TimeCost TradeOffs in ResourceConstraint Project Scheduling Problems with Overlapping Modes


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1 TimeCost TradeOffs in ResourceConstraint Proect Scheduling Problems with Overlaing Modes François Berthaut Robert Pellerin Nathalie Perrier Adnène Hai February 2011 CIRRELT Bureaux de Montréal : Bureaux de Québec : Université de Montréal Université Laval C.P. 6128, succ. Centreville 2325, de la Terrasse, bureau 2642 Montréal (Québec) Québec (Québec) Canada H3C 3J7 Canada G1V 0A6 Téléhone : Téléhone : Télécoie : Télécoie :
2 TimeCost TradeOffs in ResourceConstraint Proect Scheduling Problems with Overlaing Modes François Berthaut 1,2,*, Robert Pellerin 1,2, Nathalie Perrier 1,2, Adnène Hai 1,3 1 Interuniversity Research Centre on Enterrise Networks, Logistics and Transortation (CIRRELT) 2 Deartment of Mathematics and Industrial Engineering, École Polytechnique de Montréal, P.O. Box 6079, Station Centreville, Montréal, Canada H3C 3A7 3 Deartment of Oerations and Decision Systems, Université Laval, 2325, de la Terrasse, Québec, Canada G1V 0A6 Abstract. In comanies, overlaing is commonly regarded as a romising strategy to accelerate roect execution. Overlaing consists in executing in arallel two sequential activities by allowing a downstream activity to start before the end of an ustream activity based on reliminary information. However, overlaing entails rework in downstream activity caused by alteration of information exchanged until finalized information is available and additional coordination and communication. Rework and coordination/communication require additional resources and costs. We investigate the timecost tradeoffs in resourceconstrained roect scheduling roblem with different feasible modes of overlaing including rework and coordination/communication. The roblem is formulated as a linear integer rogram. An examle of a 30 activity roect is rovided to illustrate the utility and efficiency of the model. Our results highlight the closed interaction between resource constraints and overlaing modes and confirm the relevance of ointly consider them. Keywords. Activity overlaing, concurrent engineering, roect management, roect scheduling. Acknowledgements. This work was suorted by the Jarislowsky Foundation, the SNC Lavalin firm and by the Natural Sciences and Engineering Council of Canada (NSERC). This suort is gratefully acknowledged. Results and views exressed in this ublication are the sole resonsibility of the authors and do not necessarily reflect those of CIRRELT. Les résultats et oinions contenus dans cette ublication ne reflètent as nécessairement la osition du CIRRELT et n'engagent as sa resonsabilité. * Corresonding author: Déôt légal Bibliothèque et Archives nationales du Québec Bibliothèque et Archives Canada, 2011 Coyright Berthaut, Pellerin, Perrier, Hai and CIRRELT, 2011
3 TimeCost TradeOffs in ResourceConstraint Proect Scheduling Problems with Overlaing Modes 1 Introduction The RCPSP (ResourceConstrained Proect Scheduling Problem) has been addressed in numerous aers. Various models attemt to minimize roect time comletion while considering limited resources (Hartmann 1999). Hartmann and Briskorn (2010) have also resented an overview of different RCPSP extensions. Among extensions addressed in the literature, different ractices have been develoed to reduce time of roect execution in order to establish a baseline schedule or to modify it following roect delay during its execution through overlaing, crashing and substitution of activities (Gerk and Qassim 2008). In engineering roect, overlaing is considered as a core technique for saving develoment time (Smith and Reinertsen 1998). It consists in starting a downstream activity before receiving all the final information required. It has been demonstrated to be a owerful tool for reducing roduct develoment times from concetual design to roduction startu in a wide range of industries, such as software (Blackburn and al. 1996), mobile hones (Lin et al 2010) automobiles (Clark and Fuimoto 1991) and airlanes (Sabbagh 1996). Overlaing has also been alied in design and build hases of construction roects (PenaMora and Li 2001; Dzeng 2006). Indeed, a common ractice in construction roects is to reach 20% of build hase while design is comleted at 80%. Overlaing activities or roect hases, and the surrounding organizational activities required to suort it, are often referred as simultaneous or concurrent engineering (Terwiesch and Loch 1999). As reliminary information rovided by ustream activities may evolve until it becomes final information, overlaing often causes additional rework and modification in downstream activities. Such rework may outweigh the overla benefices of arallel activity execution in terms of cost and time, articularly if develoment uncertainty is not resolved early during the roect (Terwiesch and Loch 1999). Indeed, if the develoment uncertainty is high, most of the tasks done on reliminary information will be reworked, which make overlaing unfavorable. Frequent information exchange between the develoment teams reduces the negative effect of overlaing, but requires additional time and cost for communication and coordination. However, roect lanners and managers determine overlaing strategies on an ad hoc basis without always considering rework and interaction between activities (Lin et al. 2010), yielding inefficient roect management. A growing body of literature in oerations management has investigated the question of when and to which extent overlaing should be alied. Two grous of models have been develoed in the literature to analyze overlaing interactions. First, many authors consider only a coule of activities or roect hases and no resource constraints to establish the best tradeoff between overlaing and rework. Krishnan et al. (1997) introduced the concet of information evolution and downstream sensitivity to describe interactions in overlaed activities with the assumtion that communication and coordination are instantaneous and costless. Information evolution refers to the ustream generated information useful for downstream activities. Downstream sensitivity refers to the imact of a change in ustream activity on the downstream activity. They develoed a model to determine when to start the downstream activity so as to minimize the develoment cycle time. Roemer et al. (2000) adated the concets of evolution and sensitivity to model the robability of rework as a function of the overla duration and studied the timecost tradeoffs in overlaing. Loch and Terwiesch (1998), who studied the timetomarket minimization roblem, and Lin et al. (2010), who studied the timecost tradeoffs roblem, investigated the integrated roblem of overlaing and CIRRELT
