Optimal Resource-Constraint Project Scheduling with Overlapping Modes

Transcription

1 Optial Resource-Constraint Proect Scheduling with Overlapping Modes François Berthaut Lucas Grèze Robert Pellerin Nathalie Perrier Adnène Hai February 20 CIRRELT Bureaux de Montréal : Bureaux de Québec : Université de Montréal Université Laval C.P. 628, succ. Centre-ville 2325, de la Terrasse, bureau 2642 Montréal (Québec) Québec (Québec) Canada H3C 3J7 Canada GV 0A6 Téléphone : Téléphone : Télécopie : Télécopie :

2 Optial Resource-Constraint Proect Scheduling with Overlapping Modes François Berthaut,2, Lucas Grèze 2, Robert Pellerin,2,*, Nathalie Perrier,2, Adnène Hai,3 Interuniversity Research Centre on Enterprise Networks, Logistics and Transportation (CIRRELT) 2 Departent of Matheatics and Industrial Engineering, École Polytechnique de Montréal, P.O. Box 6079, Station Centre-ville, Montréal, Canada H3C 3A7 3 Departent of Operations and Decision Systes, Université Laval, 2325, de la Terrasse, Québec, Canada GV 0A6 Abstract. Overlapping activities is widely used to accelerate proect execution. Overlapping consist in executing in parallel two sequential activities by allowing a downstrea activity to start before the end of an upstrea activity based on preliinary inforation. In copanies, overlapping is exained in resource constraints scheduling without considering interaction between activities and rework caused by alteration of inforation exchanged until finalized inforation is available at the copletion of the upstrea activity. By contrast, the literature deals with overlapping of couple of activities without considering a whole proect with resource constraints. We here investigate the resource-constrained proect scheduling proble with different feasible odes of overlapping including associated rework. We forulate the optiization proble as an integer linear prograing proble. An exaple of a 30 activity proect is provided to illustrate the utility and efficiency of this odel. An optial solution is reached within reasonable coputation tie. Our results also highlight the closed interaction between resource constraints and overlapping odes and confirs the relevance of ointly consider the. Keywords. Activity overlapping, concurrent engineering, proect anageent, proect scheduling. Acknowledgeents. This work was supported by the Jarislowsky Foundation, the SNC- Lavalin fir and by the Natural Sciences and Engineering Council of Canada (NSERC). This support is gratefully acknowledged. Results and views expressed in this publication are the sole responsibility of the authors and do not necessarily reflect those of CIRRELT. Les résultats et opinions contenus dans cette publication ne reflètent pas nécessaireent la position du CIRRELT et n'engagent pas sa responsabilité. * Corresponding author: Robert.Pellerin@cirrelt.ca Dépôt légal Bibliothèque et Archives nationales du Québec Bibliothèque et Archives Canada, 20 Copyright Berthaut, Grèze, Pellerin, Perrier, Hai and CIRRELT, 20

3 Optial Resource-Constraint Proect Scheduling with Overlapping Modes François Berthaut Lucas Grèze Robert Pellerin Nathalie Perrier Adnène Hai February 20 CIRRELT Bureaux de Montréal : Bureaux de Québec : Université de Montréal Université Laval C.P. 628, succ. Centre-ville 2325, de la Terrasse, bureau 2642 Montréal (Québec) Québec (Québec) Canada H3C 3J7 Canada GV 0A6 Téléphone : Téléphone : Télécopie : Télécopie :

