions are compared with optimal solutions for small networks. Also the efﬁciency of the proposed algorithm, for real world problems, in terms of solution quality and CPU time, is compared to one of the well-known metaheuristic algorithms, namely Genetic Algorithm of Hartmann (GAH). The computational results reveal that the proposed method provides appropriate results for small networks and real world problems. © 2011 Elsevier B.V. All rights reserved. 1. Introduction A number of recent papers present various algorithms for the Resource-Constrained Project Scheduling Problem (RCPSP) but a smaller number of published algorithms consider stochastic net- work projects, e.g., PERT type projects. However, in practice, PERT projects are usually carried out with limited resources. Thus, the need for proper resource constrained project scheduling models for PERT network projects is very great [15]. The generalization of RCPSP where multi-mode is taken into account is called the Multi-Mode Resource-Constrained Project Scheduling Problem or brieﬂy MRCPSP and is denoted by MPS|prec|C max [18]. Table 1 pro- vides an overview of some of the most important related problem types encountered in the multi-mode projects scheduling liter- ature. The problems are classiﬁed with respect to whether they allow for generalized precedence relations (CPM precedence con- straints with zero time lag), multiple activity modes, (multiple) renewable resource types and (multiple) non-renewable resource types. ∗ Corresponding author. Tel.: +98 21 66413034; fax: +98 21 66413025. E-mail address: fatemi@aut.ac.ir (S.M.T. Fatemi Ghomi). The MRCPSP contains activities interrelated by ﬁnish-start-type precedence constraints with a time lag of zero, which require one or more constrained renewable resources in order to minimize the project duration. Each activity has speciﬁed work content, e.g. in terms of person-days, and can be performed in different modes, i.e. with different durations and resource requirements; as long as the required work content is met. A set of allowable execution modes can then be speciﬁed for each activity, each characterized by dura- tion and associated constant resource requirements such that their product should provide the activity’s speciﬁed work content. This paper considers the MRCPSP with PERT type network for which activity durations d j are independent continuous random variables with given distribution function. We present a stochas- tic model for the MRCPSP problem with zero completion-to-start time lags for pairs of activities. Their precedence relationships and resource requirements may vary depending on the chosen mode. All resources considered are renewable. The objective is to deter- mine a mode and a start time for each activity so that all constraints are obeyed and the makespan is minimized. The problem is well recognized as important and difﬁcult since Blazewicz et al. [3] have proven that the RCPSP is NP-hard in a strong sense as a generalization of the job shop scheduling problem. As a generalization of the RCPSP, the MRCPSP is strongly NP-hard [21]. Thus, many heuristics and meta-heuristic approaches have 1568-4946/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2011.09.007