Evaluation of the Resource-Constrained Critical Path Method Algorithms

Evaluation of the Resource-Constrained Critical Path Method Algorithms Kyunghwan Kim1 and Jesús M. de la Garza, A.M.ASCE2

Abstract: This study evaluates the resource-constrained critical path method 共RCPM兲, which the writers have recently proposed. RCPM establishes a critical path method 共CPM兲-like, resource-constrained schedule by resource-dependent activity relationships 共or resource links兲 that the five-step RCPM technique identifies. With its CPM-like feature, RCPM provides the critical path and float data that are not available in traditional resource-constrained scheduling techniques. In addition, RCPM provides more flexibility to the schedule through identified alternative schedules, which allow certain activities to be executed beyond their late finish times without delaying the project completion. This paper evaluates the RCPM’s performance by comparing it with five related previous studies. A brief review of each study is also included in this paper. This comparison shows that RCPM performs well in identifying resource links and alternative schedules, compared to other methods. This study is of interest to academics because it highlights the advantages and disadvantages of different algorithms that have attempted to overcome present problems in traditional resource-constrained scheduling techniques. DOI: 10.1061/共ASCE兲0733-9364共2005兲131:5共522兲 CE Database subject headings: Scheduling; Critical path method; Resource allocation; Construction management.

Introduction Applying resource constraints is required in construction scheduling. Otherwise, the schedule is not realistic, since some resources are highly limited in most construction projects and the startability of certain activities is determined by the limited resources. Many resource constrained scheduling 共RCS兲 techniques have been developed to apply resource constraints 共Kelly 1963; Moder et al. 1983兲. These traditional RCS techniques successfully generate resource-constrained schedules, in which all activities can be executed without resource constraints. However, they do not provide correct float data and the critical path of the schedules 共Wiest 1964; Woodworth and Shanahan 1988; Fondahl 1991; Just and Murphy 1994; Bowers 1995; Lu and Li 2003兲. A resource-constrained schedule contains resource dependencies between activities as well as technological relationships. The traditional RCS techniques do not consider the resource dependencies, resulting in incorrect activity float data and then incorrect critical path. Correct float data and the critical path are prerequisites in construction scheduling and control. Without this information, the project would be very hard to control, since every activity should be treated as a critical activity. In addition, the total float of an activity is very important in delay impact analyses because of its sharing property with other activities and the amount of impact on the project completion time 共de la Garza et al. 1991; Callahan et al. 1992; Bartholomew 1998兲. The other drawback of a resource-constrained schedule is in1

Research Scientist, Construction Research Institute, Hanyang Univ., Seoul, 133-791, Korea. E-mail: [email protected] 2 Vecellio Professor, Charles E. Via, Jr., Dept. of Civil and Environmental Engineering, Virginia Polytechnic Institute and State Univ., Blacksburg, VA 24061-0105. E-mail: [email protected], Program Director, Information Technology and Infrastructure Systems, Civil and Mechanical Systems Division, National Science Foundation, Arlington, VA 22230. E-mail: [email protected]

flexibility in activity schedules. Traditional RCS techniques normally generate a single fixed early start schedule. However, depending on the resource usage and the network conditions, there could be alternative start and finish times for certain activities in the same schedule without delaying the project completion time 共Bowers 2000兲. If these alternative schedules are comprehended initially, the schedule will be more flexible and thus better able to deal with unanticipated events, such as equipment failure, delays in material delivery, etc. A few studies, including the writers’, have proposed heuristic algorithms to correctly identify float data, the critical path, and alternative activity schedules in resource-constrained scheduling. This paper briefly reviews the resource-constrained critical path method 共RCPM兲 that the writers have proposed through Kim and de la Garza 共2003兲, and evaluates it by comparing it with other studies provided in Wiest 共1964兲; Woodworth and Shanahan 共1988兲; Bowers 共1995, 2000兲; and Lu and Li 共2003兲.