4 TimeCost TradeOffs in ResourceConstraint Proect Scheduling Problems with Overlaing Modes information exchange olicy assuming that information exchange usually requires time and cost. In addition to quantifying the amount of rework as a function of the overla duration, these aers showed that the otimal overlaing strategy is to overla as much as ossible when information exchange is instantaneous and costless, while there exists an otimal overla duration which differs from the maximum ossible overla duration when information exchange requires nonnegligible time and cost. Other aroaches have considered whole roects instead of couled of activities under the assumtion that the relation between overlaing amount and rework is reliminary known for overlaable activities. They mostly use design structure matrix (DSM) to reresent deendencies, to minimize feedbacks, and to identify overlaing oortunities between activities. Among other models, Gerk and Qassim (2008) develoed an analytic roect acceleration linear model via activity crashing, overlaing and substitution with resource constraints. Wang and Lin (2009) develoed a stochastic overlaing rocess model to assess schedule risks. Their simulation model considers iterations and robabilities of rework. Iterations are mostly defined as interaction between design activities which lead to rework in ustream activities caused by feedbacks from downstream activities. However, their model does not take into account resource constraints. Cho and Einger (2005) introduced a simulation model with stochastic activity durations, overlaing, iterations, rework and considered resource constraints for some activities. They showed that theses constraints can delay some overlaed activities and delay the roect. All these aers assume a simle linear relationshi between rework and overlaing amount with an uer and lower bound and consider that information exchange is instantaneous and costless. In summary, most contributions in the related literature fail to consider a realistic relationshi between overlaing, rework and communication/coordination in the RCPSP. The obective of this aer is to extend the classical RCPSP with a realistic overlaing model that deals with additional workloads and costs incurred by rework and coordination/communication. We assume that the information flow is unidirectional from ustream to downstream activities. Consequently, the rework caused by overlaing is only assigned to the downstream activities and there is no activity iteration. The main difference with the aforementioned overlaing models is that overlaing is restricted to a set of feasible overla durations for each coule of overlaable activities, instead of considering a continuous and bounded interval for the overla duration. These overlaing modes are characterized by different overla durations, rework durations and costs, and communication/coordination durations and costs. For convenience, the overlaing modes are subsequently converted into activity modes, each of which reresenting a combination of overlaing modes of an activity with the associated overlaable activities. This transformation enables to easily formulate the RCPSP with overlaing modes as a linear integer rogramming roblem, which shares similarities with the classical multimode RCPSP model (Hartmann 1999). This model allows finding an otimal makesan in reasonable calculation time. The remainder of the aer is organized as follows. Section 2 first describes the roblem statement and assumtions. The gain maximization roblem and the makesan minimization roblem are formulated in Section 3. An illustrative examle and comutational results of the timecost tradeoffs are then resented in section 4. Section 5 concludes with recommendations for future work. 2 CIRRELT
5 TimeCost TradeOffs in ResourceConstraint Proect Scheduling Problems with Overlaing Modes 2 Problem Statement and assumtions The roect scheduling literature largely focuses on the generation of a recedence and resource feasible schedule that otimizes the schedule obective(s) (most often the roect makesan or cost) and that should be used as a baseline schedule for executing the roect. This schedule serves as a basis to allocation of resources, lanning of material rocurement, communication and coordination within the roects and with external entities (client, subcontractors, ), etc. Here, we assume that all information required for the scheduling of the roect is known in advance, and consequently the roblem is formulated and solved in a deterministic environment. This section is then devoted to resent the roect model and the information required to solve the roect scheduling roblem with resource constraints and overlaable activities. A roect is defined by a set of activities, S, including two fictitious activities 0 and n+1, which corresond to the roect start and roect end, resectively, with zero rocessing time. We denote by d the estimated nominal rocessing time of activity considering that all the final information required from receding activities are available at its start; in other word, if activity is rocessed without overlaing. All the symbols and their definitions used along this aer are resented in Table Precedence constraints Frequently used roectlanning methods rovide grahic descritions of task workflows in the form of the socalled activityonnode or activityonarc networks. These networks deict the logical execution sequence of