4 Optial Resource-Constraint Proect Scheduling with Overlapping Modes François Berthaut,2, Lucas Grèze 2, Robert Pellerin,2,*, Nathalie Perrier,2, Adnène Hai,3 Interuniversity Research Centre on Enterprise Networks, Logistics and Transportation (CIRRELT) 2 Departent of Matheatics and Industrial Engineering, École Polytechnique de Montréal, P.O. Box 6079, Station Centre-ville, Montréal, Canada H3C 3A7 3 Departent of Operations and Decision Systes, Université Laval, 2325, de la Terrasse, Québec, Canada GV 0A6 Abstract. Overlapping activities is widely used to accelerate proect execution. Overlapping consist in executing in parallel two sequential activities by allowing a downstrea activity to start before the end of an upstrea activity based on preliinary inforation. In copanies, overlapping is exained in resource constraints scheduling without considering interaction between activities and rework caused by alteration of inforation exchanged until finalized inforation is available at the copletion of the upstrea activity. By contrast, the literature deals with overlapping of couple of activities without considering a whole proect with resource constraints. We here investigate the resource-constrained proect scheduling proble with different feasible odes of overlapping including associated rework. We forulate the optiization proble as an integer linear prograing proble. An exaple of a 30 activity proect is provided to illustrate the utility and efficiency of this odel. An optial solution is reached within reasonable coputation tie. Our results also highlight the closed interaction between resource constraints and overlapping odes and confirs the relevance of ointly consider the. Keywords. Activity overlapping, concurrent engineering, proect anageent, proect scheduling. Acknowledgeents. This work was supported by the Jarislowsky Foundation, the SNC- Lavalin fir and by the Natural Sciences and Engineering Council of Canada (NSERC). This support is gratefully acknowledged. Results and views expressed in this publication are the sole responsibility of the authors and do not necessarily reflect those of CIRRELT. Les résultats et opinions contenus dans cette publication ne reflètent pas nécessaireent la position du CIRRELT et n'engagent pas sa responsabilité. * Corresponding author: Robert.Pellerin@cirrelt.ca Dépôt légal Bibliothèque et Archives nationales du Québec Bibliothèque et Archives Canada, 20 Copyright Berthaut, Grèze, Pellerin, Perrier, Hai and CIRRELT, 20

5 Optial Resource-Constraint Proect Scheduling with Overlapping Modes Introduction The RCPSP (Resource-Constrained Proect Scheduling Proble) has been addressed in nuerous papers. Various odels attept to iniize proect tie copletion while considering liited resources []. Hartann and Briskorn [2] have also presented an overview of different RCPSP extensions. Aong extensions addressed in the literature, different practices have been developed to reduce tie of proect execution in order to establish a baseline schedule or to odify it following proect delay during its execution through overlapping, crashing and substitution of activities [3]. In engineering proects, overlapping is considered as a core technique for saving developent tie [3][4] [5]. Overlapping consists in starting an activity before receiving all the final inforation required. This practice often causes future rework and odification as new inforation is gained in subsequent activities. As such, the total reduction of tie is the difference between overlapped tie and rework. Nevertheless, ost odels in the literature consider crashing as a strategy to reduce the proect akespan without taking into account possible rework. In practice, overlapping depends not only on dependency between activities but also on inforation exchange policy between upstrea and downstrea activity and progress evolution. Two groups of odels have been developed in the literature to analyze overlapping interactions. First, any authors consider only couples of activities and no resource constraints to establish the best trade-off between overlapping and rework. For instance, Krishnan et al. [7] developed a odel-based fraework to anage the overlapping of coupled activities. This odel introduced the concept of inforation evolution and downstrea sensitivity to describe interaction between both activities. Inforation evolution refers to the upstrea generated inforation useful for downstrea activities. Downstrea sensitivity refers to the ipact of a change in upstrea activity on the downstrea activity. The ore the ipact is significant, the ore the sensitivity is high. Relying on these concepts, Krishnan [8] defined different types of appropriated overlapping strategies: iterative, preeptive, distributive and divisive overlapping. In a siilar anner, Bogus et al. [9] identified appropriate strategies to efficiently ipleent overlapping in practice. Lin et al. [6] also iproved the overlapping odel by incorporating the downstrea progress evolution and deterined the optial overlap aount. These odels assue that overlapping paraeters can be derived fro historical data. Other approaches have considered whole proects instead of coupled of activities under the assuption that relation between overlapping aount and rework is preliinary known for overlappable activities. They ostly use design structure atrix (DSM) to represent dependencies, to iniize feedbacks, and to identify overlapping opportunities between activities. DSMs were introduced by Steward [0]. Aong other odels, Gerk and Qassi [3] developed an analytic proect acceleration linear odel via activity crashing, overlapping and substitution with resource constraints. Wang and Lin [] developed a stochastic overlapping process odel to assess schedule risks. Their siulation odel considers iterations and probabilities of rework. Iterations are ostly defined as interaction between design activities which lead to rework caused by feedbacks fro downstrea activities. However, their odel does not take into account resource constraints. Cho and Eppinger [2] also introduced a siulation odel with stochastic activity durations, overlapping, iterations, rework and considered resource constraints for soe activities. They showed that theses constraints can delay soe overlapped activities and delay the proect. All these papers assue a siple linear relationship between rework and overlapping aount with an upper and lower bound. It is also iportant to note that Lin et al. [6] stated that copanies deterine overlapping strategies on an ad hoc basis without always considering rework and interaction between activities. In suary, there are no RCPSP odel that yet considers a realistic relationship between rework and overlapping aount. The obective of this paper is to extend the classic RCPSP with a realistic overlapping odel in order to partially fill this gap. We here assue that the inforation flow is unidirectional fro upstrea to downstrea activities. Consequently, the rework caused by execution of CIRRELT-20-09

