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1 Oen Archive TOULOUSE Archive Ouverte (OATAO) OATAO is an oen access reository that collects the work of Toulouse researchers and makes it freely available over the web where ossible. This is an author-deosited version ublished in: htt://oatao.univ-toulouse.fr/ Erints ID: To link to this article: DOI: /j.rocs URL: htt://dx.doi.org/ /j.rocs To cite this version: Cherkaoui, Kaouthar and Pellerin, Robert and Batiste, Pierre and Haït, Alain A Time Driven RCCP Model with Two Levels of Planning and a Reactive Planning Aroach for Tactical Project Planning. (2015) Procedia Comuter Science, vol ISSN Any corresondence concerning this service should be sent to the reository administrator: staff-oatao@listes-diff.in-toulouse.fr

2 Available online at ScienceDirect Procedia Comuter Science 64 (2015 ) Conference on ENTERrise Information Systems / International Conference on Project MANagement / Conference on Health and Social Care Information Systems and Technologies, CENTERIS / ProjMAN / HCist 2015 October 7-9, 2015 A time driven RCCP model with two levels of lanning and a reactive lanning aroach for tactical roject lanning Kaouthar Cherkaoui a, Robert Pellerin a, Pierre Batiste a, Alain Haït b * a École Polytechnique de Montréal, C.P. 6079, Succ. Centre-ville, Montreal, H3C 3A7, Canada b University of Toulouse, Institut Suérieur de l Aéronautique et de l Esace, 10 av. Edouard Belin, Toulouse, France Abstract This aer rooses a mixed-integer linear rogramming (MILP) time driven RCCP model that handles different lanning levels by varying the length of the time eriods. The model is based on the continuous time reresentation of work ackages start and end events and the discrete time reresentation of resource constraints. The model considers shorter eriod lengths at the beginning of the lanning horizon to roduce a detailed lan, while further eriods cover a larger san. The roosed model is comared to a RCCP model with fixed eriod lengths. Both models were tested on medium-sized roject instances. The comutational results roved that the roosed model erforms considerably better in terms of comutational times. A reactive lanning aroach is also roosed to analyze the erformance and robustness of both models. The results analysis showed that reducing resource caacities of aggregated eriods by a ercentage of 20% allowed us to obtain comarable results for the two models The The Authors. Published Published by by Elsevier Elsevier B.V. B.V. This is an oen access article under the CC BY-NC-ND license (htt://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under resonsibility of SciKA - Association for Promotion and Dissemination of Scientific Knowledge. Peer-review under resonsibility of SciKA - Association for Promotion and Dissemination of Scientific Knowledge Keywords: Rough-Cut Caacity Planning; reactive roject lanning; aggregation. 1. Introduction Hierarchical roject lanning is used in ractice to break down roject lanning into more manageable arts. Indeed, in most roject environments, it is unrealistic to lan the whole roject at only one level since the collection * Corresonding author. Tel.: address: alain.hait@isae-suaero.fr The Authors. Published by Elsevier B.V. This is an oen access article under the CC BY-NC-ND license (htt://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under resonsibility of SciKA - Association for Promotion and Dissemination of Scientific Knowledge doi: /j.rocs

3 258 Kaouthar Cherkaoui et al. / Procedia Comuter Science 64 ( 2015 ) of the required information is very time consuming and subject to many uncertainties [1]. Using aggregate lanning techniques is more aroriate for the long and medium-term. Rough Cut Caacity Planning (RCCP) is often suggested for tactical lanning, as roosed by De Boer [1] and adoted by many other authors [2-5]. In such models, work ackages (WPs) that consist of arts of the roject to be later divided into more detailed activities are lanned on a horizon discretized into time buckets. Rough estimates are made concerning resource requirements and caacities that can usually be extended by external hours. The resource consumtion rate of WPs may change from one eriod to another. As a result, RCCP serves as a tool to define realistic constraints for the underlying detailed scheduling roblem [2], such as due dates and resource caacity levels. However, all RCCP models roosed in the literature divide the horizon into eriods of equal lengths, assuming that the accuracy of data is the same for all eriods. This is unfortunately unrealistic for large rojects that last for a long eriod. Working with more aggregate data for the remote eriods is more aroriate as caacity estimates of larger eriods tend to be more