International Construction Specialty Conference of the Canadian Society for Civil Engineering (ICSC) (5th : 2015)

One relation to rule them all : the point-to-point precedence relation that substitutes the existing… Hajdu, Miklos Jun 30, 2015

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5th International/11th Construction Specialty Conference 5e International/11e Conférence spécialisée sur la construction    Vancouver, British Columbia June 8 to June 10, 2015 / 8 juin au 10 juin 2015   ONE RELATION TO RULE THEM ALL: THE POINT-TO-POINT PRECEDENCE RELATION THAT SUBSTITUTES THE EXISTING ONES Miklos Hajdu1,2,3  1 Ybl Miklos Faculty of Architecture and Civil Engineering, Szent István University, Hungary 2 Faculty of Architecture, Budapest University of Technology and Economics, Hungary 3 miklos.hajdu63@gmail.com Abstract: Precedence Diagram Method (PDM) has gained the widest acceptance in the scheduling practice in the last decades due to its modeling flexibility over other existing techniques, and to the relative simplicity of its mathematical background. The four basic precedence relationships have been serving planners for more than half a century. However, even this model is not flexible enough; proper modeling of overlapped activities seems to be a never-ending debate. Different practical and theoretical solutions have been proposed during the years for better modeling overlapped activities. The most promising among them is the development of a new type of relation that can connect any two arbitrary points of the related activities. These relations can be called point-to-point relations. Different authors in various ways have proposed similar solutions. To the best of our knowledge, the literature on the mathematical model of PDM using this new relation is lacking. Main results of the paper are: 1) standardized discussion of the different approaches to point-to-point relations; 2) proper mathematical model of PDM with point-to-point relations; 3) introduction of the algorithm that can handle point-to-point relations with both minimal and maximal lag to define the earliest and latest feasible time policy. 1 INTRODUCTION AND LITERATURE REVIEW Widely used network techniques are more than half a century old. The results of Fondahl, (Fondahl 1961), Roy (Roy 1959), (Roy 1960), IBM (IBM 1964) and many others have led to the present form of the Precedence Diagram Method, the prevailing network technique of our times. PDM has hardly changed during the decades in spite of the critiques it has received about its modeling capabilities. Proper modeling of overlapping activities seems to be a never-ending debate when traditional precedence relations are used, (Douglas et al. 2006) because traditional endpoint relations are simply not suitable for describing this kind of logic. Different solutions have been proposed using the traditional precedence relations; but fragmentation of activities and developments based on this idea (Tarek & Menesi 2010) seem to be the best theoretical solution despite the arising practical problems, namely the multiplication of the number of activities and precedence relations.  Probably the fragmentation technique has given the idea of connecting the inner points of the activities, which will be discussed in this paper. These point-to-point relations connecting the internal points of the activities seem to be theoretically more suitable for modeling overlapping activities – especially if continuous activities are assumed - as multiple relations are allowed between the activities. To the best of our knowledge four partly parallel works regarding point-to-point relations can be found:  Kim (Kim 2010, 2012) calls his new relations bee-line relations and the graphical representation Bee-line Diagram (BDM), while Francis and Miresco (Francis & Mireco 2000, 2002) call their new relations temporal functions and they call their graphical representation method 340-1 chronographic approach. Plotnick (Plotnick 2004) calls his method Relationship Diagramming method (RDM) using the term of ‘event’ for the internal points. Ponce de Leon (Ponce de Leon 2010) uses the term Graphical Diagramming Method (GDM) and connected internal points are called embedded nodes. Despite the differences in terminology and definitions, e.g. bee-line and RDM relation does not allow a lag between the connected inner points, maximal lags are defined only in the work of Francis and Miresco, the concept behind all these works is the same. All these improvements regarding the relationships can be seen as a new type of precedence relation that can substitute all traditional precedence relations, as it will be shown later. The goal of this paper is twofold: firstly, to remedy a common shortcoming of these works (authors have failed to present the mathematical model); secondly, to show that traditional precedence relationships can be derived from the general point-to-point relations discussed in this paper. A proper mathematical model and the algorithm will be introduced using standardized technical terminology. The new point-to-point relation can be seen as a generalization of traditional precedence relations. It will be shown that the existing traditional precedence relations are special cases of the point-to-point relations; they connect the endpoints of the activities instead of the internal points.  2 THE MATHEMATICAL MODEL 2.1 Notations Let a directed acyclic graph be given with one start (s) and one finish node (f). Let N = {1,2,…i…j…n} stand for the set of nodes also called activities.  A will define the set of arcs, also called precedence relations. The ‘super’ relation defined later can have minimal or maximal lags, therefore Amin and Amax subsets are introduced for differentiating relations with minimal and maximal lags. In the algorithm, relations with maximal lags will be transformed into relations with minimal lags. In this case A* denotes  the set of relations . An activity i is defined by its start and finish points Pis, PiF, shortly Si or Fi, or by any of the two aforementioned points and its duration di. Additional points of the activities can also be defined. Pik stands for the kth point of activity i. The relative place of the kth point of activity i is defined by the time span (ti k) from the start point of activity i (PiS). A relation can be defined between any lth internal point of activity j and any kth internal point of activity i by defining the time that must elapse between the two points (zi kjl). Therefore this ‘super’ precedence relation can be defined either by the points and the lag as (Pik;Pjl, zikjl) or by the relative positions of the points and the lag (ti k; tjl , zikjl). Explanation can be seen in Fig. 1. Lags can be defined using production volumes as well.  Figure 1: Explanation of notations and the the ‘super’ precedence relation. (t ik ; t jl ; zikjl) 340-2 Let the time when a point is accomplished be called point or event time and denoted by Tik. This way TiS stands for the start, and TiF stands for the finish of activity i. Table 1 shows how the traditional relations and bee-line relations can be derived from the new ‘super’ relation. Transformation of temporal functions (Francis & Miresco 2002) and RDM relations is obvious so this is not presented here. If instead of time, the names of the points of the activities are used for describing the relation, then even the same notation can be used. (e.g. instead of SS100 days (S; S; 100 days) can be used.) Based on the above notations the following model can be defined. Table 1: ‘Super’ relation can be used instead of traditional and bee-line precedence relations   Known precedence relations Equivalent point-to-point relation PDM Start-to-Start zi j ( 0 ; 0; ziSjS )   or ( S; S; ziSjS)  Finish-to-Start zi j ( d i ; 0; ziFjS )   or ( F; S; ziFjS)  Finish-to-Finish zi j ( d i ; d j; ziFjS )   or ( F; F; ziFjF)  Start-to-Finish zi j ( 0 ; d j; ziSjS )   or ( S; F; ziSjF) BDM (t ik, t jl) (t ik; t jl  ; 0) 2.2 The model The first two conditions tell that all precedence relations must be satisfied. [1] describes the precedence relations with minimal lags, while [2] describes the precedence relations with maximal lags. [1]    Tjl –Tik ≥ zikjl            ∀ (Pik;Pjl)  ∈ Amin    [2]    Tjl –Tik ≤ zikjl            ∀ (Pik;Pjl)  ∈ Amax     By definition Tik  = TiS+tik and  Tjl  = TjS+tjl   therefore [1] and [2] can be modified as:  [1*]    TjS – TiS ≥ zikjl  – tjl   + tik     ∀ (Pik;Pjl)  ∈ Amin    [2*]    TjS – TiS ≤ zikjl  – tjl  +tik       ∀ (Pik;Pjl)  ∈ Amax   The finish of the activities can be calculated according to [3]. Activities are assumed to be continuous [4]. Let’s set the start of the project to zero. [5]   [3]    TiS  +di = TiF     ∀ i  ∈ N     [4]    Tik  - tik = TiS     ∀ i  ∈ N    and  ∃ Pik   (Pik exist) [5]    TsS  = 0      The T policy that satisfies [1*], [2*], [3], [4], and [5] is called a feasible time policy. An infinite number of feasible time policies exist, but the objective of the model is to find that/those time policy/policies where the project duration is the minimum, that is  [6]   TfF  − TsS → min    that is      TfF  − 0 → min    that is TfF   → min     This model is an LP model, so any LP solver can be used for solving it, furthermore, based on the simplistic structure of this LP problem different efficient primal dual algorithms can be developed. The solution shown below is based on the modification of the simplistic and widely used CPM/PDM time analysis. This approach is probably the easiest to digest for planning engineers.  