6 Optial Resource-Constraint Proect Scheduling with Overlapping Modes activities based on preliinary inforation is only assigned to the downstrea activities of the identified overlappable activities. Inforation exchange is assued to be costless and instantaneous. The ain difference with the aforeentioned overlapping odels is that overlapping is restricted to a set of feasible overlap aounts, instead of considering a continuous and linear relation between overlap aount and rework. We forulate this resource-constrained proect scheduling proble with overlapping odes as an integer linear prograing proble, which is closed to the classical ulti-ode RCPSP odel []. This odel allows finding an optial akespan in reasonable calculation tie. The reainder of the paper is organized as follow. Section 2 first describes the proble stateent and assuptions. An illustrative exaple and coputational results are then presented in section 3. The paper concludes with recoendations for future work in section 4. 2 Proble Stateent A proect is defined by a set of activities, S, including two fictitious activities 0 and n+, which correspond to the proect start and proect end, respectively, with zero processing tie. We denote by d the noinal processing tie of activity considering that all the final inforation required fro preceding activities are available at its start; in other word, if activity is processed without overlapping. All the sybols and their definitions used along this paper are presented in Table. Table : Sybols and definitions Sybol Definition S Set of activities n Nuber of non-duy activities E Set of teporal or precedence constraints i (i, ) Precedence constraint d Processing tie of activity A Set of couples of overlappable activities A Set of iediate predecessors of activity that are overlappable with activity P Set of iediate predecessors of activity that are not overlappable with activity Pred() Set of iediate predecessors of activity R Set of renewable resources R k Constant aount of available units of renewable resource k R k Per period usage of activity of renewable resource k n i Nuber of precedence odes of the couple (i,) β in Aount of overlap duration between activities i and in precedence ode n, expressed as a fraction of d μ in Expected aount of rework in the downstrea activity when activities i and are overlapped in precedence ode n Nuber of execution odes of activity α i Aount of overlap duration between activities i and in execution ode, expressed as a fraction of d r Expected aount of rework in activity in execution ode T Upper bound of the proect s akespan t = 0,..,T Periods EF Earliest possible finish tie of activity Latest possible finish tie of activity 2. Precedence constraints Frequently used proect-planning ethods provide graphic descriptions of task workflows in the for of the so-called activity-on-node or activity-on-arc networks. These networks depict the logical execution 2 CIRRELT-20-09