Resource-Constrained Critical Path Method Review RCPM establishes a CPM-like, resource-constrained schedule by introducing resource-dependent activity relationships 共resource links兲 in addition to the existing technological relationships in the original CPM schedule. The RCPM process consists of five steps. Step 1 performs traditional CPM calculations. Based on the CPM data of Step 1, Step 2 performs serial resource-constrained scheduling 共Kelly 1963; Moder et al. 1983兲, during which resource links are created if an activity is delayed due to resource constraints. One or more relationships to the delayed activity are created from certain activities that have caused the delay and that have released the delay-causing resources after their completion. Step 3 calculates late start and finish times through a backward pass, considering both technological and resource links identified in Step 2. In Step 4, every activity that has a nonzero total float, based on Step 3, is checked to determine if the total float is

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Fig. 1. Critical path method 共CPM兲, resource constrained scheduling 共RCS兲, and resource-constrained critical path method 共RCMP兲 schedules

available for the whole period. Resource links are created if the total float is not available due to resource constraints. The output of Step 4 becomes the final schedule of RCPM, but one more step is performed to provide more flexibility to the schedule. Step 5 identifies alternative schedules for certain activities that can be scheduled beyond late finish times, determined by the resource links. Fig. 1 illustrates the results of CPM, RCS, and RCPM. The CPM schedule cannot be executable with a given resource availability of three units. In the RCS schedule, resource conflicts are removed, but most total floats should not be trusted. For example, Activity B has a 2-day total float, but a 1-day completion delay of it should delay the start time of Activity D by 1 day because of resource constraint, and then the completion time of this schedule should be delayed by 1 day. Hence the 2-day total float of Activity B does not exist; this nonexisting total float in the RCS schedule is referred to as Phantom Float. Through the RCPM procedure, three resource links are identified; resource links A to B and B to

D are created in Step 2 and resource link C to B is created in Step 4 of the RCPM procedure. With these resource links, the schedule has correct total floats and a critical path considering both technological and resource dependent activity relationships. In addition, as an alternative schedule, Activity C can be scheduled for day 6 or 7, which are beyond its late finish time 共day 2兲 determined by resource links. More detailed algorithm description and examples are provided in Kim and de la Garza 共2003兲.

Resource-Constrained Critical Path Method Evaluation against Other Algorithms The RCPM algorithm has been evaluated by comparing it with previous algorithms proposed by Wiest 共1964兲; Woodworth and Shanahan 共1988兲; Bowers 共1995兲; and Lu and Li 共2003兲, respectively. This evaluation focuses on the feasibility of identified resource links, the existence of required resource links, and the

Fig. 2. Comparison with Wiest’s algorithm

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Table 1. Float Comparison with Wiest’s Algorithm RCPM ID

TF

Wiest’s FF

1 0 0 2 2 0 3 2 0 4 2 2 5 1 1 6 0 0 Note: RCPM⫽resource-constrained critical path applicable.

TF

FF

0 3 2 NA 2 1 0 method; and NA⫽not

amount of floats. For the evaluation of alternative schedules, RCPM is compared with Bowers 共2000兲.

Wiest’s Study Review Wiest 共1964兲 proposes a systematic procedure to identify the total float of each activity on a resource-constrained schedule. The total float of an activity is the start time difference between the leftjustified schedule 共early start schedule兲 and the right-justified schedule 共late start schedule兲. In the left-justified schedule, no activity can start earlier by left shifting of that activity alone without violating technological relationships and resource constraints. The traditional RCS result can be a left-justified schedule because it finds the earliest possible start time for each activity, considering technological relationships and resource availability 共Moder et al. 1983; Ritchie 1985兲. Given the left-justified schedule and resource constraints, the right-justified schedule can be established based on the following priority rules: 1. Right shift activities with the latest EFT first; 2. In the case of a tie in the EFT, right shift activities with the latest possible LST first; 3. In the case of a tie in the LST, right shift activities with the smallest resource requirement first; and 4. In the case of a tie in the resource requirement, right shift activities with the smallest ID first. In this right-justified schedule, some activities can start later than the left-justified schedule, and the start time difference of an activity between two schedules becomes the total float of the activity. Since the “critical path” in CPM normally reflects technological relationships only, Wiest 共1964兲 refers to the chain of activities with a zero total float as the “critical sequence.” Comparison The schedule shown in Fig. 2共a兲 is used for the comparison. The RCPM-generated schedule is shown in Fig. 2共b兲, in which the resource availability is five units and a curved link represents a resource link. RCPM also generates alternative schedules for Activities 2 and 3; either or both of them can be scheduled for day 4, which is beyond their late times. Wiest 共1964兲 does not provide a specific RCS technique, so the RCPM-generated schedule 关Fig. 2共b兲兴 without resource links is used for a Wiest’s left-justified schedule, as shown in Fig. 2共c兲. Fig. 2共d兲 is the right-justified schedule from Fig. 2共c兲. Again, no activity in the left-justified schedule can start earlier by left shifting it without violating technological relationships and resource constraints. Similarly, no activity in the right-justified schedule can start later by right shifting

