Time coupling methods project scheduling and time ...

Zdzisław Hejducki Magdalena Rogalska

Time coupling methods project scheduling and time/cost optimization

Oficyna Wydawnicza Politechniki Wrocławskiej Wrocław 2011

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Drukarnia Oficyny Wydawniczej Politechniki Wrocławskiej. Zam. nr ???/2011.

Contents List of symbols .................................................................................................................................... 1. Introduction .................................................................................................................................. 2. Time Coupling Method (TCM) .................................................................................................... 2.1.Introduction ............................................................................................................................ 2.2.Principles of TCM application ................................................................................................ 2.3.TCM I ..................................................................................................................................... 2.4.TCM II ................................................................................................................................... 2.5.TCM III .................................................................................................................................. 2.6.TCMs IV, V and VI ................................................................................................................ 2.7.TCM computations ................................................................................................................. 2.8.Directions in TCM development ............................................................................................ 3. Time optimization in TCM by B&B algorithms ........................................................................... 3.1.Algorithms for establishing construction work sequence ....................................................... 3.2.Work sequencing in TCM I .................................................................................................... 3.3.Organization methods enabling modelling of work complexes with complex internal structure .......................................................................................................................................... 4. Time/cost optimization by hybrid evolutionary algorithm ............................................................ 4.1.Basic properties of hybrid evolutionary algorithm ................................................................. 4.2.Computation experiment – case study 1 ................................................................................. 4.3.Computation experiment – case study 2 ................................................................................. 5. Metaheuristc algorithms in TCM .................................................................................................. 5.1.Introduction ............................................................................................................................ 5.3.Relationship between TCM and queuing theory ..................................................................... 5.3.Selected metaheuristic algorithms .......................................................................................... 5.4.Application of selected metaheuristic algorithms ................................................................... 5.5.Application of evolutionary algorithms .................................................................................. 5.5.1. Basic properties of genetic algorithms .......................................................................... 5.5.2. Basic properties of hybrid evolutionary algorithm ....................................................... 5.5.3. Computing experiment – case study ............................................................................. 5.6.Case study .............................................................................................................................. 6. Conclusion .................................................................................................................................... References ..........................................................................................................................................

5 7 9 9 16 18 21 23 25 28 34 36 39 49 54 60 60 62 65 71 71 72 73 74 75 78 79 80 81 85 87

List of symbols LT (least time) – the shortest time between an activity in progress and the next activity, LD (least distance) – the shortest distance between an activity in progress and the next activity, CPH (controlling path) – the critical path of activities [days], CL (controlling link) – a graphic link between successive activities (on the critical path), CP (controlling point) – a critical point corresponding to the start of the next activity, T – the duration of an activity, TT (total time) – the total time of the execution of an activity, m – the number of activities, n – the number of sectors. n – the number of rows in matrix T, m – the number of columns in matrix T, T – matrix of work durations, – the earliest work commencement date, twr – the earliest work completion date, twz – the latest work commencement date, tpr tpz – the latest work completion date, – time of execution of work j in sector i, tij – duration of critical work j in sector i diagonal q, tijq – time of expansion of partial flow j, t rj tSsw Sfw Ssp Ssfp H P h, k Oi Pj R LPM N T jz

– couplings between realization means for early dates, – couplings between sectors for early dates, – couplings between realization means for late dates, – couplings between sectors, – a subset of the earliest work commencement dates, – a set of work commencement dates, – labour inputs, – object (structure) i, – work process j, – a set of work commencement dates on the matrix diagonal, – the limit possible minimum (an index), – a partial flow, – a vector of partial flow completion,

T jR

– a vector of partial flow expansion,

A B

– a two-column matrix, – a two-column matrix modified by the Johnson algorithm.

1. Introduction This publication presents a systematically ordered set of problems relating to the time coupling method (TCM) for scheduling construction projects. It uses an approach different than the one represented by methods such LSM, RSM, LOB and CPM/PERT found in the English language literature on the subject. It is the result of the authors’ latest research on scheduling methods, particularly TCM. An attempt was made to exploit new scheduling theory developments, i.e. metaheuristic algorithms, mainly the evolutionary HEA algorithm created by W. Bożejko and M. Wodecki. This work is also the result of seminars on improving metaheuristic algorithms for the scheduling of production undertakings, with a special focus on TCM. The discussions and creative inspirations resulted in many papers representing interdisciplinary attempts to solve technical difficulties encountered in construction work scheduling. The participants in the seminars come from the Institute of Building Engineering and the Institute of Automation, Robotics and Informatics at Wrocław University of Technology, the Institute of Building Engineering at Lublin Technical University and the Institute of Informatics at Wrocław University. This team is part of a group of people who for many years have been collaborating in research on the optimization of discrete processes occurring directly in production and construction, including scheduling (with single- and multi-criteria objective functions) from both the theoretical and practical points of view, focusing mainly on the application of artificial intelligence methods in solving NP-difficult combinatorial optimization problems and on the realization of algorithms in the (parallel and distributed) multiprocessor computation environment. The research being the subject of the seminars has been conducted for years and some of its results have already been published. When considering construction project scheduling problems in terms of classic queuing theory one encounters difficulties connected with the choice of a proper model and a proper algorithm for the completely new, complicated and NP-difficult task-scheduling problems that often arise as a result. The following techniques are currently used: • simulating annealing, Krikpatrik et al. (1983); • the genetic algorithm, Davis (1987);

8

Chapter 1

• the tabu search, Glover (1989); • neural networks, Hertz et al. (1991). Thanks to the big advances made in both the theory and technology of algorithms, heuristic algorithms are now the only way of arriving at feasible solutions. The main aim of this publication was to collect and systematize research on improving TCM for construction project scheduling. The publication presents the results of studies and computing tests relating to the modelling and implementation of original computing algorithms for construction project scheduling taking into account minimum-time, optimum-time and cost objective functions. * Special thanks are due to our colleagues Dr. Wojciech Bożejko and Dr. Mieczysław Wodecki for their collaboration in computations. Gratitude is due the late Professors Juliusz Mrozowicz and Victor Alexeyevich Afanasjev for the inspiration for this research which is the continuation of their scientific ideas and their desire to propagate time coupling methods.

2. Time Coupling Method (TCM) The Time Coupling Method (TCM)1 devised by Victor A. Afanasjev is described briefly below to acquaint the general public of scientists in the world with TCM. So far this subject has been presented in an academic way, which made its comprehension difficult and constituted a barrier for a larger readership. In contrast, TCM computations performed using the ORGANIZATOR program are simple and quick. Considering the lack of international publications on TCM, we present the general idea of the method, the stages in its development and the way of computing the main values: CP, LT, TT and determining the course of the critical path (CPH). A calculation example is provided. The current development of TCM with regard to new applications is described. We dedicate this chapter to Prof. Victor A. Afanasjev, honouring his wish to propagate the Time Coupling Method, unfortunately no longer with his guidance2.

2.1. Introduction Linear Scheduling Methods (LSM), referred to as TCM (Time Coupling Method) in Central and East European countries, stem from the course of construction work, based on the division of a project into sectors. In the form of four-sector work division they were already employed in ancient Egypt in the construction of pyramids, irrigation canals and other structures having a sectoral or linear character. In our times they are used for the building of motorways, waterworks, sewage systems, telephone lines, skyscrapers and other structures. The two varieties of linear scheduling methods have been developing in parallel (despite the iron curtain between the East of Europe and the rest of the world) because of the very nature of construction work. It is precisely the natural connection between practical methods and their scientific description that caused the parallel development of the methods in Russia and Poland and in such countries as Canada and the USA. 1 2

The Time Coupling Method (TCM) is an alternative method to CPM/PERT. Victor A. Afanasjev died in 2002.

10

Chapter 2

Already in the 1930s M.S. Budnikov described for the first time in the Soviet block the concept of stream work (LSM), assuming sectors of equal size and the same work realization time in the sectors and consequently, gangs of appropriate size to simultaneously carry out the tasks. One of the followers of M.S. Budnikov (1961)[11] was V.A. Afanasjev (1957, 1988, 2000) [2]–[4] who in a scientific way described work carried out in accordance with LSM, assuming sectors of different size and so different task realization times for particular gangs. All the necessary calculations could be done manually, but they were laborious and time-consuming. In the late 1970s the methods were adapted for use on RIAD computers. Juliusz Mrozowicz (1982, 1997)[33], [34] was the first one to propagate LSM in Poland. He introduced a LSM systematics in Polish and presented LSM assumptions, terminology and calculation methods. Zdzisław Hejducki (1999, 2000, 2001, 2003, 2004) [23]– [28] improved LMS by introducing task optimisation and reduction. He described in a scientific way the process commencement time, duration and completion time computing algorithms for different constraints. Scheduling of construction projects by the Time Coupling Method (time couplings are interdependences between activities and sectors, which take resource and technical limitations into account), first developed by Victor A. Afanasjev and applied by him to a military airfield construction project in the USSR in 1946, has not been presented internationally because of the Soviet Union’s separatistic policies and secretiveness and because this was not required by that country’s scientific community. Despite its numerous applications in Eastern Europe the method is practically unknown in the world. Due to the lack of a basic publication which would explain the method’s fundamentals, papers describing its advanced aspects tend to be not accepted for publication in international journals. TCM’s main advantage is that time couplings (not only total project time TT but also all the interconnections) can be computed using the ORGANIZATOR software. The drawbacks are that only one contractor can work in one sector at the same time and the ORGANIZATOR program’s capacity is limited to 20 sectors and 20 activities. It is, however, possible to combine two or more computing sequences. In the West, linear scheduling methods have developed mainly in Canada and the USA. In (1969) O’Brien introduces the notion of ‘Line of Balance’ [5], [35], [36]. The subject is also investigated by Carr and Meyer (1974) [10] and Halpin and Woodhead (1976) [20]. Peer (1974) [37] and Selinger (1980) [46], Handa and Barcia (1986) [21] and Chrzanowski and Johnston (1986) [15] develop a ‘Construction Planning Technique’, O’Brien (1975) [36] and Barrie and Paulston (1978) [8] – a ‘Vertical Production Method’, Birrell (1980) [9] – a ‘Time-Location Matrix Model’, Johnston (1981) [29] and Stradl and Cacha (1982) [48] – a ‘Time Space Scheduling Method’. Whiteman and Irwig (1988) [50] publish works on ‘Disturbance Scheduling’ and Thabet and Beliveau (1994) [49] – on ‘Horizontal and Vertical Logic Scheduling for Multistory

Time Coupling Method (TCM)

11

Projects. O’Brien (1975) [36], Arditi and Albulak (1979, 1986) [6], [7] and Melin and Whiteaker (1981) [31] develop linear scheduling tools such as: line-of-balance diagrams, bar-charts, vertical production charts and velocity diagrams. Russel and Caselton (1988)[43] introduce extensions to linear scheduling optimisation, in the form of interrupt duration vector choices. Russel and Wong (1993) [44] describe the development and use of a new generation of planning structures providing a basis for a superset for the traditional critical path method and over the years introduce various extensions needed to represent the realities of construction. Senouci and Eldin (1996) [49] present a nonserial dynamic programming formulation for the scheduling of linear projects with nonsequential activities. El-Rayes and Moselhi (1998, 2001) [16], [17] optimise resources in repetitive construction projects. Harris and Ioannou (1998) [22] work out a repetitive scheduling method with technical constraints. Adeli and Karim (1997) [1] develop a neural dynamics cost optimisation model for construction projects. In contradistinction to the above methods, in TCM the way in which the project is to be carried out: method 1 – no work stoppages, method 2 – no stoppages in sectors, method 3 – minimum lead time and possible work and sector stoppages, methods 4, 5, 6 – minimum time and additional constraints is assumed for calculations. Currently research focuses on hybrid methods of optimising complex construction processes. Packages of processes described by TCM, LSM, LOB, genetic algorithms, neural networks and bar-charts are incorporated into one schedule. The paper presents the fundamentals of TCM so that further papers on this subject can be acceptable for publication in international journals. History of TCM In 1946 Victor A. Afanasjev was 25 years old and he was a lieutenant of the Engineering Forces in the Soviet Union Army. Because of the Cold War all kinds of logistic activities were treated as a military secret. Hence the Time Coupling Method developed by V.A. Afanasjev was first published as late as in 1973 when the author retired from the army and started working at the Petersburg (then Leningrad) University of Architecture and Civil Engineering. In 1946 lieutenant V.A. Afanasjev got an order to build a military airport. The design included: the organization of the whole project and the construction of flying strips, underground fuel tanks, hangars, service workshops, an airdrome control tower, buildings for headquarters, a command station, barracks (apartment buildings, a kitchen, a canteen, storehouses), internal roads, a railway siding, a fence, watchtowers and a temporary camp for soldiers who were to build the military airport. The huge project had to be completed within 8 months. Because of the project’s priority character, the government assigned 15 000 soldiers and allocated unlimited supplies of building materials and machines to the construction of the military airport (supplies were guaranteed despite the drastic shortages of building materials and machines after World War II which had finished the year before).

12

Chapter 2

If the project had not been completed in time, V.A. Afanasjev would have been taken to a court martial, which meant capital punishment in Stalin’s times. Therefore, finishing the project in time was not a matter of profit or loss, but a matter of life or death. The project became a real challenge for the young officer who did not have any experience in this field. V.A. Afanasjev tried to apply the knowledge about scheduling which he had acquired during his studies. He applied the method devised by M.S. Budnikov (the Stream Method of Organization (SMO) (1961) [11]. The method consists in dividing a project into sectors, assuming that contractors do their activities in their sectors at the same time. The activity in each sector has to take the same number of time units for each contractor. The total time of activity in each sector has to be the same as well. The above factors determine the number of workers in a contractor’s gang of workers. Hence the method assumptions are as follows: S = constant, t = constant, w=0 where: S – a sector (a part of a project), t – time of activity execution by a contractor’s gang, w – time for which the contractor’s gang waits to start an activity in the next sector. A simple example of the application of SMO is shown in Fig. 1. The total time of activity execution (TT) in sectors 1 and 2 amounts to 40 time units.