7 Optial Resource-Constraint Proect Scheduling with Overlapping Modes sequence of dependent (sequential) activities and independent (parallel or concurrent) activities. However, these tools fail to incorporate interdependent-type relation, activities iterations and to odel inforation flows between activities. The Design Structure Matrix (DSM) representation can handle these additional relations between activities with the broader concept of inforation sharing [4]. Inforation exchange between activities can occur at the beginning, the iddle or the end of an activity and includes both tangible and intangible types such as parts, part diensions, and bill of aterials, which constitute the outputs fro an upstrea activity and are required to begin the work of a downstrea activity. A DSM is a square atrix where rows and coluns represent activities. The DSM ais to represent the inforation flows for a given subset of activities and constitutes the first step in analyzing potential feedbacks. Indeed, feedback inforation exchanges fro downstrea to upstrea activities correspond to design odification requests due to inability to eet target design requireents or design flaws detected in downstrea stages []. Any feedback inforation exchange fro downstrea activities lead to odifications and reworks perfored by the upstrea activities to accoodate these changes and iterations between upstrea and downstrea activities can virtually occur to fix the probles identified. In order to iniize feedbacks, the DSM can be partitioned using block triangularization algorith to obtain a unidirectional sequence of inforation exchange [4]. As a last resort, activities can be aggregated or decoposed into lower-level activities to eliinate feedbacks. In this paper, we assue that such preliinary studies have been conducted to identify the nature of relations between activities and to deterine a feasible sequence of activities without any feedback fro downstrea activities. The proect is then only coposed of independent and dependent activities and the resulting inforation flow within the proect between activities is assued to be unidirectional fro upstrea to downstrea activities. The analysis of inforation exchanges between dependent couple activities enables to categorize the into non-overlappable and overlappable ones. The forer represents the case where a downstrea activity requires the final output inforation fro an upstrea activity to be executed or the copletion of the upstrea activity. The latter represents the case where a downstrea activity can begin with preliinary inforation and receives final update at the end of the upstrea activity. This relation provides the opportunity to overlap two activities so that a downstrea activity can start before an upstrea activity is finished. While the non-overlappable activities are connected with the classical finish-to-start precedence constraint, the overlappable ones are connected with a finish-to-start-plus-lead tie precedence constraint where the lead-tie accounts for the aount of overlap. In the reainder of the paper, we denote by A and P the sets of couples of overlappable and nonoverlappable activities, respectively. Siilarly for each activity, A and P represent the set of iediate predecessors that are overlappable and non-overlappable with activity, respectively. The set of precedence constraints in the proect and the set of iediate predecessors of activity are defined by: E = A U P Pred ( ) = A U P, S () 2.2 Model of the overlapping process Figure shows the overlapping process of two activities i and in A. The downstrea activity starts with preliinary inputs fro the downstrea activity i. The aount of overlap, α i, is expressed as a fraction of the downstrea activity s duration. As the upstrea activity proceeds, its inforation evolves to its final for and is released to the downstrea activity at its copletion. This approach iplies that the traditional pattern of exchange of finalized inforation at the end of the upstrea activity is altered to a ore frequent exchange of evolving inforation during the overlap period. However, additional rework is often necessary to accoodate the changes in the upstrea inforation in the downstrea developent. CIRRELT