accurate than estimates of small eriods. Furthermore, the majority of RCCP models are intended for a deterministic environment. Projects are, however, subject to many uncertainties, esecially on the tactical level where only reliminary estimates have been established for a large art of the roject. Recognizing these limitations, we roose a RCCP model that integrates different levels of lanning by dividing the lanning horizon into eriods of variable lengths. The eriods become larger as we advance in time. We also roose a reactive lanning aroach to test the robustness and erformance of our model in the case of uncertain data. Our model is comared to a RCCP model with fixed eriod lengths as suggested by Haït and Baydoun [6]. The reminder of the aer is organized as follows. Section 2 rovides a literature review of existing RCCP models and reactive scheduling aroaches. Section 3 rooses a time driven RCCP model handling different lanning levels. Section 4 then reorts our comutational results. Section 5 resents a reactive lanning aroach for the time driven RCCP and resents comarative results obtained by alying the reactive rocedure. Finally, concluding remarks are given in section Literature review The RCCP roblem is described by a number of WPs with given recedence constraints and a number of indeendent resource tyes. Each WP has a release date, a required number of units from each resource tye and a minimal duration resulting from the maximum amount of resources that can be allocated in a eriod. The time-horizon is discretized into eriods, for which each resource has an internal caacity. The roblem consists of determining the eriods in which each WP is erformed and the fraction of the WP that is erformed. WPs may start and/or end during a eriod [2]. Two kinds of RCCP roblems can be distinguished in the literature: resource driven and time driven RCCP. In the resource driven RCCP, the availabilities of the resources are constrained to the internal caacities and the aim is to meet due dates as much as ossible or to minimize the roject duration. In time driven RCCP, a time limit on the roject is given, caacity can be exanded by external hours and the aim is to minimize the use of external caacity. Among existing RCCP models, Hans [2] rooses an exact solution aroach based on the concet of the roject lan that indicates for each WP the eriods in which it is allowed to be erformed. He imlicitly deals with recedence constraints by forbidding work ackages with a recedence relation to be rocessed in the same eriod or by allowing a WP to start during the eriod where its redecessor ends without ensuring their succession. Haït and Baydoun [6] roose a MILP RCCP model that combines continuous variables reresenting a WP s start and end, and discrete variables for a eriodic consideration of the resource constraints. Their formulation allows WPs linked by recedence constraints to be executed within the same eriod while maintaining their succession. Some Resource-Constrained Project scheduling roblem (RCPSP) models roosed in the literature for the oerational level are also suitable for the RCCP roblem, as they do not consider redefined activity durations and ermit a variable resource usage between eriods. These models are denominated RCPSVP (RCPSP with variable intensity activities) like the MILP model roosed by Kis [7] or FRCPSP (RCPSP with flexible resource rofiles) like the MILP models roosed by Naber and Kolisch [8]. Several heuristic methods have also been develoed to solve the RCCP roblem, including constructive heuristics and linear based heuristics [1, 9, 10]. However, all the RCCP models roosed in the literature assume that the detail level of data and the recision of lanning is the same for the entire roject horizon, which is unrealistic for large rojects. Furthermore, the majority

4 Kaouthar Cherkaoui et al. / Procedia Comuter Science 64 ( 2015 ) of RCCP aroaches are intended for a deterministic environment, although many uncertainties characterize the tactical lanning level. There are two main aroaches roosed to coe with uncertainty in a scheduling environment: roactive and reactive scheduling. Proactive scheduling consists of constructing a redictive schedule that accounts for statistical knowledge of uncertainty [11]. Reactive scheduling consists of revising or re-otimizing a schedule when an unexected event occurs [11]. Some authors roose roactive RCCP models [10, 12]. Many reactive scheduling aroaches were also roosed for the oerational level, but they are not suitable for the tactical lanning level. Indeed, the majority of the reactive scheduling methods are based on scheduling olicies. A scheduling olicy is a decision rocess that defines which set of activities begin at certain decision oints corresonding to the comletion of activities [13]. However, the rescheduling decisions in the tactical level should be made on a eriodic basis since resource caacities are global over the eriods and resource rates of WPs can change from one eriod to another. Based on these observations, in the next section we roose a time driven RCCP model that integrates different levels of lanning by dividing the horizon into eriods of variable lengths. The eriods are larger as we advance in time and the detail level of data is therefore adated to the length of the eriods. In Section 5, we also roose a reactive lanning rocedure adated to the tactical lanning level. 