Fig. 1. Gantt’s chart of activities organisation (according to SMO devised by Budnikov), TT = 40 time units

The SMO approach did not allow completion in a predetermined period. The human resources were not fully used in the above approach. In order to cut down the execution time of a project it was necessary to use the human resources to the maximum. Initially, V.A. Afanasjev made an assumption that sectors had constant size, whereas individual contractors’ activity times in sectors were not constant. As a result,


13

it became possible to reduce a contractor’s activity time by maximally increasing (under given physical constraints) the number of workers in the contractor’s gang. The consequences were: 1) the continuity of activities in particular sectors was maintained and 2) stoppages appeared for contractors. The waiting time for the contractor to enter the next sector was no longer equal to zero. The above can be noted as follows: S = constant, t ≠ constant, w ≠ 0. In his later works, V.A. Afanasjev referred to the above approach as METHOD II. The method is illustrated in Figure 2.

Fig. 2. Gantt’s chart of activities organisation (according to TCM II developed by Afanasjev) for constant size of sectors and TT = 30 time units

According to Fig. 2, the total project execution time (TT) (in sectors 1 and 2) has been cut down from 40 to 30 time units. But there are some stoppages and the waiting time for contractor’s gang A2 and contractor’s gang A3 to start sector 2 is respectively 4 and 6 time units. The above solution was unacceptable from the military point of view: the stoppages would have a demoralising influence on the soldiers and this would be considered sabotage in Stalin’s times. From the organisation point of view, the human resources were still not fully used. Thus the TMC II approach had to be further modified. The modification consisted in the assumption that the waiting time for a contractor’s gang to enter the next sector must equal zero. In this approach (later called METHOD I – TCM I), sectors still have constant sizes but the time of activity execution by particular contractors’ gangs in the same sector can be different and the waiting time for a contractor’s gang to start the next sector must be equal to zero. The above can be noted as follows: S = constant, t ≠ constant, w = 0.

14

Chapter 2

The TCM I approach is presented in Figure 3.

Fig. 3. Gantt’s chart of activities organisation (according to TCM I developed by Afanasjev) for constant size of sectors and TT = 30 time units

The total project execution time (TT) did not change in comparison with TCM II – it still amounted to 30 time units. But contractor gang stoppages were suppressed. Still the continuity of activities in sectors was not maintained and so the commanding staff might have had some reservations about this solution. It appeared, however, that activities could be done much faster because there were some free sectors in which the next contractors’ gangs could start work at once. Therefore the sizes of the sectors and the sequence in which the activities were to be executed in the particular sectors had to be changed. The result was METHOD III (TCM III). The above can be noted as follows: S ≠ constant, t ≠ constant, w = 0. The proper selection of sector sizes and sector sequences allows one to cut down the execution time of a project. The above is illustrated in Figures 4 and 5.

Fig. 4. Gantt’s chart of activities organisation (according to TCM III developed by Afanasjev) for TT = 32 time units with differentiation of sector size: S1′ – changed size of sector S1, S2′ – changed size of sector S2


15

Fig. 5. Gantt’s chart of activities organisation (according to TCM III developed by Afanasjev) with differentiation of sector size, TT = 28 time units

Table 1 shows a project division into sectors and the obtained total project execution times. It is assumed that the sum of activities execution times in sectors is a constant value, i.e.: S1 + S2 = S1′ + S2′. The size of a sector for contractor’s gang A1 is assumed whereas the times for the other contractor’s gangs are calculated using linear interpolation. Table 1. Set of project durations calculated for sequence of activities execution in sectors and different sizes of sectors

No. 1 2 3 4 5 6 7 8 9 10

Time of activity Time of activity Time of activity Total time of execution by execution by execution by Order activity execution of sectors contractor’s gang A1 contractor’s gang A2 contractor’s gang A3 in sectors S1′ + S2′ S1′ S2′ S1′ S2′ S1′ S2′ 11 9 6.6 5.4 4.4 3.6 29.0 S1′ S2′ S2′ S1′ 9 11 5.4 6.6 3.6 4.4 30.6 S1′ S2′ 12 8 7.2 4.8 4.8 3.2 28.0 S2′ S1′ 8 12 4.8 7.2 3.2 4.8 32.0 S1′ S2′ 13 7 7.8 4.2 5.2 2.8 28.8 S2′ S1′ 7 13 4.2 7.8 2.8 5.2 33.0 14 6 8.4 3.6 5.6 2.4 32.4 S1′ S2′ 6 14 3.6 8.4 2.4 5.6 34.0 S2′ S1′ 15 5 9 3 6 2 32.0 S1′ S2′ 5 15 3 9 2 6 35.0 S2′ S1′

According to Table 1, the total time (TT) of activity execution in sector S1′ is different than in S2′ and it depends on the sequence of activities execution in the sectors. The shortest execution time, equal to 28 time units, is obtained in case 3. The longest execution time, equal to 35 time units, is obtained in case 10. As a result of the search for new divisions into sectors the project execution time has been reduced from 30 (TCM II) to 28 time units.

16

Chapter 2

V.A. Afanasjev applied TCM to the military airport project. The method proved correct and it was later applied many times to the construction of military facilities. Unfortunately, no details about the projects are available since the latter were considered to be military secrets. The calculation scheme presented above is simplified version. In reality, the number of gangs was 150 and there were 200 sectors, which made the calculations very difficult (there were no computers then). TCM can be applied to construction projects divisible into sectors. A sector can be a part of a structure (a building floor, 100 m of a flying strip, 200 m2 of a roof, one underground fuel tank, etc.) or a whole structure (if several structures are to be built simultaneously). But the method has a serious limitation – only one gang can work in one sector at the same time. The following symbols are used in TMC: LT (least time) – the shortest time between an activity in progress and the next activity, LD (least distance) – the shortest distance between an activity in progress and the next activity, CPH (controlling path) – the critical path of activities [days], CL (controlling link) – a graphic link between successive activities (on the critical path), CP (controlling point) – a critical point corresponding to the start of the next activity, T – the duration of an activity, TT (total time) – the total time of the execution of an activity, m – the number of activities, n – the number of sectors.

2.2. Principles of Time Coupling Method TCM application As described in [2]–[4], [23], [33], [34], in order to begin scheduling by TCM (Fig. 6) one should have: • input data specifying the division of a project into sectors S1 ... Sn (a sector is a part of land or building, a fragment of a pipeline, etc.) and the size of the latter, • specified kinds of activities and the technological sequence (determined by the technology used) in which they are to be carried out, • deterministically fixed (determined on the basis of catalogue data which in Poland are collected in the Catalogue of Labour Standards) times of performing the activities in each sector.

17

Time Coupling Method (TCM) INPUT

OUTPUT

S1..........Sn

TT

TCM

P1....…..Pm

LT1.1….LTn.m

T1.1 ......Tn.m

LD1.1….LDn.m

Fig. 6. Computational scheme and input and output conditions for TCM

As a result of TCM computations the following are obtained: • a work schedule, • networks of dependences, • a table of internal dependences between the start and finish times of the particular activities in the sectors. TCM I

TCM II

P1 S1

P1 S2

P1 S1

P1 S2

P2 S1

P2 S2

P2 S1

P2 S2

TCM III

TCM IV

P1 S1

P1 S2

P1 S1

P1 S2

P2 S1

P2 S2

P2 S1

P2 S2

TCM V

TCM VI

P1 S1

P1 S2

P1 S1

P1 S2

P2 S1

P2 S2

P2 S1

P2 S2

One-way dependences with continuity Two-way dependences with continuity Two-way dependences with possible continuity

Fig. 7. Diagram of priority dependences in TCM

18

Chapter 2

The values from the table of dependences are represented by symbols LT in the schedule (if in the schedule there is an LT symbol in a given time unit before the line representing the time of carrying out activity Px in sector Sy, this means that the activity can start in this time unit). Thus LT stands for an interruption in carrying out the activity but with the possibility of starting activity Px earlier – already in the time unit in which LT appears. The time of carrying out activity Px is constant (previously deterministically fixed). LD in the schedule represents a situation when no tasks are performed in a given sector. TT represents the total project realization time whose minimization can be the priority. With the desired LT, LD and TT input data in mind one selects the most suitable method from the six scheduling TCMs. Figure 7 shows priority dependences for each of the six methods. Priority dependences are such dependences which must occur in a given method. For example, in TCM I the continuity of work of gangs is the priority dependence. It may happen, however, that as a result of a change in the composition of the gangs, additional dependences, such as the continuity of sectors, will appear. The six methods are discussed in detail below.

2.3. Time Coupling Method TCM I TCM I is applied when the priority is to assure the continuity of activities P1 ... Pm in sectors S1 ... Sn. This usually entails the assurance of work continuity for the contractors or the equipment. Activity continuity assurance may become a priority because of the adopted technology (e.g. seamless concreting). A similar situation arises in the case of a leading activity, i.e. one which enables the commencement of all the tasks contingent on it. It so happens that hiring equipment or a given contractor becomes unprofitable if stoppages are taken into account. Also repeated undertaking of work by the same contractor may be unprofitable because of the involved equipment transport and assembly. Some highly specialized contractors tend to undertake a task on condition that work continuity will be assured. In some services demand exceeds supply, which makes the assurance of work continuity for contractors the priority. TCM I is a proper tool for scheduling such tasks. The output data generated by the ORGANIZATOR TCM computer program are shown in Fig. 8. ORGANIZATOR (1997) for Windows is a computer program developed by Soft Project based in Wrocław, Poland. The computing algorithms were written by Z. Hejducki and K. Czapliński. The ORGANIZATOR’s earlier versions were OPTY (1982) written in Fortran and POTOK (1993) for DOS. The program is not commercially available at the moment, but it is to be released soon.

19

Time Coupling Method (TCM) INPUT

OUTPUT

S1..........Sn

TT ? TT min

TCM I

P1....…..Pm

LT1.1….LTn.m = 0

T1.1 ......Tn.m

LD1.1….LDn.m ? 0

Fig. 8. Computational scheme and output conditions for TCM I

Deterministically calculated sectors S1, S2 ... Sn–1, Sn and activities P1, P2 ... Pm.–1, Pm in a technological sequence and their performance times T1.1 ... Tn.m are used as the input for the computations. It is assumed that at this stage no final sequence of activities in particular sectors has been fixed. It will be ultimately fixed after basic computations during the first optimisation – searching for such a sequence of activities in sectors which will result in the shortest project realization time (the Organizator program performs this search when the ‘optimise’ option is selected). Therefore a certain sequence of sectors should be assumed, however changing it later will pose no problem. One simply must select the ‘optimise’ option and the operation will be performed in a few seconds without it being necessary to re-enter the data or analyse them in detail. The computations usually yield the best results when the duration of the activities in the final sector is the shortest in comparison with that of the other activities. As a result of the computations one obtains the total time (TT) in which the project will be carried out. The time is longer than or equal to the minimum project duration TT min (in minutes) obtained by the other five methods using the ORGANIZATOR program. Time intervals LT1.1 ... LTn.m between the activities in progress and the next ones are always equal to zero, i.e. the continuity of activities P1 ... Pm is ensured. Probably no continuity of work in the sectors will be assured (sometimes it can be obtained, but this is not a priority). The priority dependences, i.e. the ones for which LTs must always be equal to zero, used as the basis for drawing up a schedule are shown in Fig. 9. INPUT S1..........Sn

TCM I

OUTPUT

P1S1

P1S2

P1S3

P2S1

P2S2

P2S3

P3S1

P3S2

P3S3

P1....…..Pm T1.1 ......Tn.m

TT ? TT min LT1.1….LTn.m = 0

Fig. 9. Priority dependences in TCM I

LD1.1.....LDn.m ? 0

20

Chapter 2

To illustrate the differences between the particular TCMs (I-VI), computations are made for all the methods using the same data shown in Table 1, but for the different assumptions and priority dependences characteristic of the particular methods. Table 2. General table of activity durations Sector no. S1 S2 S3 S4

P1 2 5 2 1

Activity P2 P3 3 1 2 4 2 1 3 4

P4 4 1 3 2

After entering the data from Tab. 1 and performing TCM I computations, TT = 20 time units was obtained. To check if time TT could be shortened, optimisation (by changing the order of carrying out the tasks in the sectors) was performed. As a result of changing the order of the sectors to S3, S4, S1, S2, TT = 18 units was obtained. The schedule, which takes into account LD, is shown in Fig. 10.

Fig. 10. TCM I schedule, TT = 18 units

According to Fig. 10, in the case of TCM I there are no activity stoppages, and so no contractor stoppages, but sector stoppages (nobody works in a given sector during this time) appear in: S1 – 4 time units, S2 – 1 time unit, S3 – 3 time units and S4 – 1 time unit (two time units between activities 1 and 2, one unit between activities 2 and 3 and one unit between activities 3 and 4). Thus the conditions specified in Fig. 4 are fulfilled.

21


2.4. Time Coupling Method TCM II TCM II is applied when the priority is to ensure the continuity of work in sectors S1 ... Sn for performed activities P1 ... Pm. It is assumed that in no sector can stoppages occur during the whole project, i.e. in each case LD must be equal to zero. Whereas interruptions in the work of individual contractors are allowable, i.e. the continuity in carrying out the particular activities is not preserved. Hence a given contractor may be asked to perform work several times during the realization of the project. This entails waiting time LT different from zero for the given contractor or the transfer of the resources to another construction site. The assurance of work continuity in the particular sectors can be given priority when repair work is to be done in a building where production in progress has to be maintained and the assignment of a sector for repair work entails stopping the production process (then it is often impossible to assign several sectors simultaneously. TCM II (often unnamed) is used to speed up works: it often seems that when a given sector has already been freed, then the immediate introduction of the next contractor will speed up construction no matter how long the contractor will have to wait to enter the next sectors or how high the costs of waiting or transporting and installing the necessary equipment will be. There must be good reasons, e.g. organizational, production or economic constraints (dates of payments), for applying TCM II since a schedule drawn up by this method usually generates total project time TT longer than the ones calculated by the other methods. The output data generated by the ORGANIZATOR TCM II program are shown in Fig. 11. INPUT

OUTPUT

S1..........Sn P1....…..Pm T1.1 ......Tn.m

TT ? TT min

TCM II

LT1.1….LTn.m ? 0 LD1.1….LDn.m = 0

Fig. 11. Computational scheme and output conditions for TCM II

Similarly as in TCM I, deterministically fixed sectors S1 ... Sn and activities P1 ... Pm in a technological order and their performance times T1.1 ... Tn.m are assumed. It is also assumed that at this stage the order of sectors is not important – it will be modified through the ‘optimisation II’ option in the ORGANIZATOR program. Since priority is given to the maintenance of work continuity in the particular sectors, the time of waiting for taking up work in the next sector by the given con-

22

Chapter 2

tractor will be longer or equal (in some cases) to zero: LTn.m ≥ 0, whereas LD1.1 ... LDn.m will always be equal to zero. The priority dependences shown in Fig. 7 form the basis for the remaining computations. INPUT S1..........Sn

TCM II P1S1

P1S2

OUTPUT P1S3

P1....…..Pm T1.1 ......Tn.m

TT ? TT min LT1.1….LTn.m ? 0

P2S1

P2S2

P2S3

P3S1

P3S2

P3S3

LD1.1….LDn.m = 0

Fig. 12. Priority dependences in TCM II

Similarly as in the case of TCM I, using the data shown in Table 1 and giving priority to the maintenance of work continuity in the particular sectors, a project schedule was drawn up (Fig. 13).