deendent (sequential) activities and indeendent (arallel or concurrent) activities. However, these tools fail to incororate interdeendenttye relation, activities iterations and to model information flows between activities. The Design Structure Matrix (DSM) reresentation can handle these additional relations between activities with the broader concet of information sharing (Browning 2001). Information exchange between activities can occur at the beginning, the middle or the end of an activity and includes both tangible and intangible tyes such as arts, art dimensions, and bill of materials, which constitute the oututs from an ustream activity and are required to begin the work of a downstream activity. A DSM is a square matrix where rows and columns reresent activities. It aims to reresent the information flows for a given subset of activities and constitutes the first ste in analyzing otential feedbacks. Feedback information exchanges from downstream to ustream activities corresond to design modification requests due to inability to meet target design requirements or design flaws detected in downstream stages (Wang and Lin 2009). Any feedback information exchange from downstream activities lead to modifications and reworks erformed by CIRRELT
6 TimeCost TradeOffs in ResourceConstraint Proect Scheduling Problems with Overlaing Modes Table 1 Symbols and definitions Symbol Definition Symbol Definition S Set of activities Cr im Rework cost in the downstream activity when (i, ) are overlaed in recedence mode m n Number of nondummy activities Cc im Total cost for coordination and communication when (i, ) are overlaed in recedence mode m E Set of temoral or recedence constraints Number of execution modes of the activity i Precedence constraint m i Precedence mode of the coule (i, ) in execution mode (i, ) (of activity ) d Processing time of activity β i Amount of overla duration between activities i and in execution mode (of activity i) A Set of coules of overlaable activities μ i Exected amount of rework in activity in execution mode (of activity i) P Set of coules of nonoverlaable activities ρ i Exected total amount of time for coordination and communication when (i, ) are overlaed in execution mode (of activity i) Po Pn Set of immediate redecessors of activity that are overlaable with activity Set of immediate redecessors of activity that are not overlaable with activity P Set of immediate redecessors of activity μ i So Sn Set of immediate successors of activity that are overlaable with activity Set of immediate successors of activity that are not overlaable with activity S Set of immediate successors of activity δ m i Precedence mode of the coule (i, ) in execution mode (of activity i) Amount of overla duration between activities i and in β i execution mode (of activity ) Exected amount of rework in activity in execution mode (of activity ) Exected total amount of time for coordination and ρ i communication when (i, ) are overlaed in execution mode (of activity ) Exected total amount of rework in activity in execution μ mode Exected total amount of time for coordination and communication in activity in execution mode R Set of renewable resources CR Rework cost of activity in execution mode R k R k Constant amount of available units of renewable resource k Per eriod usage of activity of renewable resource k CC Co Total cost for coordination and communication of activity in execution mode Oortunity cost (cost of increasing/decreasing the makesan by one unit of time) m i Number of recedence modes of the coule (i, ) D Proect Due date α i Amount of overla duration between activities i and T Uer bound of the roect makesan r i Exected amount of rework in the downstream activity when (i, ) are overlaed C lim Uer bound of the total overlaing cost σ i Exected total amount of time for coordination and communication when (i, ) are overlaed t = 0,..,T Periods α im Amount of overla duration between activities i and in recedence mode m, exressed as a fraction of d Earliest ossible finish time of activity r im Exected amount of rework in the downstream activity when (i, ) are overlaed in recedence Latest ossible finish time of activity mode m σ im Exected total amount of time for coordination and communication when (i, ) are overlaed in recedence mode m the ustream activities to accommodate these changes, and iterations between ustream and downstream activities can virtually occur to fix the roblems identified. In order to minimize feedbacks, the DSM can be artitioned using 4 CIRRELT
7 TimeCost TradeOffs in ResourceConstraint Proect Scheduling Problems with Overlaing Modes block triangularization algorithm to obtain a unidirectional sequence of information exchange (Browning 2001). As a last resort, activities can be aggregated or decomosed into lowerlevel activities to eliminate feedbacks. In this aer, we assume that such reliminary studies have been conducted to identify the nature of relations between activities and to determine a feasible sequence of activities without any feedback from downstream activities. The roect is then only comosed of indeendent and deendent activities and the resulting information flow within the roect between activities is assumed to be unidirectional from ustream to downstream activities. The analysis of information exchanges between deendent coule activities enables to categorize them into nonoverlaable and overlaable ones. The former reresents the case where a downstream activity requires the final outut information from an ustream activity to be executed or the comletion of the ustream activity. The latter reresents the case where a downstream activity can begin with reliminary information and receives final udate at the end of the ustream activity. This relation rovides the oortunity to overla two activities so that a downstream activity can start before an ustream activity is finished. While the nonoverlaable activities are connected with the classical finishtostart recedence constraint, the overlaable activities are connected with a finishtostartluslead time recedence constraint where