it. Total and free floats of each method are shown in Table 1. The main difference between the two methods is the existence of resource links. RCPM provides resource links, while Wiest’s does not. Through resource links, the RCPM schedule provides more useful information than Wiest’s. First, the RCPM schedule shows how total floats are shared by activities on a path through the technological and the resource links, while Wiest’s does not. There is no resource availability violation for the TF periods of each activity in the RCPM schedule, while Wiest’s has possible violations on day 2 and 3 because no information is provided about the shared total floats among activities. Second, the alternative schedules in the RCPM schedules can give more flexibility, which would compensate the less TF in Activity 2 of the RCPM schedule. Finally, the RCPM schedule shows activity free floats, again through the technical and the resource links, while Wiest’s does not. On the other hand, a limitation of RCPM is that it does not show the exact resource constraint condition with schedule changes. For instance, from Fig. 2共b兲, when the finish time of Activity 1 is delayed by one day, there is no need to delay the start time of Activity 5, which should be delayed only if both Activities 1 and 4 are delayed. This limitation results from the lack of dynamic features in RCPM with schedule changes. RCPM mainly focuses on the early times of activities in creating resource links and generates at least one resource link by a priority order 共resource requirement兲 in Step 2, when multiple activities are involved. A resource link is created even if the resource requirement of two resource-linked activities is less than the maximum availability. Without this link, representing the delayed activity in a CPM-like schedule is not available within the current RCPM algorithm. For instance, from Fig. 2共b兲, if the resource link between Activities 1 and 5 does not exist, then the early start time of Activity 5 in the RCPM schedule with resource links will be the beginning of day 1, which does not correctly represent the schedule. In addition, all resource links in RCPM are permanent, but some resource links should be temporal at the initial schedule. For instance, the resource link between Activities 1 and 5 should be temporal. Then, if Activity 4 is delayed later, the temporal link needs to become permanent. If Activity 1, on the other hand, is delayed later, then the temporal link between Activities 1 and 5 should be removed and a new permanent link between Activities 4

Table 2. Activity Data in Woodworth’s Example Activity

Duration

Resource

2–3 6 A 2–4 3 A 3–11 4 B 4–5 6 A 4–6 3 B 5–10 8 A 6–7 4 B 6–8 3 B 7–9 6 A 8–9 2 B 9–10 8 A 10–12 5 B 11–12 2 A 12–13 2 A Note: Adapted from Woodworth and Shanahan 共1998, p. 90兲.

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Quantity 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Fig. 3. Comparison with Woodworth’s algorithm

Table 3. Comparison with Woodworth’s Algorithm Comparison item

Woodworth’s

RCPM

Project duration

44

46

Number of resource links

8

5 Table 4. Bowers’ Algorithm Overview

Total float

Free float

10 days in one path 1 day for 共4–6, 6–7, 3–11, 6–8, 8–9兲 1 day for 3 days for activity 11–12 1 day for 1 day for 10 days for activity 8–9, 3 days for activity 11–12

activity activity activity activity

4–5, 6–8, 8–9, 10–12

1 day for activity 4–5, 1 day for activity 8–9, 1 day for activity 10–12 Note: RCPM⫽resource-constrained critical path method.

Step

Procedure

1 Forward pass, ignoring resources 2 Backward pass, ignoring resources 3 Forward pass, with resources 4 Note resource utilization history 5 Backward pass, with resources Note: Reproduced from Bowers 共1995, p. 81兲.

Output EST, EFT LST, LFT EST, EFT Resource links LST, LFT

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and 5 needs to be created. More research on this dynamic feature in resource links is required.