340-3 3 ALGORITHMS 3.1 Point-to-point relations with minimal lag The goal of the algorithm is to find the optimal time policy, the earliest and the latest out of the existing –   sometimes millions of – optimal time policies. The earliest optimal time policy is denoted by EiS and EiF. The latest optimal policy is denoted by LiS and LiF. The applied algorithm has two phases. The result of the first phase is the earliest optimal time policy, while the result of the second phase is the latest optimal time policy.  Let’s suppose for the sake of simplicity that only relations with minimal lags are allowed in the network. In this case, the earliest start and finish of an activity j can only be calculated, if the earliest start dates for all its predecessors are known. As all precedence relations must be satisfied, the early start of a given j can be defined by the maximum of the shifts caused by the preceding relations of activity j, that is:  [7]   EjS =  max   {   EiS + t ik + zikjl – t jl       ∀ (Pik;Pjl)  ∈ Amin }  To start, an activity with known predecessors has to be found. In the beginning, only the start activity satisfies this condition: all of its predecessors are known because it does not have any. After these introductory thoughts, the steps of the first phase, that is the steps aiming to find the earliest time policy, can be summarized as follows:  Step 1 Let EiS= -∞ and EiF=-∞  ∀ i ∈ N;  Let ESS=0 Let g:=1 Step 2  REPEAT   g:=g+1   Choose an activity j from the unknowns (EiS=-∞) with known predecessors only.   IF there is no such activity then GO TO Step 3    EjS =  max   {   EiS + t ik + zikjl – t jl       ∀ (Pik;Pjl)  ∈ Amin };     EjF=EjS+d j UNTIL  g=n Step 3  IF g<n                    THEN        STOP (There is a loop in the network.) ELSE        p=EfF  (Project duration is the same as the early finish of the finish activity.) During the backward pass, the latest optimal time policy will be defined. It is completed by working from the terminal activity to the initial activity in reverse direction of the arrows. It is based on the observation that the late activity times of an activity can only be calculated, if these dates are known for all of its successors.  As all successor relations must be satisfied, the late start of a given i can be defined by the maximum of the shifts caused by the succeeding relations of activity i, that is:  [8]   LiS =  min   {   LjS + t jk - zikjl – t il       ∀ (Pik;Pjl)  ∈ Amin }  The rules of the backward pass can be summarized as follows: Step 1 Let LiS= ∞ and LiF= ∞  ∀ i ∈ N;  Let Li=FS=p- df ; Let g:=1 Step 2  REPEAT   g:=g+1   Choose an activity i from the unknowns (LiS=∞) with known successors only.   LiS =  min   {   LjS + t jk - zikjl – t il       ∀ (Pik;Pjl)  ∈ Amin } ;       LiF=LiS+d i UNTIL  g=n (Note: Loop detection is not necessary during the backward pass.) 340-4 3.2 Point-to-point relations with mixed lags Calculations with mixed lags require more computational steps. Maximal relations have to be transformed into minimal relations first. Comparing conditions [1] and [2], it can be seen that the difference between a relation with minimal or maximal lags lies in the direction of the operand. Transforming a relation with maximal lag into a relation with minimal lag requires a simple multiplication by -1.  [2]    Tjl –Tik ≤ zikjl            ∀ (Pik;Pjl)  ∈ Amax      /    *(-1) [9]    Tik –Tjl ≤ - zjlik          ∀ (Pik;Pjl)  ∈ Amax      This is nothing else but a relation from j to i with a negative minimal lag (see Fig. 2).  Figure 2: Point-to-point relation with maximal lag a) and its minimal equivalent b)   Traditional precedence relations with maximal lags, and their transformed equivalent minimal lags can be found in Table 2. Table 2: Traditional precedence relations and their equivalent minimal versions.  Traditional precedence relations with maximal lag Equivalent  point-to-point relation with maximal lag Transformed equivalent precedence relations with minimal lag* Transformed equivalent  point-to-point relation with minimal lag* maxSSz max(0; 0; z) SS-z (0; 0; -z) maxFSz max(di; 0; z) SF-z (0, dj; -z) maxFFz max(di; dj; z) FF-z (di; dj; -z) maxSFz max(0, dj; z) FS-z (di; 0; -z) * Transformed equivalents go in the opposite direction However, converting relations with maximal lags into their equivalent minimal relations can result in so-called transformation loops. In Fig. 3 it can be seen that there is a loop between B and C.   The simple time analysis presented in 3.1 cannot be used in case of loops. One can easily check it by selecting the activities. E.g. during the forward pass activity A has to be chosen first. After that none of the activities can be selected, B has two predecessors but only A is known and C is not; C has two predecessors but only A is known and B is not, so the algorithm will stop here. In case of loops, different algorithms exist to find the longest path. We will use the modified version of the algorithm developed by Bellman (Bellman, 1958) and Ford (1956) for finding the shortest path between any two points of a cyclic graph. The algorithm is based on the idea that during the forward pass all activities are calculated using the dates of their predecessors even if those have not taken their final dates yet. When all activities have been calculated this way, it has to be checked whether there have been changes in activity dates or not. If the answer is yes, then the entire calculation must be repeated again and again, until we come to the results we had in the previous iteration. In every iteration, at least 340-5 one activity takes its final value, so after maximum n iterations we get the results. Usually much fewer iterations are necessary. If the nth iteration still brings changes, then the network cannot be solved due to the maximal relations that probably impose logic on the network, which contradicts the minimal or other maximal lags. E.g. imagine that B can start minimum five days after the finish of A (F;S; 5) point-to-point relation but another relation describes that B should start maximum 4 days after the finish of A (F;S;max4).  Figure 3: Transformation of relations with maximal lag into their minimal equivalent can result in loops.   This definite contradiction cannot be solved. In this case, the value of the transformation loop will be positive, and the result for the project duration will increase by at least this value in every iteration even after the nth iteration. The steps below summarize the forward pass:  Step 1 Let EiS= -∞ and OLD_EiS= -∞   ∀ i ∈ N;  Let  Ej=SS=0 and  OLD_Ej=SS=0; Let h=0;  Let No_of_Iter=0 Step 2  REPEAT   There_were_changes:=FALSE; No_of_Iter:= No_of_Iter+1   REPEAT    h:=h+1    Select any j activity that has not been selected in this iteration yet                  EjS =  max   { OLD_Ejs;  (EiS + t ik + zikjl – t jl       ∀ (Pik;Pjl)  ∈ A*) };  EjF=EjS+d j IF EjS> OLD_EjS THEN There_were_changes:=TRUE UNTIL  h=n Let OLD_EiS=Eis ∀ i ∈ N UNTIL No_of_iter>n or There_were_changes:=FALSE Step 3  IF No_of_Iter>n THEN There is no solution. (There is a loop with positive value.)              IF There_were_changes:=FALSE THEN we arrived to a feasible optimal time policy (All            activity dates remained unchanged after two iterations.)  The rules of backward pass can be summarized as follows: Step 1 Let LiS= -∞ and OLD_LiS= -∞   ∀ i ∈ N;  Let  Lj=SS=Ej=sS and  OLD_Lj=SS= Lj=SS; Let h=n;   Step 2  REPEAT               There_were_changes:=FALSE;    REPEAT 340-6    Select any i activity that has not been selected in this iteration yet                  LiS =  min   { OLD_LiS;  (LjS + t jk - zikjl – t il       ∀ (Pik;Pjl)  ∈ A*) }; LiF=LiS+d i                  IF LiS> OLD_LiS THEN There_were_changes:=TRUE                      h:=h-1 UNTIL  h=0 Let OLD_LiS=Lis ∀ i ∈ N UNTIL There_were_changes:=FALSE  Notes to the algorithm: • Loop detection was done during the forward pass, so there is no need for that during the backward pass. • Any order of activities can be used during the algorithm, which can largely modify the number of iterations. Here we used the ascending order of activities during the forward pass and the descending order during the backward pass. In the optimal case, the first iteration presents the results and the second iteration will validate this, in the pessimistic case, n+1 iterations are necessary.  • Following the optimal order of the forward pass will be the worst during the backward pass, and vice versa.   