An efficient method for scheduling construction projects with resource constraints

An efficient method for scheduling construction projects with resource constraints

International Journal of Project Management 19 (2001) 29±45 www.elsevier.com/locate/ijproman An ecient method for scheduling construction projects ...

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International Journal of Project Management 19 (2001) 29±45

www.elsevier.com/locate/ijproman

An ecient method for scheduling construction projects with resource constraints M. Chelaka L. Abeyasinghe, David J. Greenwood *, D. Eric Johansen Department of the Built Environment, University of Northumbria at Newcastle, Ellison Building, Newcastle Upon Tyne, NE1 8ST, UK Received 28 September 1999; received in revised form 12 April 2000; accepted 27 April 2000

Abstract In construction scheduling, con¯icts can arise when activities require common resources that are available only in limited quantities. To overcome this, while retaining minimum project durations, mathematical techniques have been developed for allocating resources. However, these produce a `hard' in¯exible approach to resource-constrained schedules. The authors propose an ecient resource allocation algorithm (LINRES) which o€ers a more ¯exible approach. To study its performance, an experiment was conducted on 10 small network examples (6 to 29 activities) and the results were compared with those generated by a total of 32 existing heuristic rules. The results show that the LINRES algorithm outperformed most other heuristic rules, including the widely used MINSLK rule in both single- and multi-resource networks. It also provides a reasonable trade-o€ between the resourceaggregation pro®les and the durations. # 2000 Elsevier Science Ltd and IPMA. All rights reserved. Keywords: Heuristic rules; Planning and scheduling; Precedence networks; Resource-constrained scheduling

1. Introduction Construction industry projects involve complex packages of work for which the design and contracting organisations are responsible; the product is generally large, discrete and prototypical. These and other characteristics of the industry make particular demands upon the planning and scheduling techniques that have to be developed to serve it. It has been argued that the more sophisticated planning methods used in other industries do not suit the construction industry. For example, Johansen [1] in a study of small and mediumsized UK building projects found that site managers tended to discard the formal systematic schedules they inherited from head oce, which they mistrusted as `theoretical', and adopted their own more `¯exible' approach to scheduling work. To dismiss this as unenlightened site management, or as a response to the inevitable uncertainty of construction projects is to

* Corresponding author. Tel.: +44-191-22-74-691; fax: +44-19122-73-167. E-mail address: [email protected] (D.J. Greenwood).

ignore the evidence that many network-derived schedules simply do not work in the ®eld. Woodworth and Shanahan [2] have shown that schedules based on timeoriented networks are exceeded by an average of around 38%. 1.1. Schedules based on time-oriented networks Conventional critical path method (CPM), and programme evaluation and review technique (PERT) scheduling procedures start with an assumption of unlimited availability of resources for each project activity [3±5]. In other words, the analysis is based solely on the time requirements of the activities regardless of the resource needs of each activity. The early and late dates calculated with the critical path algorithm are based on the duration of the activities in the project and the relationships or technological constraints between them. It produces a schedule that minimises the overall project duration. In these calculations, the issue of resource availability is either not taken into account or neglected until after the initial time-oriented calculations [6]. Many of the problems with real-life projects arise when activities require resources that are available only in

0263-7863/00/$20.00 # 2000 Elsevier Science Ltd and IPMA. All rights reserved. PII: S0263-7863(00)00024-7

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limited quantities and the demands of concurrent activities cannot be satis®ed [2]. Recognition of this limitation has directed many researchers towards the problem of scheduling activity networks under resource constraints. 1.2. Resource scheduling Resource scheduling explicitly and systematically incorporates decisions about the capacity into the scheduling process. Gordon and Tulip [7] have outlined its history and development and describe the basic steps of aggregation, smoothing and levelling of resources. Matthews [8] identi®es two approaches to resource scheduling as: . Time-constrained scheduling . Resource-constrained scheduling Both methods begin with a time analysis (CPM) (which fails to consider capacity and therefore contains `unrealistic' dates) and proceed to resolve resource overloads by moving activities into periods when the capacity to undertake the activity exists. Time-constrained resource scheduling assumes that time constraints are ®xed, and seeks to resolve capacity overloads by manipulating the timing of activities within their total ¯oat, and without a€ecting the initial project completion time. Resource-constrained scheduling accepts the priority of ®xed resource availability, and permits not only sequencing and ¯oat times to be altered, but (if necessary) the project duration to be increased beyond the initial non-constrained project duration. In many cases therefore, time analysis and time-constrained scheduling should be considered only as intermediate steps in the process of schedule development [8]. In terms of performing resource scheduling, Gordon and Tulip [7] identify two main approaches, the `serial approach' (where priority indices are determined once, before starting the scheduling operation) and the `parallel approach' (where priority indices are updated each time an activity is scheduled). A parallel approach with allowable relaxation of `total resource limitation' and `total time limitation' was considered suitable for `construction type' projects since they tend to contain a high proportion of activities which can be split. 1.3. Optimising the results of resource-constrained scheduling The general resource-constrained project scheduling problem (RCPSP) arises when a set of interrelated activities (precedence relations) is given and when each activity can be performed in one of the several ways (modes). Questions arise regarding which resourceduration mode should be adopted, and when should each activity begin so as to optimise some pre-speci®ed