Woodworth’s Study

Review The main features of the algorithms by Woodworth and Shanahan 共1988兲 are a resource label for each activity and resourceconstrained linkages between activities. Basic procedures to establish them are as follows: 1. CPM forward and backward passes 2. Forward resource-constrained scheduling • Prior to scheduling, create a resource pool for each resource type. • During scheduling 共parallel method兲, an activity will be delayed if there is not enough resources in the resource pool at the time when scheduling is attempted. • During scheduling, a resource sequence label is given to each activity for an individual resource requirement. Each label consists of a resource ID and a usage order. For example, a label C3 indicates that the activity is the third activity to use Resource C. In addition, the sequence of the activity is also kept in each resource category, so the order of activities that utilize the resource type can be easily identified. 3. Backward resource-constrained scheduling • This procedure starts from the terminal activity and visits predecessors like the CPM backward pass except that several procedures are added to make the resource linkages and to find LFT and LST as described below. • Search backward from the current activity 共in-progress activity兲, considering technological dependencies and/or resource usage history, and find the immediate predecessor

Fig. 4. Resource index of Fig. 2共b兲 schedule

activities of the current activity. The search by resource usage history is available, since each activity has a resource index and each resource category has a list of activities. An identified previous activity could be either a technological predecessor or a resource-constrained predecessor, or both. The LFT of each identified activity is set equal to the LST of the current activity. A resource-constrained link is created if the identified activity is not a technological predecessor of the current activity.

Comparison The scheduling example provided in Woodworth and Shanahan 共1988兲 is used for the comparison. Table 2 shows the activity data and Fig. 3共a兲 shows a time-scaled CPM network. With availability

Fig. 5. Comparison with Bowers’ algorithm: Example-I 526 / JOURNAL OF CONSTRUCTION ENGINEERING AND MANAGEMENT © ASCE / MAY 2005

Table 5. Activity Data in Bowers’ Example 共II兲 Resource ID

Duration

Successors

1 12 2, 3, 4 2 12 5, 6, 7, 8 3 9 10 4 18 11, 12 5 12 9, 14 6 12 18 7 12 18 8 12 18 9 12 18 10 9 15 11 12 20 12 12 13, 16 13 12 20 14 12 17, 25 15 12 17, 28 16 12 17, 32 17 12 22, 30 18 6 19 19 9 22 20 6 21 21 6 30 22 9 23, 27 23 6 24 24 9 26 25 18 26 26 12 34 27 6 29 28 18 29 29 9 34 30 6 27, 31 31 9 33 32 18 33 33 12 34 34 12 35 35 3 Resource availability Note: Reproduced from Bowers 共1995, pp. 84

A

B

C

0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 and 85兲.

0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 1

of one unit for each resource, the RCPM-generated schedule is shown in Fig. 3共b兲 and Woodworth’s result is in Fig. 3共c兲. The Woodworth’s method employs a parallel RCS method, while RCPM does a serial RCS method. In addition, Woodworth and Shanahan 共1988兲 do not provide a specific priority rule when activities compete for a limited resource; instead, any activity can be selected based on the user’s decision. Thus the sequences of activities in Figs. 3共b and c兲 are different and thus resource links are different, resulting in different completion times. Table 3 shows an overall comparison between the two schedules. Since the resource-constrained scheduling techniques applied are different, there are no important issues in this comparison. However, this comparison shows how a resource scheduling technique, serial versus parallel, can affect a schedule in terms of the activity sequence, resource links, and then floats. Although Woodworth’s method performs well with the given example, which is simple in terms of resource requirement 共one unit for all activities兲 and availability 共one unit for all resources兲,

there is no detailed procedure provided for handling multiple resources with multiple activities like the Fig. 2共b兲 schedule. Based on the given procedure in Woodworth and Shanahan 共1988兲, Activity 4 may have resource links with Activities 5 and 6, which are unnecessary in that schedule. Fig. 4 shows the resource index condition of the Fig. 2共b兲 schedule. With this condition, a search from Activity 6 may find Activities 1 and 4 as its predecessors because they are immediate predecessors based on the resource index. Then, a resource link between Activities 4 and 6 will be created; no resource link is required between Activities 1 and 6, since Activity 1 is a technological predecessor of Activity 6. Similarly, a search from Activity 5 may also find Activities 1 and 4 as its predecessors, making a resource link between Activities 1 and 5 and another link between Activities 4 and 5.