8 Optial Resource-Constraint Proect Scheduling with Overlapping Modes The expected duration of this rework is denoted by r i. The total aount of tie required to execute both activities, D i, is expressed as follows: i i ( i ) ri D = d + d α + (2) If d d, the aount of overlap is usually bounded by the fraction d i /d in order to prevent the i downstrea activity to start before the upstrea activity. If overlapping was not applied, the total aount of tie required to execute both activities would siply be D = d + d. Depending on the nature of the activities, there ay exist a trade-off between tie gains fro overlapping and rework. i i d i i Preliinary inforation Final inforation α i. d r i d tie Fig.. Overlapping process of two activities The ain issue with the overlapping proble is to quantify the aount of rework as a function of the aount of overlap. A significant part of the literature in overlapping is dedicated to the deterination of the optial overlap aount for two activities without resource constraints. Indeed, the overlapping proble requires exploring the behavior and interaction of activities during their processes. Krishnan et al. [7] presented a pioneer paper in this field. They proposed a odel of dependency based on the upstrea inforation evolution which characterizes the refineent of inforation fro its preliinary for to a final value and the downstrea sensitivity which represents the duration of a downstrea iteration to incorporate upstrea changes. Loch and Terwiesch [5] adapted these concepts by considering the upstrea evolution as the rate of odifications in the upstrea and the downstrea sensitivity as the ipact of a odification on downstrea rework, and ointly analyzed overlapping and counication policies between two activities. Roeer et al. [6] introduced the concept of probability of rework as a function of the overlap duration which encopasses both the evolution and sensitivity odels proposed in [7]. While Krishnan et al. [7] and Loch and Terwiesch [5] addressed the proect s akespan iniization proble, Roeer et al. [6], Lin et al. [7] and Lin et al. [6] extended the odel of Loch and Tewiesch [5] to exaine the tie-cost tradeoffs in overlapping. Counication policies should be considered along overlapping if the inforation exchange between activities require non-negligible tie and cost [6]. We here assue that inforation exchange is costless and instantaneous. An iportant finding of the aforeentioned papers is that the duration of rework is a convex increasing function of the aount of overlap and that the tie to coplete the upstrea and downstrea activities is convex in the aount of overlap. The forer stateent is intuitive: if the aount of overlap increases, then the preliinary inforation at the downstrea activity s start will be ore unreliable and ore downstrea changes ust be incorporated. The latter stateent entails that there exists a unique aount of overlap such that the gain fro overlapping exceeds the loss due to rework in an optial anner. According to the upstrea evolution, the downstrea dependency, the learning effect and the activity durations, the additional rework ay not evolve faster than the overlap duration when overlapping increases, which indicates that the total aount of tie required to execute activities is onotonic 4 CIRRELT-20-09

9 Optial Resource-Constraint Proect Scheduling with Overlapping Modes increasing and that coplete overlapping is optial. The conditions for optiality of coplete overlapping are derived in [6][6][7] under different assuptions. 2.3 Precedence and overlapping odes In order to study the interaction between overlapping and resource constraints in the scheduling optiization proble with ultiple activities including several overlapping opportunities, the relation between rework and overlap aount is required for a range of overlap aounts for each couple of overlappable activities. Indeed, the optial overlap aounts for a resource-constraints proect coposed of several couples of overlappable activities are not necessarily set to the optial values found for each couple of activities [2][3][8]. In this paper, overlapping is assued to be defined for discrete values of overlap durations. First, this assuption is ore realistic considering that scheduling is perfored in practice on a period-by-period basis (i.e., hour, day, week): resource availabilities and allocations are estiated per period, while activity durations are discrete ultiples of one period []. Second, activity progress is easured in practice according to the copletion of internal ilestones which corresponds to iportant events, such as design criteria frozen, detailed design copleted, drawings finalized, or any activity deliverables. This preliinary inforation is issued at interediate points and used as input for a downstrea activity. Therefore, the start tie of an overlapped downstrea activity is restricted to a finite nuber of instants corresponding to upstrea activities ilestones which constitutes different feasible odes for the execution of overlapping activities. Each overlapping ode is characterized by an aount of overlap expressed as a fraction of the downstrea activity s duration and a rework duration. Overlapping odes can be generalized to precedence odes in order to describe all precedence relationships between activities. For each couple of precedence constraints i, there exists at least one precedence ode n i which corresponds to a basic finish-to start relation without overlapping. When ( i, ) A, there exist additional precedence odes associated with the different overlapping strategies. Thus, precedence odes can be expressed as follows: n i =, β i= 0, μ i= 0, S, i P (3) n i >, β i= 0, μ i= 0 and 0 < β in <, in> 0 n 2,ni, S, i A (4) where β in and μ in denote the aount of overlapped tie between activities i and and the expected aount of rework in the downstrea activity when activities i and are executed in precedence ode n=,.., n i. μ, [ ] When i and are overlappable, they can be either overlapped and executed in ode n, n=2,,n i, or sequentially perfored in ode n= without overlapping. As depicted in Figure 2, it is iportant to note that the precedence constraints on the finish tie of activities i and will defer depending on the overlapping ode: when not overlapped, the downstrea activity start tie is superior or equal to the upstrea activity finish tie, whereas the downstrea activity start tie is equal to the upstrea activity finish tie inus one of the feasible overlap duration in the case of overlapping. 2.4 Multiple overlapping and activity odes We assue that there is no restriction concerning the nuber of overlappable or non-overlappable predecessors. If an activity is overlapped by ultiple upstrea activities, feasible overlapping odes are assued to be copatible. Consider for exaple the case of a downstrea activity with two upstrea activities, denoted by i and i 2. If both couples (i, ) and (i 2, ) are overlapped, the aount of rework in downstrea activity is between CIRRELT