3. RCCP model with variable eriod lengths This section resents a time-driven RCCP model with variable eriod lengths based on the RCCP model roosed by Haït and Baydoun [6]. The notation is given in Table 1 and Table 2. The objective of the model (1) is to minimize the cost of non-regular caacities used over all resource grous and eriods. Minimize rr P yext The main constraints of the model are the following: r tsi D 1, k 1 k zsi i I P ts i D 1, 1 k H D zs i I P 1 k i i k k zs zs i I P i1, di D zsi zfi 1 i I, P di D zsi 1 zfi i I, P d i tf i D 1 k D zs H i 1 1 k i i I, P d i D 1 k 1 zs k i 1 i i i I, P d P i tfi tsi ii (9) ts RD i I i i (1) (2) (3) (4) (5) (6) (7) (8) (10) ts tf i I, j Pred i j i x Q Q d ii, P max i rr ri i i x i 1 i I P yint yext x Q r R, P r r ii i ri (11) (12) (13) (14)

5 260 Kaouthar Cherkaoui et al. / Procedia Comuter Science 64 ( 2015 ) yint K r R, P r r (15) Table 1. Nomenclature: sets and arameters. Sets / Parameter Descrition P Set of time eriods ( P) I Set of work ackages (i I) R Set of resource grous (r R) D Duration of eriod (in time units) H Time horizon: H P D RD i max Q i Q ri Pred i K r Ready date of WP i Maximum workload that can be assigned to i during one time unit Required workload of WP i on resource grou r Set of redecessors of WP i Available caacity of resource grou r during eriod Table 2. Nomenclature: variables. Variable Descrition ts i, tf i Starting time and finishing time of WP i zs i zf i Binary variable that equals 1 if ts i is in eriod or before Binary variable that equals 1 if tf i is in eriod or before d i Duration of WP i within the eriod (0 d i D ) x i Intensity of WP i in eriod (0 x i 1) yint r internal load of resource r in eriod yext r external load of resource r in eriod Constraints (2) to (4) ensure the link between continuous and binary variables in order to situate WP starting times. Variable zsi is equal to 1 if WP i begins before or during eriod, 0 otherwise. The same mechanism alies for WP ending times tfi and binary variables zfi. Constraints (5) to (9) give the relationshi between WP durations over eriods di from one side and binary variables zsi and zfi and starting and finishing time events tsi and tfi on the other side. Constraints (10) ensure that WPs cannot start before their release dates. Constraints (11) ensure that recedence relations are resected. Constraints (12) ensure that the workload assigned to a WP during a eriod does not exceed the maximum allowed workload for this WP during its execution duration within this eriod. Constraints (13) ensure that the total required workload of WP i is fulfilled. Constraints (14) and (15), together with the objective function (1), define the internal and external loads. The external load yextr reresents the ortion of load over the internal caacity limit. 4. Comutational results In this section, we comare our model (denoted by VP) to the RCCP model by Haït and Baydoun [6] (denoted by FP). In model VP, we consider that the first four eriods have a duration equal to one time unit, the fifth eriod has a duration equal to the rest of the Euclidean division of the time horizon by four (if this rest is different from zero) and the remaining eriods have a duration equal to four time units each. The resource caacities for large eriods are set equal to the sum of the caacities of the corresonding small eriods in model FP. All the eriods in model FP have a duration equal to one time unit. To comare the models, we used a subset of the instances generated by De Boer [1] that are commonly used to test RCCP models. The instances are characterized by three arameters: the number of WPs, the number of resource tyes and the average slack. The latter arameter, being based on due dates of WPs, is not considered in our tests since we make the assumtion that WPs do not have redefined due dates at this stage of