Fig. 13. TCM II schedule, TT = 20 units

As shown in Fig. 13, there are no stoppages in the sectors (no LD) but there are interruptions in the activities: LT4.1 = 2, LT4.3 = 2, LT4.4 = 2, LT2.2 = 2, LT2.4 = 2, LT3.1 = 2, LT3.2 = 2, which after optimisation gives total project time TT = 20 time units – 2 units more than in TCM I. Therefore, as mentioned above, there must be good reasons for applying TCM II.

23


2.5. Time Coupling Method TCM III TCM III is used when the assurance of the minimum project realization time is given priority. This is achieved by assuming such internal priority dependences that in each case both LD and LT approach zero (Fig. 14). INPUT

OUTPUT

S1..........Sn P1....…..Pm

TT = TT min

TCM III

LT1.1….LTn.m ? 0

T1.1 ......Tn.m

LD1.1….LDn.m ? 0

Fig. 14. Computational scheme and output conditions for TCM III

If TCM III is applied, work stoppages may occur in the particular sectors and the continuity of the activities, and thus the continuity of work of the contractors, may be interrupted. The method, as implemented in the ORGANIZATOR program, generates a minimum number of stoppages in the sectors and a minimum number of interruptions in the activities. An ideal solution is a schedule in which both LT and LD are equal to zero. It is a rare case, but possible. It occurs for the Budnikov method, i.e. when the sectors are identical and the durations of all the activities are the same. To reduce LD and LT, optimisation computations are performed through the ‘optimisation III’ option. In this way the order of the sectors is changed to obtain the best solution, i.e. TT = TT min. In TCM III all internal dependences are regarded as having priority (Fig. 10) and, unlike in TCM I and TCM II, there is no one kind of priority dependency. INPUT S1..........Sn

TCM III P1S1

P1S2

OUTPUT P1S3

P1....…..Pm T1.1 ......Tn.m

TT = TT min LT1.1….LTn.m ? 0

P2S1

P2S2

P2S3

P3S1

P3S2

P3S3

Fig. 15. Priority dependences in TCM III

LD1.1….LDn.m ? 0

24

Chapter 2

TCM III is currently the most often used method. Many projects are carried out assuming the minimum project realization time. Formerly, project realization time minimization was applied only to major projects, but today, considering that there is unrestricted access to resources and that construction costs are very high, TCM III is used increasingly more frequently. And so very often TT = TT min determines the choice of a method. In TCM III most of the activities are carried out without any time reserves (without interruptions) and so the activities may happen to be on a controlling path. Hence usually more than one controlling path can be determined, which is essential for the application of TOC (theory of constraints, formulated by E.M. Goldratt). This is being investigated by the authors. A project schedule drawn up using TCM III with the ‘optimisation III’ option and the data from Table 1 is shown in Fig. 16.

Fig. 16. TCM III schedule, TT = 18 time units = TT min

In the case of TCM III, as Fig. 16 shows, there are stoppages in sectors LD, interruptions in carrying out activities LT and interruptions LDT (during the latter there are no tasks in a given sector and at the same time an activity interruption occurs). After optimisation a sequence of tasks different than the one for TCM II, but identical as the one for TCM I, was obtained. But this is not always the case since for different input data (Tab. 1) the sequence of sectors after optimisation may be different. For the assumed data, total project time TT amounts to 18 time units, i.e. it is the same as for TCM I. In comparison with the latter method, there are differences in the LT and LD arrangements. The following are noted: LD3.3 = 2 and LD3.4 = 1 in sector 3, LT4.1 = 1 in sector 4 and LD1.2 = 1, LDT1.2 = 1 and LDT1.3 = 1 in sector 1 and no activity or sector interruptions occur in sector 2. A comparison of total project time TT for method I and III shows that for the assumed input data the same result (TT = 18 time

25


units) was obtained. Thus organizational considerations are the factor which determines the choice of a project realization method. Having analysed the locations of LD and LDT, one should select an option advantageous from the organizational and financial points of view.

2.6. Time Coupling Method TCMs IV, V and VI TCMs IV, V and VI have the same computational scheme and identical output conditions (Fig. 17). INPUT

OUTPUT

S1..........Sn

TT ? TTmin

TCM IV TCM V TCM VI

P1....…..Pm T1.1 ......Tn.m

L T1.1….LTn.m ? 0 L D1.1….LDn.m ? 0

Fig. 17. Computational scheme and output conditions for TCMs IV, V and VI

It is assumed that the total project realization time will be longer or equal to TT min and that sector and activity interruptions may occur. TCMs IV, V and VI differ in their priority dependences. It is the additional priority dependences, matched to the situation on the building site (to the organizational and resource constraints), which can lead to the most proper and effective scheduling of the project. INPUT S1..........Sn

TCM IV P1S1

P1S2

OUTPUT P1S3

P1....…..Pm T1.1 ......Tn.m

TT = TT min BP1.1….BPn.m ? 0

P2S1

P2S2

P2S3

P3S1

P3S2

P3S3

Fig. 18. Priority dependences in TCM IV

BS1.1….BSn.m ? 0

26

Chapter 2

In method IV, the priority dependences which occurred in TCM III and also dependences (Fig. 18) for the simultaneous commencement of the particular activities are assumed. The latter dependences are called diagonal since all computations in the ORGANIZATOR program are based on matrix analysis. The use of TCM IV is justified when the dates of starting tasks are precisely fixed. This occurs, for example, during repairs of railway traction systems. In order to repair a part of a railway traction system the dates in the sectors involved must be harmonized because of transport constraints, i.e. the railway traffic must be stopped at a particular time and place. In TCM IV, the date of leaving the final sector by the last contractor (the activity carried out by the latter is the final one on the controlling path) is fixed while the other contractors may leave the final sector earlier since there is no way of ensuring work continuity for them in another sector, i.e. no other section of the railway traction system can be closed. TCM IV can be applied when after contractor A1 has finished work in a given sector, contractor A2 can immediately enter this sector (no technological break is needed after contractor A1 has finished work). A typical work schedule drawn up using TCM IV and the data given in Tab. 1 is shown in Fig. 19.

Fig. 19. Schedule TCM IV, TT = 18 time units

The obtained schedule (Fig. 19) is similar to the one for TCM III (Fig. 11). This is due to the fact that the diagonal dependences have been dominated by orthogonal dependences. Stoppages LD occur in sectors: S3 – 3 time units and S1 – 3 time units and stoppages LT in activities: P1 – 1 time unit, P2 – 1 time unit and P3 – 1 time unit. In TCM V the same priority dependences as in TCM III and also dependences (Fig. 20) for the simultaneous completion of the particular activities are assumed. The use of TCM V is justifiable when the dates of completion of tasks for all the contractors have been fixed and it happens to be the same date, i.e. each of the con-

27


tractors has to start work at the latest possible date. This solution allows for technological breaks or resource constraints. A typical example here is a major repair of a railway line sector. The repair includes the following activities: the disassembly of the track, the excavation and disposal of the old breakstone, the laying of a new track, the backfilling of the track with breakstone, the tamping and horizontal and vertical adjustment of the rails and the sleepers, the disassembly of the electric traction and the assembly of a new electric traction. The works are performed using special trains. The technological dependences determine the latest work commencement dates to ensure the least conflict between the realizations of the activities (usually one or two railway tracks are available). The date of completion of the tasks is precisely fixed and it is considered to be the deadline by which all the contractors have to leave the construction site simultaneously. A work schedule drawn up and optimised using TCM V and the data given in Tab. 1 is shown in Fig. 21. INPUT S1..........Sn

TCM V

OUTPUT

P1S1

P1S2

P1S3

P2S1

P2S2

P2S3

P3S1

P3S2

P3S3

P1....…..Pm T1.1 ......Tn.m

TT = TT min BP1.1….BPn.m ? 0

Fig. 20. Priority dependences in TCM V

Fig. 21. TCM V schedule, TT = 19 time units

BS1.1….BSn.m ? 0

28

Chapter 2

In the case of TCM V, as Fig. 21 shows, there are sector, activity and sectoractivity shortages. As mentioned above, TCM V’s distinguishing feature is the latest possible, simultaneous completion of tasks by the different contractors, which is reflected in the schedule’s time units: 4, 7, 12, 14 and 18. The method’s assumptions determine the total project time which is seldom longer than TT min = 18 time units of TCM I, III and IV. Sector and activity interruptions are so located as to satisfy the priority conditions, i.e. to ensure the latest possible commencement of the successive activities. TCM VI is a combination of TCM IV and V. It is assumed that besides orthogonal dependences (such as the ones in TCM III) there are diagonal dependences typical for TCMs IV and V. The priority dependences are shown in Fig. 21.

2.7. Time Coupling Method TCM computations So far the idea of TCM and the successive stages in its development have been presented in a simplified graphic form. Now such quantities as: activity start and finish times, sectors sizes, the number of contractors, sequences of sectors, the critical path and the project execution time which are needed for a graphic schedule should be calculated. The strict rules which are obligatory in the army are not obligatory in civilian life where contractor stoppages and situations when activities are not done in a continuous way in a sector are allowable. Also some limitations can be introduced. This can be done without affecting the total time of project execution (TT). The computer program ORGANIZATOR is used for the computations. Its latest version takes into account weather conditions in highway engineering and TOC (Goldratt’s theory of constrains). ORGANIZATOR generates cyclograms and networks of dependences (arrangement of activities in sectors and interrelations among them), but it does not generate Gantt’s charts. The times of execution of particular activities in sectors and control points CP are the main values in the cyclograms. The computation procedure consists of the following steps: • elaboration of a table of times Tn.1, • calculation of the elements of matrix Mmn • calculation of the main values: CP, LT and TT. Elaboration of table of times Tn.1 Times Tn.1 – the times of activities execution by contractor 1 in sectors n, where T1.1 is the time of activity execution by contractor 1 in sector 1, T2.1 is the time of activity execution by contractor 1 in sector 2, T3.2 is the time of activity execution by contractor 2 in sector 3 and so on. The above notation is used in a matrix of activity execution times. The matrix consists of rows which represent sectors (marked from

29


1 to n) and columns which represent contractors (marked from 1 to m) executing activities. The elements of the matrix of activities (the matrix is a model of a project) are the times of activities execution in sectors Ti,j, where: i = 1, 2, ..., n and j = 1, 2, ..., m. The elaboration of the table of times consists in assigning activities, in such a way that the rows correspond to sectors and the columns to activities. Hence: • T1.1. is placed at the intersection of column 1 and row 1. This deterministic time is introduced from the commencement of activities. Then time of activity T2.1 is entered at the intersection of column 1 and row 2. The latter time is also assigned in a deterministic way. • Then the execution time of activity 2 is entered at the intersection of row 1 and column 2, which represents sector 1 and the activity done by contractor 2. The remaining rows in column 2 are filled by the execution times of activity 2 done by contractor 2 in the next sectors. The drawn up table of times is shown below. Table 3. General table of activity times Number of sector 1 2 ... i … n

Activity 1

Activity 2

Activity ...

Activity j

Activity ...

Activity m

T1.1 T2.1 ... Ti.1 ... Tn.1

T1.2 T2.2 ... Ti.2 … Tn.2

... ... ... … … …

T1.j T2.j ... Ti.j … Tn.j

... ... ... … … …

T1.m T2.m ... Ti.m … Tn.m

Calculation of elements of matrix Mm–1,m In a cyclogram, activities are represented by a broken continuous or discontinuous line depending on the continuity of an activity in particular sectors. In order to obtain the minimum total execution time (TTmin) broken lines should be situated as close to each other as possible. The distance, denoted as LT, is equal to one of matrix Mm–1,m elements – the element with the highest value. The matrices must be calculated for each two consecutive activities. Thus for m activities it is necessary to calculate (m–1) matrices. Then time matrices M1,2, M2,3, ..., Mm–1,m must be elaborated. If activities are done in n sectors, the elements of matrix M1.2 have the values as in formula (1):

M1.2

⎛ a1 ⎞ ⎛ T 1.1 ⎞ ⎞ ⎛ a1 ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎜ a2 ⎟ ⎜ T 1.1 + T 2.1 − T 1.2 ⎟ ⎟ ⎜ a1 + T 2.1 − T 1.2 ⎟ = ⎜ a3 ⎟ = ⎜ T 1.1 + T 2.1 + T 3.1 − T 1.2 − T 2.2 ⎟ = ⎜ a2 + T 3.1 − T 2.2 ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎜ . ⎟ ⎜. ⎟ ⎟ ⎜. ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎠ ⎝ an −1 + Tn.1 − T (n − 1).2 ⎠ ⎝ an ⎠ ⎝ .

(2.1)

30

Chapter 2

The way in which the successive elements of matrix M1.2 are calculated is shown Figs. 22 and 23.

Fig. 22. Graphic model of LT calculation method – step 1

Fig. 23. Graphic model of LT calculation method – step 2

The calculation of matrix Mm–1,m elements consists in summing the elements in column 1 and subtracting column 2 elements from the sum (e.g. to determine the 3rd element (a3) of matrix M1.2, it is necessary to add three values in column 1 and subtract two values from the top of column 2). The element with the maximum value is indicated for each calculated matrix.