the leadtime accounts for the amount of overla. Note that the finishtostart recedence constraint is the most conventional tye of relationshi used in ractice and in roect management tools such as MS Proect or Primavera (Cho and Einger 2005). In the remainder of the aer, we denote by A and P the sets of coules of overlaable and nonoverlaable activities, resectively. Similarly, for each activity, Po and Pn reresent the set of overlaable and nonoverlaable redecessors, resectively, while So and Sn denote the set of overlaable and nonoverlaable succesors, resectively. The set of recedence constraints in the roect, E, and the set of immediate redecessors, P, and immediate successors, S, of each activity are defined by: E = AU P (1) P = Po U Pn, S = So U Sn, S (2) 2.2 Model of the overlaing rocess Figure 1 shows the overlaing rocess of two activities (i, ) in A. The downstream activity starts with reliminary inuts from the ustream activity i. The amount of overla, α i, is exressed as a fraction of the downstream activity s duration. As the ustream activity roceeds, its information evolves to its final form and is released to the downstream activity at its comletion. This aroach imlies that the traditional attern of exchange of finalized information at the end of the ustream activity is altered to a more frequent exchange of evolving information during the overlaing rocess. However, additional rework is often necessary to accommodate the changes in the ustream information in the downstream develoment. The exected duration of the sum of rework is denoted by r i. Moreover, frequent information exchange allows the downstream team to be aware of the latest ustream change to be incororated in their work. For each time information CIRRELT
8 TimeCost TradeOffs in ResourceConstraint Proect Scheduling Problems with Overlaing Modes d i + σ i i Preliminary information exchange Final information exchange d + r i + σ i Fig 1. Overlaing rocess of two activities D i Coordination and communication Rework time is exchanged, ustream and downstream teams have to dro everything they are doing and commit themselves to set a crossfunctional team meeting and discuss the latest changes for downstream incororation (Loch and Terwiesch 1998; Lin et al. 2010). The exected duration of the sum of information exchange durations for communication and coordination due to overlaing is denoted by σ i. The total amount of time required to execute both activities, D i, is exressed as follows: D i = d + d ) i ( 1 α i + ri + σ i (3) If d di, the amount of overla is usually bounded by the fraction d i / d in order to revent the downstream activity to start before the ustream activity. If overlaing was not alied, the total amount of time required to execute both activities would simly be D + i = di d. Deending on the nature of the activities, there may exist a tradeoff between time gains from overlaing, rework, and communication and coordination. In addition, if we consider the costs associated with rework, communication and coordination, and the oortunity costs for finishing the roect earlier (remium) or later (enalty for delay), there may also exists a tradeoff between additional cost for overlaing and oortunity cost for finishing earlier. These give rise to the three following main otimization roblems roosed in literature for the overlaing of two activities without resource constraints, with or without the assumtion of instantaneous and costless information exchange: timetomarket minimization roblem, with or without a maximum costs constraint (Loch and Terwiesch 1998; Roemer et al. 2000; Lin et al. 2010), cost minimization roblem, subect to a maximum timetomarket constraint (Roemer et al. 2000; Roemer an Ahmadi 2004), gain maximization roblem, subect to a maximum timetomarket constraint (Roemer et al. 2000; Lin et al. 2009; Lin et al. 2010). The main issue with the overlaing roblem is to quantify the amount of rework as a function of the amount of overla. Indeed, the overlaing roblem requires exloring the behavior and interaction of activities during their rocesses. Krishnan et al. (1997) resented a ioneer aer in this field. They roosed a model of deendency based 6 CIRRELT
9 TimeCost TradeOffs in ResourceConstraint Proect Scheduling Problems with Overlaing Modes on the ustream information evolution, which characterizes the refinement of information from its reliminary form to a final value, and the downstream sensitivity, which reresents the duration of a downstream iteration to incororate ustream changes. Roemer et al. (2000) and Roemer and Ahmadi (2004) introduced the concet of robability of rework as a function of the overla duration, which encomasses both the evolution and sensitivity models roosed in Krishnan and al. (1997). When information exchange requires nonnegligible time and cost, Loch and Terwiesch and Lin et al. (2010) assumed that each information exchange (i.e., meeting) has a setu time and a setu cost considered as constant. The communication/coordination olicy is then characterized by the frequency and the number of information exchange. Loch and Terwiesch (1998) adated the concets roosed by Krishnan et al. (1997) by considering the ustream evolution as the rate of modifications in the ustream and the downstream sensitivity as the imact of a modification on downstream rework, and ointly analyzed overlaing and communication olicies between two activities. Lin and al. (2010) investigated the evolution of the downstream rogress and refined the model roosed by Loch and Terwiesch (1998). An imortant finding of the aforementioned aers is that the duration of rework is a convex increasing function of the amount of overla. The former statement is intuitive: if the amount of overla increases, then the reliminary information at the downstream activity s start will be more unreliable and more downstream changes must be incororated. In addition, when information exchange is considered, the otimal coordination and communication olicy for a given amount of overla is such that the resulting duration of coordination and communication is concave or convex