Bowers’ Study „for Resource Link… Review Bowers 共1995兲 also suggests creating resource links for a solution to find correct floats and the critical path in a resource-constrained schedule. The Activity-On-Node 共AON兲 network is adopted to represent resource links more efficiently than the Activity-OnArrow 共AOA兲 network, which is employed in Woodworth and Shanahan 共1988兲. The AOA is awkward, creating many dummy activities to represent the resource links. Bowers’ procedure is conceptually similar to that of Woodworth and Shanahan 共1988兲, but the resource link is created before the backward resource pass as shown in Table 4. However, Bowers 共1995兲 does not provide explicit information of how the resource link is created, so it may not be available to handle complex concurrent activities based on information provided in the paper.

Comparison The two examples provided in Bowers 共1995兲 are used in this comparison. The first example is relatively simple with 11 activities as shown in Fig. 5共a兲. Although Bowers’ applies a parallel method while RCPM does a serial method, both methods coincidentally generate the same resource-constrained schedule for this example and then identical resource links as shown in Fig. 5共b兲. The other example schedule provided is in Table 5. Fig. 6共a兲 shows the RCPM schedule, and Fig. 6共b兲 shows Bowers’. Since the density of resource constraints in the schedule is relatively low as shown in Table 5, both Bowers’ and RCPM create the same activity sequence except for Activities 5–9. These differences can be disregarded because each of the activities has the same duration and resource requirement, and the delay impact through the resource links on Activities 14 and 18 is the same in both methods. Bowers’, however, creates two unnecessary resource links in this example. One unnecessary resource link is created between Activities 11 and 13. This resource link is not required because Activities 11 and 13 can be executed concurrently, since each activity requires one unit of Resource B and the maximum availability of Resource B is two. Given the topology of this example schedule, this resource link is unnecessary from the RCPM’s point of view. With this unnecessary resource link, the total float of Activity 11 is 3 days in Bowers’, while 15 days in RCPM. The other unnecessary resource link is created between Activities 22 and 23. There is no total float difference for Activity 22 from this

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Fig. 6. Comparison with Bowers’ algorithm: Example-II

resource link, since the technological link between Activities 22and 23 overlaps the resource link. These two unnecessary resource links lead to a conclusion that Bowers’ does not consider technological relationships while identifying resource links.

Lu’s Study

Review Lu and Li 共2003兲 propose an algorithm, called the resourceactivity critical-path method 共RACPM兲, to schedule activities based on the usage condition of each resource unit. After the forward pass with resources is completed, resource links are identified from the resource usage condition, and the resource links are considered during the backward pass, like in other previously described methods. The forward pass in this method has several unique features, which are briefly explained as follows:

• Time-scaled resource-activity bar chart: In this chart, as shown in Fig. 7, every unit of available resources in a project is listed on the vertical axis and then one activity 共a dashed rectangle兲 is represented as a bundle of time-scaled bars, each of which stands for the resource requirement of the activity. • Ready-to-serve time 共RST兲: Each resource unit has an RST, which is the latest available time of the resource unit in a scheduling phase or the time when it is ready to serve to an activity. All RSTs are normally set to zero at the beginning of scheduling and the RST of each resource unit is increased with completion of an activity that requires the resource unit. For example, if a resource unit is used for an activity, then the RST of the resource unit will become the completion time of the activity, but if a resource unit is never used, the RST will still be zero. • Idle time 共IDT兲: Each resource unit assigned to an activity has a non-negative IDT. The IDT of a resource unit indicates how long the resource unit has been idle before it is assigned to the

Fig. 7. Resource-activity bar chart in Lu’s method 528 / JOURNAL OF CONSTRUCTION ENGINEERING AND MANAGEMENT © ASCE / MAY 2005

Table 6. Activity Data in Lu’s Example Activity

Duration

Table 7. Resource Links in Lu’s Method

Successors

A 2 D, E B 3 F, G C 5 G D 4 H E 4 — F 3 I G 6 I H 2 — I 3 — Resource availability Note: Reproduced from Lu and Li 共2003, p. 415兲.