managerial goal. The general version of the problem is that each activity could be performed in one of the several ways, i.e. a continuous duration-resource function. For simplicity, this study has been restricted to a discrete duration-resource function where only one execution mode for each activity will be assumed. It also operates with a version of the problem where resources are renewable, activities cannot be interrupted, and the managerial objective is to minimise project duration. Numerous modi®cations to the original critical path method algorithm have been proposed for solving the scheduling problem under resource constraints. Two broad approaches have been used, namely (a) optimisation by mathematical programming techniques, and (b) heuristic techniques [9,10]. 1.4. Solutions based on mathematical optimisation These techniques seek to de®ne the problem as a mathematical programming problem. The best solution is the one that gives the shortest project duration or the one which provides the smoothest resource pro®le. In some cases, these two objective functions are combined, resulting in preferred trade-o€s; for example, in a slight increase in project duration with a decrease in resource level variation. However, such optimisation techniques remain computationally impractical for most real-life large projects because of the enormous number of variables and constraints. 1.5. Solutions based on heuristics The alternative approach is to develop heuristics which allow a process of choosing between activities that are competing for the use of a scarce resource. The inherent variables in any project scheduling process are time and resources. When these are constrained, the resulting outcomes can vary along a constraint continuum Ð a spectrum of combinations of time and resources with an assumption of unlimited availability at each extremity. The various heuristics and their algorithms assist in deciding upon the point on the continuum at which the schedule should end up. Many heuristic models have been developed and are available as computer packages. Each works di€erently, produces di€erent schedule outcomes, and is likely to be better in some situations than in others. 1.6. Research objectives The research described in this paper has two objectives: 1. to develop a new heuristic, as close as possible to the CPM output, for scheduling activities under resource constraints in which the project duration

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compares favourably with the results of other heuristic rules; 2. to test the performance of the new heuristic by investigating the resources-aggregation pro®les corresponding to each of the project milestones enhanced in terms of their availabilities, requirements, actual utilisation and levels of idleness, and thereby anticipating future resource demands. There follows a review of heuristic techniques from the RCPSP literature, followed by the presentation of a new single- and multi-resource planning and scheduling model (LINRES). The model is then tested on example problems and the results are compared with those that result from other heuristic rules published in research journals. 2. Heuristic techniques from the literature A common feature of all heuristics is that they provide a criteria for prioritisation. Accordingly, in attempting to perfect algorithms for scheduling activities under resource constraints, theorists have designed various values to determine which activity will get the resources and which activity will be delayed. One of the simpler examples is the work of Matthews [8] who tackled the RCPSP problem using resource allocation algorithms with di€erent prioritisation criteria for activities. Thus, activities were allocated a resource and scheduled during the earliest available period, based on one or the other of the following values: 1. 2. 3. 4. 5. 6. 7. 8. 9.

Lowest total ¯oat time Earliest start prioritisation Latest start prioritisation Earliest ®nish prioritisation Latest ®nish prioritisation Activity duration (ascending) Activity duration (descending) Activity number (ascending) Activity number (descending)

The results showed that the above heuristics 1, 3 and 7 performed best (i.e. gave the shortest project duration) compared with heuristic 9 which was the worst (i.e. longest, giving a duration approximately 13% greater than the best value). Elsayed and Nasr [9] examined a number of heuristics for allocating resources in a single-project under singleresource constraints. Each heuristic involves the prioritisation of activities based on values (ACTIM, ACTRES, TIMRES, GENRES, ROT, ACROS, TIMROS and TIMGEN) calculated for each activity. Some of these values are weighted and some are more complex

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combinations of several simpler ones. The ACTIM value of an activity is calculated as the maximum length of time that the activity `controls' throughout the critical path network;1 the ACTRES value is the sum of the products of the time and resources that the activity `controls' through the network; TIMRES and GENRES are weighted combinations of the ACTIM and ACTRES values;2 ROT is determined by the maximum sum, on any one path that the activity controls, of the ratios of resources to time; ACROS is the maximum resource value on any one path that the activity controls through the network; TIMROS and TIMGEN are respectively, weighted combinations of the ACROS and ACTIM, and TIMROS and GENRES values. The experiments carried out by Elsayed and Nasr [9] demonstrated that TIMROS and TIMGEN outperformed the others, in terms of meeting resource constraints within the shortest project duration. The previously referenced work compared the relative performance of di€erent heuristics against one another; however, Davis and Patterson [11] compared the e€ectiveness of alternative heuristic sequencing rules relative to a mathematically optimised solution using `bounded enumeration' procedure [12±14]. The comparison between optimum and heuristic-based solutions was performed over a group of (single-project, multiresource) test problems. The heuristic sequencing rules selected were used in a parallel approach, in which activity priority is determined during scheduling rather than before. The heuristics were: . . . . . . . .