4 SAMPLE PROJECT 4.1 Sample project: only minimal lags are allowed A small sample project is shown in Fig. 4 a) consisting only of relations with minimal lags. Results can be seen in Figure 4b), calculations can be tracked below. Forward pass: Only activity A can be selected.  EAS=0; EAF=EAS+dA=0+6=6 Only activity B can be selected. EBS =  {(EAS +t Ak + zA3B0 – t B0) }={(0+3+0-0)}=3;    EBF=EBS+dB=3+6=9 Only activity C can be selected. ECS=max{(EAS+tA6+zA6C0–tC0);(EBS+tB2+zB2C0–tC0)}={(0+6+4-0);(3+2+2-0)}= 10; ECF=ECS+dC=15 Only activity D can be selected. EDS=max{(EBS+tB6+zB6D0–tD0);(ECS+tC3+zC3D0–tD0);(ECS+tC4+zC4D1–tD1); (ECS+tC5+zC5D2–tD2)}={(3+6+3-0);(10+3+0–0);(10+4+0–1);(10+5+0–2)}={12;13;13;13}=13;  EDF=EDS+dD=17 Early dates for all activities have been calculated, the forward pass is finished. (See Fig. 4)  340-7 Figure 4a) Sample project with minimal lags.                          Figure 4b) Results of the calculations.   Backward pass: Only activity D can be selected. LDF=17; LDS=LDF-dD=17-4=13 Only activity C can be selected. LCS =min{(LDS+tD2 –zC5D2 –tC5); (LDS+tD1 –zC4D1 –tC4); (LDS+tD0 –zC3D0 –tC3)}= min{(13+2-0-5);(13+1-0-4);(13+0-0-3)}=min{(10;10;10)}=10       LCF=LCS+dc=10+5=15 Only activity B can be selected. LBS =min{(LDS+tD0 –zC6D0 –tC6);(LCS+tC0 –zB2C0 –tB2)= min{(13+0-3-6);(10+0-2-2)}=min{(4;6)}=4       LBF=LBS+dB=4+6=10 Only activity A can be selected. LAS=min{(LBS+tB0–zA3B0 –tA3);(LCS+tC0 –zA6C0-tA6)=min{(4+0-0-3);(10+0-4-6)} =min{(1;0)}=0       LAF=LAS+dA=0+6=6 Late dates for all activities have been calculated, the backward pass is finished. (See Fig. 4) 4.2 Sample project: both minimal and maximal lags are allowed A small sample project is shown on Fig. 5a) consisting of relations with minimal and maximal lags. The network with the transformed maximal relation and with the results is shown in Fig. 5b). Due to the transformation loop, the iterative algorithm has to be used.  Any order of the activities can be used. Here we use the A;B;D;C order for the forward pass. The calculations can be followed in Figure 6. Boxes of those activities that have been changed during the iteration are filled with grey. Iteration #1 EAS= max  { OLD_EAs }=0; EAF=EAS+dA=0+6=6 EBS=max{OLD_EBS;(EAS+tA3+zA3B0-tB0);(ECS+tC0+zC0B2-tB2)}={-∞;(0+3+0-0};(-∞+0-2-2)}=3    EBF=EBS+dB=9 EDS=max{OLD_EDS;(EBS+tB6+zB6D0-tD0);(ECS+tC3+zC3D0-tD0);(ECS+tC4+zC4D1-tD1)(ECS+tC5+zC5D2-tD2)}={-∞; (3+6+3-0}; (-∞+3+0-0); (-∞+4+0-1); (-∞+5+0-2)}=12          EDF=EDS+dD=16 ECS=max{OLD_ECS;(EBS+tB2+zB2C0-tC0);(EAS+tA6+zA6C0-tC0)={-∞;(3+2+2-0};(0+6+4-0)}=10    ECF=ECS+dC=15  Figure 5a) Sample project with mixed lags.                      Figure 5b) Transformed network with the results  340-8 Iteration #2 EAS= max  { OLD_EAs }=0; EAF=EAS+dA=0+6=6 EBS=max{OLD_EBS;(EAS+tA3+zA3B0-tB0);(ECS+tC0+zC0B2-tB2)}={3;(0+3+0-0};(10+0-2-2)}=6    EBF=EBS+dB=12 EDS=max{OLD_EDS;(EBS+tB6+zB6D0-tD0);(ECS+tC3+zC3D0-tD0);(ECS+tC4+zC4D1-tD1)(ECS+tC5+zC5D2-tD2)}={12; (6+6+3-0}; (10+3+0-0); (10+4+0-1); (10+5+0-2)}=15          EDF=EDS+dD=19 ECS=max{OLD_ECS;(EBS+tB2+zB2C0-tC0);(EAS+tA6+zA6C0-tC0)={10;(6+2+2-0};(0+6+4-0)}=10    ECF=ECS+dC=15 Iteration #3 EAS= max  { OLD_EAs }=0; EAF=EAS+dA=0+6=6 EBS=max{OLD_EBS;(EAS+tA3+zA3B0-tB0);(ECS+tC0+zC0B2-tB2)}={3;(0+3+0-0};(10+0-2-2)}=6    EBF=EBS+dB=12 EDS=max{OLD_EDS;(EBS+tB6+zB6D0-tD0);(ECS+tC3+zC3D0-tD0);(ECS+tC4+zC4D1-tD1)(ECS+tC5+zC5D2-tD2)}={12; (6+6+3-0}; (10+3+0-0); (10+4+0-1); (10+5+0-2)}=15          EDF=EDS+dD=19 ECS=max{OLD_ECS;(EBS+tB2+zB2C0-tC0);(EAS+tA6+zA6C0-tC0)={10;(6+2+2-0};(0+6+4-0)}=10    ECF=ECS+dC=15  Figure 6: Forward pass computation. No activity dates have changed in the course of iteration #3; therefore the forward pass is finished. For the backward pass the D; B; C; A sequence is selected. Calculations can be followed below. Iteration #1 LDS= min{OLD_LDS; (nil) }=15;    LDF=LDS+dD=19 LBS=min{OLD_LBS;(LDS+tD0-zB6D0-tB6);(LCS+tC0-zB2C0-tB2)}=min{∞;(15+0-3-6);( ∞+0-2-2)}=6   LBF=LBS+tB=12 LCS=min{OLD_LCS;(LBS+tB2-zC0B2-tC0);(LDS+tD0-zB3D0-tB3); (LDS+tD1-zB4D2-tB4) (LDS+tD2-zB5D2-tB5)}=min{∞;(6+2-(-2)-0);(15+0-0-3);(15+1-0-4);(15+2-0-5)}=10        LCF=LCS+tC=15 LAS=min{OLD_LAS;(LCS+tC0-zA6C0-tA6);(LBS+tB0-zA3B0-tA3)}=min{∞;(10+0-4-6);(6+0-0-3)}=0   LAF=LAS+tA=6 Iteration #2 340-9 LDS= min{OLD_LDS; (nil) }=15;    LDF=LDS+dD=19 LBS=min{OLD_LBS;(LDS+tD0-zB6D0-tB6);(LCS+tC0-zB2C0-tB2)}=min{6;(15+0-3-6);(10+0-2-2)}=6   LBF=LBS+tB=12 LCS=min{OLD_LCS;(LBS+tB2-zC0B2-tC0);(LDS+tD0-zB3D0-tB3); (LDS+tD1-zB4D2-tB4) (LDS+tD2-zB5D2-tB5)}=min{10;(6+2-(-2)-0);(15+0-0-3);(15+1-0-4);(15+2-0-5)}=10        LCF=LCS+tC=15 LAS=min{OLD_LAS;(LCS+tC0-zA6C0-tA6);(LBS+tB0-zA3B0-tA3)}=min{0;(10+0-4-6);(6+0-0-3)}=0   LAF=LAS+tA=6 Activity dates have not changed during the iteration. Calculations are finished. Results of the backward pass (and the forward’s as well) can be seen in Fig. 5b.  5 DISCUSSIONS AND FURTHER RESEARCH A new ‘super’ precedence relationship based on the results of Francis & Mireco and Kim has been discussed in the paper. It has been shown that the traditional precedence relations (SS, SF, FS and FF with either minimal or maximal lags) could be derived from the new relation. It has also been shown that different results of parallel works e.g. bee-line relations can also be derived from this new ‘super’ relation; therefore claiming these results as a new technique is a wrong approach. This new relation forms a connection between two arbitrary points of two activities; therefore the name point-to-point relation fully describes the nature of this new relation. Following this logic, the traditional precedence relations, which could be called endpoint relations, form a subset of point-to-point relations as they connect the endpoints of the activities.  Point-to-point relations affect the very fundaments of network techniques, therefore all definitions, generalizations, problems based on the ‘old’ precedence relations must be checked and modified accordingly, if necessary, including the definitions and calculations of floats, the definition of  the critical path, the classification of critical activities (Weist 1981) (Hajdu 1996) (Walls & Lino 2001), the algorithms for resource optimization etc. To our best knowledge, this work has not been done yet. References Bellman, R.E. 1958 On a Routing Problem. Quarterly of Applied Mathematics 16 1959. 87-90 Douglas, E.E.; Calvery T. T. McDonald, D.F.; Winter, R.M. 2006. The Great Negative Lag Debate. 2006 AACE International Transactions PS 02.01.- PS 02.07 Fondahl, J.W 1961. A non-computer approach to the critical path method for the construction industry. Technical Report #9 The Construction Institute, Department of Civil Engineering, Stanford University, Stanford California Ford, L.R. 1956. Network Flow Theory. The Rand Corporation 1956. Francis, A., Miresco, E.T. 2000. Decision Support for Project Management Using a Chronographic Approach. Proceedings of the 2nd International Conference on Decision Making in Urban and Civil Engineering, 2000 Lyon, France, 845-856. Francis, A., Miresco, E.T. 2002. Decision Support for Project Management Using a Chronographic Approach. Journal of Decision Systems, Special issue JDS-DM in UCE: Decision Making in Urban and Civil Engineering, 11(3-4): 383-404.  Hajdu, M. 1996.  Network Scheduling Techniques For Construction Project Management. Kluwer Academic Publishers, ISBN 0-7923-4309-3 Hegazy, T & Menesi, W. 2010 Critical Path Segments Scheduling Technique. Journal of Construction Engineering and Management, 136(10): 1078-1085 IBM 1964. Users Manual for IBM 1440 Project Control System (PCS). 1964 Kim, S. 2010. Advanced Networking Technique Kimoondang, South Korea 2010 Kim, S. 2012. CPM Schedule Summarizing Function of the Beeline Diagramming Method. Journal of Asian Architecture and Building Engineering, 11(2) November 2012; 367-374 Plotnick FL, 2004 Introduction to Modified Sequence Logic, Conference Proceedings, PMICOS (first annual) Conference, April 25, 2004, Montreal, Canada  Ponce de Leon, G. 2008 Graphical Planning method. PMICOS Annual Conference, Chicago, IL, 2008 340-10 Roy, G.B., 1959 Théorie des Graphes: Contribution  de la théorie des graphes á l1 étude de certains problémes linéaries, Comptes rendus des Séances de l1 Acedémie des Sciences. séence du Avril 1959, s2437-2449, 1959 Roy, G.B. 1960. Contribution de la théorie des graphes à l’étude de certains problems d’ordonnancement",  Comptes rendus de la 2ème conférence internationale sur la recherché opérationnelle, Aix-en-Provence, English Universities Press, Londres 171-185. Valls, V and Lino, 2001. Criticality Analysis in Activity-on-Node Networks with Minimal Time Lags. Annals of Operations Research, 102(1-4) 17-37 Wiest, J.D. 1981 Precedence diagramming method: Some unusual characteristics and their implications for project managers, Journal of Operations management, Volume 1/3 Feb. 1981. 