10 Optial Resource-Constraint Proect Scheduling with Overlapping Modes i Finish tie of activity i rework n=4 n=3 n=2 n= Possible values of the Finish tie of activity tie Fig. 2. Precedence constraints on the finish ties of two overlappable activities i and the axiu of single rework and the su of the, depending on the duplicate rework, as stated in [2]. Without loss of generality, the latter is considered in the odel. In typical proects involving engineering phases, the nuber of precedence and overlapping relationships ay largely exceeds the nuber of activities. As each activity can have several overlappable or nonoverlappable predecessors, we introduce the notion of execution odes associated to activities. Each activity ode represents a cobination of possible precedence or overlapping odes of an activity with its overlappable or non-overlappable predecessors. Consequently, the set of activity odes for each activity is generated by a full factorial design of the precedence and overlapping odes with its predecessors. Tables 2a, 2b and 2c show the activity odes in the case of non-overlappable predecessors, only one overlappable predecessor (with four overlapping odes), and two predecessors (each with three overlapping odes), respectively. Table 2a: execution ode of activity in the case of non-overlappable predecessors α i, i P r 0 0 Table 2b: execution ode of activity in the case of one overlappable predecessor i A k P n i α k n k α i r β k2 0 μ k2 3 3 β k3 0 μ k3 4 4 β k4 0 μ k4 Table 2c: execution ode of activity in the case of two overlappable predecessors i, i k P 2 A n i, n i2, α i,, α i2,, n k α k r β i2,,2 0 μ i2,, β i2,,3 0 μ i2,,3 4 2 β i,,2 0 0 μ i,, β i,,2 β i2,,2 0 μ i,,2 +μ i2,, β i,,2 β i2,,3 0 μ i,,2 +μ i2,,3 7 3 β i,,3 0 0 μ i,, β i,,3 β i2,,2 0 μ i,,3 +μ i2,, β i,,3 β i2,,3 0 μ i,,3 +μ i2,,3 6 CIRRELT-20-09

11 Optial Resource-Constraint Proect Scheduling with Overlapping Modes 2.5 The 0- Integer Linear Prograing odel Each activity ust finish within the tie window { EF,..., } with respect to the precedence relations and the activity durations. They can be derived fro the traditional forward recursion and backward recursion algoriths considering that the proect ust start at tie 0 and that T constitutes an upper bound of the proect s akespan (i.e., the su of processing ties of all activities) []. We define the decision variables (i.e., the finish ties and the overlapping odes) as follows: if activity is executed in ode and finsished at tiet X t = 0 otherwise S, t [ 0,T ], (5) and [ ] i The decision on the activity odes can be classed into three cases. On the one hand, if activities (i,) are not overlappable, the decision is siply not to overlap. On the other hand, if activities (i, ) are overlappable, these activities can be either overlapped ( > ) or executed in series ( =). The resource-constrained scheduling proble with overlapping can then be forulated with the 0- integer non-linear prograing odel as follow: n+ n+ Miniize t X n +,t, (6) Subect to : = t= EFn+ i n+ n+ X t = then Xit ( t d ) If t= EF = t = EFn+ t= EFn+ i n+ n+ If t = t X it = t = 2 t= EF = t= EFn+ = 2 t= EFn+ n t+ d + r R k X b Rk, k R = 2 = b= t i i = t= EFi = t= EF t X, S, i Pred( ) (7) X then ( d ) r ) t X it = t= EF t X t t t [ 0,T ] ( α X, A, i A (8) i t (9), S, i A( ) (0) X =, S () t { 0, } X t =, S and [, ], t [ 0,T ] (2) i The obective (6) iniizes the finish tie of the duy sink activity and therefore, the proect s akespan. Constraints (7) represent the finish-to-start precedence constraints when activities are not overlapped. If activities are overlapped, constraints (8) state that the downstrea activity ust start at the upstrea activity finish tie inus one of the feasible overlap duration. Constraints (7) and (8) reflect the precedence and overlapping constraints presented in Figure 2. Constraints (9) define the resource constraints. Constraints (0) guarantee that the downstrea activity of a couple of overlappable activities can not finish before the upstrea activity s finish tie. Constraints () ensure that each activity is assigned one activity ode and one finish tie. Finally, constraints (2) define the aforeentioned binary decision variables. The 0- integer non-linear prograing odel given by the obective (6) and the constraints (7)-(2) can be transfored to a 0- integer linear prograing odel. Constraints (7) and (8) are reforulated as follows: CIRRELT