Minimum job slack (MINSLK) Resource scheduling method (RSM) Minimum late ®nish time (MINLFT) Greatest resource demand (GRD) Greatest resource utilisation (GRU) Shortest imminent operation (SIO) Most jobs possible (MJP) Select jobs randomly (RAN)

In the resource scheduling method (RSM), priority is given to the activity with the minimum time between its own earliest ®nish time (EFT) and the latest start time (LST) of the succeeding activity. The remainder of the values are fairly self-explanatory; the reader is referred to the original paper for a more detailed explanation of each. In Davis and Patterson's experiment [11], MINSLK, MINLFT and RSM performed the best, whereas GRU, 1

Thus, the ®rst critical activity in a network would have a higher ACTIM value than the last. 2 The TIMRES value is an average of an activity's normalised ACTIM and ACTRES values, whereas GENRES is calculated as a speci®c weight w …0  w  1† of the sum of the weighted values of normalised ACTIM and ACTRES values.

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GRD, SIO and MJP were poor performers (compared even to RAN) in minimising project duration. The results also revealed that certain project and resource characteristics can a€ect the performance of a particular heuristic sequencing rule. These characteristics include: . The ratio of average resource requirements per activity to the amount available . The ratio of average total ¯oat per activity to the critical path length (average ¯oat ratio) . The ratio of number of activities to number of precedence relationships (project complexity) Further investigation into the performance of heuristic sequencing rules was carried out by Boctor [15], whose intention was to introduce some ecient multi-heuristic procedures. Tests were performed on a number of small (5±20 activities) and large (38±111 activities) projects, using up to three resource types. The measures used to assess the eciency of each heuristic were: 1. the average percentage increase in project duration above the optimum duration or the critical path length; 2. the number of times each heuristic produced the optimum or the shortest duration. The results con®rmed earlier ®ndings that MINSLK, MINLFT and RSM were the most ecient rules. When combinations of heuristics were employed sequentially, the most ecient combinations were always found to include the MINSLK rule, and the probability of getting the best solution was proportional to the number of heuristics combined. Boctor [15] also observed that serial heuristics are, on an average, ®ve times faster to perform than parallel ones, although RSM requires three times the processing time of other parallel heuristics. Kolisch [14] was critical of the logic behind the commonly used RSM priority rule and suggested an improved model known as improved RSM (IRSM). He also developed two new priority rules: worst case slack (WCS) and average case slack (ACS). Kolisch showed that on an average, the three new rules outperformed the other heuristics, including LFT and MINSLK. Moselhi and Lorterapong [13] proposed a model for assigning scarce resources to activities collectively rather than individually. The new algorithm was coded in BASIC language along with four highly regarded heuristic rules: MINSLK, MINLFT, GRD, and Shortest Duration (SHD). A total of 31 CPM-type multiresource-constrained project networks were taken from the literature and used to examine the performance of the new algorithm against existing heuristic rules

(including RSM) and a number of available optimal solutions. The proposed algorithm proved superior (i.e. resulted in shorter project durations) in the majority of cases. In response to the problems associated with a multiplicity of resource constraints, Nkasu [6] developed an iterative heuristic scheduling method known as COMputer Sequencing Approach to multi-Resource-constrained Scheduling (COMSARS). The method attempts to produce schedules that minimise the total project completion time as well as minimise the total idle resources, thereby facilitating project-cost savings. It reports a chronological listing of all the activities that start and ®nish in conformity with the constraints imposed by both, the activity precedence relationships and the resource availabilities. Further, COMSARS produces the resource aggregation pro®les in terms of their availabilities, requirements, actual utilisation, and levels of idleness at various identi®able milestones throughout the entire project duration. As Davis and Patterson [11] have noted (see above), the performance of a particular heuristic should be investigated with regard to the project and resource characteristics that may a€ect it. This issue is addressed in a study by Ulusoy and Ozdamar [16] in which a new heuristic rule, weighted resource utilisation and precedence (WRUP) is compared with the widely used rules. The authors examine the relationship between the performance of seven heuristics (MINSLK, LFT, RSM, GRD, SIO, RAN and the new proposed rule WRUP) and four network/resource characteristics: aspect ratio, complexity, resource utilisation factor and dominant obstruction value. WRUP prioritises activities based upon a combination of two weighted values Ð the number of immediate successors and the resource utilisation ratio Ð for each activity. Thus Priority ˆ w…p†n…i† ‡ w…r†

X r…ik†=R…k† k

where w(p) w(r) n(i) r(ik) R(k)

precedence weight, resource utilisation weight, [1 ÿ w…p†] number of immediate successors of activity i required units of resource type k per period by activity i units available of resource type k per period

The balance of weighting depends upon the nature of the problem: high resource requirement problems require a higher weighting for the resource utilisation ratio; for moderate resource utilisation networks, a 1:1 ratio between the two weights are recommended.