121-130 340-11  5th International/11th Construction Specialty Conference 5e International/11e Conférence spécialisée sur la construction    Vancouver, British Columbia June 8 to June 10, 2015 / 8 juin au 10 juin 2015   ONE RELATION TO RULE THEM ALL: THE POINT-TO-POINT PRECEDENCE RELATION THAT SUBSTITUTES THE EXISTING ONES Miklos Hajdu1,2,3  1 Ybl Miklos Faculty of Architecture and Civil Engineering, Szent István University, Hungary 2 Faculty of Architecture, Budapest University of Technology and Economics, Hungary 3 miklos.hajdu63@gmail.com Abstract: Precedence Diagram Method (PDM) has gained the widest acceptance in the scheduling practice in the last decades due to its modeling flexibility over other existing techniques, and to the relative simplicity of its mathematical background. The four basic precedence relationships have been serving planners for more than half a century. However, even this model is not flexible enough; proper modeling of overlapped activities seems to be a never-ending debate. Different practical and theoretical solutions have been proposed during the years for better modeling overlapped activities. The most promising among them is the development of a new type of relation that can connect any two arbitrary points of the related activities. These relations can be called point-to-point relations. Different authors in various ways have proposed similar solutions. To the best of our knowledge, the literature on the mathematical model of PDM using this new relation is lacking. Main results of the paper are: 1) standardized discussion of the different approaches to point-to-point relations; 2) proper mathematical model of PDM with point-to-point relations; 3) introduction of the algorithm that can handle point-to-point relations with both minimal and maximal lag to define the earliest and latest feasible time policy. 1 INTRODUCTION AND LITERATURE REVIEW Widely used network techniques are more than half a century old. The results of Fondahl, (Fondahl 1961), Roy (Roy 1959), (Roy 1960), IBM (IBM 1964) and many others have led to the present form of the Precedence Diagram Method, the prevailing network technique of our times. PDM has hardly changed during the decades in spite of the critiques it has received about its modeling capabilities. Proper modeling of overlapping activities seems to be a never-ending debate when traditional precedence relations are used, (Douglas et al. 2006) because traditional endpoint relations are simply not suitable for describing this kind of logic. Different solutions have been proposed using the traditional precedence relations; but fragmentation of activities and developments based on this idea (Tarek & Menesi 2010) seem to be the best theoretical solution despite the arising practical problems, namely the multiplication of the number of activities and precedence relations.  Probably the fragmentation technique has given the idea of connecting the inner points of the activities, which will be discussed in this paper. These point-to-point relations connecting the internal points of the activities seem to be theoretically more suitable for modeling overlapping activities – especially if continuous activities are assumed - as multiple relations are allowed between the activities. To the best of our knowledge four partly parallel works regarding point-to-point relations can be found:  Kim (Kim 2010, 2012) calls his new relations bee-line relations and the graphical representation Bee-line Diagram (BDM), while Francis and Miresco (Francis & Mireco 2000, 2002) call their new relations temporal functions and they call their graphical representation method 340-1 chronographic approach. Plotnick (Plotnick 2004) calls his method Relationship Diagramming method (RDM) using the term of ‘event’ for the internal points. Ponce de Leon (Ponce de Leon 2010) uses the term Graphical Diagramming Method (GDM) and connected internal points are called embedded nodes. Despite the differences in terminology and definitions, e.g. bee-line and RDM relation does not allow a lag between the connected inner points, maximal lags are defined only in the work of Francis and Miresco, the concept behind all these works is the same. All these improvements regarding the relationships can be seen as a new type of precedence relation that can substitute all traditional precedence relations, as it will be shown later. The goal of this paper is twofold: firstly, to remedy a common shortcoming of these works (authors have failed to present the mathematical model); secondly, to show that traditional precedence relationships can be derived from the general point-to-point relations discussed in this paper. A proper mathematical model and the algorithm will be introduced using standardized technical terminology. The new point-to-point relation can be seen as a generalization of traditional precedence relations. It will be shown that the existing traditional precedence relations are special cases of the point-to-point relations; they connect the endpoints of the activities instead of the internal points.  2 THE MATHEMATICAL MODEL 2.1 Notations Let a directed acyclic graph be given with one start (s) and one finish node (f). Let N = {1,2,…i…j…n} stand for the set of nodes also called activities.  A will define the set of arcs, also called precedence relations. The ‘super’ relation defined later can have minimal or maximal lags, therefore Amin and Amax subsets are introduced for differentiating relations with minimal and maximal lags. In the algorithm, relations with maximal lags will be transformed into relations with minimal lags. In this case A* denotes  the set of relations . An activity i is defined by its start and finish points Pis, PiF, shortly Si or Fi, or by any of the two aforementioned points and its duration di. Additional points of the activities can also be defined. Pik stands for the kth point of activity i. The relative place of the kth point of activity i is defined by the time span (ti k) from the start point of activity i (PiS). A relation can be defined between any lth internal point of activity j and any kth internal point of activity i by defining the time that must elapse between the two points (zi kjl). Therefore this ‘super’ precedence relation can be defined either by the points and the lag as (Pik;Pjl, zikjl) or by the relative positions of the points and the lag (ti k; tjl , zikjl). Explanation can be seen in Fig. 1. Lags can be defined using production volumes as well.  Figure 1: Explanation of notations and the the ‘super’ precedence relation. (t ik ; t jl ; zikjl) 340-2 Let the time when a point is accomplished be called point or event time and denoted by Tik. This way TiS stands for the start, and TiF stands for the finish of activity i. Table 1 shows how the traditional relations and bee-line relations can be derived from the new ‘super’ relation. Transformation of temporal functions (Francis & Miresco 2002) and RDM relations is obvious so this is not presented here. If instead of time, the names of the points of the activities are used for describing the relation, then even the same notation can be used. (e.g. instead of SS100 days (S; S; 100 days) can be used.) Based on the above notations the following model can be defined. Table 1: ‘Super’ relation can be used instead of traditional and bee-line precedence relations   Known precedence relations Equivalent point-to-point relation PDM Start-to-Start zi j ( 0 ; 0; ziSjS )   or ( S; S; ziSjS)  Finish-to-Start zi j ( d i ; 0; ziFjS )   or ( F; S; ziFjS)  Finish-to-Finish zi j ( d i ; d j; ziFjS )   or ( F; F; ziFjF)  Start-to-Finish zi j ( 0 ; d j; ziSjS )   or ( S; F; ziSjF) BDM (t ik, t jl) (t ik; t jl  ; 0) 2.2 The model The first two conditions tell that all precedence relations must be satisfied. [1] describes the precedence relations with minimal lags, while [2] describes the precedence relations with maximal lags. [1]    Tjl –Tik ≥ zikjl            ∀ (Pik;Pjl)  ∈ Amin    [2]    Tjl –Tik ≤ zikjl            ∀ (Pik;Pjl)  ∈ Amax     By definition Tik  = TiS+tik and  Tjl  = TjS+tjl   therefore [1] and [2] can be modified as:  [1*]    TjS – TiS ≥ zikjl  – tjl   + tik     ∀ (Pik;Pjl)  ∈ Amin    [2*]    TjS – TiS ≤ zikjl  – tjl  +tik       ∀ (Pik;Pjl)  ∈ Amax   The finish of the activities can be calculated according to [3]. Activities are assumed to be continuous [4]. Let’s set the start of the project to zero. [5]   [3]    TiS  +di = TiF     ∀ i  ∈ N     [4]    Tik  - tik = TiS     ∀ i  ∈ N    and  ∃ Pik   (Pik exist) [5]    TsS  = 0      The T policy that satisfies [1*], [2*], [3], [4], and [5] is called a feasible time policy. An infinite number of feasible time policies exist, but the objective of the model is to find that/those time policy/policies where the project duration is the minimum, that is  [6]   TfF  − TsS → min    that is      TfF  − 0 → min    that is TfF   → min     This model is an LP model, so any LP solver can be used for solving it, furthermore, based on the simplistic structure of this LP problem different efficient primal dual algorithms can be developed. The solution shown below is based on the modification of the simplistic and widely used CPM/PDM time analysis. This approach is probably the easiest to digest for planning engineers.  