6 Kaouthar Cherkaoui et al. / Procedia Comuter Science 64 ( 2015 ) lanning. The subset considered is comosed of 50 instances, each having 20 WPs and 10 resource tyes. We ran both models on a PC with four 2.4 GHz Intel(R) Core(TM) i7 Processors under Windows 8, using IBM ILOG CPLEX Otimization Studio 12.6, and we terminated the search after 5000 seconds. Table 3 shows for each model the average CPU times and the average CPU times to otimality. The latter only considers the instances that are solved to otimality for both models. We observe that model VP erforms considerably better than model FP in terms of comutational times. Table 4 shows the average ga, standard deviation, minimum and maximum ga in ercentage between the objective value obtained with model VP and the objective value obtained with model FP. The ga is defined as follows: ga = (cost VP - cost FP)/cost FP. Variables cost VP and cost FP refer resectively to the roject cost (objective value) obtained with model VP and model FP. We observe that the objective values obtained with model VP are very otimistic in comarison to those of model FP. This observation can be justified by the higher flexibility in allocating resources in model VP, since internal caacity limits are given for larger eriods. However, once faced with the real disaggregated caacity limits during roject execution, the cost of the roject could increase considerably (if we suose that the roject time limit cannot be extended or can be extended at the rice of high enalties). Assuming that caacity limits of the detailed eriods used in model FP are accurate, the objective values found by model FP are robably more realistic than those of model VP. To find more comarable results, we reduced the caacity limits of aggregated eriods used to resolve model VP gradually until we found satisfying results. The best results in terms of comarability to the objective values found with model FP were found when reducing caacity limits by 20% for model VP. We denote by VP-20% the model VP after reducing caacity limits of aggregated eriods by 20%. Table 5 shows that the objective values obtained with model VP-20% are very close to those of model FP. In order to test the erformance and robustness of models VP, VP-20% and FP in case of uncertain data, in the next section we roose a reactive lanning rocedure that will be alied to the three models. Table 3. Comutational times (in seconds). Avg. CPU time* Avg. CPU time to otimality VP FP * The maximum time limit of 5000 seconds is used for instances that could not reach otimality Table 4. Comarison of the objective value obtained with model VP vs. model FP. Average ga -31,69% Standard deviation 9,34% Minimum ga -66,06% Maximum ga -18,51% Table 5. Comarison of the objective value obtained with model VP-20% vs. model FP. Average ga 0,72% Standard deviation 8,82% Minimum ga -21,35% Maximum ga 19,24% 5. A reactive lanning rocedure for the time driven RCCP roblem 5.1. Descrition of the reactive lanning aroach The roosed reactive lanning aroach reacts to the uncertainties of WP work contents. It consists of eriodically regenerating a new schedule at the end of each one-time unit eriod while simulating the roject execution between each two consecutive rescheduling oints. The rocedure is illustrated in Fig. 1.

7 262 Kaouthar Cherkaoui et al. / Procedia Comuter Science 64 ( 2015 ) A new schedule is generated after each one-time unit eriod, using the same exact aroach alied to generate the baseline schedule but on a modified roject network. The baseline schedule corresonds to the schedule obtained for the whole roject with the estimated data. The roject network is modified as follows: WPs that are already comleted are omitted from the roject network and WPs that have started but are not yet finished are ket in the network with the real reminder of their work content. The WPs not yet started are considered with their estimated work contents as in the baseline schedule. The time horizon (H) is decreased by one time unit and the duration vector is udated for model VP and VP-20% using the same mechanism alied for the original roject instances. Resource caacities are also adated in consequence for model VP. Resource caacities of the aggregated eriods for model VP-20% corresond to the reduced caacities of aggregated eriods in model VP by a ercentage of 20%. Between each two consecutive rescheduling oints, the roject execution is simulated by allocating resources to the active WPs on the simulated eriod based on the u-to-date information about their work contents (ste 2 in Fig. 1). The available resources for the simulated eriod corresond to the internal caacities and the external resources redicted to be used according to the last generated schedule (the schedule generated at t = -1). The WPs that are in rogress at the beginning of the simulated eriod are executed in this eriod as decided by the last generated schedule, since their lanning was based on the real reminder of their work content. A riority