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The tests revealed that WRUP performed best, with LFT and MINSLK being second and third-best respectively. Furthermore, since the use of LFT, MINSLK and RSM heuristics requires the critical path network to be resolved and priorities reassigned at each event completion, they take longer to perform than WRUP. In terms of the signi®cance of di€erent network/ resource characteristics on the performance of the heuristics, the e€ect of di€erent combinations of the four characteristics from a total of 64 test problems was examined. The interaction of the resource utilisation factor and complexity was found to be signi®cant in almost all the cases. 3. Proposed new resource scheduling algorithm 3.1. Introduction The previous section has reviewed the problem of project scheduling under resource constraints, and described some of the e€orts of researchers to solve it by developing and testing heuristics. The present section presents a new heuristic approach to the solution of resource-constrained scheduling problems. The new heuristic (LINRES) was developed manually using precedence network (activity-on-node) and Gantt chart representations of a project. The method can cope equally with networks conceived in the activity-onarrow format. The aim is to establish a resource schedule which operates within given resource constraints, but with the shortest possible project duration, that is, a duration as close as possible to that of the initial unconstrained version. Unlike most of the heuristics previously reviewed, LINRES does not employ a priority dispatching approach, thus, it is neither a serial nor a parallel method of schedule construction. The LINRES algorithm uses conventional CPN and Gantt charts to create an unconventional type of ancillary network (containing a number of new rules and concepts) as a tool for solving resource-constrained scheduling problems. The ancillary network will be introduced and developed in the following sections. Newly coined de®nitions and concepts appear in italics in the text. They are explained by footnotes and are summarised schematically in the following section. 3.2. LINRES applied to a simple project For simplicity's sake, the process will ®rst be demonstrated on the small example project (seven activities plus `dummy' start and ®nish activities) illustrated in Fig. 1. Only such rules as are pertinent to the demonstration project will be introduced at this stage. For the rationale behind the fuller, generalised model see Abeyasinghe [17].

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As with traditional approaches, the following initial information is required: 1. 2. 3. 4. 5.

the project activities; their precedence relationships; the estimated duration of each activity; the resource requirements of each activity; the limitation on the availability of each of the resources speci®ed.

The method begins with a standard critical path analysis (Fig. 1) and the ®rst step in LINRES is to determine the activity links3 based on it. The resulting network, the link-structure4 is then manipulated to avoid any problem arising from resource constraints. This involves consideration of both the time and resource dimensions simultaneously and introduces the concept of companion activities.5 The link-structure is then compressed as far as possible, according to certain rules, and the resulting solution can then be translated back into a bar chart or CPN format. 3.3. A simple example A small example project (seven activities plus dummy start and ®nish activities) is shown in Fig. 1. The same example is presented in the form of a Gantt chart in Fig. 2. There is only one resource, and its limit is 5 units at any one time. Step 1: Create a network using the standard critical path procedure (see Fig. 1). Step 2: Create a Gantt chart based on the early startand-®nish dates (see Fig. 2). Step 3: Using the Gantt chart obtained in Step 2 draw the LINRES link-structure for the project. The ®rst stage is to proceed systematically through the project's time-scale and determine the activity links. These correspond to the various logic paths in the project network (A±C±E, A±D±F, and B±G in the example in Fig. 1). The interconnection of logic paths is accommodated by using two forms of link: those between on-line activities (for example A±C±E) are shown by right-handed arrows, whilst o€-line links between companion activities (activities that start at the same time, such as A±B and C±D) are shown by vertical lines drawn between them. An annotated illustration of this step is given in Fig. 3. Where an activity has more than one immediate successors (with no start-gap6 between them) to ®ll the role of companion activity, they are positioned in descending order of their durations. 3 Geometry of activity relationships represented by using vertical lines and right-handed arrows. 4 Combined relationships of both on- and o€-line activities. 5 Activities which are o€-lined to each other in the link-structure. 6 Start-to-start time di€erence between companion activities.

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Fig. 1. A typical CPM project network with limited resources.

Step 4: This step involves manipulating the model's structure to eliminate resource con¯icts whilst retaining the minimum duration. To achieve this, LINRES requires (i) the altering of companion activity links that result in resource con¯icts, and (ii) the compression of the link structure by removing any on-line links that are not on the bottom link and substituting feasible (i.e. non-con¯icting) companion activity links. This is best shown with reference to Fig. 4. Moving from left to right along the link-structure obtained in Step 3, start with the companion activities and check for resource con¯icts. When a con¯ict is discovered, move the top activity in the link to the right, until it ®nds the nearest feasible companion activity (i.e. one with which it would not produce a resource con¯ict). Where this is not possible, go to Step 5. Thus, in Fig. 4(a), companion activities A and B are non-con¯icting (i.e. their combined resource require-

Fig. 2. Gantt chart (with time constraints) based on early dates.

ment is 5, and thus within the limit imposed) whereas companion activities C and D con¯ict (i.e. their combined resource requirement is 6 units and they cannot feasibly begin at the same time). Their link must therefore be broken (see Fig. 4(b)) and activity D (and with it link D-F) moved to the right to seek a non-con¯icting companion activity. In case of the simple example, however, this is not feasible. There is only one other potential companion activity (Activity E), and combining D and E would cause a resource con¯ict (8 units), thus, the rule in Step 5 must be invoked (see below). The remaining task in Step 4 is the compression of the link structure by removing any on-line links that are not on the bottom link and substituting feasible

Fig. 3. Link-structure for the simple example.

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Fig. 4. Illustration of Steps 4±7 of the LINRES algorithm.