340-3 3 ALGORITHMS 3.1 Point-to-point relations with minimal lag The goal of the algorithm is to find the optimal time policy, the earliest and the latest out of the existing –   sometimes millions of – optimal time policies. The earliest optimal time policy is denoted by EiS and EiF. The latest optimal policy is denoted by LiS and LiF. The applied algorithm has two phases. The result of the first phase is the earliest optimal time policy, while the result of the second phase is the latest optimal time policy.  Let’s suppose for the sake of simplicity that only relations with minimal lags are allowed in the network. In this case, the earliest start and finish of an activity j can only be calculated, if the earliest start dates for all its predecessors are known. As all precedence relations must be satisfied, the early start of a given j can be defined by the maximum of the shifts caused by the preceding relations of activity j, that is:  [7]   EjS =  max   {   EiS + t ik + zikjl – t jl       ∀ (Pik;Pjl)  ∈ Amin }  To start, an activity with known predecessors has to be found. In the beginning, only the start activity satisfies this condition: all of its predecessors are known because it does not have any. After these introductory thoughts, the steps of the first phase, that is the steps aiming to find the earliest time policy, can be summarized as follows:  Step 1 Let EiS= -∞ and EiF=-∞  ∀ i ∈ N;  Let ESS=0 Let g:=1 Step 2  REPEAT   g:=g+1   Choose an activity j from the unknowns (EiS=-∞) with known predecessors only.   IF there is no such activity then GO TO Step 3    EjS =  max   {   EiS + t ik + zikjl – t jl       ∀ (Pik;Pjl)  ∈ Amin };     EjF=EjS+d j UNTIL  g=n Step 3  IF g<n                    THEN        STOP (There is a loop in the network.) ELSE        p=EfF  (Project duration is the same as the early finish of the finish activity.) During the backward pass, the latest optimal time policy will be defined. It is completed by working from the terminal activity to the initial activity in reverse direction of the arrows. It is based on the observation that the late activity times of an activity can only be calculated, if these dates are known for all of its successors.  As all successor relations must be satisfied, the late start of a given i can be defined by the maximum of the shifts caused by the succeeding relations of activity i, that is:  [8]   LiS =  min   {   LjS + t jk - zikjl – t il       ∀ (Pik;Pjl)  ∈ Amin }  The rules of the backward pass can be summarized as follows: Step 1 Let LiS= ∞ and LiF= ∞  ∀ i ∈ N;  Let Li=FS=p- df ; Let g:=1 Step 2  REPEAT   g:=g+1   Choose an activity i from the unknowns (LiS=∞) with known successors only.   LiS =  min   {   LjS + t jk - zikjl – t il       ∀ (Pik;Pjl)  ∈ Amin } ;       LiF=LiS+d i UNTIL  g=n (Note: Loop detection is not necessary during the backward pass.) 340-4 3.2 Point-to-point relations with mixed lags Calculations with mixed lags require more computational steps. Maximal relations have to be transformed into minimal relations first. Comparing conditions [1] and [2], it can be seen that the difference between a relation with minimal or maximal lags lies in the direction of the operand. Transforming a relation with maximal lag into a relation with minimal lag requires a simple multiplication by -1.  [2]    Tjl –Tik ≤ zikjl            ∀ (Pik;Pjl)  ∈ Amax      /    *(-1) [9]    Tik –Tjl ≤ - zjlik          ∀ (Pik;Pjl)  ∈ Amax      This is nothing else but a relation from j to i with a negative minimal lag (see Fig. 2).  Figure 2: Point-to-point relation with maximal lag a) and its minimal equivalent b)   Traditional precedence relations with maximal lags, and their transformed equivalent minimal lags can be found in Table 2. Table 2: Traditional precedence relations and their equivalent minimal versions.  Traditional precedence relations with maximal lag Equivalent  point-to-point relation with maximal lag Transformed equivalent precedence relations with minimal lag* Transformed equivalent  point-to-point relation with minimal lag* maxSSz max(0; 0; z) SS-z (0; 0; -z) maxFSz max(di; 0; z) SF-z (0, dj; -z) maxFFz max(di; dj; z) FF-z (di; dj; -z) maxSFz max(0, dj; z) FS-z (di; 0; -z) * Transformed equivalents go in the opposite direction However, converting relations with maximal lags into their equivalent minimal relations can result in so-called transformation loops. In Fig. 3 it can be seen that there is a loop between B and C.   The simple time analysis presented in 3.1 cannot be used in case of loops. One can easily check it by selecting the activities. E.g. during the forward pass activity A has to be chosen first. After that none of the activities can be selected, B has two predecessors but only A is known and C is not; C has two predecessors but only A is known and B is not, so the algorithm will stop here. In case of loops, different algorithms exist to find the longest path. We will use the modified version of the algorithm developed by Bellman (Bellman, 1958) and Ford (1956) for finding the shortest path between any two points of a cyclic graph. The algorithm is based on the idea that during the forward pass all activities are calculated using the dates of their predecessors even if those have not taken their final dates yet. When all activities have been calculated this way, it has to be checked whether there have been changes in activity dates or not. If the answer is yes, then the entire calculation must be repeated again and again, until we come to the results we had in the previous iteration. In every iteration, at least 340-5 one activity takes its final value, so after maximum n iterations we get the results. Usually much fewer iterations are necessary. If the nth iteration still brings changes, then the network cannot be solved due to the maximal relations that probably impose logic on the network, which contradicts the minimal or other maximal lags. E.g. imagine that B can start minimum five days after the finish of A (F;S; 5) point-to-point relation but another relation describes that B should start maximum 4 days after the finish of A (F;S;max4).  Figure 3: Transformation of relations with maximal lag into their minimal equivalent can result in loops.   This definite contradiction cannot be solved. In this case, the value of the transformation loop will be positive, and the result for the project duration will increase by at least this value in every iteration even after the nth iteration. The steps below summarize the forward pass:  Step 1 Let EiS= -∞ and OLD_EiS= -∞   ∀ i ∈ N;  Let  Ej=SS=0 and  OLD_Ej=SS=0; Let h=0;  Let No_of_Iter=0 Step 2  REPEAT   There_were_changes:=FALSE; No_of_Iter:= No_of_Iter+1   REPEAT    h:=h+1    Select any j activity that has not been selected in this iteration yet                  EjS =  max   { OLD_Ejs;  (EiS + t ik + zikjl – t jl       ∀ (Pik;Pjl)  ∈ A*) };  EjF=EjS+d j IF EjS> OLD_EjS THEN There_were_changes:=TRUE UNTIL  h=n Let OLD_EiS=Eis ∀ i ∈ N UNTIL No_of_iter>n or There_were_changes:=FALSE Step 3  IF No_of_Iter>n THEN There is no solution. (There is a loop with positive value.)              IF There_were_changes:=FALSE THEN we arrived to a feasible optimal time policy (All            activity dates remained unchanged after two iterations.)  The rules of backward pass can be summarized as follows: Step 1 Let LiS= -∞ and OLD_LiS= -∞   ∀ i ∈ N;  Let  Lj=SS=Ej=sS and  OLD_Lj=SS= Lj=SS; Let h=n;   Step 2  REPEAT               There_were_changes:=FALSE;    REPEAT 340-6    Select any i activity that has not been selected in this iteration yet                  LiS =  min   { OLD_LiS;  (LjS + t jk - zikjl – t il       ∀ (Pik;Pjl)  ∈ A*) }; LiF=LiS+d i                  IF LiS> OLD_LiS THEN There_were_changes:=TRUE                      h:=h-1 UNTIL  h=0 Let OLD_LiS=Lis ∀ i ∈ N UNTIL There_were_changes:=FALSE  Notes to the algorithm: • Loop detection was done during the forward pass, so there is no need for that during the backward pass. • Any order of activities can be used during the algorithm, which can largely modify the number of iterations. Here we used the ascending order of activities during the forward pass and the descending order during the backward pass. In the optimal case, the first iteration presents the results and the second iteration will validate this, in the pessimistic case, n+1 iterations are necessary.  • Following the optimal order of the forward pass will be the worst during the backward pass, and vice versa.   4 SAMPLE PROJECT 4.1 Sample project: only minimal lags are allowed A small sample project is shown in Fig. 4 a) consisting only of relations with minimal lags. Results can be seen in Figure 4b), calculations can be tracked below. Forward pass: Only activity A can be selected.  