Resource 4L 4L 4L 3L 1L 2L 2L 2L 2L 6L

activity. Thus the IDT of a resource unit is the difference between the EST of the activity and the RST. • Earliest-ready, first-serving rule: This rule is applied when an activity requires multiple units of the same resource type. The priority is given to the resource unit that has a smaller RST. In the case of a tie, any resource unit can be randomly selected. In this way, all resources can be uniformly assigned to all activities, avoiding overworked and underworked resources. With these unique features, the forward pass is performed as follows: 1. Initialize the RSTs of all available resource units. 2. Select an activity 共current activity兲 based on a given activity selection priority rule. Lu and Li 共2003兲 applied the ResDur 共=resource requirement⫻ duration兲 order, but any priority rule can be employed. 3. Assign resource units to the current activity based on the earliest-ready, first-serving rule.

Activity A B C D E G H Note: Reproduced from Lu and Li 共2003, p. 414兲.

Successors D, E A B, G H F F I

4.

Determine the EST of the current activity. The EST is the maximum of the EFTs of all predecessors of the current activity in the CPM network and the RSTs of all assigned resource units. 5. Calculate IDT of the assigned resource units, i.e., IDT = EST− RST. 6. Check whether the resource units assigned to the current activity can be utilized to other activities during their IDT. If that is available, schedule those activities first and adjust the IDTs accordingly. 7. Determine the EFT of the current activity by adding the EST and its duration, and update the RSTs of all resource units upon the completion of the current activity. 8. Go to Step 2 and repeat the same procedures until all activities are scheduled. Resource links can be identified from the usage condition of each resource unit. Since one resource unit is available for only one activity at one time, an activity that first utilizes the same resource unit assigned to a prescheduled activity becomes a resource-constrained successor of the prescheduled activity. This

Fig. 8. Comparison with Lu’s algorithm JOURNAL OF CONSTRUCTION ENGINEERING AND MANAGEMENT © ASCE / MAY 2005 / 529

Table 8. Alternative Activity Schedules with Flexible Floats Initial schedule

Fig. 9. Missing resource link in resource-constrained critical path method

relationship can be identified from the time-scaled resourceactivity bar chart. Once all resource links are identified, Lu’s method performs the backward pass, creating more reliable float data and the critical path.

Comparison One example schedule provided in Lu and Li 共2003兲, as shown in Table 6, is used for comparison. With this schedule, RCPM generates Fig. 8共a兲 output, creating four resource links and Lu’s method generates Fig. 8共b兲 output, creating five resource links. The forward pass of Lu’s method generates a time-scaled resource-activity bar chart as previously illustrated in Fig. 7. From this chart, nine resource links are identified as listed in Table 7. Some of the resource links 共A-D, A-E, C-G, and D-H兲 are redundant because of the existing technological links in the original CPM network. Fig. 8共b兲 is the schedule in a fenced-bar chart format 共Melin and Whiteaker 1981兲 after those redundant links are removed. There are several differences in the project completion time, the activity sequence, and the resource links between RCPM and Lu’s method, but direct comparison is not suitable because the priority rules applied to each method are different. However, there are several drawbacks in Lu’s method and its output compared to RCPM’s. Lu’s method may generate a large number of redundant resource links as demonstrated in the example because it does not

Float

ID

Duration

ES

LS

F1⬘

F2

F3

EES

LLS

5 6 7 8 9 11

12 12 12 12 12 12

24 36 24 36 48 30

24 36 36 48 48 33

0 0 12 12 0 3

9 12 12 12 12 12

9 24 24 24 12 15

24 24 24 24 36 30

33 48 48 48 48 45

Note: F1⬘: Float 1 共F1兲 of the original schedule 关reproduced from Bowers 共2000, p. 860兲兴.