(i.e. non-con¯icting) companion activity links. Thus, in Fig. 4(b), the link B±G is broken and G is moved to the right to ®nd the nearest non-con¯icting companion activity (in this case C) as shown in Fig. 4(c). Step 5: An activity or activity link which has found no companion in Step 4 (as with activity link D±F above) will be transferred to the next available position on the bottom link (taking into account logic and timing). Thus, activity link D±F is transferred as shown in Fig. 4(c). In the simple example, the two processes Ð removing resource con¯icts, and compression of the link structure by moving on-line links to the bottom link are now complete. In larger networks, Steps 4 and 5 would be repeated until there

are no resource con¯icts and the structure has been compressed as described. Step 6: The situation in Fig. 4(c) is now feasible (it does not contravene resource constraints) but has increased the overall duration from 19 to 30 units. The next step is to further compress the link structure. Existing on-line links may be replaced with companion relationships provided that neither logic nor resource limits are violated. Thus, in Fig. 4(d), a companion relationship between F±E has been substituted for the previous on-line linkage (see Fig. 4(c)) as there is neither a logical nor resourcebased objection to E and F commencing simultaneously.

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Step 7: Repeat Steps 4±6 until the minimum-duration baseline activity linkage is obtained (see Fig. 4(d) where A±C±D±F is the baseline activity linkage). The e€ect has been to reduce the overall duration to 28 units. Before proceeding to obtain the baseline activity linkage in Step 7, any activity with more than one on-line link should have it reduced to one by the use of dummy activities with zero duration. This phenomenon is not in evidence in the simple example but its use is illustrated in Abeyasinghe [17]. Step 8: Beside each activity, indicate its duration and resource requirements (in parentheses) then consider all companion activities. For any set of companion activities, if the duration of the baseline activity is greater (or equal) to that of its connector activity or activities (taking into account any start-gap) then the baseline will require no amendments and it becomes the critical path. Where this is not the case, go to Step 10. This is the case in the example, where, for companion activities G±C, the duration of G (16 units) is greater than that of the baseline activity C (10 units). This means that the baseline obtained in Step 7 will need to be amended in line with Steps 10±15 (below). Step 9: Having obtained the ®nal baseline activity linkage, calculate all the possible critical paths to obtain the overall project duration.

Step 10: If any set of companion activities does not comply with the rule in Step 8 (above), then the baseline obtained in Step 7 must be amended. Consider such companion activities one by one (from left to right), and go to Step 11. Step 11: Does any connector cause resource con¯icts with any neighbouring activity or activity group to the right? If so, then go to Step 12. If not, then return to Step 9 and no amendments are required. In the example, it has been noted that connector activity G (16 units) has a longer duration than its companion on the baseline, C (10 units). If G and C were free to start simultaneously, and D was to follow directly after C (the situation as it stands in Fig. 5a) then there would be a resource con¯ict between the ongoing activity G and the commencing activity D. This can be more clearly seen in the Gantt chart version of Fig. 5(a). Thus, in the example it is necessary to proceed to Step 12. Step 12: Does any connector cause resource con¯icts with a neighbouring successor activity on the baseline? If so, then go to Step 14. If not, then go to Step 13. In the example, activity D is on the baseline, therefore it is necessary to proceed to Step 14. Step 13: Possible critical path(s) may occur along the baseline and/or across the connector(s) and onto either the resource con¯icting neighbour(s) or the

Fig. 5. Illustration of Steps 8±14 of the LINRES algorithm.

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Fig. 6. Main steps in algorithm of LINRES.

shortest activity in a `con¯icting group7 scenario. Go back to Step 9; the path(s) that give(s) the longest unique project duration will become critical. The illustration of this rule using the simple example will be left until its link structure is completed (below). Step 14: Move the baseline activity one step backward provided that it does not contravene any logic. The backward movement should be continued until the connector(s) ®nd(s) a resource non-con¯icting neighbour activity on the baseline. Having completed this step return to Step 13; where it is not possible, go to Step 15. In the example, activities G and D were found to be incompatible (see Step 8) and D was on the baseline (see Step 12). D is moved one step backward, and contravenes no logic in its new position 7

Any companion relationship which yields a resource con¯ict.

between A and C (see Fig. 5(b)). We can now return to Step 13 and determine the critical path(s). In the example, critical paths occur along the baseline (A±D±C±F) and also across connectors, as represented in the example by the arrows D±G and G±E in Fig. 5(b). The two resulting critical paths (A± D±C±F and A±D±G±E) can be more clearly seen in the Gantt chart version of Fig. 5(b). Step 15: Where the backward movement is about to contravene logic relationships, (see Step 14) it is terminated. Go back to Step 9, and recalculate all possible critical paths. The path(s) that give(s) the longest project duration will be critical. This step is not required in the example, as the backward movement of D in Fig. 5(b) does not contravene logic. In certain situations, depending on network structure, the LINRES methodology requires the incorporation of