EAS=0; EAF=EAS+dA=0+6=6 Only activity B can be selected. EBS =  {(EAS +t Ak + zA3B0 – t B0) }={(0+3+0-0)}=3;    EBF=EBS+dB=3+6=9 Only activity C can be selected. ECS=max{(EAS+tA6+zA6C0–tC0);(EBS+tB2+zB2C0–tC0)}={(0+6+4-0);(3+2+2-0)}= 10; ECF=ECS+dC=15 Only activity D can be selected. EDS=max{(EBS+tB6+zB6D0–tD0);(ECS+tC3+zC3D0–tD0);(ECS+tC4+zC4D1–tD1); (ECS+tC5+zC5D2–tD2)}={(3+6+3-0);(10+3+0–0);(10+4+0–1);(10+5+0–2)}={12;13;13;13}=13;  EDF=EDS+dD=17 Early dates for all activities have been calculated, the forward pass is finished. (See Fig. 4)  340-7 Figure 4a) Sample project with minimal lags.                          Figure 4b) Results of the calculations.   Backward pass: Only activity D can be selected. LDF=17; LDS=LDF-dD=17-4=13 Only activity C can be selected. LCS =min{(LDS+tD2 –zC5D2 –tC5); (LDS+tD1 –zC4D1 –tC4); (LDS+tD0 –zC3D0 –tC3)}= min{(13+2-0-5);(13+1-0-4);(13+0-0-3)}=min{(10;10;10)}=10       LCF=LCS+dc=10+5=15 Only activity B can be selected. LBS =min{(LDS+tD0 –zC6D0 –tC6);(LCS+tC0 –zB2C0 –tB2)= min{(13+0-3-6);(10+0-2-2)}=min{(4;6)}=4       LBF=LBS+dB=4+6=10 Only activity A can be selected. LAS=min{(LBS+tB0–zA3B0 –tA3);(LCS+tC0 –zA6C0-tA6)=min{(4+0-0-3);(10+0-4-6)} =min{(1;0)}=0       LAF=LAS+dA=0+6=6 Late dates for all activities have been calculated, the backward pass is finished. (See Fig. 4) 4.2 Sample project: both minimal and maximal lags are allowed A small sample project is shown on Fig. 5a) consisting of relations with minimal and maximal lags. The network with the transformed maximal relation and with the results is shown in Fig. 5b). Due to the transformation loop, the iterative algorithm has to be used.  Any order of the activities can be used. Here we use the A;B;D;C order for the forward pass. The calculations can be followed in Figure 6. Boxes of those activities that have been changed during the iteration are filled with grey. Iteration #1 EAS= max  { OLD_EAs }=0; EAF=EAS+dA=0+6=6 EBS=max{OLD_EBS;(EAS+tA3+zA3B0-tB0);(ECS+tC0+zC0B2-tB2)}={-∞;(0+3+0-0};(-∞+0-2-2)}=3    EBF=EBS+dB=9 EDS=max{OLD_EDS;(EBS+tB6+zB6D0-tD0);(ECS+tC3+zC3D0-tD0);(ECS+tC4+zC4D1-tD1)(ECS+tC5+zC5D2-tD2)}={-∞; (3+6+3-0}; (-∞+3+0-0); (-∞+4+0-1); (-∞+5+0-2)}=12          EDF=EDS+dD=16 ECS=max{OLD_ECS;(EBS+tB2+zB2C0-tC0);(EAS+tA6+zA6C0-tC0)={-∞;(3+2+2-0};(0+6+4-0)}=10    ECF=ECS+dC=15  Figure 5a) Sample project with mixed lags.                      Figure 5b) Transformed network with the results  340-8 Iteration #2 EAS= max  { OLD_EAs }=0; EAF=EAS+dA=0+6=6 EBS=max{OLD_EBS;(EAS+tA3+zA3B0-tB0);(ECS+tC0+zC0B2-tB2)}={3;(0+3+0-0};(10+0-2-2)}=6    EBF=EBS+dB=12 EDS=max{OLD_EDS;(EBS+tB6+zB6D0-tD0);(ECS+tC3+zC3D0-tD0);(ECS+tC4+zC4D1-tD1)(ECS+tC5+zC5D2-tD2)}={12; (6+6+3-0}; (10+3+0-0); (10+4+0-1); (10+5+0-2)}=15          EDF=EDS+dD=19 ECS=max{OLD_ECS;(EBS+tB2+zB2C0-tC0);(EAS+tA6+zA6C0-tC0)={10;(6+2+2-0};(0+6+4-0)}=10    ECF=ECS+dC=15 Iteration #3 EAS= max  { OLD_EAs }=0; EAF=EAS+dA=0+6=6 EBS=max{OLD_EBS;(EAS+tA3+zA3B0-tB0);(ECS+tC0+zC0B2-tB2)}={3;(0+3+0-0};(10+0-2-2)}=6    EBF=EBS+dB=12 EDS=max{OLD_EDS;(EBS+tB6+zB6D0-tD0);(ECS+tC3+zC3D0-tD0);(ECS+tC4+zC4D1-tD1)(ECS+tC5+zC5D2-tD2)}={12; (6+6+3-0}; (10+3+0-0); (10+4+0-1); (10+5+0-2)}=15          EDF=EDS+dD=19 ECS=max{OLD_ECS;(EBS+tB2+zB2C0-tC0);(EAS+tA6+zA6C0-tC0)={10;(6+2+2-0};(0+6+4-0)}=10    ECF=ECS+dC=15  Figure 6: Forward pass computation. No activity dates have changed in the course of iteration #3; therefore the forward pass is finished. For the backward pass the D; B; C; A sequence is selected. Calculations can be followed below. Iteration #1 LDS= min{OLD_LDS; (nil) }=15;    LDF=LDS+dD=19 LBS=min{OLD_LBS;(LDS+tD0-zB6D0-tB6);(LCS+tC0-zB2C0-tB2)}=min{∞;(15+0-3-6);( ∞+0-2-2)}=6   LBF=LBS+tB=12 LCS=min{OLD_LCS;(LBS+tB2-zC0B2-tC0);(LDS+tD0-zB3D0-tB3); (LDS+tD1-zB4D2-tB4) (LDS+tD2-zB5D2-tB5)}=min{∞;(6+2-(-2)-0);(15+0-0-3);(15+1-0-4);(15+2-0-5)}=10        LCF=LCS+tC=15 LAS=min{OLD_LAS;(LCS+tC0-zA6C0-tA6);(LBS+tB0-zA3B0-tA3)}=min{∞;(10+0-4-6);(6+0-0-3)}=0   LAF=LAS+tA=6 Iteration #2 340-9 LDS= min{OLD_LDS; (nil) }=15;    LDF=LDS+dD=19 LBS=min{OLD_LBS;(LDS+tD0-zB6D0-tB6);(LCS+tC0-zB2C0-tB2)}=min{6;(15+0-3-6);(10+0-2-2)}=6   LBF=LBS+tB=12 LCS=min{OLD_LCS;(LBS+tB2-zC0B2-tC0);(LDS+tD0-zB3D0-tB3); (LDS+tD1-zB4D2-tB4) (LDS+tD2-zB5D2-tB5)}=min{10;(6+2-(-2)-0);(15+0-0-3);(15+1-0-4);(15+2-0-5)}=10        LCF=LCS+tC=15 LAS=min{OLD_LAS;(LCS+tC0-zA6C0-tA6);(LBS+tB0-zA3B0-tA3)}=min{0;(10+0-4-6);(6+0-0-3)}=0   LAF=LAS+tA=6 Activity dates have not changed during the iteration. Calculations are finished. Results of the backward pass (and the forward’s as well) can be seen in Fig. 5b.  5 DISCUSSIONS AND FURTHER RESEARCH A new ‘super’ precedence relationship based on the results of Francis & Mireco and Kim has been discussed in the paper. It has been shown that the traditional precedence relations (SS, SF, FS and FF with either minimal or maximal lags) could be derived from the new relation. It has also been shown that different results of parallel works e.g. bee-line relations can also be derived from this new ‘super’ relation; therefore claiming these results as a new technique is a wrong approach. This new relation forms a connection between two arbitrary points of two activities; therefore the name point-to-point relation fully describes the nature of this new relation. Following this logic, the traditional precedence relations, which could be called endpoint relations, form a subset of point-to-point relations as they connect the endpoints of the activities.  Point-to-point relations affect the very fundaments of network techniques, therefore all definitions, generalizations, problems based on the ‘old’ precedence relations must be checked and modified accordingly, if necessary, including the definitions and calculations of floats, the definition of  the critical path, the classification of critical activities (Weist 1981) (Hajdu 1996) (Walls & Lino 2001), the algorithms for resource optimization etc. To our best knowledge, this work has not been done yet. References Bellman, R.E. 1958 On a Routing Problem. Quarterly of Applied Mathematics 16 1959. 87-90 Douglas, E.E.; Calvery T. T. McDonald, D.F.; Winter, R.M. 2006. The Great Negative Lag Debate. 2006 AACE International Transactions PS 02.01.- PS 02.07 Fondahl, J.W 1961. A non-computer approach to the critical path method for the construction industry. Technical Report #9 The Construction Institute, Department of Civil Engineering, Stanford University, Stanford California Ford, L.R. 1956. Network Flow Theory. The Rand Corporation 1956. Francis, A., Miresco, E.T. 2000. Decision Support for Project Management Using a Chronographic Approach. Proceedings of the 2nd International Conference on Decision Making in Urban and Civil Engineering, 2000 Lyon, France, 845-856. Francis, A., Miresco, E.T. 2002. Decision Support for Project Management Using a Chronographic Approach. Journal of Decision Systems, Special issue JDS-DM in UCE: Decision Making in Urban and Civil Engineering, 11(3-4): 383-404.  Hajdu, M. 1996.  Network Scheduling Techniques For Construction Project Management. Kluwer Academic Publishers, ISBN 0-7923-4309-3 Hegazy, T & Menesi, W. 2010 Critical Path Segments Scheduling Technique. Journal of Construction Engineering and Management, 136(10): 1078-1085 IBM 1964. Users Manual for IBM 1440 Project Control System (PCS). 1964 Kim, S. 2010. Advanced Networking Technique Kimoondang, South Korea 2010 Kim, S. 2012. CPM Schedule Summarizing Function of the Beeline Diagramming Method. Journal of Asian Architecture and Building Engineering, 11(2) November 2012; 367-374 Plotnick FL, 2004 Introduction to Modified Sequence Logic, Conference Proceedings, PMICOS (first annual) Conference, April 25, 2004, Montreal, Canada  Ponce de Leon, G. 2008 Graphical Planning method. PMICOS Annual Conference, Chicago, IL, 2008 340-10 Roy, G.B., 1959 Théorie des Graphes: Contribution  de la théorie des graphes á l1 étude de certains problémes linéaries, Comptes rendus des Séances de l1 Acedémie des Sciences. séence du Avril 1959, s2437-2449, 1959 Roy, G.B. 1960. Contribution de la théorie des graphes à l’étude de certains problems d’ordonnancement",  Comptes rendus de la 2ème conférence internationale sur la recherché opérationnelle, Aix-en-Provence, English Universities Press, Londres 171-185. Valls, V and Lino, 2001. Criticality Analysis in Activity-on-Node Networks with Minimal Time Lags. Annals of Operations Research, 102(1-4) 17-37 Wiest, J.D. 1981 Precedence diagramming method: Some unusual characteristics and their implications for project managers, Journal of Operations management, Volume 1/3 Feb. 1981. 121-130 340-11  ONE	  RELATION	  TO	  RULE	  THEM	  ALL	  Miklós	  Hajdu	  Budapest	  University	  of	  Technology	  The	  point-­‐to-­‐point	  relaDon	  that	  subsDtute	  the	  exisDng	  ones	  AND	  OTHER	  INTERESTING	  ISSUES	  Such	  as	  the	  characterisDcs	  of	  criDcal	  acDviDes	  	  when	  point-­‐to-­‐point	  relaDons	  are	  in	  use	  or	  conDnious	  relaDons	  or	  relaDons	  without	  direcDon.	  	  