deending on the shae of the ustream information evolution (Loch and Terwiesch 1998). When the ustream evolution is linear, the duration coordination and communication is a nondecreasing function with resect to the amount of overla (Loch and Terwiesch 1998; Lin et al. 2010). Another imortant finding of the aforementioned aers is that the time to comlete the ustream and downstream activities is a convex increasing function of the amount of overla when information exchange has negligible cost and duration. Therefore, the otimal amount of overla is the maximum feasible amount of overla. With nonnegligible information exchange cost and duration, the time to comlete the ustream and downstream activities is either convex, concave or concaveconvex with resect to the amount of overla deending on the shae of the ustream information evolution (Loch and Terwiesch 1998). In articular, the time to comlete the ustream and downstream activities is convex when the ustream evolution is linear, such that the otimal amount of overla may be greater that the maximum feasible amount of overla (Loch and Terwiesch 1998; Lin et al. 2010). To sum u, the amount of overla which minimizes the time to comlete the ustream and downstream activities has been derived under different conditions and lead to the conclusion that the otimal overla amount is not necessarily the maximum feasible amount of overla and that the time to comlete the ustream and downstream activities is either convex, concave or concaveconvex with resect to the amount of overla. However, when overlaing is considered for more than two activities for a whole roect with several overlaable coules of activities, the models of overlaing rocess roosed in the current literature consider a simlistic linear relation between the rework and the amount of overla. We roose in the next section an overlaing model for roect with several overlaable coules of activities, which both relaxes this assumtion and encomasses any overlaing rocess roosed so far for two activities. CIRRELT
10 TimeCost TradeOffs in ResourceConstraint Proect Scheduling Problems with Overlaing Modes The overlaing costs have also received much attention in the literature in order to formulate the cost minimization and the gain maximization roblems. The overlaing costs are comosed of the cost of rework and the communication/coordination cost. The cost of rework are usually considered as a linear function of the rework duration with or without a fixed cost (Roemer et al. 2000; Roemer and Ahmadi 2004; Gerk and Qassim 2008; Lin et al. 2010). For examle, Roemer et al. (2000) argued that the rework cost corresonds to the hours of engineering send on rework multilied with the average wage of engineers er unit of time. Similarly, Lin et al. (2010) assumed that communication and coordination has a constant setu cost er meeting or information exchange. Therefore, the communication/coordination cost is roortional to the amount of time sent for communication/coordination. 2.3 Precedence and overlaing modes In order to study the interaction between overlaing and resource constraints in the scheduling otimization roblem with multile activities including several overlaing oortunities, the relations between the amount of overla, rework duration and cost and communication/coordination duration and cost are required for a range of amount of overla for each coule of overlaable activities. Indeed, the otimal overla amounts for a resourceconstraints roect comosed of several coules of overlaable activities are not necessarily set to the otimal values found for each coule of activities (Browning and Einger 2002; Cho and Einger 2005; Gerk and Qassim 2008). In this aer, overlaing is assumed to be defined for discrete values of overla durations. First, this assumtion is more realistic considering that scheduling is erformed in ractice on a eriodbyeriod basis (i.e., hour, day, week): resource availabilities and allocations are estimated er eriod, while activity durations are discrete multiles of one eriod (Hartmann 1999). Second, activity rogress is measured in ractice according to the comletion of internal milestones which corresonds to imortant events, such as design criteria frozen, detailed design comleted, drawings finalized, or any activity deliverables. This reliminary information is issued at intermediate oints and used as inut for a downstream activity. Therefore, the start time of an overlaed downstream activity is restricted to a finite number of instants corresonding to ustream activities milestones which constitutes different feasible modes for the execution of overlaing activities. Each overlaing mode is characterized by an amount of overla exressed as a fraction of the downstream activity s duration, rework duration and cost and communication/coordination duration and cost. These arameters can be either derived from models of overlaing rocess resented in revious section when historical data are available or estimated by engineers for each overlaing mode. Table 2a Precedence modes for a nonoverlaable coule of activities (i, ) in P Overlaing mode of coule (i, ), m Amount of overla, α im Rework duration, r im Coordination/ communication duration, σ im Rework cost, Cr im Coordination/ communication cost, Cc im CIRRELT