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certain further general principles. These were super®cially covered in the previous section for simplicity's sake, but are described in full in Abeyasinghe [17]. The main steps of the complete LINRES algorithm is shown in Fig. 6. 3.4. Establishing the critical path(s) using scheduled dates Once the `Baseline Activity Linkage' (BAL) has been established, the project's critical path can take any one of the four routes, depending on the circumstances. This is illustrated in Fig. 7 by introducing the concept of scheduled dates. These are the dates that can be calculated as a result of the operation of the LINRES algorithm. The cumulative start and ®nish dates obtained by the LINRES algorithm are referred-to as Early Scheduled Start (ESS) and Early Scheduled Finish (ESF), respectively. As shown in Fig. 7(b), the ESS of activity P is 0. The ESF will be obtained by adding the duration to the ESS, i.e. (0 ‡ dp ). The ESF of an activity will then be the ESS of its immediate successors, thus activity A will ÿ  have the ESS, m ˆ 0 ‡ dp . Since the connector activity B has a start-gap of x days, the ESS of this activity becomes (y+x), where y = m. Thereby, the ESF of activities A and B become (m ‡ da ) and (y ‡ x ‡ db ), respectively. As noted earlier, there are four possible situations, each of which could a€ect the routing of the critical path: 1. When da  …db ‡ x†, the critical path occurs along the baseline itself; no change is necessary for BAL. The ESSs of activities Q and R become n ˆ 1 ˆ …m ‡ da †, respectively.

2. If da < …db ‡ x† and the connector (B) has no resource con¯icts with its neighbours, then the situation is that in case 1 above. 3. If da < …db ‡ x† and the connector (B) has resource con¯icts with its neighbours (R) except the baseline activity, then the critical path(s) may occur along the baseline and/or across the connector and onto R. In this case, the BAL does not require amendments, although the ESSs of activities Q and R become n ˆ …m ‡ da † and 1 ˆ …y ‡ x ‡ db †, respectively. 4. If da < …db ‡ x† and the connector (B) has resource con¯icts with its neighbouring baseline activity (Q), then the necessity to amend the BAL arises. The critical path(s) may occur either as in case 3 above Ð in which case the BAL must have changed, or across the connector alone and with or without a€ecting the BAL. In both cases, the ESS of activity Q and R are determined upon their new positions if they were to be taken. Otherwise, both Q and R will have ESS of n ˆ 1 ˆ …y ‡ x ‡ db † each. The purpose of the foregoing section was to introduce LINRES by applying it to a simple example. The resulting resource-aggregation pro®les are shown in Fig. 8. 4. Experimental programme The aim of this section is to demonstrate the use of the proposed LINRES algorithm, and to compare its performance with the various other heuristics that have been proposed by researchers over the last three dec-

Fig. 7. Calculation of activity scheduled dates.

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ades, and which have been described in the earlier part of this paper. The LINRES heuristic technique can be used to solve real-life large project networks. However, to facilitate

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demonstration, ten small examples of single- and multiresource networks with resource constraints have been adopted from the literature. Each project was subjected to resource scheduling tests using the LINRES

Fig. 8. LINRES aggregate resource pro®les for the simple example [limit: 5 units].

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Table 1 Characteristics of the network examples used Network number

Source

Number of activitiesa

Number of resource types

CPM duration

1 2 3 4 5 6 7 8 9 10

Elsayed and Nasr [9, p. 301] Ulusoy and Ozdamar [16, p. 1148] Gordon and Tulip [7, p. 364] Abeyasinghe [17, p. 60] Davis and Patterson [11, p. 947] Bell and Han [12, p. 319] Moselhi and Lorterapong [13, p. 297] Matthews [8, p. 46] Nkasu [6, p. 187] Raz and Marshall [18, p. 244]

7 20 6 9 25 14 14 13 29 6

1 1 1 1 3 4 5 10 6 2

19 days 30 days 6 days 45 days 31 days 21 days 18 days 120 days 33 weeks 7 days

a

Excluding dummy activities.

algorithm. The performance of LINRES on each example was then assessed against results produced by a variety of other existing heuristic techniques and obtained from the same published sources as the primary project data. The number and size of the networks were such that manual computation was feasible. The number of activities in the sample ranged from 6 to 29, the number of resource types varied between one and ten, and the project durations varied from 6 days to 33 weeks. The sources and the characteristics of the network examples are summarised in Table 1. The table shows three singleresource and six multi-resource examples chosen from publications between 1975 and 19978 The 10th example was introduced to widen the variety of logic relationships between activities and to investigate the implication of this on the LINRES model.

in project duration over the initial critical path length (also expressed as the lowest normalised duration.9 Where a mathematically optimised duration was available from the literature source, this too, was used as a benchmark for comparing the performance of the heuristics. The results are summarised in Table 3. Further, a detailed comparison of the results obtained is shown in Fig. 9 From these results, it can be seen that the LINRES algorithm produced identical or better results than

4.1. Analysis of results

It was noted by Ulusoy and Ozdamar [16] that the success of di€erent heuristics varied under di€erent test conditions. This was supported by the current experiments, and, although a rigorous investigation was not undertaken, initial observations appeared compatible with Ulusoy and Ozdamar's account that project variables such as resource utilisation, project complexity and aspect ratio have an e€ect on heuristic performance. A detailed description of each test, and the resulting resource-aggregation pro®les given by LINRES can be found in Abeyasinghe [17].