12 Optial Resource-Constraint Proect Scheduling with Overlapping Modes i n+ = t= EFn+ n+ t X i = t= EFn+ it n+ ( t d i ) r ) = t= EFn+ t i ( α X, S, i Pred( ) (3) t α X Y, S, i A( ) (4) i i i t X it ( t d ( i ) r ) X t T ( Yi ) α, S, i A( ) (5) = t= EFi = t= EFi Y i = 0, S, i Pred( ) (6) { } Note that Y i is an additional binary variable. Constraints (3) represent the finish-to-start precedence constraints, with a negative lead tie in the case of overlapping. According to constraints (4), if two overlappable activities (i, ) are overlapped, then Y i = and thus the union of constraints (3) and (5) is equivalent to the equality constraints (8). If activities (i, ) are not overlapped, then Y i is unrestricted and constraints (5) are not restrictive. In view of significant efficiency of linear prograing solutions ethod [9][20], and the ipleentation of these ethods in coercial software packages, the linearization of the aforeentioned precedence constraints is extreely useful. 3 Illustrative exaple 3. Data We consider a proect instance generated by Kolisch and Sprecher [2] coposed of 30 non-duy tasks and 4 renewable resources. The activity durations, resource consuptions, resource availabilities and precedence relations are siilar to those presented in [22]. As no overlapping was defined in the original instance, the additional overlapping data have been generated (i.e., A, M i, α i,r ). Eight couples and two triplets of overlappable activities have been considered, as depicted in Tables 3 and 4. As a reinder, the overlapping aount and rework for non-overlappable activities are null. Table 3 : Overlapping data for the couples of overlappable activities Upstrea activity i Downstrea activity Mode α i r Upstrea activity i Downstrea activity Mode / / / / /7 0 2 / / /7 4 4/ / / / / α i r 8 CIRRELT-20-09

13 Optial Resource-Constraint Proect Scheduling with Overlapping Modes Table 4: Overlapping data for the triplets of overlappable activities Upstrea Upstrea activities Downstrea Mode α activity i α i2 r activities i i 2 i i 2 Downstrea activity Mode α i α i2 r / / / /9 4 / / /7 /7 0 5 /7 2/9 6 /7 2/7 6 /7 3/9 7 3/ / /7 /7 8 3/7 2/ /7 2/ /7 3/ Results The illustrative case was ipleented in AMPL Studio v.6. and solved with Cplex 2.2. In the original proect extracted fro PSPLIB, the optial proect akespan is 38 and 43 tie units with and without resource constraints, respectively. Because of the overlapping opportunities defined in our illustrative exaple, soe activities can start before the end of soe of their predecessors based on preliinary inforation and with additional rework. Consequently, soe overlappable activities can start earlier than scheduled in the original optial schedule. The resulting optial akespan with overlapping odes is 34 and 39 tie units with and without resource constraints, respectively, as shown in Table 5. Table 5: Effects of resource constraints and overlapping on the optial akespan and the coputation tie Case Resources constraints Overlapping Modes Nuber of overlapped activities Nuber of overlappables activities Optial akespan No No Yes No No Yes Yes Yes CPU s Tie For the scheduling proble without resource constraints and with overlapping odes (case 3), the overlapping odes of the non-critical overlappable activities obtained in the optial schedule are not the overlapping odes with the higher gain when overlappable activities are considered separately. Therefore, overlappable activities on the critical path are overlapped at their local optiu, as any reduction of the tie to execute critical activities will decrease the proect akespan. As highlighted in Table 5, only half of the set of overlappable activities are indeed overlapped for the scheduling proble without resource constraints and with overlapping odes. Even though the coputational ties are reasonable in all cases, Table 5 also reveals that overlapping odes significantly increase it, as it adds further coplexity to the already coplex case of resource-constrained scheduling proble, which is known to be a NP-hard optiization proble [20]. When resource constraints are considered, overlapping is less perfored than without resource constraints. As expected, overlapping lead to additional workload and to ore resource consuptions. Overlapping is thus less attractive and only one third of the set of overlappable activities are overlapped with resource constraints and overlapping odes (case 4). The overlapping odes obtained with the optial akespan are detailed in Table 6. This confirs that overlapping and resource constraints are closely interrelated. CIRRELT