31


Calculation of main values: CP, LT and TT The element with the maximum value in matrix Mm–1,m has a value of LT(m–1).m which represents the shortest time between an activity in progress (m–1) and the next activity (m). Critical point values CP(m–1).m (which correspond to the next activity (m) after the activity in progress (m–1)) are calculated as a sum of times LT of the preceding activities, according to (2): CP (m − 1).m = LT 1.2 + LT 2.3 + ... + LT (m − 2).(m − 1) + LT (m − 1).m

(2.2)

The total project execution time (TT) is equal to the length of the critical path. It often happens that calculations yield several critical paths in one cyclogram. In such a case it is necessary to choose one of them and calculate TT by summing the duration of the activities in the sectors which belong to the chosen critical path. Another way of calculating TT consists in summing CP(n–1).m and Tn.m (the last activity time in all sectors n). The critical path (CPH) starts at the end of the last activity and then it goes from the current activity schema to the previous one. The course of the critical path from the current activity to the previous one is determined taking into account the minimum distance between activities (LT). This means that it is necessary to find such sectors in which successive activities are done in a continuous way (without any breaks). The activities executed in the sectors (from the first activity to the last one) have a critical character and they form a critical path. Calculation example The calculation method is illustrated below with a simple example. Data for the calculations were taken from the TCM III graphics presented earlier in the paper. The calculations are made for two cases: Gantt’s charts in Fig. 4 and Fig. 5, differing in sector size. Three contractors executing three activities in two sectors are assumed. Elaboration of table of times Tn.1 Case 1 T1.1 = 8 time units The execution times are shown in Table 4. Matrix Mm–1.m defining the project structure is created. Calculations are performed for the data presented in Figure 4. Table 4. Table of execution times of activities 1, 2, 3 in sectors 1, 2 Number of sector 1 2

Activity 1

Activity 2

Activity 3

T 1.1 = 8 T 2.1 = 12

T 1.2 = 4.8 T 2.2 = 7.2

T 1.3 = 3.2 T 2.3 = 4.8

32

Chapter 2

Case 2 T1.1 = 12 time units The execution times are shown in Table 4. Matrix Mm.–1.m defining the project structure is created. Calculations are performed for the data presented in Figure 5. Table 5. Table of execution times of activities 1, 2, 3 in sectors 1, 2 Number of sector 1 2

Activity 1

Activity 2

Activity 3

T 1.1 = 12 T 2.1 = 8

T 1.2 = 7.2 T 2.2 = 4.8

T 1.3 = 4.8 T 2.3 = 3.2

Calculation of elements of matrix Mm–1,m Case 1 T1.1 = 8 time units M1.2 and M2.3 are calculated from formula (1).

8 ⎛ ⎞ ⎛ 8 ⎞ ⎟⎟ = ⎜⎜ ⎟⎟ M1.2 = ⎜⎜ ⎝ 8 + 12 − 4,8 ⎠ ⎝15,2 ⎠

(2.3)

The second element (which is equal to 15.2) is the matrix M1.2 element with the maximum value. Hence it is assumed as LT1.2. 4 .8 ⎛ ⎞ ⎛ 4.8 ⎞ ⎟⎟ = ⎜⎜ ⎟⎟ M 2.3 = ⎜⎜ ⎝ 4.8 + 7.2 − 3.2 ⎠ ⎝ 8.8 ⎠

(2.4)

The second element (which is equal to 8.8) is the matrix M2.3 element with the maximum value. Hence it is assumed as LT2.3. Case 2 T1.1 = 12 time units M1.2 and M2.3 are calculated from formula (1). 12 ⎛ ⎞ ⎛ 12 ⎞ ⎟⎟ = ⎜⎜ ⎟⎟ M1.2 = ⎜⎜ ⎝ 8 + 12 − 7.2 ⎠ ⎝12.8 ⎠

(2.5)

The second element (which is equal to 12.8) is the matrix M1.1 element with the maximum value. Hence it is assumed as LT1.2. 7 .2 ⎛ ⎞ ⎛ 7 .2 ⎞ ⎟⎟ = ⎜⎜ ⎟⎟ M 2.3 = ⎜⎜ ⎝ 4.8 + 7.2 − 4.8 ⎠ ⎝ 7.2 ⎠

(2.6)

Both elements (each equal to 7.2) are the matrix M2.3 elements with the maximum values. Hence they are assumed as LT2.3.

33


Calculation of main values: CP, LT and TT Case 1 T1.1 = 8 time units According to formula (3): LT1.2 = 15.2 and according to formula (4): LT2.3 = 8.8. CP1.2 = 15.2; CP2.3 = 24 According to formula (2): TT = 32. Case 2 T1.1 = 12 time units According to formula (5): LT1.2 = 12.8 and according to formula (6) LT2.3 = 7.2. CP1.2 = 12.8; CP2.3 = 20 According to the formula (2): 2TT = 28. Cyclograms In order to show the differences in the course of critical path CPH, two solutions are presented here. The course of activities 1, 2, 3 in sectors 1, 2 (corresponding to Gantt’s chart in Fig. 4, TCM III and TT = 32 time units) is shown in Fig. 24. The course of activities 1, 2, 3 in sectors 1, 2 (corresponding to Gantt’s chart in Fig. 5, TCM III and Time units

50

CPH

40

Activity 3

LD 2.3

30

LT2.3

Activity 2

LD 1.2

20

Activity 1

10 LT1.2 1

2

3

Fig. 24. Case 1: course of activities 1, 2, 3 in sectors 1, 2 for data from Fig. 4, TCM III

34

Chapter 2

TT = 28 time units) is shown in Fig. 25. The element with the highest value is selected from the calculated elements of matrix M1.2. The above value is marked on the time axis and it is a starting point of activity 2. Then, after drawing the broken line for activity 2, LD1.2 (the shortest distance between activity 1 and activity 2) is determined by the graphic method. Finally, the critical path (CPH) should be determined. It starts at the end of the last activity and then returns to the preceding activity. Time units

50

CPH

40

activity 3 30

CP 2.3

activity 2

LD 2.3 20 LT2.3

activity 1

LD 1.2 10

CP 1.2

LT1.2 1

2

3 Sector

Fig. 25. Case 2: course of activities 1, 2, 3 in sectors 1, 2 for data from Fig. 5, TCM III

According to Figures 24 and 25, the main values on the two critical paths are different. Control points CP1.2 have the following values: CP1.2 = 15.2 in Fig. 24 while in Fig. 25 CP1.2 = 12.8. Similarly, the values of points CP2.3 are respectively 24 and 20. The shortest times between the activity in progress and the next activity in the same sector have the following values: LT1.2 = 15.2 in Fig. 24 while LT1.2 = 12.8 in Fig. 25, LT2.3 = 8.8 in Figure 24 whereas LT2.3 = 7.2 in Fig. 25.

2.8. Directions in Time Coupling Method TCM development Victor A. Afanasjev’s Time Coupling Method is being further developed. Many scientists have based their dissertations on TCM’s assumptions, e.g. J. Mrozowicz (1984) [33], Z. Hejducki (2001) [23], M. Sawicki (1997) [45], I.E. Rumianceva


35

(1981) [42], A.V. Afanasjev (son) (2000) [4] and others. The above scientific works have not been presented in international publications yet. They deal with, among other things, internal connections among activities and sectors under technical and resources limitations. Also interdependences for the execution of activities in several projects at the same time are investigated. Calculation algorithms taking into account limitations, e.g. the minimum execution time of an activity, have been developed. The development of TCM is aimed at adapting it to TOC (Goldratt’s theory of constrains) and CCS/BM (Critical Chain Scheduling/Buffer Management), taking into account randomness, risk and activity duration uncertainty, M. Rogalska (2003) [41], improving B&B algorithms and extending TCM to enable work of more than one contractor in one sector.

3. Time optimization in Time Coupling Method TCM by B&B algorithms It has been presented the algorithm of construction work graphic schedule which takes into account the proper sequencing problem. The algorithm has been described step by step i.e. the construction work organization methods with zero couplings between realisation means. It has been also shown the calculation process of “Limit Possible Minimum” LPM with taking into account technical and organization limits. The complexity of realization processes in construction, where new technological solutions are introduced continuously, requires the continuous improvement of construction work organization methods. This applies to all stages of the investment process from planning, through design, to realization. Therefore increasingly sophisticated methods must to be developed and applied in practice to solve optimally organizational-technological problems. In construction, the process of planning and scheduling has typically been accomplished using network scheduling techniques. The most common is the critical path method (CPM). Challenges in scheduling method has been offered as a solution to these problems by: Arditi (1986, 2001, 2002), Hegazy 1993, Johnston 1984. Line of balance (LOB) is a variation of linear scheduling method and present by Arditi, Psaros (1987), Hamerlink and Rowings (1986), Rabhar, Harris and Ioannou (1998), provide an algorithm for determining the controling activity path in a linear schedule. Solutions of many particular problems have been presented by: Chrzanowski i Johnson (1986), Harris i Ioannou (1998). Solutions of choice of optimal scheduling variants with application of many criteria have been presented by: Zavadskas (2002). It has been presented the algorithm of construction work graphic schedule which takes into account the proper sequencing problem Afanasjev (1994, 2000), Mrozowicz (1997, 1999, 2001), Hejducki (1999, 2000, 2001). One of the factors which determine the effectiveness with which construction works are carried out is their proper sequence taking account of technological and organizational constraints. The problem belongs to the deterministic theory of scheduling tasks and it has permutational character and n! possible variants. Because of the large number of its possible solutions, the problem is considered to be highly complex

38

Chapter 3

computationally. In practice, one can establish a rational sequence in which works are to be carried out by changing the order of works on the particular work fronts. However, it would be difficult, or even impossible, to check all the possible realization variants because of the computational complexity of the combinatorial problems. Therefore review division and limitation algorithms, enabling the focused surveying of the space of solutions, are used. Because of the relatively high computing speed they are more effective, but less accurate, than a comprehensive review of realization variants would be. They allow one to find a suboptimal solution. To organize building works, methods of division and limitation allowing one to establish a rational sequence of the works are employed. They take into account the specificity of construction processes through an index called “Limit Possible Minimum – LPM” which characterizes nodes in a tree diagram constructed according to the rules of the division and limitation method. The present shows the algorithms of construction work scheduling with a calculation example. This part has been made within the project of The International School of Flow Organisation Methods, which has been founded and developed by Professor Victor A. Afanasevs’ team from St. Petersburg State University of Architecture and Civil Engineering. A lot of projects have been made according to “The Principles of St. Petersburg’s School of Flow Work Organisation” elaborated by Professor V. A. Afanasevs’ team. Work organization method with zero couplings between realisation means (TCM I) This method ensures the continuity of work of gangs, machines and equipment. Specialist gangs move without breaks from one lot to another keeping up work continuity in the partial flows, which is a common organisational constraint. A flow is modelled according to the above method in two stages: first a rational sequence in which jobs are to be performed on lots (building structures) is determined and then time characteristics needed for graphic models (the Gantt chart, the cyclic line diagram and the network diagram) are calculated. A sequencing problem can be solved by means of the modified S.M. Johnson algorithm. Together with the branch and bound method it allows one to determine a rational sequence of jobs on lots with uninterrupted continuity of work in partial flows and minimised realisation time. Although this method enables the planning of the continuous work of gangs, it often results in work sector downtimes. The latter can be eliminated only if rhythmic flows are modelled. In a flow designed in this way, the time of deployment of partial flows (the time which passes between the entrance of successive gangs onto a building site) increases as the number of work lots increases. Research has shown that this may considerably prolong the task lead time. The latter may become longer than the time of realisation

Time optimization in Time Coupling Method TCM by B&B algorithms

39

of a work complex planned using the method that takes into account couplings between both realisation means and work sectors. Work organization method with zero couplings between work sectors (TCM II) The main feature of this method is the possibility of planning continuous work in work sectors. The minimum work complex realisation time can be obtained if such a sequence of jobs is determined at which the sum of first construction process realisation times and times of winding down the remaining processes is the smallest. If we look at graphical representations (cyclograms) of the flow we can see that the middle block of construction processes has no effect on the work complex lead time. This fact was exploited in an algorithm searching for a sequence of objects to be realised, which would ensure the shortest lead time. The larger the number of jobs “covered” by the first process, the smaller the sum of winding down times. A review of possible combinations involving the use of the tree diagram and an auxiliary index referred to as the lowest possible minimum (LPM) will lead to a rational sequence of performing jobs in the work sectors. Then the basic time characteristics of the flow modelled by means of this method are determined. Continuous work on lots is possible if breaks in the work of gangs are permitted. The gangs can pass from one lot to another immediately after the preceding gang leaves the lot. This condition can be fulfilled if a given work gang has finished the job on the preceding (in a fixed order) lot and waits for work. This constraint affects the work complex realisation time which is usually longer than the time determined taking into account the couplings between the realisation means and the work sectors. Work organization method taking into account couplings between realisation means and work sectors (TCM III) This method ensures the shortest lead time for a work complex. Since the two kinds of time couplings are taken into account at the same time, a critical path appears in the model. By adding up the lead times of the jobs along the critical path the work complex lead time (T) is obtained. If the sequence of jobs in work sectors (building structures) is changed, the order of jobs along the critical path also changes and so does the work complex lead time. The minimum lead time for a complex of structures can be achieved if such an order of their realisation is established that the sum of the lead times of the jobs that precede and close the leading partial flow or work complex will be the smallest, assuming that the critical path crosses the leading partial flow or complex.

40

Chapter 3

The sequencing problem is solved by means of the branch and bound method which allows one to search the solution space. After a rational order of jobs on lots (structures) is established the time characteristics of the flow are determined taking into account two kinds of couplings: couplings between realisation means and couplings between work sectors. In this method, breaks in the work of gangs and work sector idle time are allowed in order to arrive at such a critical path which ensures the shortest lead time for the investment task. The work sector idle time or the gang’s time of waiting for employment are reduced substantially through the immediate availability of gangs for employment. But the achievement of the shortest lead time for a complex of structures through this method poses difficulties in the planning of uniform and continuous use of realisation means. Work organization methods taking into account couplings between realisation means and work sectors, diagonal couplings and reverse diagonal couplings (TCM IV) The algorithm used for searching for the sequence of building structures to be realised by these methods is similar to the one described above. The difference is in the way the flow is modelled. For each of the methods, flow time characteristics are determined taking into account appropriate couplings. Tests have shown that work complex lead times in this case are longer or equal to the ones obtained by the above method. Work organization methods taking into account diagonal couplings and reverse diagonal couplings (TCM V) In this method, jobs having the same rank form diagonals whose duration is equal to that of critical jobs. A change in the order of jobs in the particular sectors alters the set of critical jobs and consequently changes the work complex lead time. The minimum job lead time can be achieved by putting jobs in such an order that jobs of longest duration are located on one diagonal or on as few diagonals as possible. Basic time characteristics are determined taking into account appropriate diagonal and reverse diagonal couplings.