Two groups of projects were tested; the ®rst (examples one to four) consisted of four single-resource problems containing 6 to 20 activities, and given a maximum resource availability of 5 units. The second group (examples ®ve to ten) comprised six multi-resource problems composed of between 6 and 29 activities, with up to ten resource types and varying constraints on resource availability. Each problem was solved using the LINRES method, and its performance compared with di€erent heuristic rules found in the literature. The main characteristics of these problems (and the results obtained) are given in Table 2. The performance criterion used to judge solutions was that of the least increase

8 Such that one example is chosen from the 1970s, two examples from the 1980s, and six examples from the 1990s. 9 The normalised duration was obtained by dividing the duration after resource allocation by the duration before allocation.

. . . . . .

all of the other heuristics tested on Example 1; six out of the seven heuristics tested on Example 2; two out of three heuristics tested on Example 3; ®ve out of the eight heuristics tested on Example 5; nine out of the 10 heuristics tested on Example 8; the only other heuristic tested on Example 10.

5. Discussion of the results The investigation, based on hypothetical projects, revealed that the proposed LINRES algorithm outperformed the most widely used existing heuristic techniques for scheduling projects under resource constraints. Despite the relatively small number of tests performed, LINRES demonstrated an equal rate of

M. C. L. Abeyasinghe et al. / International Journal of Project Management 19 (2001) 29±45

success with both single- and multi-resource problems. Throughout the scheduling experiments, the proposed model showed itself to have the potential for practical, ¯exible and ecient solutions and appeared to show a

41

better `understanding' of the resource constraints than most serial and parallel heuristics. Unlike the highly regarded LFT, MINSLK and RSM rules, LINRES has the advantage of needing to solve the critical path

Table 2 Comparison of project durations Description

Characteristics No. of activities No. of resource types Resource limits Critical path length Optimum durationa Serial heuristicsb ACTIM ACTRES TIMRES GENRES ROT ROT-ACTIM ROT-ACTRES TIMROSc TIMGEN WRUPd Late start timee Early start time Total ¯oatf Late ®nish time Early ®nish time Activity number Activity number (descending) Duration Duration (descending) Unknown (based on total ¯oat) Parallel heuristics MINSLKg MINSLKh LFT RSM GRD SIO RAN GRU MJP Other heuristics Proposed heuristic rulei Proposed heuristic rulej COMSARS methodology LINRES a

Network No. 1

2

3

4

5

6

7

8

9

10

7 1 5 19

20 1 3 30

6 1 2 6

9 1 5 45

25 3 each 6 31 64

14 4 each 10 21 31

14 5 15,4,2,2,2 18 22

13 10 each 1 120

29 6 10, 5, 4, 2, 12 33 35

6 2 each 1 7

34 34 34 34 36 34 34 28 28

44

46 46 46 46 46 46

3

195 205 195 210 215 200 220 215 195 190

8 7

74 67 74 80 71 74 68 76 31

28

45

8

60

73

35

As given in the literature. Activities are sorted in the ascending order of the rule unless otherwise stated. c Ties were broken by choosing activity having the longest duration. d Ties were broken by choosing activity having the lowest number. e Ties were broken by choosing activity having the least total ¯oat. f Ties were broken by choosing activity having the least late start time. g Activities cannot be split. h Activities can be split. i Bell and Han [12]. j Moselhi and Lorterapong [13]. b

9

22 28

195

35 52

9

42

M. C. L. Abeyasinghe et al. / International Journal of Project Management 19 (2001) 29±45

network only once throughout the entire procedure. A further consideration is that LINRES appears to deal more elegantly with the problem of `¯oat dependency' as introduced by Raz and Marshal [18]. Although this is not covered in the present paper, the issue would merit future consideration.

6. Conclusions and recommendations for further research In conclusion, the general performance of the proposed LINRES algorithm is found to be satisfactory in the sense that it enables a realistic but uncomplicated

Table 3 Percentage increase above the critical path length and the optimum durationa Heuristics

Network No. 1

Serial ACTIM ACTRES TIMRES GENRES ROT ROT-ACTIM ROT-ACTRES TIMROS TIMGEN WRUP Late start time Early start time Total ¯oat Late ®nish time Early ®nish time Activity number Activity number (descending) Duration Duration (descending) Unknown (based on total ¯oat) Parallel MINSLK(1)

78.95 78.95 78.95 78.95 89.47 78.95 78.95 47.37 47.37

2

46.67

53.33

MINSLK(2) LFT

53.33

RSM

53.33

GRD

53.33

SIO

53.33

RAN

53.33

3

4

5

6

33.33

33.33 16.67

47.62 0.00

Proposed heuristic rule(4) COMSARS methodology

a

33.33

10

28.57

116.13 4.69 138.71 15.63 158.06 25.00 129.03 10.94 138.71 15.63 119.35 6.25 145.16 18.75

Other Proposed heuristic rule(3)

50.00

9

138.71 15.63

MJP

47.37

8

62.50 70.83 62.50 75.00 79.17 66.67 83.33 79.17 62.50 58.33

GRU

LINRES

7

33.33

135.48 14.06

Given in italics. (1), (2), (3), and (4) refer to g, h, i, and j footnotes in Table 2, respectively.