11 TimeCost TradeOffs in ResourceConstraint Proect Scheduling Problems with Overlaing Modes Table 2b Overlaing modes for an overlaable coule of activities (i, ) in A Overlaing mode of coule (i, ), m Amount of overla, α im Rework duration, r im Coordination/ communication duration, σ im Rework cost, Cr im Coordination/ communication cost, Cc im α i2 r i2 σ i2 Cr i2 Cc i2 M M M M M M m i α i,,mi r i,,mi σ i,,mi Cr i,,mi Cc i,,mi Overlaing modes can be generalized to recedence modes in order to describe all recedence relationshis between activities. For each coule of recedence constraints i, there exists at least one recedence mode which corresonds to a basic finishto start relation without overlaing. When ( i, ) A, there exist additional recedence modes associated with the different overlaing strategies. The recedence modes can be exressed as resented in Tables 2a and 2b. When activities i and are overlaable, they can be either overlaed and executed in mode m = 2,,m i, or sequentially erformed in mode m = 1 without overlaing. As deicted in Figure 2, it is imortant to note that the recedence constraints on the finish time of activities i and will defer deending on the overlaing mode: when not overlaed, the downstream activity start time is suerior or equal to the ustream activity finish time, whereas the downstream activity start time is equal to the ustream activity finish time minus one of the feasible overla duration in the case of overlaing. i Finish time of activity i m =4 m =3 m =2 m =1 Possible values of the Finish time of activity time Coordination and communication rework Fig. 2 Precedence constraints on the finish times of two overlaable activities i and deending on the overlaing modes m 2.4 Multile overlaing and activity modes We assume that there is no restriction concerning the number of overlaable or nonoverlaable redecessors. If an activity is overlaed by multile ustream activities, feasible overlaing modes are assumed to be comatible. Consider for examle the case of a downstream activity with two ustream activities, denoted by i 1 and i 2. If both CIRRELT
12 TimeCost TradeOffs in ResourceConstraint Proect Scheduling Problems with Overlaing Modes coules (i 1, ) and (i 2, ) are overlaed, the amount of rework in downstream activity is between the maximum of single rework and the sum of them, deending on the dulicate rework, as stated in Cho and Einger (2005). Without loss of generality, the latter is considered in the model. Similarly the amount of time sent for communication/coordination in activity is assumed to be the sum of the communication/coordination durations with its overlaed redecessors and successors. In tyical roects involving engineering hases, the number of recedence and overlaing relationshis may largely exceeds the number of activities. As each activity can have several overlaable or nonoverlaable redecessors and successors, we introduce the notion of execution modes associated to activities. Each activity mode reresents a combination of ossible recedence or overlaing modes of an activity with its overlaable or nonoverlaable redecessors and successors. Consequently, the set of activity modes {1,, } for each activity is generated by a full factorial design of the recedence and overlaing modes with its redecessors and successors. Table 3a shows the activity modes in the case of nonoverlaable redecessors and successors. Similarly, Tables 3b and 3c resents the activity modes in the case of only one overlaable redecessor (with four overlaing modes) and no overlaable successor, and one overlaable redecessor and one overlaable successor (each with three overlaing modes), resectively. Table 3a Activity modes of activity in the case of nonoverlaable redecessors and successors i Pn k Sn m i β i μ i ρ i m k β k μ ik ρ k μ δ CR CC Table 3b Activity modes of activity in the case of one overlaable redecessor (with 4 overlaing modes) and no overlaable successors i Po i' Pn k Sn m i β i μ i ρ i m i β i μ i ρ i m k β k μ ik ρ k μ δ CR CC α i2 r i2 σ i r i2 σ i2 Cr i2 Cc i2 3 3 α i3 r i3 σ i r i3 σ i3 Cr i3 Cc i3 4 4 α i4 r i4 σ i r i4 σ i4 Cr i4 Cc i4 10 CIRRELT
13 TimeCost TradeOffs in ResourceConstraint Proect Scheduling Problems with Overlaing Modes Table 3c Activity modes of activity in the case of one overlaable redecessor (with 3 overlaing modes) and one overlaable successor (with 3 overlaing modes) i Po i' Pn k So k' Sn m i β i μ i ρ i m i β i μ i ρ i m k β k μ ik ρ k m k β k μ k ρ k μ δ CR CC α k2 r k2 σ k r k2 σ k α k3 r k3 σ k r k3 σ k3 0 Cc k2 2 Cc k α i2 r i2 σ i r i2 σ i2 Cr i2 2 Cc i2 Cc i + Cc 5 2 α i2 r i2 σ i α k2 r k2 σ k r i2 + r k2 σ i2 + σ k2 Cr i2 2 2 k2 Cc i + Cc 6 2 α i2 r i2 σ i α k3 r k3 σ k r i2 + r k3 σ i2 + σ k3 Cr i2 2 2 k3 7 3 α i3 r i3 σ i r i3 σ i3 Cr i3 Cc i3 Cc i + Cc 8 3 α i3 r i3 σ i α k2 r k2 σ k r i3 + r k2 σ i3 + σ k2 Cr i3 2 3 k 2 Cc i + Cc 9 3 α i3 r i3 σ i α k3 r k3 σ k r i3 + r k3 σ i3 + σ k3 Cr i3 2 3 k3 m i, β i, μ i and ρ i denote the recedence/overlaing mode, the amount of overla, the rework duration and the communication/coordination duration of the coule (i, ) in activity mode = 1,, (i.e., the activity modes of the downstream activity ), resectively. The same symbols with the rime symbol reresent the same definitions exress in activity modes of the ustream activity i. δ, μ, CR and CC are the total communication/coordination duration (i.e., the sum of communication/coordination duration with overlaable redecessors and successors), the total rework duration, the total rework cost and the total communication/coordination cost of activity in mode = 1,,. Note that the communication/coordination cost associated with a coule (i, ) in A must be slit between activity i and. Without loss of generality, the communication/coordination cost is assumed to be equally slit between ustream and downstream activities. 3 Performance otimization models Each activity in S must finish within the time window {,..., } with resect to the recedence relations, the overlaing oortunities and the activity durations. As stated in Hartmann (1999), they can be derived from the traditional forward recursion and backward recursion algorithms considering that the roect must start at time 0 and that T constitutes an uer bound of the roect s makesan (i.e., the sum of rocessing times of all activities). We define the decision variables (i.e., the finish times and the overlaing modes) as follows: CIRRELT