consider original technological links of the CPM network when resource links are identified. Those redundant activity relationships do not cause any time calculation errors, but the complexity of the scheduling network will be significantly increased. Because of this reason, Lu and Li 共2003兲 suggest that redundant relationships be removed before the backward pass, but they do not provide a procedure as to how to remove them. Another disadvantage is that Lu’s method may create wrong resource links because they are identified based on the resource unit usage. For example, Activities H and I from Fig. 7 are using the same resource units and Activity I is the first activity that utilizes the same resource units of Activity H, so a resource link is created between them as shown in Fig. 8共b兲. However, in the sense of the traditional resource-constrained scheduling concept, Activities H and I can be concurrently executed, since the maximum resource availability is six. A similar case happens for the resource links of Activity E-F and Activity G-F. Only one resource link will be required if either Activity E or G is delayed and concurrently performed with Activity F. The other drawback is that Lu’s method with multiple types of resources could generate longer project duration than a conventional RCS technique or RCPM because of the earliest-ready, first-serving rule and ready-to-serve time 共RST兲. For example, assuming all other data are the same, if Activity A requires 2L instead of 4L and one unit of a different resource type, i.e., one carpenter that has a limit of one, RCPM generates a 17-day schedule, 2 days shorter than the original in Fig. 8共a兲. This shorter duration is available because Activity A can be scheduled to start

Fig. 10. Initial schedule of activities 5–9 in Bowers’ method 530 / JOURNAL OF CONSTRUCTION ENGINEERING AND MANAGEMENT © ASCE / MAY 2005

Table 9. Alternative Activities Schedules in Resource-Constrained Critical Path Method Initial schedule

Alternative period

ID

Duration

ES

LS

Start

End

6 7

12 12

24 36

24 48

36 48

60 60

on day 0 and its subsequent activities can be scheduled earlier than the original. However, the schedule in Lu’s method will be the same as in Fig. 7 or Fig. 8共b兲 because of the earliest-ready, first-serving rule and RSTs. When Lu’s method attempts to schedule Activity A after scheduling of Activities C, B, and G, the RSTs all 6L are still 14, 14, 8, 8, 8, and 8, respectively 共see Fig. 7兲. In addition, scheduling Activity A using Activity B’s IDT 共in step 6 of the forward pass in Lu’s method兲 is not available, since Activity B does not require any carpenter. A similar case happens on the other scheduling example provided in Lu and Li 共2003兲. On the other hand, there is a disadvantage in RCPM compared to Lu’s method. RCPM does not detect all resource links required to show correct resource dependencies between activities. For example, as shown in Fig. 9, a delay of Activity A should delay Activity D because of the resource limit. In order to represent this condition correctly, there should be a resource link between Activities A and D. RCPM does not detect this resource link, but Lu’s method generates this resource link because Activity D immediately utilizes the same resource unit assigned to Activity A. This missing resource link is the result of the deliberately simplified heuristic RCPM algorithm to save the computation time in real projects with a large scheduling network. The TF will be the same no matter how many successors by resources are created from certain activities, Activity A in this case. Instead, rescheduling with RCPM is required whenever the schedule is changed. For example, if the duration of Activity A is increased, then RCPM will detect the resource link between Activities A and D.

Bowers’ Study „for Alternative Schedules… Review Bowers 共2000兲 proposed a set of procedures to identify alternative schedules and to measure the start time flexibility of each activity

Table 10. Algorithm Comparison Summary Wiest’s Woodworth’s Bowers’ Lu’s RCPM Creation of RL — Consideration of O multiple resources and activities Validity of identified — RL Completeness of — RL Dynamic feature of — RL Note: RCPM⫽resource-constrained ⫽resource links.

O —

O —

O O

O O

O

—

—

O

—

—

O

—

—

—

—

—

critical path method; and RL

in a resource-constrained schedule. The method proposed in Bowers 共1995兲, which identifies resource links in RCS, is applied to calculate late start and finish times. In order to find alternative schedules, the algorithm generates 4n + 2 resource-constrained schedules per project, where n equals the number of activities in the project. The original resource-constrained schedule is used for a basis to create two schedules for each activity, attempting to schedule the project after setting the earliest possible start time of the activity as its original LST 共first schedule兲 and later than the original LST 共second schedule兲. In the same way, the reversed schedule of the original schedule is used as the basis to create another two schedules. Thus four scheduling processes are performed for each activity, creating additional 4n resourceconstrained schedules per project. Among the 4n schedules, only equivalent schedules, which have the same duration of the original schedule, are eligible for consideration of alternative schedules. In order to measure the start time flexibility, three float values are compared for each activity. Float 1 共F1兲 belongs to each particular schedule of the eligible schedules, Float 2 共F2兲 is the maximum of all F1 of an activity, and Float 3 共F3兲 is the difference of the latest late start time 共LLS兲 and the earliest early start time 共EES兲 of an activity among all eligible schedules. If F2 or F3 of an activity is greater than the F1 of the original schedule, the activity can be scheduled at an alternative time, which can be found from the eligible schedules.