3.1. Algorithms for establishing construction work sequence One of the factors which determine the effectiveness with which construction works are carried out is their proper sequence within technological and organisational


41

constraints. The problem belongs to the deterministic theory of scheduling tasks and it has permutational character and n! possible variants. Because of the large number of its possible solutions, the problem is considered to be highly complex computationally. In practice, one can establish a rational sequence in which works are to be carried out by changing the order of works in the particular work sectors. However, it would be difficult, or even impossible, to check all the possible realisation variants because of the computational complexity of the combinatorial problems. Therefore, review division and limitation algorithms, enabling the focused surveying of the space of solutions, are used. Because of the relatively high computing speed they are more effective, but less accurate, than a comprehensive review of realisation variants would be. They allow one to find a suboptimal solution. To organise building works, branch and bound methods allowing one to establish a rational sequence of the works are employed. They take into account the specificity of construction processes through an index called “Limit Possible Minimum – LPM” which characterises nodes in a tree diagram constructed according to the rules of the branch and bound method. Elimination and selection rules A set of branch & bound algorithms which are used to solve the permutation problem can be characterised by a set of parameters. Among them the selection rule (allowing one to determine the search tree’s next vertex at which division is to take place) and the elimination rules (enabling the reactivation of the tree’s generated nodes) should be singled out. In construction work organization methods the following matrix notation of input data is used: T = [ti,j]

i = 1, ..., m

j = 1, ..., m where: ti,j – is the time of carrying out the j-th work in the i-th sector. The notation facilitates the division of works into subsets comprising matrix rows and the formation of tree diagram levels with a decreasing number of nodes. The first level will include n nodes and the last level – two nodes. The elimination and selection rules allow one to move, via calculation procedures, up the successive levels of the tree diagram and to discard some of the generated vortices. For this purpose an algorithm for determining the nodes’ numerical characteristics by means of LPM (Limit Possible Minimum) is used. Since the construction work organization methods have different properties, the tree’s nodes are described in different ways according to the peculiarities of the methods. Algorithms for determining LPM for a construction work organization method which takes into account couplings

42

Chapter 3

between realisation means, couplings between work sectors, and their combinations are presented below. The so-called tree of variants is used to analyse the different possible sequences of work sectors. In the first stage, n variants are considered. First the j-th ( j = 1, …, n) variant is realized while the order of the other sectors has not been established yet. For each of the variants one calculates the so-called limit possible time minimum (LPM) which has the property of being less than time T1 calculated for any sequence of the sectors whose order has not been fixed. In this way one makes sure that with further expansion, time T1 will not be less than LPM1. At the end of stage I it is checked for which variant the calculated LPM1 is the least. In the second stage the next variant for which LPM1 is currently the least is expanded. The expansion of the variant (with a certain number of rows s) whose sequence has been fixed consists in the forming of new variants (n–s) using one row with a fixed position from the rows whose positions have not been established yet. The whole procedure of searching for a sequence of work sectors is completed when the least of the LPM1s calculated so far is realised only by the variants whose all sectors have been determined, i.e. at the very bottom of the variant tree. Since the LPM1 calculated for the above final variants is equal to T1 and LPM1 cannot be less after the expansion of a given variant, the above procedure will yield all the possible sequences of work sectors with shortest time T of carrying out the whole work complex. Algorithms for establishing sequence in construction work organization method with zero couplings between realisation means (TCM I) The following formula for the duration of the realisation of a whole work complex for the continuous use of realisation means is applied:

T=

m

∑

t rj +

j =2

n

∑t

i,m

(3.1)

i =1

where t rj is the duration of the expansion of the next j-th partial stream, i.e. the difference between the time of commencement of the j-th partial stream and that of the j-1-th partial stream. The second summand, i.e. n

∑t i =1

i,m

(3.2)


43

does not depend on the sequence in which works are carried out in the particular sectors. Thus a search for the shortest duration T of the realisation of a whole complex of works can be limited to the minimum-time optimisation of the first summand, i.e. T1 =

m

∑t

r j

(3.3)

j =2

Expansion time t rj depends on the mutual synchronisation of two adjacent partial streams and so it depends on the sequence in which works are carried out in the particular work sectors. This can be expressed as follows:

⎡ k t rj = max ⎢ ti , j −1 + 1≤k ≤n ⎣ i=1

∑

k −1

∑t

⎤

i, j ⎥

i =1

⎦

(3.4)

If one solution is sufficient, one needs to test the variant tree only until LPM1 is currently minimum for any fully expanded variant. In order to represent fully the way in which the right sequence of carrying out works is established it is necessary to define a calculation procedure for LPM1. For this purpose one should use a modified Johnson algorithm which makes it possible to establish the proper sequence of work sectors for two partial streams. Johnson algorithm which takes into account technological and organisational relationships between construction works The input data are in the form of a two-column matrix: A = [ai,k] i = 1, ..., n

k = 1, 2 corresponding to two adjacent partial streams. Then one creates a three-column matrix: B = [bi,k] i = 1, ..., n

k = 1, 2, 3 The first two columns of matrix B consist of the same rows as in matrix A, but in the order established by the Johnson algorithm [Johnson (1954)]. The shifted rows’ initial numbers bearing information about which work sector they come from are in the third column. The modified Johnson algorithm, taking account of relationships specific for construction works, is presented below.

44

Chapter 3

Instructions I

Commentary

a) Make substitutions: sg:1; Sd: n.

sg (ds) – number of currently highest (lowest) free row.

b) Find such indices i0, k0 that max ai,0, k0 = 1 ≤ i ≤ n k = 1, 2

Searching for lowest value in table A.

c) If k0 = 1, make substitutions: bsg,1 = ai,0,1; bsg,2: = ai,0,2; bsg,3: = i0; sg : = sg + 1. If k0 = 2, make substitutions: bsd,1 : = ai0,1; bsd,2 : = ai0,2; bsd,3 : = i0; sd : = sd – 1.

Shifting upwards row with minimum element on left side.

Shifting downwards row with minimum element on right side.

d) Let M be large number – larger than all elements of matrix A, e.g. M = 1030. Make substitutions: ai0,1 : = M; ai0,2 : = M. e) If sg < sd, go back to point b) and if sg = ds, end algorithm.

Testing if all rows of matrix A have been exhausted.

Prior to the testing of the variant tree one should successively calculate auxiliary matrices Bj, j = 1, ..., (m – 1). Matrix Bj is created by applying the Johnson algorithm to the data in matrix A consisting of columns with numbers j, j + 1, i.e. A = [ti,j] i = 1, ..., n

k = 1, j + 1 Now LPM1 is calculated for the case with an established sequence of sectors (rows) bearing numbers (u1, ..., us), s ≤ n. Let us denote set (u1, ..., uS), consisting of the numbers of the rows whose sequence has been established, by D.


45

Instructions II Commentary a) For each j = 1, ..., (m – 1) create Matrix Cj shows relationship between matrix adjacent columns. Cj = [Cji,k] i = 1, ..., n First s rows have already been deterk = 1, 2 as follows: mined. i) For all indices i = 1, ..., s; k = 1, 2 substitute Introduction of remaining rows in order Cji,k: = tui, j+1–1 established by auxiliary matrix jB. ii) Substitute: p: = s + 1; iii) For all in turn i = 1, ..., n do if bi ,j3 ∈ D, substitute:

p: = p +1; C pj ,1 = bi ,jk for k = 1, 2 Period of expansion of matrix Cj’s second column relative to first one is calculated.

b) For all j = 1, ..., (m – 1) calculate k −1 ⎤ ⎡ k t rj = max ⎢ ci ,1 + c1, 2 ⎥ 1≤ k ≤ n i =1 ⎦ ⎣ i =1

∑

m −1

c) LPM1 =

∑t

∑

r j

j =1

Algorithm for establishing order in construction work organization method with zero couplings between work sectors (TCM II) The total duration of the realisation of a work complex for work organization where works in the particular sectors are carried out continuously can be represented as follows: T=

m −1

∑t

1, j

j =1

+

n

∑t

i ,m

+

i =1

m −1

∑Δ

(3.5)

r

r =1

where: Δr is a positive difference between the “hidden” part of sector (r + 1) and the “hiding” part of sector r, expressed by the following formula

Δ r = max(Δ r ,0)

⎡ m −1 Δ r = max ⎢ t r +1, j + 1≤ k ≤ n −1 ⎢⎣ n = k

∑

m

∑t

j = k +1

⎤

r, j ⎥

⎥⎦

(3.6)

46

Chapter 3

Numbers Δr are interruptions (downtimes) caused by the fact that it was impossible for each of the previous gangs to work simultaneously in the two sectors: r and (r + 1). Similarly as in the previous method, a variant tree is used to test different possible sequences of work sectors. The numerical value of LPM, which is less than realisation time T for the rows whose sequence has already been established, should be determined for any order of the rows whose sequence has not been established. For this purpose let us define two matrices P and R. Matrix P = Pi Jk=1,…,n contains the following numbers (in a descending order): m

∑t

i = 1, ..., n

i, j

(3.7)

j =2

The above are sums of all, except the first one, the elements in a given row. Similarly the matrix: Q = [qi,j]

i = 1, ..., n

j = 1, 2

contains in its first column the following numbers (in a descending order): m −1

∑t

i = 1, ..., n

i, j

(3.8)

j =1

which are sums of all, except the last one, the elements in a given row. Whereas the second column holds the numbers of rows (fronts) prior to ordering. The sequence of rows bearing numbers (u1, ..., us) has been established. Let nk denote the number of the k-th row of matrix Q, containing in the second place a number which does not belong to set (u1, ..., us), i.e. a row whose position has not been established (qnk,2 ε < u1, ..., us>). Let us define dk = pk – qnk,1,

for k = 1, ..., (n – s).

Numbers dk are hypothetical quantities Δr for the rows whose sequence has not been established. When the sequence of all the rows has been established, possible interruptions (downtimes), i.e. Δk, will be longer. If when calculating LPM one replaces Δr in the formula for work complex realisation time T by appropriate dk, one will obtain LPM not higher than the value of time T. For any order of rows whose sequence has not been established dk allows one to find the optimum solution by means of the variant tree.

LPM =

m −1

∑

j =1

t u 1, j +

n

∑ i =1

ti , m +

s −1

∑

r =1

Δr +

n−s

∑d k =1

d

(3.9)


47

where: Δ r = max(Δ r , 0)

⎡ m -1 Δ r = max ⎢ tur +1, j + 1≤ k ≤ n −1 ⎢⎣ k = k

∑

⎤

m

∑t

ur , j ⎥

⎥⎦

j = k +1

Similarly as in the previous method, the variant tree is searched by expanding the variant whose LPM is currently the least and the search ends when the minimum is realised by only the variants whose sequence of all rows has been established. Algorithms for establishing sequence in construction work organization method which takes into account couplings between realisation means and couplings between work sectors (TCM III)

This method differs fundamentally from the other two. It assumes that the realisation time is equal to the sum of the durations of the works along the critical path. This is a set of works which must be carried out sequentially, i.e. one after another, whereby it becomes impossible to find a time of the realisation of a whole work complex which would be shorter than the critical path. First the sums of the elements in the particular rows and columns of matrix T are calculated, i.e. Wi =

m

∑t

i = 1, ..., n

i, j ,

j =1

Kj =

n

∑t

j = 1, ..., m

i, j ,

i =1

Then the highest of the values is determined, i.e.

M = max which is done for a matrix row or column. It is assumed that the critical path runs through a selected row or column determined in the above way. The value of LPM corresponds to the minimum critical path passing through a selected row or column for the established sequence of rows. If the sequence of rows bearing numbers u1, ..., us is established, then for the critical path running through row i0 one determines LPM as follows

LPM =

s

∑ k =1

tuk ,1 +

m

∑ j =1

tio, j +

n

∑t

i, m i =1 i∈( u1 , ..., u s )

(3.10)

48

Chapter 3

But if the critical path runs through column j0, LPM is given by this formula: j −1 o

LPM =

∑ j =1

tu1, j +

n

∑

min

ti , jo + 1 ≤ i ≤ n

i∈( u1 , ..., u s )

i =1

m

∑

ti , j j = j o +1

(3.11)

If maximum M is realised for a column, the variant tree will grow very much and usually almost a whole subtree (branch), corresponding to a variant with one row’s position established, will be expanded. LPM has such a property that as the tree is expanded the LPM value does not decrease and the sought variant will be located at the tree’s bottom. If maximum M is realised for a matrix row, LPM may both increase or decrease as the variant tree’s branch is expanded, which makes it impossible to find a minimum LPM without searching all the nodes and branches of the tree diagram. Algorithm for determining minimum LPM

If M = Wi,0, take step a). If M = Kj,0, take step b). If M = Wi,0 = Kj,0, take both step a) and b). Instructions i) For all i =1, ..., n, except i = i0, substitute bi = ti,1 – ti,m ii) Substitute bi,0 : = 0 iii) Put vector (bi) i = 1, ..., n into increasing sequence and into first column of matrix [Ci,k] i = 1, ..., n j = 1, 2 Place list of initial numbers prior to sorting into second column, i.e. c1,2 is number of next numeral ci,1 in sequence (bi) i = 1, ..., n

Commentary Calculation in each row, except i0, of differences between first column elements and last column elements. Above row i0 there are rows for which first element is smaller than last one and the other rows are below it.

Sequence (C1,2) i = 1, ..., n is the sought sequence of sectors. It should be noted that the same LPM value will be obtained also for other sequences. The rows above the critical row can be reordered in any way and also the rows below the critical row can be reordered. If i1, c1,1,2 = i0 (i.e. the critical row has been placed as ii), one can arbitrarily rearrange the first (i1 – 1) elements in sequence /(c1,2) i = 1, ..., n and also (c – i1), which will not affect the LPM value.