66.67 12.90

22.22 0.00 55.56 27.27

62.50

6.06 0.00 57.58 48.57

28.57

M. C. L. Abeyasinghe et al. / International Journal of Project Management 19 (2001) 29±45

Fig. 9. Comparison of results of single- and multi-resource networks.

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44

M. C. L. Abeyasinghe et al. / International Journal of Project Management 19 (2001) 29±45

approach in making better scheduling and resource allocation decisions: it also appears to perform better on test problems than most comparable existing techniques. Further investigation is required as to when and why this is the case, and for this, tests on real-life projects would be an appropriate and logical development. The paper describes tests carried out experimentally on hypothetical projects and any claims regarding the bene®ts of the algorithm are subject to the testing of LINRES on real-life projects. In embarking upon this, a natural question is whether the LINRES model can be extended to problems of larger size and complexity. Though no extensive study has yet been performed, Bell and Han [12] suggest that the relative performance of heuristics in small-sized (20 activities) problems is matched in large-sized (100 activities) projects. The LINRES model can accept any number of activities, with no upper limit to the number of resources. This becomes increasingly signi®cant in the case of larger networks. Furthermore, the performance tests carried out on the model were computed manually. Development of a computer program would reduce the execution time considerably, allowing the possible expansion of sample size to better insight into the model's capabilities, as well as facilitating its use in the ®eld. Once ®eld testing on real-life individual projects is undertaken, an examination of the performance of LINRES on problems dealing with the demand for common, constrained resources by more than one project (multi-project scheduling) may be considered worthwhile.

[6] Nkasu MM. COMSARS: a computer-sequencing approach to multiresource-constrained scheduling Ð Part 1: Deterministic networks. International Journal of Project Management 1994;12(3):183±92. [7] Gordon J, Tulip A. Resource scheduling. International Journal of Project Management 1997;15(6):359±70. [8] Matthews M. Resource scheduling: incorporating capacity into schedule construction. Project Management Journal 1994;25(2):44±54. [9] Elsayed EA, Nasr NZ. Heuristics for resource-constrained scheduling. International Journal of Production Research 1986;24(2):299±310. [10] Tsai DM, Chiu HN. Two heuristics for scheduling multiple projects with resource constraints. Construction Management and Economics 1996;14:325±40. [11] Davis EW, Patterson JH. A comparison of heuristic and optimum solutions in resource-constrained project scheduling. Management Science 1975;21(8):944±55. [12] Bell CE, Han J. A new heuristic solution method in resourceconstrained project scheduling. Naval Resource Logistics 1991;38(3):315±31. [13] Moselhi O, Lorterapong P. Near optimal solution for resourceconstrained scheduling problems. Construction Management and Economics 1993;11:293±303. [14] Kolisch R. Ecient priority rules for the resource-constrained project scheduling problem. Journal of Operations Management 1996;14:179±92. [15] Boctor FF. Some ecient multi-heuristic procedures for resource-constrained project scheduling. European Journal of Operational Research 1990;49:3±13. [16] Ulusoy G, Ozdamar L. Heuristic performance and network/ resource characteristics in resource-constrained project scheduling. Journal of the Operational Research Society 1989;40(12):1145±52. [17] Abeyasinghe MCL. Scheduling under resource constraints of activity networks in project management. MSc Thesis (unpublished), University of Northumbria at Newcastle, UK, 1998 [18] Raz T, Marshall B. E€ect of resource constraints on ¯oat calculations in project networks. International Journal of Project Management 1996;14(4):241±8.

References [1] Johansen DE. Hard or soft: planning on medium size construction projects. Proceedings of The 12th Annual Conference of the Association of Researchers in Construction Management, ARCOM 96, Sheeld Hallam University, UK, 11±13 Sep.:73±82. [2] Woodworth BM, Shanahan S. Identifying the critical sequence in a resource-constrained project. International Journal of Project Management 1998;6(2):89±96. [3] Lester A. Project planning and control. 2nd ed. London: Butterworth±Heinemann, 1991. [4] Mawdesley M, Askew W, O'Reilly M. Planning and controlling construction projects: the best laid plans. England: Addison Wesley/Longman, 1997. [5] Cooke B, Williams P. Construction planning, programming and control. London: Macmillan, 1998.

Chelaka Abeyasinghe gained his ME in Civil Engineering and MSc in Construction Project Management at University College London and at University of Northumbria at Newcastle, respectively. He is currently doing his Ph.D. at Napier University, Edinburgh, dealing with the use of recycled materials in road construction.

M. C. L. Abeyasinghe et al. / International Journal of Project Management 19 (2001) 29±45 David Greenwood has been working as a Senior Lecturer in Construction Management at University of Northumbria at Newcastle since 1980 before which he worked for a national contractor. He is a Fellow of the Chartered Institute of Building, a visiting lecturer at the Universities of Reading and The Universite d'Artois, France.

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Eric Johansen is a senior lecturer at University of Northumbria at Newcastle which he joined in 1990 after 18 years in project management with one of the largest contractors in UK. A member of the Chartered Institute of Building and the Association for Project Management, he obtained an M.Phil. between 1994 and 1996. His research interests include project planning, total quality and lean construction.