list is then generated for the WPs lanned to start during eriod (according to the last generated schedule) by ordering them in increasing order of their scheduled start time. These WPs are selected and executed on eriod based on their order in the riority list. The assigned workload to a WP in the riority list is roortional to its duration within the simulated eriod. For instance, let us suose that a WP i was lanned to start at t = -1 and finish at t = -1/4 with an estimated work content of 30 men.eriod. If its earliest start time corresonds to t = -1 and its real work content equals 50 men.eriod, the assigned workload to the WP i on the eriod is 40 men.eriod, and it will be still in rogress at the end of eriod. Otherwise, if the real work content of the WP i equals 20 men.eriod, it will end at t = -1/2 with a total assignment of its required work content. These affectations are subject to the condition of availability of resources. Indeed, if the requested resources are not comletely available, we only allocate to the WP i the available resources while maintaining the roortionality of resources. If at least one requested resource is not available, the WP is then delayed to a future eriod. Before moving to the next WP in the riority list, we udate if necessary the earliest starting times Ste 1: Aly an exact resolution aroach to generate a new schedule for t [-1 H] Ste 2: Simulate the roject execution on the interval [-1 ] = +1 Ste 3 Udate the data of the new roject network Unscheduled WPs? yes No Save and analyze results Fig. 1. Concetual flow chart of the reactive lanning aroach. of WP i successors that are elements of the riority list in order to revent them from beginning before the end of WP i. In order to maintain stability, we aly the railway scheduling constraint commonly used in reactive scheduling rocedures [13, 14]. It consists of forbidding WPs to start earlier than their scheduled starting times (according to the last generated schedule).

8 Kaouthar Cherkaoui et al. / Procedia Comuter Science 64 ( 2015 ) Stes 1 to 3 of the reactive lanning rocedure described in Fig. 1 are reeated until all WPs are scheduled. The simulated cost of the roject corresonds to the sum of the real amounts of external resources used on the simulated eriods Comutational results of the reactive lanning aroach The reactive lanning rocedure was coded in Matlab R2013a interfaced with CPLEX Otimization Studio by using the CPLEX class API in Matlab. The tests were conducted on the same subset of instances described in section 4. A time limit of 2000 seconds has been set for the CPLEX resolution of the models at each decision oint. For each model (VP, VP-20% and FP) and for each roject network instance, 50 executions are simulated by drawing triangular-distributed WP work contents. The triangular distribution of a WP work content is characterized by a minimum value equal to half the exected value of work content, a maximum value equal to 1.5 times the exected work content and a most likely value equal to the exected work content. The exected work content refers to the estimated value of the work content used to generate the baseline schedule. The three models were tested on the same simulated scenarios for each roject instance. A small change has been made to the instances by increasing the horizon by 20% since the original horizon may be insufficient to rogram WPs with higher work content than redicted. The objective function of the three models was also adated in consequence by adding to the cost of the roject a term that highly enalizes the tardiness of the roject over its original horizon. The tests for the reactive lanning rocedure were erformed on a comutational grid consisting of 26 PCs with two 3.07 GHz Intel(R) Xeon(R) X5675 Processors using Linux. Three measures are used for the erformance and robustness evaluation of the models. The first measure denoted by Var X refers to the variation in ercentage between the simulated cost (cost' X) and estimated cost (cost X) of the roject according to model X, where X is either FP, VP or VP-20%: Var X = (cost' X - cost X)/ cost X. The smaller this measure is, the more robust the model. The second measure denoted by VarInit X corresonds to the difference in ercentage between the estimated cost obtained with model X (VP or VP-20%) vs. the one obtained with model FP: VarInit X = (cost X - cost FP)/cost FP. Finally, the third measure denoted by VarFinal X corresonds to the difference in ercentage between the simulated cost obtained with model X (VP or VP-20%) vs. the one obtained with model FP: VarFinal X = (cost' X - cost' FP)/cost' FP. Table 6 shows that the robustness of the model in terms of the variability between the simulated and estimated cost of the