14 TimeCost TradeOffs in ResourceConstraint Proect Scheduling Problems with Overlaing Modes X t 1 if activity is executed in mode and finished at time t = 0 otherwise and [ 1, ] S, t [ 0,T ] (4) The decision on the activity modes can be classed into three cases. On the one hand, if activity is not overlaable with any immediate redecessor or successor, the decision is simly not to overla. On the other hand, if activity is overlaable, this activity can be either overlaed with at least one of its overlaable redecessor or successor ( > 1) or not overlaed ( = 1). The resourceconstrained scheduling roblem with overlaing is formulated in this section with the obective of maximizing the roect gain. Next, we resent a variation of this roblem with the obective of minimizing the roect makesan. 3.1 Proect gain maximization roblem The resourceconstrained scheduling roblem of maximizing the roect gain with overlaing can be formulated with the nonlinear 01 integer rogramming model as follows: n+ 1 n+ 1 Maximize Co D t X ( CR + CC ) = 1 t= n+ 1 n n+ 1,t, X t = 2 = 1t= (5) Subect to If X t1 = 1 then t = i i = 1t= ( t d ) t X X 1 i it If X t = 1 then = 2t= i i = 1t= t= t, S, i P ( t ) X = t d ( 1 β ) ( μ + δ ) i [ i ] = 1t= (6) ρ i it X t, S, i Po (7) n R k = 2 = 1 t+ d 1+ μ + δ i i = 1 t= t X i it b= t = 1 t= X b t X R t k, k R, t [ 0,T ] (8), S, i Po (9) 12 CIRRELT
15 TimeCost TradeOffs in ResourceConstraint Proect Scheduling Problems with Overlaing Modes i i = 1t= = 1t= β ' i Xit = βi X t, S, i Po (10) i = 1t= X = 1, S (11) t { 0, 1}, t [ 0,T ] X t =, S and [ 1, ] (12) The obective function (5) maximizes the roect gain, which is comosed of the oortunity cost for finishing earlier or later than a target time, D (i.e., the roect makesan without overlaing or any roect due date), and the overlaing costs (rework and communication/coordination costs). Constraints (6) reresent the finishtostart recedence constraints when activities are not overlaed. If activities are overlaed, constraints (7) state that the downstream activity must start at the ustream activity finish time minus one of the feasible overla duration. Constraints (6) and (7) reflect the recedence and overlaing constraints resented in Figure 2. Constraints (8) define the resource constraints. Constraints (9) guarantee that the downstream activity of a coule of overlaable activities can not finish before the ustream activity s finish time. Constraints (10) state that the activity modes of two overlaable activities are such that the overlaing modes are the same for both activities. Constraints (11) ensure that each activity is associated with one activity mode and one finish time. Finally, constraints (12) define the aforementioned binary decision variables. The nonlinear 01 integer nonlinear rogramming model given by the obective function (5) and the constraints (6) (12) can be transformed into a linear 01 integer rogramming model. Constraints (6) and (7) are reformulated as follows: i i = 1 t= ( t ) X t d ( 1 β ) ( μ + δ ) i [ i ] ρ X, S, i P (13) i it = 1 t= t β i Xit Yi, S, i Po (14) = 1t= i i = 1 t= ( t ) X [ t d ( 1 β ) ( μ + δ )] X T ( 1 Y ) i i it ρ, = 1 t= i S, i Po (15) { 0, 1} Y i =, S, i Po (16) Note that Y i is an additional binary variable. Constraints (13) reresent the finishtostart recedence constraints, with a negative lead time in the case of overlaing. According to constraints (14), if two overlaable activities (i, ) are overlaed, then Y i = 1 and thus the union of constraints (13) and (15) is equivalent to the equality constraints (7). If activities (i, ) are not overlaed, then Y i is unrestricted and constraints (15) are not restrictive. t i CIRRELT
16 TimeCost TradeOffs in ResourceConstraint Proect Scheduling Problems with Overlaing Modes The roect gain maximization roblem can then given be formulated as a linear 01 integer rogramming roblem given by the obective function (5) and the constraints (8)(16). 3.2 Proect makesan minimization roblem The roect makesan minimization roblem can be formulated by relacing the obective function (5) by the following obective function: n Minimize + n = 1t= t X n + 1,t, (17) n+ 1 The obective function (17) minimizes the finish time of the dummy sink activity and therefore, the roect s makesan. In addition, we introduce the following constraint to ensure that the total overlaing cost will not exceed a given value: n = 2 = 1t= ( CR + CC ) X Clim t (18) The roect makesan minimization roblem can then be formulated as a linear 01 integer rogramming roblem given by the obective function (17) and the constraints (8)(16) and (18). The roect gain maximization and makesan minimization roblems allow to study the timecost tradeoffs between roect duration and overlaing costs with resource constraints. To our knowledge, such scheduling roblems with resource constraints and overlaing oortunities has only been treated by Gerk and Qassim (2008). However, these authors used a simlified overlaing model with a linear relation between rework and overla duration and they assumed that information exchange is instantaneous and costless. Our formulation allow any overlaing rocess to be alied, with the condition that overlaing can only be executed at feasible modes. 4 Illustrative examle 4.1 Proect data We consider a roect instance generated by Kolisch and Srecher (1996) comosed of 30 nondummy activities and 4 renewable resources. The activity durations, resource consumtions and recedence relations are summarized in Table 4. The availability of the resources are set to R k = 20, k = 1,,4. As no overlaing was defined in the original instance, the additional overlaing data have been generated. Fifteen coules of overlaable activities and their resective overlaing modes have been considered, as deicted in Table 5. The number of overlaing mode has been restricted to 3 for each coule of overlaable activities. As assumed in Roemer et al. (2000), Roemer and Ahmadi (2004) and Lin et al. (2010), the rework and 14 CIRRELT
17 TimeCost TradeOffs in ResourceConstraint Proect Scheduling Problems with Overlaing Modes Table 4 Proect data comosed of 30 nondummy activities Activit Duration d Resource consumtion R 1 R 2 R 3 R 4 Set of nonoverlaable redecessors, Pn Set of overlaable redecessors, Po {} {} {1} {} {} {2} {3} {} {2} {} {} {3} {3} {} {} {7} {2} {} {} {8} {4,8} {} {5} {10} {} {10} {} {11} {4, 7, 9} {} {13} {} {10} {11} {4} {6} {13, 15, 18} {} {7, 18} {} {} {13, 20} {8, 15} {} {5, 9, 21} {} {12, 19, 23} {} {16, 17} {24} {9} {} {14, 17, 21} {} {15, 26, 27} {} {11, 21} {19} {5, 29} {28} {22, 30} {25} {31} {} communication/coordination costs are considered as linear functions of the time sent on rework and communication, where the linear factors are the average wages of the teams er unit time (i.e., $200 er unit time for each resource). As a reminder, the overla amount, the rework and the communication/coordination data for nonoverlaable activities are set to zero. For the sake of conciseness, the activity modes of each activity are not all resented in this aer, as they are easily derived from Table 5. We only resent in Table 6 the activity modes of activity 8 as an CIRRELT