2015-­‐07-­‐23	  2:57	  PM	   Miklós	  Hajdu	  	  ICSC2015	  Vancouver,	  Canada	   2	  SS0	   	  	  SS0	   	  START	   SS0	   SS3	  A1	  	  3d	  FS	  0	  B1	  	  3d	  FS	  0	  C1	  	  5d	   SS3	   SS0	   F1	  	  7d	  FS0	  G1	  	  4d	   H1	  	  4d	  FS-­‐2	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  FS	  0	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  max	  FS2	  	  	  	  	  	  50m,0m,	  0d	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  100	  m,	  	  50	  m,0d	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  150	  m,	  100m,0d	  	  	  	  F	  S	  3	  FS5	   	  FS3	   SS	  	  3	  A2	  	  5d	  FS2	  	  B2	  	  2d	   SS	  5	   E2	  	  4d	  50m	  F2	  	  5d	  FS0	  G2	  	  5d	   	  	  	  	  	  	  	  FF	  3	  	  	  SS2	  	  	  	  FF2	   50	  m	  50m,0m,0d	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  100	  	  	  m,50m,0d	   FS-­‐3	   	  FS-­‐3	  A3	  	  6d	  FS	  0	  B3	  	  5d	   SS3	  FF3	  C3	  	  5d	   D3	  	  4d	  FS0	  E3	  	  4d	  10d	  F3	  	  4d	   50m	  max50	   G3	  	  6d	  FS0	  H3	  	  4d	  FS0	   50	  	  ,m	   	  	  50	  	  m	   50	  	  m	   10	  	  	  d	   	  	  	  	  	   10	  	  	  d	  	  	  	  	  	  	  	  	  	  	  	  FS	  0	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  max	  FS0	  A5	  	  4d	  FS	  0	  B4	  	  7d	  FS0	  C4	  	  6d	   FS0	  maxFS0	  D4	  	  3d	  50m	  E4	  	  5d	  FS0	  G4	  	  4d	   H4	  	  5d	  FS0	  	  	  	  	  	  	  	  	  	  50m,0m,	  0d	  	  100m,50m,0d	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  150m,100m,0d	  	  50	  	  m	   50	  	  m	   	  50	  	  m	   50	  	  m	   	  10	  	  d	   50m	  A5	  	  7d	   C5	  	  4d	   FS0	   D5	  	  7d	  F,S,0	  E5	  	  5d	  F,S,0	  F5	  	  5d	  FS0	  G5	  	  5d	  FS0	  H5	  	  3d	  F,S,	  	  0	  	  50	  	  m	   50m	   50m	   50m	   50m	  A6	  	  5d	   	  	  	  FS	  0	  maxFS	  0	  B6	  	  6d	   FS0	   C6	  	  8d	   FS3	   D6	  	  2d	  FS0	  E6	  	  4d	  FS0	  F6	  	  4d	   H6	  	  5d	  FF0	  FF0	   FF0	   FF0	   FF0	   FF0	  FF0	  FF0	  END	  Things	  we	  do	  not	  like	  in	  this	  network...	  2015-­‐07-­‐23	  2:57	  PM	   Miklós	  Hajdu	  	  ICSC2015	  Vancouver,	  Canada	   3	  A1	  	  	  	  	  3d	  A2	  	  	  	  	  5d	  FS	  -­‐2d	  NegaDve	  lag	  Why	  we	  do	  not	  like	  this?	  2d	  Dme	  locaDon	  Only	  one	  point	  is	  under	  control…..	  	  and	  we	  can	  control	  it	  only	  in	  the	  future	  	  …	  2015-­‐07-­‐23	  2:57	  PM	   Miklós	  Hajdu	  	  ICSC2015	  Vancouver,	  Canada	   4	  SS0	   	  	  SS0	   	  START	   SS0	   SS3	  A1	  	  3d	  FS	  0	  B1	  	  3d	  FS	  0	  C1	  	  5d	   SS3	   SS0	   F1	  	  7d	  FS0	  G1	  	  4d	   H1	  	  4d	  FS-­‐2	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  FS	  0	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  max	  FS2	  	  	  	  	  	  50m,0m,	  0d	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  100	  m,	  	  50	  m,0d	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  150	  m,	  100m,0d	  	  	  	  F	  S	  3	  FS5	   	  FS3	   SS	  	  3	  A2	  	  5d	  FS2	  	  B2	  	  2d	   SS	  5	   E2	  	  4d	  50m	  F2	  	  5d	  FS0	  G2	  	  5d	   	  	  	  	  	  	  	  FF	  3	  	  	  SS2	  	  	  	  FF2	   50	  m	  50m,0m,0d	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  100	  	  	  m,50m,0d	   FS-­‐3	   	  FS-­‐3	  A3	  	  6d	  FS	  0	  B3	  	  5d	   SS3	  FF3	  C3	  	  5d	   D3	  	  4d	  FS0	  E3	  	  4d	  10d	  F3	  	  4d	   50m	  max50	   G3	  	  6d	  FS0	  H3	  	  4d	  FS0	   50	  	  ,m	   	  	  50	  	  m	   50	  	  m	   10	  	  	  d	   	  	  	  	  	   10	  	  	  d	  	  	  	  	  	  	  	  	  	  	  	  FS	  0	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  max	  FS0	  A5	  	  4d	  FS	  0	  B4	  	  7d	  FS0	  C4	  	  6d	   FS0	  maxFS0	  D4	  	  3d	  50m	  E4	  	  5d	  FS0	  G4	  	  4d	   H4	  	  5d	  FS0	  	  	  	  	  	  	  	  	  	  50m,0m,	  0d	  	  100m,50m,0d	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  150m,100m,0d	  	  50	  	  m	   50	  	  m	   	  50	  	  m	   50	  	  m	   	  10	  	  d	   50m	  A5	  	  7d	   C5	  	  4d	   FS0	   D5	  	  7d	  F,S,0	  E5	  	  5d	  F,S,0	  F5	  	  5d	  FS0	  G5	  	  5d	  FS0	  H5	  	  3d	  F,S,	  	  0	  	  50	  	  m	   50m	   50m	   50m	   50m	  A6	  	  5d	   	  	  	  FS	  0	  maxFS	  0	  B6	  	  6d	   FS0	   C6	  	  8d	   FS3	   D6	  	  2d	  FS0	  E6	  	  4d	  FS0	  F6	  	  4d	   H6	  	  5d	  FF0	  FF0	   FF0	   FF0	   FF0	   FF0	  FF0	  FF0	  END	  Things	  we	  do	  not	  like	  in	  this	  network...	  2015-­‐07-­‐23	  2:57	  PM	   Miklós	  Hajdu	  	  ICSC2015	  Vancouver,	  Canada	   5	  A2	  	  	  	  	  5d	  A3	  	  	  	  	  6d	  FF	  2d	  SS	  2d	  CombinaDon	  of	  SS	  and	  FF	  ...	  Only	  two	  points	  are	  under	  control…..	  	  	  	  	  anything	  can	  happen	  between	  them	  2d	  2d	  Dme	  locaDon	  Why	  we	  do	  not	  like	  this?	  2015-­‐07-­‐23	  2:57	  PM	   Miklós	  Hajdu	  	  ICSC2015	  Vancouver,	  Canada	   6	  SS0	   	  	  SS0	   	  START	   SS0	   SS3	  A1	  	  3d	  FS	  0	  B1	  	  3d	  FS	  0	  C1	  	  5d	   SS3	   SS0	   F1	  	  7d	  FS0	  G1	  	  4d	   H1	  	  4d	  FS-­‐2	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  FS	  0	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  max	  FS2	  	  	  	  	  	  50m,0m,	  0d	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  100	  m,	  	  50	  m,0d	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  150	  m,	  100m,0d	  	  	  	  F	  S	  3	  FS5	   	  FS3	   SS	  	  3	  A2	  	  5d	  FS2	  	  B2	  	  2d	   SS	  5	   E2	  	  4d	  50m	  F2	  	  5d	  FS0	  G2	  	  5d	   	  	  	  	  	  	  	  FF	  3	  	  	  SS2	  	  	  	  FF2	   50	  m	  50m,0m,0d	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  100	  	  	  m,50m,0d	   FS-­‐3	   	  FS-­‐3	  A3	  	  6d	  FS	  0	  B3	  	  5d	   SS3	  FF3	  C3	  	  5d	   D3	  	  4d	  FS0	  E3	  	  4d	  10d	  F3	  	  4d	   50m	  max50	   G3	  	  6d	  FS0	  H3	  	  4d	  FS0	   50	  	  ,m	   	  	  50	  	  m	   50	  	  m	   10	  	  	  d	   	  	  	  	  	   10	  	  	  d	  	  	  	  	  	  	  	  	  	  	  	  FS	  0	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  max	  FS0	  A5	  	  4d	  FS	  0	  B4	  	  7d	  FS0	  C4	  	  6d	   FS0	  maxFS0	  D4	  	  3d	  50m	  E4	  	  5d	  FS0	  G4	  	  4d	   H4	  	  5d	  FS0	  	  	  	  	  	  	  	  	  	  50m,0m,	  0d	  	  100m,50m,0d	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  150m,100m,0d	  	  50	  	  m	   50	  	  m	   	  50	  	  m	   50	  	  m	   	  10	  	  d	   50m	  A5	  	  7d	   C5	  	  4d	   FS0	   D5	  	  7d	  F,S,0	  E5	  	  5d	  F,S,0	  F5	  	  5d	  FS0	  G5	  	  5d	  FS0	  H5	  	  3d	  F,S,	  	  0	  	  50	  	  m	   50m	   50m	   50m	   50m	  A6	  	  5d	   	  	  	  FS	  0	  maxFS	  0	  B6	  	  6d	   FS0	   C6	  	  8d	   FS3	   D6	  	  2d	  FS0	  E6	  	  4d	  FS0	  F6	  	  4d	   H6	  	  5d	  FF0	  FF0	   FF0	   FF0	   FF0	   FF0	  FF0	  FF0	  END	  Things	  we	  do	  not	  like	  in	  this	  network…	  	  unders and	  yet…	  maxFS2	  2015-­‐07-­‐23	  2:57	  PM	   Miklós	  Hajdu	  	  ICSC2015	  Vancouver,	  Canada	   7	  A2	  	  	  5d	  B1	  	  	  3d	  FS	  2d	  FS0	  Maximal	  relaDons...	  B2	  	  	  2d	  Dme	  acDviDes	  B1	  B2	  A2	  maxFS2	  B1	  -­‐	  	  excavaDon	  for	  basements	  B2	  -­‐	  	  retaining	  wall	  2015-­‐07-­‐23	  2:57	  PM	   Miklós	  Hajdu	  	  ICSC2015	  Vancouver,	  Canada	   8	  SS0	   	  	  SS0	   	  START	   SS0	   SS3	  A1	  	  3d	  FS	  0	  B1	  	  3d	  FS	  0	  C1	  	  5d	   SS3	   SS0	   F1	  	  7d	  FS0	  G1	  	  4d	   H1	  	  4d	  FS-­‐2	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  FS	  0	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  max	  FS2	  	  	  	  	  	  50m,0m,	  0d	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  100	  m,	  	  50	  m,0d	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  150	  m,	  100m,0d	  	  	  	  F	  S	  3	  FS5	   	  FS3	   SS	  	  3	  A2	  	  5d	   FS2	   B2	  	  2d	   SS	  5	   E2	  	  4d	  50m	  F2	  	  5d	  FS0	  G2	  	  5d	   	  	  	  	  	  	  	  FF	  3	  	  	  SS2	  	  	  	  FF2	   50	  m	  50m,0m,0d	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  100	  	  	  m,50m,0d	   FS-­‐3	   	  FS-­‐3	  A3	  	  6d	  FS	  0	  B3	  	  5d	   SS3	  FF3	  C3	  	  5d	   D3	  	  4d	  FS0	  E3	  	  4d	  10d	  F3	  	  4d	   50m	  max50	   G3	  	  6d	  FS0	  H3	  	  4d	  FS0	   50	  	  ,m	   	  	  50	  	  m	   50	  	  m	   10	  	  	  d	   	  	  	  	  	   10	  	  	  d	  	  	  	  	  	  	  	  	  	  	  	  FS	  0	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  max	  FS2	  A5	  	  4d	  FS	  0	  B4	  	  7d	  FS0	  C4	  	  6d	   FS0	  maxFS0	  D4	  	  3d	  50m	  E4	  	  5d	  FS0	  G4	  	  4d	   H4	  	  5d	  FS0	  	  	  	  	  	  	  	  	  	  50m,0m,	  0d	  	  100m,50m,0d	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  150m,100m,0d	  	  50	  	  m	   50	  	  m	   	  50	  	  m	   50	  	  m	   	  10	  	  d	   50m	  A5	  	  7d	   C5	  	  4d	   FS0	   D5	  	  7d	  F,S,0	  E5	  	  5d	  F,S,0	  F5	  	  5d	  FS0	  G5	  	  5d	  FS0	  H5	  	  3d	  F,S,	  	  0	  	  50	  	  m	   50m	   50m	   50m	   50m	  A6	  	  5d	   	  	  	  FS	  0	  maxFS	  0	  B6	  	  6d	   FS0	   C6	  	  8d	   FS3	   D6	  	  2d	  FS0	  E6	  	  4d	  FS0	  F6	  	  4d	   H6	  	  5d	  FF0	  FF0	   FF0	   FF0	   FF0	   FF0	  FF0	  FF0	  END	  Things	  we	  do	  not	  	  understand	  yet…	  2015-­‐07-­‐23	  2:57	  PM	   Miklós	  Hajdu	  	  ICSC2015	  Vancouver,	  Canada	   9	  C1	  	  	  	  	  5d	  C3	  	  	  	  	  5d	  50m,0m,0d	  Point-­‐to-­‐point	  relaDons...	  