roject is considerably imroved in model VP-20% comared to model VP. Model FP has the best results in terms of robustness. The variation in the cost of the roject is, however, accetable in model VP-20%, contrary to model VP. The variations between the estimated costs by models VP and VP-20% and the estimated cost by model FP confirm our observations made in section 4. Finally, the simulated costs obtained with models VP and VP-20% are comarable to the costs simulated with model FP. Fig. 2 illustrates three curves. The dark curves reresent the variation between the simulated and estimated cost obtained with model VP-20% and model FP. The curve with the lighter color reresents the variation between the estimated cost obtained with model VP-20% vs. model FP. Instances are classified based on the increasing values of the last measure: VarInit VP-20%. We observe that model FP is considerably more robust than model VP-20% for instances where the estimated cost by model VP-20% is very otimistic in comarison to the estimated cost by model FP. However, as the ga between the estimated costs by the two models becomes smaller, the robustness of the two models becomes comarable. When the ga between the estimated costs by the two models is inside the interval [6% 23%], model VP-20% becomes more robust than model FP. It would then be interesting to lower the caacity levels of aggregated eriods by more than 20% and to test the influence on the robustness of the model. Table 6. Performance and robustness measures. Var VarInit VarFinal VP Mean Value 64,6% -30,5% 4,6% Standard deviation 45,2% 9,9% 3,7% VP-20% Mean Value 10,0% 1,3% 4,8% Standard deviation 11,7% 9,5% 3,2%

9 264 Kaouthar Cherkaoui et al. / Procedia Comuter Science 64 ( 2015 ) FP Mean Value 5,4% - - Standard deviation 3,7% Conclusion Fig. 2. Comarison of the robustness of models FP and VP-20%. In this aer, we roose a time driven RCCP model intended for tactical lanning of large rojects. The model integrates different levels of lanning by varying the length of the eriods. The eriods become larger as we advance in time and the level of data and lanning is therefore adated to the length of the eriods. Our model is comared to a RCCP model handling only one level of lanning. Comutational results show that our model erforms much better in terms of comutational times since distant eriods are more aggregated. By lowering the caacity limits of aggregated eriods by a ercentage of 20%, we found comarable erformance and robustness results for the two models. Tests should, however, be conducted on more roject instances to analyze whether the ercentage of reduction of caacities could be generalized, or if it is influenced by some instance arameters. References 1. De Boer, R., Resource-constrained multi-roject management, a hierarchical decision suort system. University Of Twente: The Netherlands; Hans, E.W., Resource loading by branch-and-rice techniques. University Of Twente: The Netherlands; Hans, E.W., et al., A hierarchical aroach to multi-roject lanning under uncertainty. Omega 2007;35(5): Leus, R., The generation of stable roject lans. Comlexity and exact algorithms. Katholic University Leuven: Belgium; Masmoudi, M., Tactical and oerational roject lanning under uncertainties: alication to helicoter maintenance, in Université de Toulouse. France; Haït, A. and G. Baydoun. A new event-based MILP model for the resource-constrained roject scheduling roblem with variable intensity activity (RCPSVP). in Proceedings of the IEEE International Conference on Industrial Engineering and Engineering Management. Hong Kong; Kis, T., A branch-and-cut algorithm for scheduling of rojects with variable-intensity activities. Mathematical rogramming 2005;103(3): Naber, A. and R. Kolisch, MIP Models for Resource-Constrained Project Scheduling with Flexible Resource Profiles. Euroean Journal of Oerational Research 2014;239(2): Gademann, N. and M. Schutten, Linear-rogramming-based heuristics for roject caacity lanning. Iie Transactions 2005;37(2): Wullink, G., Resource loading under uncertainty. University of Twente: The Netherlands; Van de Vonder, S., E. Demeulemeester, and W. Herroelen, A classification of redictive-reactive roject scheduling rocedures. Journal of Scheduling 2007;10(3): Wullink, G., et al., Scenario-based aroach for flexible resource loading under uncertainty. International Journal of Production Research 2004;42(24): Van de Vonder, S., et al., Heuristic rocedures for reactive roject scheduling. Comuters & Industrial Engineering 2007;52(1): Lambrechts, O., E. Demeulemeester, and W. Herroelen, Exact and subotimal reactive strategies for resource-constrained roject scheduling with uncertain resource availabilities, in DTEW-KBI_0702. Katholic University Leuven;