Fig. 11. Initial schedule of activities 5–9 in resource-constrained critical path method JOURNAL OF CONSTRUCTION ENGINEERING AND MANAGEMENT © ASCE / MAY 2005 / 531

Comparison Bowers 共2000兲 employs the same example schedule introduced in Bowers 共1995兲, Table 5 in this paper, to demonstrate alternative schedules or scheduling flexibility. Table 8 shows activities that can be scheduled in different periods from the initial schedule with the same project completion time. Six among 35 activities could have alternative schedules. Activities 5, 7, and 11 can be scheduled later than the initial late start time, since latest late start time 共LLS兲 ⬎ initial late start time 共LS兲. Activities 8 and 9 can be scheduled earlier than the initial early start time, since earliest early start time 共EES兲 ⬍ initial early start time 共ES兲. Activity 6 can be scheduled earlier than the initial early start time or later than the initial late start time, since EES⬍ ES and LLS⬎ LS. These activity data provide a baseline to reorganize an alternative schedule when an unexpected event happens, resulting in delays of some critical activities. For example, if Activity 5 共initially critical as shown in Fig. 10兲 should be delayed, a best-fit schedule that has 33 for its LLS can be selected, and then there will be no completion time extension. With the same schedule, RCPM has identified two alternative schedules as shown in Table 9. Activity 6 is a critical activity in the initial schedule 共see Fig. 11兲, but it can be scheduled over its LFT; the period of 36–48 is available since Activity 9 can be delayed within its TF range, and the period 48–60 is available since the current daily resource requirement is one. Activity 7 is also a critical activity in the initial schedule, but it can also be scheduled over its LFT; the period 48–60 is available since the current daily resource requirement is one. However, both activities cannot be scheduled for the period 48–60. If Activity 8 starts earlier than the initial schedule, both activities can be scheduled for the period, but currently RCPM does not consider this earlier start time. Bowers’ allows an activity to start earlier than the EST from the initial schedule, so that it provides more exhaustive alternative schedules than RCPM. However, only one schedule can be selected in Bowers’. Once one schedule is selected, the other data 共EES and LLS兲 are no longer valid unless they are on the selected schedule. On the other hand, most alternative schedules in RCPM are available except some activities whose alternative schedule periods are overlapped by the same resources with resource overuse, like the period 48–60 of Activities 6 and 7 from Table 9.

Summary and Conclusion This paper has evaluated the RCPM technique by comparing it to five other algorithms. RCPM performs well in identifying resource dependencies between activities compared to other previous algorithms as summarized in Table 10. First, identified resource links show how resource dependencies are present in the scheduling network and how a change in an activity affects other subsequent activities. In addition, there is no resource limit violation for the total float period through the resource links. Second, RCPM provides a systematic procedure to handle multiple resources in multiple parallel activities. Finally, RCPM does not create unnecessary resource links, which can cause incorrect float data, because RCPM considers technological relationships when it identifies resource links. However, RCPM does not detect certain resource links when they do not affect the total floats of activities, so that RCPM procedure to find resource links is required whenever the schedule is updated. In addition, RCPM, like

the other algorithms, does not provide dynamic features of resource links with schedule changes. Alternative schedules of certain activities are additional benefits of RCPM to make the schedule more flexible against unanticipated circumstances. Most alternative schedules are available in the RCPM schedule unless they are overlapped with each other competing for the same resources beyond their availability. With these advantages, RCPM can be a more practical tool in project scheduling and control in real construction projects than other techniques. In addition, because the U.S. courts expect resource-loaded schedules for a time impact analysis 共Wickwire et al. 2001兲, RCPM will become more valuable and the usage will be increased. Continued research in this area will enhance the capability of the current RCPM technique, add new knowledge, and provide valuable assistance for construction project management.

Acknowledgments The research work described in this paper was funded by the Virginia Tech Construction Affiliates’ Center for Construction Improvement and the National Science Foundation. The opinions and findings are those of the writers and do not necessarily represent the views of the sponsors.

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