18 TimeCost TradeOffs in ResourceConstraint Proect Scheduling Problems with Overlaing Modes examle. Finally, the oortunity cost is set to $5000 er unit time, while the roect due date is set to the makesan found without overlaing modes. Table 5 Overlaing data for the coules of overlaable activities activities (i, ) mode m α im r im σ im Cr im ($) Cc im ($) activities (i, ) mode m α im r im σ im Cr im ($) Cc im ($) (2, 3) (11, 17) (3, 6) (13, 21) / / / / (6, 18) (19, 29) / / / / (7, 8) (20, 21) / / / / (8, 10) (24, 25) / / (10, 12) (25, 31) / / (10, 13) (28, 30) / / (11, 14) / Table 6 Activity modes of activity 8 Overlaable redecessor, i = 7 m 78 β 78 μ 78 ρ 78 m 8,10, β 8,10, μ 8,10, ρ 8,10, m k β k μ k ρ k Overlaable successor, k = 10 Nonoverlaable successors, k = 11, 22 μ 8 δ 8 CR 8 ($) CC 8 ($) / / / / / / / / / CIRRELT
19 TimeCost TradeOffs in ResourceConstraint Proect Scheduling Problems with Overlaing Modes 4.2 Proect gain maximization and makesan minimization roblems The illustrative case was imlemented in AMPL Studio v1.6. and solved with Clex The gain maximization roblem and the makesan minimization roblem are investigated in this section. The latter is formulated without any uer bound for the overlaing cost. For each of this otimization criterion, the scheduling roblems with or without resource constraints and with or without overlaing modes are resented. Table 7 highlights that a significant reduction of the otimal makesan is obtained with overlaing : 11.11% for the scheduling roblem with resource constraints and 15.53% for the scheduling roblem without resource constraints. For the scheduling roblem without resource constraints and with overlaing modes (case 3), almost all overlaable activities are overlaed (13 out of 15) and most of the coules of overlaed activities are overlaed at their local minimum (10 out of 13). In addition, Table 8 shows that all overlaable activities on the critical ath obtained without overlaing modes (case 1) are overlaed in the scheduling roblem with overlaing modes, as any reduction of the time to execute critical activities will decrease the roect makesan. The roect gain obtained with the makesan minimization roblem with resource constraints is slightly ositive. By contrast, the otimal makesan and the corresonding gain found in cases 1 and 3 show that minimizing the makesan using overlaing strategies can result in a lower gain, because the overlaing costs may exceed the oortunity cost. Table 7 Effects of resource constraints and overlaing on the otimal roect makesan Case Resource constraints Overlaing modes Number of overlaable activities Number of overlaed activities Otimal makesan Corresonding gain CPU s Time (s) 1 No No Yes No No Yes Yes Yes Table 8 Critical activities with and without resource constraints Critical activities Case 1 Case 3 (1, 2, 3, 7, 8, 10, 13, 21, 23, 24, 25, 31, 32) (1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 13, 18, 20, 21, 23, 24,25,31,32) When resource constraints are considered, overlaing is less erformed than without resource constraints. As exected, overlaing lead to additional workload and to more resource consumtions. Overlaing is thus less attractive and less than half of the set of overlaable activities are overlaed with resource constraints and overlaing modes (case 4). This confirms that overlaing and resource constraints are closely interrelated. The overlaing modes of the otimal makesan obtained for both the makesan minimization and the gain maximization roblems are detailed in Table 9. CIRRELT
20 TimeCost TradeOffs in ResourceConstraint Proect Scheduling Problems with Overlaing Modes Table 9 Effects of resource constraints and overlaing on the otimal roect gain coule of overlaable activities Makesan minimization roblem Gain maximization roblem Case 3 Case 4 Case 3 Case 4 overlaing mode overlaing mode overlaing mode overlaing mode (2, 3) (3, 6) (6, 18) (7, 8) (8, 10) (10, 12) (10, 13) (11, 14) (11, 17) (13, 21) (19, 29) (20, 21) (24, 25) (25, 31) (28,30) The results obtained for the roect gain maximization roblem in Table 10 show that the otimal gains in cases 3 and 4 (with overlaing modes) are significantly better than those cases 1 and 2 (without overlaing), and better than the gain obtained for the roect makesan minimization. However, the corresonding makesan is not as good as for the roect makesan minimization. Indeed, the otimal schedules in terms of roect gain lead to a 8.33% and a 12.50% reduction with and without resource constraints, resectively. Tables 7 and 10 also reveal that overlaing modes significantly increase the comutational time required to solve the otimization roblems, as it adds further comlexity to the already comlex case of resourceconstrained scheduling roblem, which is known to be a NPhard otimization roblem (Herroelen 2005). Table 10 Effects of resource constraints and overlaing on the otimal roect gain Case Resource constraints Overlaing modes Number of overlaable activities Number of overlaed activities Otimal makesan Corresonding gain CPU s Time (s) 1 No No Yes No No Yes Yes Yes In conclusion, the otimal schedules obtained for the gain maximization and the makesan minimization roblems differ in terms of resulting gain and makesan. Minimizing the roect makesan requires overlaing several coules of activities, which entails additional rework and communication/coordination costs. In order to conciliate these two contradictory targets, we roose to investigate the timecost tradeoff in the next section. 18 CIRRELT
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