49

If the critical path runs through a particular column bearing number j0, then its length depends only on the established positions of the first and last row. Thus it is reasonable to adopt the following notation: let OPT(i, j) mean that a set of LPM indices of a sequence of sectors is a set of all the sequences whose first and last element is i and j, respectively. Instructions i) For all i = 1, ..., n make substitutions: j −1 o

ai =

∑t

i, j

j =1

bi =

Commentary ai is time needed to carry out all preceding works, bearing number j0, in the i-th work sector. Similarly bi for following works bearing number j0

m

∑

ti , j j = j o +1

ii) Find such index r that ar = min ai 1≤ i ≤ n

Searching for least elements.

and such index s that bs = min bi 1≤ i ≤ n

iii) If r ≠ s, then OPT (r, s). Searching elements second in turn. If r = s, perform sequence of instructions: α) Find such index r1 that ar ,1 = min ai 1≤ i ≤ n i≠r

and such index s1 that bs ,1 = min bi 1≤ i ≤ n i≠s

β) If ar + bs,1 ≥ ar,1 + bs, then If ar + bs,1 ≤ ar,1 + bs, then OPT (r, s1)

Testing which pair (minimum element, element second in turn) yields minimum LPM.

The above procedures are applied to establish the sequence of construction works carried out on building structures or their parts (sectors, work sites). A major problem in the variant-tree calculation is the determination of the value of the tree diagram’s nodes (limit possible minimum – LPM). This index expresses the technological and organisational constraints represented by time couplings. By comparing these values one can eliminate the variants which hold no promise of a better result. Thus directions in which the space of solutions will be searched can be determined. Practical example

As an illustration of the algorithm for work planning by the work organization method with zero couplings between realisation means, an investment problem, consisting in the realisation of four structures (A-B-C-D) on which seven technological

50

Chapter 3

processes: I – earth work, II – foundation work, III – masonry work, IV – concreting work, V – roofing work, VI – plaster work and VII – finishing work are to be carried out, was formulated. Work durations, constituting elements of a work lead time matrix, were determined. A detailed calculation procedure based on the algorithm described above was worked out.

3.2. Work sequencing in Time Coupling Method TCM I STAGE I Work lead times for particular objects (building structures) – initial matrix Earth work I 4 6 3 3

Object A Object B Object C Object D

Foundation work II 2 2 4 4

Masonry work III 26 17 29 20

Concreting work IV 23 5 22 13

Roofing work V 12 5 3 11

Plaster work VI 8 10 19 13 Σ

Finishing work VII 32 37 39 34 142

STAGE II Deployment times for particular works (see: Instruction I) 1 4 6 3 3

2 2 2 4 4

trF = 9 4 23 5 22 13 trD = 43

2 2 2 4 4

3 26 17 29 20

trM = 2 5 12 5 3 11

5 12 5 3 11 trT = 12

3 26 17 29 20

4 23 5 22 13

trB = 44 6 8 10 19 13

6 8 10 19 13 TrF = 8

7 32 37 39 34


51

Lead time for structure T = trZ + trF + trM + trB + trD + trT + Σtim9w0 TS = 9 + 2 + 44 + 43 + 12 + 8 + 142 = 260 days Matrices reconstructed according to Johnson algorithm (see: Instruction Ib, c) 1′ 3 3 4 6

2′ 4 4 2 2

2′ 2 2 4 4

3′ 17 26 20 29

3′ 29 26 20 17

4′ 22 23 13 5

4′ 23 13 5 22

5′ 12 11 5 3

5′ 3 5 11

6′ 19 10 13

6′ 8 10 13

7′ 32 37 34

STAGE III Matrices with successively determined rows (see: Instruction IIa) A

1 4 3 3 6

2 2 4 4 2

3 26 2 4 4

17 20 29

4 23 29 20 17

22 13 5

5 12 13 5 22

11 5 3

6 8 3 5 11

19 10 13

10 13 19

7 32 37 34 39 142

GMM IA = 6 + 2 + 34 + 35 + 12 + 8 + 142 = 239 units B

1 4 3 3 4

2 2 4 4 2

3 17 2 4 4

26 20 29

4 5 29 26 20

22 23 13

5 5 23 13 22

12 11 3

6 10 3 11 12

19 13 8

GMM IB = 7 + 2 + 45 + 35 + 5 + 10 + 142 = 246 units

8 13 19

7 37 32 34 39 142

52

Chapter 3

C

1 3 3 4 6

2 4 4 2 2

3 29 2 2 4

17 26 20

4 22 26 20 17

23 13 5

5 3 23 13 5

12 11 5

6 19 5 11 12

10 13 8

8 10 13

7 39 32 37 34 142

GMM CI = 6 + 4 + 34 + 43 + 3 + 19 + 142 = 251 units D

1 3 3 4 6

2 4 4 2 2

3 20 2 2 4

17 26 29

4 13 29 26 17

22 23 5

5 11 23 5 22

12 5 3

6 13 3 5 12

19 10 8

8 10 19

7 34 32 37 39 142

GMM ID = 6 + 4 + 40 + 35 + 11 + 13 + 142 = 251 units Matrices constructed at intermediate stages A B

1 4 6 3 6

2 2 2 4 2

3 26 17 4 4

20 29

4 23 5 29 20

22 13

5 12 5 13 22

11 3

6 8 10 3 11

19 13

13 19

7 32 37 34 39 142

GMM IIAB = 9 + 2 + 44 + 35 + 12 + 8 + 142 = 252 units A C

1 4 3 3 6

2 2 4 4 2

3 26 29 2 4

17 20

4 23 22 20 17

13 5

5 12 3 13 5

11 5

6 8 19 11 10

13 12

10 13

7 32 39 37 34 142

GMM IIAC = 6 + 2 + 34 + 43 + 12 + 8 + 142 = 247 units A D

1 4 3 3 6

2 2 4 4 2

3 26 20 2 4

17 29

4 23 13 29 17

22 5

5 12 11 5 22

5 3

6 8 13 3 5

19 10

GMM IIAD = 6 + 2 + 34 + 35 + 15 + 8 + 142 = 246 units

10 19

7 32 34 37 39 142


53

STAGE IV Determining sequence of rows (see: Instruction II c) A C B D

1 4 3 6 3

2 2 4 2 4

3 26 29 17 20

4 23 22 5 13

5 12 3 5 11

6 8 19 10 13

7 32 39 37 34 142

GMM III ACBD = 8 + 2 + 42 + 43 + 12 + 8 + 142 = 257 units

A C D B

1 4 3 3 6

2 2 4 4 2

3 26 29 20 17

4 23 22 13 5

5 12 3 11 5

6 8 19 13 10

7 32 39 34 37 142

GMM III ACDB = 6 + 2 + 34 + 43 + 12 + 8 + 142 = 244 units

A

1 4

2 2

3 26

4 23

5 12

6 8

7 32

D

3

4

20

13

11

13

34

B C

6 3

2 4

17 29

5 22

5 3

10 19

37 39 142

GMM III ADBC = 8 + 2 + 51 + 35 + 15 + 8 + 142 = 61 units

A D C B

1 4 3 3 6

2 2 4 4 2

3 26 20 29 17

4 23 13 22 5

5 12 11 3 5

6 8 13 19 10

GMM III ADCB = 6 + 2 + 39 + 37 + 15 + 8 + 142 = 249 units

7 32 34 39 37 142

54

Chapter 3

TS=260

GMMA=239

GMMB=246

GMMAB=252

GMMAC=247

GMMACBD=257

GMMACDB=244

GMMC=251

GMMD=251

GMMAD=246

GMMADBC=261

GMMADCB=249

CYCLOGRAM OF REALIZATION OF BUILDING STRUCTURES FOR METHOD WITH ZERO COUPLINGS BETWEEN REALIZATION MEANS

Days

Fig. 26. Cyclogram for TCM I: Legend: I – Earth work, II – Foundation work, III – Masonry work, IV – Concreting work, V – Roofing work, VI – Plaster work, VII – Finishing work

As a result of the calculations a new realisation sequence: A-D-C-B was determined. This realisation sequence ensures a reduction in the task lead time from 260 units to 244 units, i.e. 6.15%. It contributes to a cost reduction and a profit increase according to the time-cost function. In building practice it means: a profit from the


55

earlier completion of a building structure, a decrease in capital, a decrease in construction work costs, etc. The algorithm for scheduling work by the work organization method with zero couplings between realisation means was applied to the practical realisation problem described above. A computer system named ORGANIZER, which incorporates the presented algorithms and enables their use in construction practice, has been developed. Recapitulation

The presented procedures allow us to establish a sequence in which to carry out works, taking into account couplings between means of realisation and couplings between work sectors. It has been shown how to establish a sequence of works in organization methods with a critical path. Since a critical path occurs in a large set of construction work organization methods, the presented procedure can be used to build organisational models.

3.3. Organization methods enabling modelling of work complexes with complex internal structure The scheduling of large complexes comprising many structures by means of the classical organization methods encounters serious difficulties. Additional technological and organizational constraints appear connected with, for example, the maintenance of the continuity of the leading processes in a complex and the performance of jobs by different organization methods or some kinds of jobs in parallel. Such problems cannot be solved by means of the existing organization methods. Therefore several algorithms enabling rational scheduling of complexes of building structures have been developed. The algorithms use the minimum-time optimization criterion and take into account the parallel or series structure of modelled complexes. The algorithms’ sequences and their basic assumptions and application areas are presented below. The terminology and definitions used are described in the preceding chapters. A complex of building structures to be realized forming a set:

O = {O1 , O2 , ..., On } is given. The jobs are performed by specialized gangs: B = {B1 , B2 , ..., Bm }

56

Chapter 3

In each building structure specified technological work processes:

O1 = 〈 Pi1 , Pi 2 , ..., Pim 〉 are to be carried out. For each process there is a specific realization: Pi , k −1 = 〈 Pik 〈 Pi , k +1

which follows from the work technology. To harmonize a complex made up of many structures the latter must be combined in such a way as to minimize the lead time for the complex: T → min. Modelling of complexes with series structure

This applies to the “vertical” division of a complex of structures necessitated by the application of different organization methods. In practice it often becomes necessary to perform some jobs by a method which takes into account couplings between realization means to ensure the continuity of work for the gangs (e.g. during the assembly of building structures). Finishing work, because of its significant effect on lead time, should be planned using the method which takes into account couplings between both realization means and work fronts and thus ensures the shortest lead time. Then the problem of synchronizing some of the structures, modelled by different organization methods and realized in different sequences, arises. An algorithm for modelling a complex with series structure has been developed (fig. 8.1). The algorithm’s sequence is as follows: Fig. 8.1. Diagram of complex with series structure. Building structures, Time scale Step 1. Determine a rational sequence of realizing the structures in the complex represented by the matrices: T1 = [tij ] ,

T2 = [ti , j +1 ]

where tij – the lead time of j-the job on the i-the structure, is determined. Step 2. After determining which kinds of jobs form the zone of junction between complexes of structures, create a set of the latest job completion dates for the work fronts. Create a sequence of values for the last kind of jobs: pz t k = [t1pzm , t 2pzm , ..., timpz , ..., t nm ]

where: k – the number of the complex of structures,


57

timpz – the latest date of completing the process in the i-th structure, n – a number of work fronts, m – a number of processes.

Step 3. Form a set of forecasted job lead times from the elements in the first column of the second matrix:

t k +1 = [t1, m +1 , t2, m +1 , ..., ti , m +1 , ..., tn, m +1 ] Once the sequence in which the structures are to be realized is fixed it often turns out that their order in the two complexes of structures is different. Then the different work organization methods and the order in which the structures are to be realized must be taken into account at the same time. Step 4. Create a set of necessary breaks for the successive structures to ensure collisionless work of gangs, according to the specified work organization method:

U = {U1 , U 2 , ..., U i , U n }, where: p

U i = timpz

−

∑t

i , m +1 ,

for k = 1, 2, ..., p; p = i – 1

k =1

p

∑t k −1

i , m +1

– the total of job lead times on the preceding fronts in a fixed realization

sequence, timpz – the latest job completion date for a given structure. Step 5. Survey the elements of set {Ui} and choose the largest one. It will indicate the period of expansion of the kinds of jobs in the junction zone:

U i = max{U }

U i = t1wr , m +1 Step 6. Determine the following earliest job commencement dates for the other elements: p

t1wr , m +1 = U i +

∑t

j , m +1

k =1

The above algorithm makes it possible to synchronize complexes of structures (or their parts) so that downtimes in the work of the gangs will be minimized in the junc-

58

Chapter 3

tion zone. The solution is sought in two steps. First a rational sequence of realizing the structures is determined and then the above algorithm is applied. Illustrative application of algorithm

A table-matrix of forecasted job lead times in the (modular) construction of a housing estate in Wrocław was drawn up. The construction jobs were integrated into eight kinds of jobs, i.e. the removal of humus, excavation, foundation work, assembly, complementary work, plaster work, painting work and parquet work, taking into account the standards in force and the resources of the contractor.

T=

2

7

5

54

12

13

15

13

5

20

15

141

34

37

35

38

2

7

5

54

12

13

15

13

5

20

15

141

34

37

35

38

The realization of the first four kinds of jobs was planned using the flow work organization method with zero couplings between realization means whereas the other complexes of structures were planned using the method which takes into account couplings between both realization means and work fronts. Computations yielded submatrices with a new sequence of the buildings to be erected. Step 1. ⎡2 7 5 54 ⎤ ⎢ 5 20 15 141⎥⎥ T1 = ⎢ ⎢2 7 5 54 ⎥ ⎢ ⎥ ⎢⎣5 20 15 141⎦

⎡12 ⎢34 T2 = ⎢ ⎢12 ⎢ ⎣34

13 15 13 ⎤ ⎥ 37 35 38⎥ 13 15 13 ⎥ ⎥ 37 35 38⎥⎦

Starting with the sequence of rows in matrix T, the scheduling of the jobs resulted in a new sequence of realization of the buildings, i.e. for matrix T1 – 1, 2, 4, 3 and for matrix T2 – 2, 1, 3, 4, whereby the lead time for the complex was reduced. Step 2. A sequence of job completion date values:

t1 = [88, 229, 370, 424] Step 3. A sequence of forecasted job lead times for the first kind of jobs in the second matrix:

t 2 = [34, 12, 12, 34] Step 4. Form set {Ui}.