100m,50m,0d	  150m,100m,0d	  C1	  	  	  	  	  	  	  	  	  	  	  	  	  5d	  	  C3	  	  	  	  	  	  	  	  	  	  	  	  5d	  Points	  can	  be	  defined	  using	  Dme	  or	  volume	  data	  Instead	  of	  0day,	  or	  0m	  S	  can	  be	  used	  	  to	  idenDfy	  the	  Start	  point	  Instead	  of	  5day,	  or	  150m	  F	  can	  be	  used	  	  to	  idenDfy	  the	  Finish	  point	  Conclusion:	  Point-­‐to-­‐point	  relaDons	  can	  replace	  the	  tradiDonal	  four	  precedence	  relaDonships	  2015-­‐07-­‐23	  2:57	  PM	   Miklós	  Hajdu	  	  ICSC2015	  Vancouver,	  Canada	   10	  C1	  	  	  	  	  5d	  C3	  	  	  	  	  5d	  50m,0m,0d	  Point-­‐to-­‐point	  relaDons...	  100m,50m,0d	  150m,100m,0d	  Francis	  and	  Miresko	  	  	  	   	  2000	  Plotnick	   	   	   	  2004	  Ponce	  de	  Leon	   	   	  2008	  Kim 	   	   	   	  2010	  Hegazy	   	   	   	  2010	  Liberzon	  (Spider	  Project) 	  2010	  	  	  The	  mathemaDcal	  model,	  that	  handle	  both	  minimal	  and	  maximal	  relaDons	  and	  the	  algorithm	  is	  in	  the	  paper.	  2015-­‐07-­‐23	  2:57	  PM	   Miklós	  Hajdu	  	  ICSC2015	  Vancouver,	  Canada	   11	  SS0	   	  	  SS0	   	  START	   SS0	   SS3	  A1	  	  3d	  FS	  0	  B1	  	  3d	  FS	  0	  C1	  	  5d	   SS3	   SS0	   F1	  	  7d	  FS0	  G1	  	  4d	   H1	  	  4d	  FS-­‐2	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  FS	  0	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  max	  FS2	  	  	  	  	  	  50m,0m,	  0d	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  100	  m,	  	  50	  m,0d	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  150	  m,	  100m,0d	  	  	  	  F	  S	  3	  FS5	   	  FS3	   SS	  	  3	  A2	  	  5d	  FS2	  	  B2	  	  2d	   SS	  5	   E2	  	  4d	  50m	  F2	  	  5d	   G2	  	  5d	   	  	  	  	  	  	  	  FF	  3	  	  	  SS2	  	  	  	  FF2	   50	  m	  50m,0m,0d	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  100	  	  	  m,50m,0d	   FS-­‐3	   	  FS-­‐3	  A3	  	  6d	  FS	  0	  B3	  	  5d	   SS3	  FF3	  C3	  	  5d	   D3	  	  4d	  FS0	  E3	  	  4d	  10d	  F3	  	  4d	   50m	  max50	   G3	  	  6d	  FS0	  H3	  	  4d	  FS0	   50	  	  ,m	   	  	  50	  	  m	   50	  	  m	   10	  	  	  d	   	  	  	  	  	   10	  	  	  d	  	  	  	  	  	  	  	  	  	  	  	  FS	  0	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  max	  FS2	  A5	  	  4d	  FS	  0	  B4	  	  7d	  FS0	  C4	  	  6d	   FS0	  maxFS0	  D4	  	  3d	  50m	  E4	  	  5d	  FS0	  G4	  	  4d	   H4	  	  5d	  FS0	  	  	  	  	  	  	  	  	  	  50m,0m,	  0d	  	  100m,50m,0d	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  150m,100m,0d	  	  50	  	  m	   50	  	  m	   	  50	  	  m	   50	  	  m	   	  10	  	  d	   50m	  A5	  	  7d	   C5	  	  4d	   FS0	   D5	  	  7d	  F,S,0	  E5	  	  5d	  F,S,0	  F5	  	  5d	  FS0	  G5	  	  5d	  FS0	  H5	  	  3d	  F,S,	  	  0	  	  50	  	  m	   50m	   50m	   50m	   50m	  A6	  	  5d	   	  	  	  FS	  0	  maxFS	  0	  B6	  	  6d	   FS0	   C6	  	  8d	   FS3	   D6	  	  2d	  FS0	  E6	  	  4d	  FS0	  F6	  	  4d	   H6	  	  5d	  FF0	  FF0	   FF0	   FF0	   FF0	   FF0	  FF0	  FF0	  END	  Things	  we	  do	  not	  	  understand	  yet…	  2015-­‐07-­‐23	  2:57	  PM	   Miklós	  Hajdu	  	  ICSC2015	  Vancouver,	  Canada	   12	  ConDnuous	  relaDons….	  C3	  	  	  	  	  5d	  C4	  	  	  	  	  6d	  50m	  C3	   50m	  C4	  Control	  all	  the	  points	  that	  belong	  to	  the	  same	  Dme	  	   	   	   	   	  (volume/locaDon	  lag)	  2015-­‐07-­‐23	  2:57	  PM	   Miklós	  Hajdu	  	  ICSC2015	  Vancouver,	  Canada	   13	  ConDnuous	  relaDons……	  C3	  	  	  	  	  5d	  C4	  	  	  	  	  6d	  1	  day	  C3	  1	  day	  C4	  …or	  	  belong	  to	  the	  same	  locaDon	  	  	  	  	  (Dme	  lag)	  2015-­‐07-­‐23	  2:57	  PM	   Miklós	  Hajdu	  	  ICSC2015	  Vancouver,	  Canada	   14	  ConDnuous	  relaDons……	  C3	  	  	  	  	  5d	  C4	  	  	  	  	  6d	  1	  day	  Model,	  algorithm	  with	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  minimal	  and	  maximal	  Dme	  and	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  volume	  lags	  can	  bee	  seen	  at	  	  2015-­‐07-­‐23	  2:57	  PM	   Miklós	  Hajdu	  	  ICSC2015	  Vancouver,	  Canada	   15	  SS0	   	  	  SS0	   	  START	   SS0	   SS3	  A1	  	  3d	  FS	  0	  B1	  	  3d	  FS	  0	  C1	  	  5d	   SS3	   SS0	   F1	  	  7d	  FS0	  G1	  	  4d	   H1	  	  4d	  FS-­‐2	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  FS	  0	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  max	  FS2	  	  	  	  	  	  50m,0m,	  0d	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  100	  m,	  	  50	  m,0d	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  150	  m,	  100m,0d	  	  	  	  F	  S	  3	  FS5	   	  FS3	   SS	  	  3	  A2	  	  5d	  FS2	  	  B2	  	  2d	   SS	  5	   E2	  	  4d	  50m	  F2	  	  5d	   G2	  	  5d	   	  	  	  	  	  	  	  FF	  3	  	  	  SS2	  	  	  	  FF2	   50	  m	  50m,0m,0d	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  100	  	  	  m,50m,0d	   FS-­‐3	   	  FS-­‐3	  A3	  	  6d	  FS	  0	  B3	  	  5d	   SS3	  FF3	  C3	  	  5d	   D3	  	  4d	  FS0	  E3	  	  4d	  10d	  F3	  	  4d	   50m	  max50	   G3	  	  6d	  FS0	  H3	  	  4d	  FS0	   50	  	  ,m	   	  	  50	  	  m	   50	  	  m	   10	  	  	  d	   	  	  	  	  	   10	  	  	  d	  	  	  	  	  	  	  	  	  	  	  	  FS	  0	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  max	  FS2	  A5	  	  4d	  FS	  0	  B4	  	  7d	  FS0	  C4	  	  6d	   FS0	  maxFS0	  D4	  	  3d	  50m	  E4	  	  5d	  FS0	  G4	  	  4d	   H4	  	  5d	  FS0	  	  	  	  	  	  	  	  	  	  50m,0m,	  0d	  	  100m,50m,0d	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  150m,100m,0d	  	  50	  	  m	   50	  	  m	   	  50	  	  m	   50	  	  m	   	  10	  	  d	   50m	  A5	  	  7d	   C5	  	  4d	   FS0	   D5	  	  7d	  F,S,0	  E5	  	  5d	  F,S,0	  F5	  	  5d	  FS0	  G5	  	  5d	  FS0	  H5	  	  3d	  F,S,	  	  0	  	  50	  	  m	   50m	   50m	   50m	   50m	  A6	  	  5d	   	  	  	  FS	  0	  maxFS	  0	  B6	  	  6d	   FS0	   C6	  	  8d	   FS3	   D6	  	  2d	  FS0	  E6	  	  4d	  FS0	  F6	  	  4d	   H6	  	  5d	  FF0	  FF0	   FF0	   FF0	   FF0	   FF0	  FF0	  FF0	  END	  Things	  we	  do	  not	  	  understand	  yet…	  2015-­‐07-­‐23	  2:57	  PM	   Miklós	  Hajdu	  	  ICSC2015	  Vancouver,	  Canada	   16	  A1	  	  	  	  	  3d	   B1	  	  	  	  	  3d	  FS	  0	  RelaDonships	  without	  direcDon…	  A1	  	  	  	  	  3d	   B1	  	  	  	  	  3d	  FS	  0	  A1	  	  	  	  	  3d	   B1	  	  	  	  	  3d	  FS	  0	  This	  can	  be	  	  this	  	  or	  this	  	  Can	  be	  seen	  at	   from	  2015-­‐07-­‐23	  2:57	  PM	   Miklós	  Hajdu	  	  ICSC2015	  Vancouver,	  Canada	   17	  Conclusions	  •  Using	  negaDve	  lag	  for	  modeling	  overlapping	  is	  the	  worst	  pracDce	  •  CombinaDon	  of	  SS	  and	  FF	  is	  a	  bit	  befer,	  at	  least	  the	  start	  and	  the	  finish	  points	  are	  under	  control	  •  Point-­‐to-­‐point	  relaDons	  are	  acceptable	  from	  pracDcal	  point	  oif	  view.	  The	  number	  of	  controlled	  points	  is	  upon	  the	  planner	  discreDon.	  (Minimal	  and	  maximal	  lags	  are	  allowed)	  •  ConDnuous	  relaDons	  are	  the	  best	  for	  modelling	  acDvity	  overlapping.	  All	  the	  point	  are	  under	  control.	  (Time	  and	  volume	  lags,	  maximal	  and	  minimal	  lags	  can	  be	  used.	  •  RelaDons	  without	  direcDons	  are	  useful	  for	  describing	  relaDons	  conencted	  to	  scrace	  resources.	  2:57	  PM	   Miklós	  Hajdu	  	  ICSC2015	  Vancouver,	  Canada	   18	  Disccussions	  •  Due	  the	  the	  new	  type	  of	  precedence	  relaDons	  such	  as	  point-­‐to-­‐point,	  conDnuous	  and	  undirected	  relaDonships	  all	  the	  former	  definiDons,	  developments,	  problems,	  algorithms	  that	  were	  based	  on	  the	  tradiDonal	  four	  relaDonships	  must	  be	  examined	  again.	  •  Examples	  are	  could	  be:	  definiDon	  of	  floats,	  criDcal	  path,	  criDcal	  acDvity	  characterisDcs,	  Dme-­‐cost	  trade	  off	  problems,	  resource	  leveling	  and	  allocaDon	  problems	  etc.	  2:57	  PM	   Miklós	  Hajdu	  	  ICSC2015	  Vancouver,	  Canada	   19	  A	  short	  example	  Is	  this	  a	  •  normal	  criDcal,	  or	  •  neutral	  criDcal,	  or	  •  reverse	  criDcal,	  or	  	  •  bi	  criDcal,	  or	  	  •  decreasing	  neutral	  increasing	  normal	  criDcal,	  or	  •  decreasing	  reverse	  incresing	  neutral	  criDcal	  	  acDvity?	  	  	  this	  characterizaDon	  valid	  in	  	  case	  of	  poin-­‐to-­‐point	  relaDons?	  Or	  in	  case	  of	  conDnuous	  relaDons?	  2015-­‐07-­‐23	  2:57	  PM	   Miklós	  Hajdu	  	  ICSC2015	  Vancouver,	  Canada	   20	  Thank	  you	  for	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  your	  afenDon!	  

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