14 Optial Resource-Constraint Proect Scheduling with Overlapping Modes Table 6: Overlapping odes obtained with the optial akespan without resource constraints (case 3) with resource constraints (case 4) Overlapped couples or triplets of activities* Mode Overlap duration Rework Local Gain Mode Overlap duration Rework Local Gain (3,8) (4,0) (2,) (0,6) (3,8) (6,7,22) *The non-displayed couples or triplets of activities are not overlapped (i.e., executed in activity ode ). 4 Conclusion and Discussion Overlapping activities is one of the ost applied strategies to accelerate a proect either in its early stage when the schedule baseline is set up or following proect delay during its execution. Overlapping is inherently risky as it entails that downstrea activities start before the inforation they require is available in a finalized for. However, additional workload required to accoodate the inforation changes transitted by upstrea activities to the overlapped downstrea activities are often ignored in practice. On the other hand, in spite of all research efforts accoplished in evaluating the relation between rework and the aount of overlap and deterining the optial overlapping strategy for two activities without resource constraints [6][7][5][6][7], only few papers have incorporated overlapping in the RCPSP [2][3]. In addition, these papers studied siplified linear rework odel that are not realistic. We investigate the oint optiization of overlapping and resource-constrained proect scheduling proble with the following assuptions: () preliinary inforation can be exchanged between identified overlappable activities, (2) the inforation flow is unidirectional fro upstrea to downstrea activities, (3) inforation exchanges are costless, (4) overlapping is restricted to a finite nuber of feasible aounts of overlap for each couple of activities, corresponding to overlapping odes, and (5) rework is preliinary estiated for each overlapping ode. The ain contribution of this paper is to present an integer linear prograing odel for the proect scheduling proble. Such a forulation shares siilarities with the traditional ulti-ode resource constraint scheduling proble (MRCPSP). Considering the liit of exact solution procedure encountered with MRCSPSP, we can anticipate that solving the RCPSP with overlapping odes for larger proects, as they usually appears in practical cases, will require the use of etaheuristics or heuristics. The relaxation of the aforeentioned assuptions also represents interesting perspectives. 5 References [] Hartann, S. Proect scheduling under liited resources. Berlin: Springer, 999. [2] Hartann, S. and Briskorn, D. A survey of variants and extensions of the resource-constrained proect scheduling proble. European Journal of Operational Research, 207(): 4, 200. [3] Bogus, S. M., Molenaar, K. R. and Diekann, J.E. Concurrent engineering approach to reducing design delivery tie. Journal of Construction Engineering and Manageent, 3():79 85, [4] Terwiesch, C. and Loch, C.H. Measuring the effectiveness of overlapping developent activities. Manageent Science, 45(4): , 999. [5] Sith, P. and Reinertsen, D. Developing Products in Half the Tie. New York: Van Nostrand Reinhold, CIRRELT-20-09

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