59

Table 8. 1. Initial sequence of realizing structures 1 2 3 4

timpz

88 229 370 424

Determined sequence of realizing structures 2 1 3 4

ti,m+1 34 1 12 34

Auxiliary job commencement dates for determined sequence 1-34 2-0 4 -34+12+12 3 -34+12

Ui 45 229 312 378

Step 5. Select the largest element of set {Ui}: U i , max = 378

Step 6. Determine commencement and completion dates for jobs in the first kind of jobs in the junction zone. Table 8.2. Determined sequence of realizing structures 2 1 3 4

tiwr , m +1

tiwz , m +1

378 + 0 = 378 378 + 34 = 412 378 + 34 + 12 = 424 378 + 34 + 12 + 12 = 436

378 + 34 = 412 412 + 12 = 424 424 + 12 = 436 436 + 34 = 470

The other time characteristics of the planned jobs can be determined using the methods described in the preceding chapters. The complexity of realization processes in construction, where new technological solutions are introduced continuously, requires the continuous improvement of construction work organization methods. This applies to all stages of the investment process from planning, through design, to realization. Therefore increasingly sophisticated methods must to be developed and applied in practice to solve optimally organizational-technological problems. Taking into account constraints which may occur in construction practice it has been observed that the presented interpretation of the notion of a time coupling is incomplete and thus limits the organization methods’ application area. In paper a proposition was made to broaden this notion and to adopt time couplings determined on the basis of the possible combinations of the earliest and latest dates of work commencement and completion. Thus a basis for a theory of time couplings was created and this made it possible to broaden the area of varying work organization methods. Research was undertaken to identify time couplings mapping the technological and organizational relationships between construction works and to indicate directions in which organization methods should be varied. The results were presented in. That was the first time this problem was brought to light thereby opening up new possibilities of

60

Chapter 3

varying and creating new work organization methods incorporating many additional technological and organizational constraints. Continuing the research into the identification of time couplings, practical possibilities of applying new kinds of such couplings and their combinations were investigated and the correctness of the mathematical notations has been verified. It has been found that the new kinds of time couplings constitute a significant element contributing to the development of new organizational methods. As presented in, new ways of determining time characteristics for the new variants of couplings make it possible to extend the methods’ application area in construction. Several kinds of time couplings are distinguished but here only questions connected with diagonal couplings linking various works in a technological sequence according to the principle that each work on a given front in a preceding technological sequence (of kinds of works) is linked to the work on the preceding front in a given technological sequence. An analysis carried out for the new cases of time couplings for the sixth work organization method has led to several practical conclusions.

4. Time/cost optimization by hybrid evolutionary algorithm The development of optimization techniques was spurred by a search for better methods of scheduling production (including construction projects). One such method is evolutionary computation (computer systems exploiting the principles of the evolution of living organisms). Scheduling of construction projects involves, among other things, the optimization of demand for realization means, e.g.: • optimize the employment level for an assigned project duration, • minimize project duration for an assigned level of means. This part concerns the optimization problem with an objective function in the form of a measure of employment level average deviation dispersion for an assigned construction project lead time. The aim is to compare, through a computation experiment, the results obtained by applying a hybrid evolutionary algorithm (HEA) and a classical genetic algorithm (GA) to the scheduling of a construction project with regard to employment level optimization. The problem belongs to a class of NP difficult problems. The optimization problem is as follows: for an assigned construction project lead time the best (according to the adopted optimization criterion) employment level must be found. One of the optimization criteria can be to include the average demand for resources in the constructed objective function. Time/cost optimization in Time Coupling Method TCM can apply to: • the minimization of resource costs for the assigned construction project lead time, • the determination of the total increase in construction project costs depending on the time buffer size for the assigned construction project lead time.

4.1. Basic properties of hybrid evolutionary algorithm The algorithm [12] starts with the (random) creation of initial population P0. The best member of population P0 is adopted as suboptimal solution π*. Let i be an algorithm iteration number. New population i + 1 (i.e. set Pi+1) is generated as follows. For

62

Chapter 4

current population Pi a set of local minima (LM i) is fixed (by carrying out the LocalOpt(π) procedure for each element π ∈ Pi). Elements which occur in the same positions at the local minima are fixed (procedure FixeSet(LMi, FSi)) and a set of fixed elements and positions (FS i+1) is created. Each permutation of new population Pi+1 has fixed elements (in fixed positions) from set FS i+1. Free elements are randomly assigned to the remaining (free) positions. If permutations β ∈ LM i and F(β) < F(π*) exist, then β is adopted for permutation π*. The algorithm ends after a predetermined number of generations have been generated. Hybrid Evolutionary Algorithm (HEA, W. Bożejko, M. Wodecki) [12–14] Initialization: randomly created population {P0 = π1, π2, ..., πη}; π* = the best member of population P0; the number of iteration i = 0; FS0 = ∅; repeat Fix local minima set LM i = {πˆ1 , πˆ 2 , ..., πˆη } , where

πˆ j = LocalOpt(πj), πj ∈ Pi; for j: = 1 to η do if F (πˆ j ) < F(π*) then π * ← πˆ j ;

Fix set FS + 1 = FixSet(LM , FSi) and generate new population Pi+1:= NewPopulation(FSi); i = i + 1; until not StopCriterion; i

i

The evolutionary algorithm’s principle of operation consists in searching for a global minimum through a random choice of a population of individuals (e.g. schedules). Different strategies, e.g. (l + 1), based on mechanisms of searching sets of feasible solutions through the self-adaptation of the mutation range are used. The genetic algorithm is a special case of the evolutionary algorithm. It has a predetermined constant population of individuals on which replications are performed. The hybrid evolutionary algorithm (HEA) first randomly creates an initial population, finds the best member and generates new populations in which local minima are sought. A set of the latter is formed and for each permutation of a new generation a better solution is adopted from the area between neighbouring local minima. The above optimization IT tools can be used to carry out practical construction project scheduling tasks, including the optimization of demand for means. They do not guarantee that optimum solutions will be found but for problems of practical size the results can be considerably improved. This can be done using classical genetic algorithms or improved evolutionary algorithms (such as HEA [12]). In order to improve the optimization tools, faster and more accurate algorithms for searching the space of

Time/cost optimization by hybrid evolutionary algorithm

63

solutions are developed . The results of a computation experiment indicating that better optimization results can be obtained through the use of a hybrid evolutionary algorithm are presented below.

4.2. Computation experiment – case study 1 The optimization problem is as follows: for an assigned construction project lead time the best (according to the adopted optimization criterion) employment level must be found. One of the optimization criteria can be to include the average demand for resources in the constructed objective function. In order to search for a minimum demand one can adopt the average deviation from the daily demand as a measure of the unevenness of the demand for workers. The objective function is:

f ( x) =

1 T

T

q j ( x) − ravg

j =1

ravg

∑

(4.1)

where:

ravg =

1 T

n

∑d r

i i

(4.2)

i =1

the average number of workers employed on each of the T days. x ∈ Rn x = (x1, x2, ..., xn) – the vector of the tasks’ starting times, xi ∈ [ai, bi] ai – the earliest time of starting task i, bi – the latest time of starting task i, qj(x) – the number of persons employed on day j, j = 1, 2, ..., T, T – a time horizon, di – duration of process i, ri – the number of persons employed to carry out process i. In order to perform optimization calculations one must calculate the objective function for each iteration. The function can be calculated as the average percentage deviation of the daily number of employed persons from the average employment during all the T days, i.e. f(x) ∈ [0, 1] for a standard objective function. The numerical data for the optimization calculations are for a complex of 12 building structures on which 9 construction processes are to be carried out. The project is represented by a matrix of construction process (r = 9) durations for the structures (f = 12).

64

Chapter 4

Fig. 27. Process of optimization by means of evolutionary algorithms [32]

The work duration matrix elements were based on bills of quantities for a housing complex. Time coupling method TCM III [4, 23, 34] was used to determine the construction project cycle (T = 216 working days). The method allows one to identify a sequence of critical processes and fix noncritical work commencement and comple-


65

tion dates. Owing to the changeable position of the jobs along the time axis one can optimize the employment level according to the adopted objective function.

⎡7 ⎢8 ⎢ ⎢8 ⎢ ⎢7 T = ⎢13 ⎢ ⎢11 ⎢9 ⎢ ⎢4 ⎢ ⎣6

4⎤ 11 8 9 9 11 8 9 8 9 8 8 ⎥⎥ 11 10 9 9 11 10 9 11 9 9 9 ⎥ ⎥ 8 7 7 8 8 7 7 8 8 8 7⎥ 6 7 7 7 7 7 7 7 7 8 15⎥ ⎥ 14 11 13 13 14 11 13 14 13 14 8 ⎥ 14 9 11 10 13 9 11 8 10 11 9 ⎥ ⎥ 8 6 7 5 7 7 8 9 9 9 5⎥ ⎥ 9 5 9 7 5 8 9 8 7 7 7⎦ 8

7

7

7

8

7

7

6

7

5

Fig.28. Gantt chart after genetic algorithm application, minimum objective function f = 1564.999

Fig. 29. Gantt chart after evolutionary algorithm application, minimum objective function f = 1508.139

66

Chapter 4

An optimization problem (12*9, T = 216 working days) concerning employment level planning for the assigned construction project lead time was computed using a genetic algorithm (GA) and a hybrid evolutionary algorithm (HEA) [12]. After 100 000 iterations using GA, objective function f = 1564.999 and the average percentage deviation of the daily number of employed persons from the average employment for all the T days amounting to 0.312375 were obtained. After 100 000 iterations using HEA, the objective function was f = 1508.999 and the average deviation amounted to 0.301198. Thanks to the use of the evolutionary algorithm the result improved by 3.57%. It has been demonstrated that evolutionary algorithms can be used for optimizing demand for resources (workers) in time coupling methods (TCM). It is also possible to employ metaheuristic (simulated annealing and taboo search) algorithms and evolutionary algorithms to solve task scheduling problems in TCM. The research into this possibility is underway. Currently, research focuses on hybrid methods of optimising complex construction processes to incorporate packages of processes described by TCM, LSM, LOB, genetic algorithms, artificial neural networks and bar-charts into one schedule.

4.3. Computation experiment – case study 2 The second optimization case consists in determining the total increase in construction project realization costs versus the time buffer size for the assigned construction project lead time. The cost analysis and control are often performed using graphical construction cost cumulation in the form of S curves, whereby one can analyze the increase in construction project costs depending on the time buffers [19, 40]. Problem formulation The starting point for the analysis is a situation when construction works are carried out as late as possible, i.e. the aim is to invoice and pay for the works as late as possible. In CPM/PERT and TCM III the positioning of works at the latest dates on the time scale makes all the works critical and devoid of time reserves (buffers) which would compensate for any disturbances in performing the works. Carrying out works at the earliest dates results in earlier (than necessary) payments to the contractors for the work done and contributes to an increase in project financing (e.g. credit) costs. An attempt was made to examine the contribution of noncritical work time buffer size to an increase in construction project (additional) costs. The interdependencies involved are expressed by a multicriterial objective function. In practice it becomes necessary to determine the best work commencement dates for the adopted criteria (in the form of an objective function) in a time interval calculated in a standard way.


67

Optimization calculations were performed using HEA. The algorithm makes it possible to determine noncritical work commencement dates for conditions expressed by an objective function. Objective function Let us assume that works assigned by permutation π are to be carried out. The works will be financed by a credit in amount D available in tranches. The objective function value corresponding to the financing of the excess (additional cost) above the available total tranche amount is: F (π ) =

T

∑ max{0, cπ (i) − t (i)} ,

(4.3)

i =1

where: cπ(i) – the total work cost on day i =1, 2, ..., T for the work order assigned by permutation π, t(i) – the tranche value on day i = 1, 2, ..., T, e.g. when the credit for amount D is divided into four equal tranches ⎧ ⎪0 ⎪D ⎪ ⎪ t (i ) = ⎨ 4 2D ⎪ ⎪ 4 ⎪ 3D ⎪⎩ 4

for i
TN, go to Step 1, otherwise STOP.

5.4. Application of selected metaheuristic algorithms The algorithms presented above are applied here to solve discrete optimisation problems encountered in the course of search for an optimum schedule, using TCM III (couplings between realisation means and couplings between work lots). As mentioned earlier, this method is very similar to problem FP⏐⏐Cmax, which is one of the most often solved problems in queuing theory where it is referred to as the permutation flow shop problem. Its assumptions are as follows:

Metaheuristic algorithms in Time Coupling Method TCM

77

• m different machines (working groups, gangs, etc.) are to do m jobs in order to carry out n production tasks (building structures, work lots, projects, etc.); • a set of machines (working groups): M = {M1, M2, M3, ..., Mk, ..., Mm}, • a set of production tasks (building structures): J = {J1, J2, J3, ..., Jj, ..., Jn}. In order to realize each task (building structure) Jj ∈ J, m different operations (jobs) from set Oj = {Oj1, Oj2, Oj3, ..., Ojk, ..., Ojm} must be performed in the same order by all the machines (working groups). It is assumed that operation (job) Ojk ∈ Oj is being performed continuously by machine (working group) Mk over time pjk > 0 (a set of durations pj of operations from set Oj: pj = {pj1, pj2, pj3, ..., pjk, ..., pjm}). It is also assumed that at any instant each machine (working group) can perform only one operation (job) and that only one operation (job) within a given task (building structure) can be performed at a given instant in the project. The optimization problem consists in finding sequence S (schedule) with specified dates of starting the operations, which would minimize the value of the objective function, i.e. the project lead time. The dates can be determined from this recurrence formula: Ci,π(j) = max{Ci, π(j–1), Ci–1, π(j)} + pi, π(j) ,

(5.1)

where: j = 1, ..., n, i = 1, ..., m, π(0) = 0, Ci,0 = 0, C0,j = 0. One seeks permutation π*∈ П for which: Cmax(π*) = min Cmax(π), π∈Π

(5.2)

where Cmax(π) = Cm, π(n). The decision variable is the order (permutation) in which the tasks (building structures) are carried out by all the machines (working groups), which is the same for all of them and it is as follows: π = (π(1), π(2), ..., π(k), ..., π(n)). The size of the set of all permutations in problem FP⏐⏐Cmax with size n × m, is n! From the optimization algorithm computing complexity point of view, already for three machines (working groups) this problem belongs to NP-difficult problems.

5.5. Application of evolutionary algorithms The problem concerns construction project planning by means of state-of-the-art computer techniques. Currently, operations research focuses on discrete optimization and improving metaheuristic algorithms such as: tabu search algorithms, genetic algorithms, evolutionary algorithms, simulated annealing algorithms and others. Results relating to the application of the algorithms, suggestions as to their improvement, case studies and general conclusions are reported in the literature on the subject. One of the potential application areas is the search for optimum construction project scheduling variants. A significant role here is played by the objective function as a tool for best

78

Chapter 5

variant evaluation and selection. In practice, it is a complex problem since often contradictory aims are to be achieved. According to [39], the criteria for evaluating and selecting schedules can be divided into: criteria based on project lead time, criteria based on required completion dates and cost based criteria. Many authors when formulating objective functions use weighted values. Exemplary objective functions are presented below. Objective function

Function form n

Weighted lead time

∑ w (c i

i

− ri )

i =1

Maximum delay

max Ti i

where: w – the assigned weight, c – the completion date, r – the date by which the work can be completed, T – the delay. The main works dealing with the application of evolutionary algorithms are those by Lawrence Fogel, Ingo Rechenberg, Hans-Paul Schewefel and John Holland. The classic genetic evolutionary algorithms are described at length in the paper by David Goldberg [18]. The fundamentals of evolutionary algorithms and their applications are presented in the publication by Zbigniew Michalewicz [32]. The development of optimisation techniques was spurred by the search for better methods of scheduling production (including construction). One of such methods are evolutionary algorithms. They are in fact computer systems using the principles of living organisms evolution. They comprise genetic algorithms, evolution strategies, evolutionary planning and genetic programming. The characteristic feature of the methods is the creation of a population of individuals (each being a schedule) through a random search among feasible solutions. The methods differ in their approach to the creation of new generation of individuals and in the selection and use of operators. Scheduling is also referred to as the tasks sequencing problem, the queuing problem or calendar planning. It belongs to basic building structure or production scheduling problems. The aim of scheduling is to interlink (in time and space) works, e.g. works whose lead time is the shortest, to be carried out according to the adopted objective function. Scheduling tasks in production (also in construction) consist in synchronizing works which are to be carried out. In a given production cell, consisting of (among other things) m workstations, production process n is carried on. The particular products are transferred between the workstation in the same fixed order. The durations of the particular technological operations are determined on the basis of standards or directively. Such an order of manufacturing the products should be determined for


79

which, for example, the total production cycle is the shortest (other objective functions are also adopted). In practice, four types of problems are distinguished: – the flow shop problem, denoted by F; – the permutational flow shop problem, denoted by P; – the job shop problem, denoted by G; – the parallel shop problem, denoted by T. Moreover, the following constraints are imposed: – the performance of a task on a given machine (work sector) cannot be interrupted, – each machine performs no more than one task at any time (this applies to each sector). The scheduling problem solution criterion applies to (among other things): – the time of execution of all the tasks: Tmax – the average time of completion of the tasks (jobs): T=

1 n Ti , n i =1

∑

where: Ti – the time of completion of the i-th task (job), n – the number of performed tasks (jobs); – delays in carrying out all the tasks (jobs):

Tmax = max{Ti } i =1, n

where: Ti = max {Ci – di} – a delay in carrying out the i-th task. In construction practice, because of the construction project features (e.g. individuality, uniqueness, prototypicality, etc.) different than those of industrial production and the necessity of taking into account special technological, material, organizational, financial and other constraints, the above problems occur in a mixed form. Scheduling from the lead time minimization point of view has been the subject of numerous publications. Construction work synchronization also depends on the availability of realization means. The optimization of demand for realization means can be regarded as a peculiar approach to construction project scheduling. In general terms, under fixed commencement and completion dates for the particular construction processes one should optimally use the limited resources. The optimization of demand for realization means includes such tasks as [30]: • under the assigned project lead time optimize the staffing level, • under the assigned level of means minimize the project lead time.

80

Chapter 5

An optimization problem with an objective function in the form of a dispersion measure of the average standard deviation of the staffing level under the assigned minimum construction project lead time is considered below. The aim is to compare (through a computing experiment) the results obtained by applying respectively the classic hybrid evolutionary algorithm (HEA) [12] and the classic genetic algorithm (GA) [18] to staffing level optimization in construction project scheduling. This problem belongs to NP-difficult problems [12].

5.5.1. Basic properties of genetic algorithms Optimization by means of GA consists of the following steps: • identifying the optimization problem solution structure; • encoding the solution in the form of combinations; • defining the objective function to be optimized; • defining GA operators and stopping criteria; • computing within the constrained area; • decoding the computed optimum solution combinations [18]. Figure 38 shows how optimization computations by means of GA are carried out. This methodology was used in [12, 18]. START Random selection of population of characteristics (signs, detectors) Development of characteristics’ usefulness in population Check stopping criteria (time, usefulness, etc.) SELECTION Selection of population of successive generations CROSSING Recombination of characteristics and selection of partners MUTATION Stochastic verfification of mated partners

EVALUATION Evaluation of new usefulness PROPAGATION Selection of next generations, based on usefulness

Fig. 38. Optimization by means of genetic algorithms [18]


81

5.5.2. Basic properties of hybrid evolutionary algorithm The algorithm [12] starts with the (random) creation of initial population P0. The best element of population P0 is assumed as suboptimal solution π*. Let i be the number of an algorithm iteration. New population i + 1 (i.e. set Pi+1) is generated as follows. A set of local minima LM i is determined (by carrying out procedure LocalOpt(π) for each element π ∈ Pi) for current population Pi. Elements occurring in the same positions in the local minima are fixed (procedure FixeSet(LM i, FSi)) whereby set FS i+1 of fixed elements and positions is created. Each permutation of new population Pi+1 has fixed elements (in the fixed positions) from set FS i+1. Free elements are randomly assigned to the remaining (unoccupied) positions. If permutation β ∈ LM i exists and F(β) < F(π*), then β is assumed as a permutation of π*. When a predetermined number of generations is generated the algorithm ends. Comparison of the properties of the algorithms The principle of operation of the genetic algorithm consists in creating a random population of n code combinations made up of binary system symbols V = {0, 1}, replicating the combinations, replacing their parts and sometimes mutating some of the bits. GA uses the schemata (describing subsets of similar combinations with good solutions) theorem [18]. As opposed to other methods, optimization computations are performed on whole populations in which decision variable values are sought, avoiding finding only a local maximum. The principle of operation of the evolutionary algorithm consists in searching for a global minimum through the random selection of a population of individuals (e.g. schedules). Various strategies (e.g. (1 + 1)), using mechanisms of searching sets of feasible solutions through the automatic adaptation of the mutation range, are employed [32]. GA is a special case of the evolutionary algorithm. It has a population of individuals fixed at the start after which replications are performed on this population. As regards he hybrid evolutionary algorithm (HEA), first an initial population is randomly created and then the best element is found and new populations are generated. Local minima are sought and a set of them is created. Then on the basis of each new generation permutation the elements which will be fixed in positions in the next generation are determined. The above IT tools of optimization can be used to solve practical construction project scheduling problems, including resource demand optimization. They do not guarantee that optimum solutions will be found but for practical size problems it is possible to considerably improve the results. Both classic genetic algorithms and improved evolutionary algorithms (e.g. HEA [12]) can be used for this purpose. Faster and more precise algorithms for searching a solution space are being developed. The computing test results presented below show that in optimization tasks the hybrid evolutionary algorithm performs better than the genetic algorithm.

82

Chapter 5 Start iter := iter+1

Determine free elements and positions

iter := 0

Yes iter > Max_iter No

Determine new elements

Random population

Delete older elements

Local optimization

Modify age of elements

End

Descending search

Automatic tuning of parameters

Fig. 39. Hybrid evolutionary algorithm [12]

5.5.3. Computing experiment – case study The scheduling of construction projects involves optimization: the best solution satisfying both the constraints and the objective function is sought. There are many optimization methods applicable to particular cases. The critical chain (CCS/BM) methodology [19, 40] as applied in construction project planning is described in [19, 40]. Carrying out a construction project involves satisfying requirements as to quality, cost, lead time and resources [40]. The results of research on improving construction project scheduling using HEA [12] are presented below. An optimum (for the adopted objective function) staffing level was sought. The aim was to minimize the deviation from the average staffing level determined using CCS/BM methods [19, 40].

Fig. 40. Classic scheme of operation of evolutionary algorithm [32]. (Environment, Initiation, Succession, Evaluation, Reproduction, Genetic operations)


83

The general scheme of operation of the evolutionary algorithm has the form of a loop encompassing reproduction, genetic operations, evaluation and succession [32]. Figure 40 shows a classic scheme of the evolutionary algorithm according to [32]. Considering the evolutionary algorithm to be the principal way of searching for better solutions (with the immediate neighbourhood taken into account) a slightly different approach than evolution modelling using binary genetic code was proposed. The approach consists in replicating randomly selected variants (individuals), where the same solution can be replicated repeatedly. The randomness of selection ensures better adaptation of variants (individuals). In operational research problems, including construction project scheduling, such an optimization method needs to be selected which will lead to the best solution at the lowest cost. The evolutionary algorithm with the (1+1) strategy, being the basis of the hybrid evolutionary algorithm (HEA) [12], is regarded as one of the most effective optimization algorithms [32]. Problem formulation

The optimization problem is to plan a uniform staffing level for a construction project, using the CCM/BM methodology [19, 40]. Time buffers introduced into the schedule increase the area of feasible solution variants and the problem size. The hybrid evolutionary algorithm (HEA) developed by W. Bożejko and M. Wodecki [12] was used to solve the optimization problem.

5.6. Case study A network model of a construction project consisting of n = 16 interlinked construction processes is analyzed. A computing test was carried out in order to determine whether time buffers in the project schedule can contribute to staffing uniformity. For the assumed constraints the objective function value was determined from the relation: f ( x) =

⎛ ⎜ q j ( x) − 1 ⎜ T j =1 ⎝ T

∑

⎞

n

∑ d r ⎟⎟ i i

i =1

⎠

where: x ∈ Rn x = (x1, x2, ..., xn) – a vector of task commencement moments; xi ∈ [ai, bi] ai – the earliest task commencement moment, bi – the latest task commencement moment,

(5.3)

84

Chapter 5

qj(x) – the number of persons employed on day j, j = 1, 2, ..., T, T – a time horizon, di – duration of process i, ri – the number of persons employed to carry out process i. A construction project network model with time buffers (supply buffers FB and project buffer PB) is shown below. Activity time and number and are given according to Goldratt’s method [19] and the anticipated resources (the number of employees) are specified.

Fig. 41. Network of dependencies

Table 6 shows the basic information on the construction project modelled by a network. The basic time parameters of the construction processes involved were computed. Table 6. Construction process data Process number 1 2 3 4 5 6 PB 7 8 FB 9 10 FB 11 12 FB

Earliest Duration commencement date 4 0 8 4 16 12 8 28 12 36 4 48 26 52 4 12 4 16 4 20 4 12 8 16 6 24 4 28 4 32 4 36

Earliest completion date 4 12 28 36 48 52 78 16 20 24 16 24 30 32 36 40

Latest commencement date 0 4 12 28 36 48 52 16 20 24 18 22 30 40 44 48

Latest completion date 4 12 28 36 48 52 78 20 24 28 22 30 36 44 48 52

Number of employees 2 5 8 6 5 4 0 6 7 0 8 10 0 6 6 0

Fig. 42. Gantt chart after optimization, minimum value of objective function f = 274 (Days, Process, Resource level, optimal dates, Number of employees, Employment graph, optimal dates, objective function value = 274)

Metaheuristic algorithms in Time Coupling Method TCM 85

86

Chapter 5

Figure 42 shows a Gantt chart of the optimum construction project schedule for the adopted objective function. The location of the critical path remains unchanged whereas the non-critical activities were placed on the time scale so as to accommodate the time buffers (TB) and to obtain a minimum objective function value. An employment graph corresponding to the planned activities is shown below the Gantt chart. It shows the best solution concerning the minimum average deviation from average staffing level. The shaded elements are time buffers, i.e. a time reserve which is used to optimally locate construction works on the time axis so as to equalize the staffing level. The genetic algorithm yielded result f = 274 after 1 000 000 iterations while the hybrid evolutionary algorithm yielded this result already after 100 iterations. The computations were performed on a 3 GHz Pentium 4 computer. In the former case the computing time was 2 seconds while in the latter case it was below 1 millisecond, i.e. the computations were over 2000 times faster. Recapitulation

Computations were performed using optimization programs based on respectively the genetic algorithm (GA) and the hybrid evolutionary algorithm (HEA). For the presented optimization problem having size n = 16 the same objective function value, i.e. min = 274 was yielded by the two algorithms. An analysis of the results for the genetic algorithm, the hybrid evolutionary algorithm and time buffers according to CCS/BM shows that the introduction of time buffers with zero resources (teams of employees) increases the optimization problem size and ensures more stable staffing level.

6. Conclusion Construction project scheduling problems, particularly the sequencing of tasks for selected objective function by means of time coupling methods (TCM), were considered. Metaheuristic algorithms, mainly the hybrid evolutionary algorithm (HEA) developed by W. Bożejko and M. Wodecki, were applied. HEA makes it possible, among other things, to solve optimization problems concerning construction project lead time minimization. Currently the world research on construction work scheduling focuses on improving scheduling methods through the use of artificial intelligence tools: genetic algorithms, neural networks and fuzzy sets. Linear scheduling methods (LSM), line of balance (LOB) scheduling methods and network scheduling (CPM/PERT) methods are analyzed and modified. Time coupling methods are the natural consequence of the adopted division of work and work technology and allow one to schedule (in time and space) construction works. Technological and organizational constraints are expressed through time couplings in a matrix model which facilitates optimization consisting in solving a sequencing problem. The time coupling methods differ from the ones (e.g. LSM, RSM, LOB, CPM/PERT) found in the western literature on the subject, mainly in that the resource demand and the priorities are set already at the start, i.e. no stoppages for work teams in TCM I, no stoppages on work lots (structures) in TCM II and possible stoppages of work teams and stoppages on work lots in TCM III. Time coupling methods IV, V and VI include additional constraints. As more effective, metaheuristic approximate algorithms (such as tabu search algorithms, genetic search algorithms and simulated annealing algorithm) which can deliver optimum (for small instance dimensions) or close to optimum solutions are proposed. The conditions for applying metaheuristic algorithms in TCMs are analyzed and the most effective algorithms are indicated in M. Podolski’s PhD dissertation (2008) [38]. The present publication collects and systematizes the authors’ research on improving the TIME COUPLING METHODS for construction project scheduling. It presents the results of studies and computing tests relating to the modelling and implementation of original computing algorithms for construction project scheduling taking into account minimum-time, optimum-time and cost objective functions. The

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potential for improving the time coupling methods through the use of AI algorithms, in particular the hybrid evolutionary algorithm, has been demonstrated. Directions for further research and possible solutions to many practical construction work scheduling problems have been indicated.

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