A research on the influence of dummy activity on float in an AOA network and its amendments

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1 : (2008) ,, (, ) : NP2hard,,..,.,,.,.,. :,,,, : TB11411 : A A research on the influence of dummy activity on float in an AOA network and its amendments WANG Qiang, LI Xing2mei, QI Jian2xun (School of Business Administration, North China Electric Power University, Beijing ,China) Abstract : As the minimal dummy activities and dummy nodes problem is NP2Hard problem, for a given project, there may be several correct AOA networks which differ in their number of dummy activities and number of nodes. The purpose of this paper is to research the influence of dummy activity on float in an AOA network and its amendments. As different drawing methods of dummy activities result in AOA networks of varied dummy nodes, therefore, the research method is based on the nodes. Firstly, Nodes are classified into three types : the normal node, out2dummy node and in2dummy node. On basis of the analysis to the influence of the node type on the activity float algorithms, the node type algorithms for activity safety float and free float are found to have error for out2dummy node and in2 dummy node separately. Finally, amendments are made and an example is presented to test the effectiveness of the algorithms. Key words : AOA networks ; time parameters ; total float ; safety float ; free float 1 (AOA ),. AOA,,.,.,.,.,,,.,, [1 ]. : : ( ) ; ( ) : ( ),,,, ; ( ), ( ),,, ; ( ),,,,.

2 CPM, CPM [2 ],.,, Battersby(1967) Thomas (1969),,,. Elmaghraby(1977) [3 ]., :, [4 ], CPM..,,. ; ;,.,,,,,,.,.,, [5 7 ]. Krishnamoorty Deo (1979) NP2hard [8 ].,., [9 11 ].,.,,,.., ( ),.,,. 2,. ; ;, ;.,. 1 2,. 1 e,f, i, j, n, o..,,. 2 j..,,. 2 i..,,

3 [12 ].. 3 ( ), ( ) ;,. 311 : ; :, [13 ].., [4,13 ],.,,,.,,. [13 ],. 1 i, ( ij) ES ij ET i. i, ET i, CPM [13 ] : ET 1 = 0 ES ij = ET i (1. 1) for i = 2 to n do (1. 2) ET i = max h P i { ET h + d hi } P i i, d hi ( hi),, ( hi), d hi = 0. 1 : 2,. 312 :..., [4,13 ],.,,,. [13 ],, : 3 j, ( ij) L F ij L T j. j, L T j, CPM : L T n L F ij = L T j (1. 3) = ET n for j = n - 1 to 1 do (1. 4) L T j = min k S j {L T k - d jk } S j j, d jk ( jk),, ( jk), d jk = 0. 3 : 4,

4 (111), (112), (113), (114),,,,, ,,.. 411,, TF ij. ( ij), [4,13] : TF ij = LS ij - ES ij = L F ij - EF ij = L F ij - ES ij - d ij (1. 5) LS ij, ES ij ; L F ij, EF ij, d ij., [4,13 ] : TF ij = L T j - ET i - d ij (116) (116) (115), (117), (116). L T j = L F ij ET i = ES ij (117) 1 3,, (117),,,. 412 ( ij),, FF ij., : FF ij = ET 3 j - EF ij ET 3 j = min ( xy) S ij { ES xy } S ij ( ij), ( xy), ET 3 j ; EF ij ( ij) [4,13 ]. (1. 8) j ET j ; ES ij. 1 : ET i = ES ij, : (118),, [4,13 ] : (1111) (1110), (1111) : EF ij = ES ij + d ij = ET i + d ij (1. 9) FF ij = ET 3 j - ET i - d ij (1110) FF ij = ET j - ET i - d ij (1111) ET j = ET 3 j (1. 12) ( ij),, ( ij), 1 j, ( ij). j, 1 ( jk), TS ij

5 ; j, 1 ( np), ( oq) ( or), DS ij. : S ij = TS ij DS ij (1113) TS ij, ET t 3 j ( jk) TS ij, j, 1, ET j.. ES jk = ET j (1114) ET t 3 j = min ( jk) TS ij { ES jk } = ET j (1. 15) DS ij, ET d 3 j. j,, j. 2, j ET j. : (118), (1113), (1115) (1116) : (1112), : ET d 3 j = min ( uv) DS ij { ES uv } ET j (1116) ET 3 j = min { ES xy } = min{ ET t 3 j, ET d 3 j } = ET j (1. 17) ( xy) S ij 5 ( ij),, ET j ET 3 j 41212,. ( ij), 2 j, ( ij) j, 2 ( np), ( nk), ( oq) ( or), j,, j. ( np) ES np = ET n ET 3 j. 2, j ET j. : (1112), : ET j ET n = ES np = ET 3 j (1. 18) 6 ( ij),, ET j ET 3 j, ET j ET 3 j,. 5 6,, j, ET j = ET 3 j, ; j, ET j ET 3 j,,,.,,. 413 ( ij),. SS ij., : SS ij = LS ij - L T 3 i L T 3 i = max ( xy) P ij {L F xy } P ij ( ij), ( xy),l T 3 i ; LS ij ( ij) [4,13]. (1119) j L T j, L F ij, 3 L T j = L F ij,

6 6 111 : (1119),, [4,13 ] : (1122) (1121), (1122) : LS ij = L F ij - d ij = L T j - d ij (1120) SS ij = L T j - L T 3 i - d ij (1121) SS ij = L T j - L T i - d ij (1. 22) L T i = L T 3 i (1. 23) ( ij), ( ij), 1 i, ( ij). i, 1 ( ai), TP ij ; i, 1 ( bf ), ( ce) ( de), DP ij. : P ij = TP ij DP ij (1124) TP ij, L T t 3 i ( ai) TP ij, i, 3, L T i.. L F ai = L T i (1. 25) L T t 3 i = max ( ai) TP ij {L F ai } = L T i (1. 26) DP ij, L T d 3 i. i,, i. 3, i L T i. : (1119), (1124), (1126) (1127) : (1123), : L T d 3 i = max ( bf) DP ij {L F bf } L T i (1. 27) L T 3 j = max {L F xy } = max{l T t 3 i,l T d 3 i } = L T i (1. 28) ( xy) P ij 7 ( ij),, L T i L T 3 i, L T i = L T 3 i, ( ij), 2 i, ( ij) i, 2 ( af ), ( bf ), ( ce) ( de), i,. i. ( af) L F af = L T f L T 3 i. 4, i L T i. : (1123), : L T i L T f = L F af = L T 3 i (1. 29) 8 ( ij),, L T i

7 L T 3 i, L T i L T 3 i,. 7 8, i,l T i = L T 3 i, ; i, L T i L T 3 i,,,.,,. 5,,., ;,., :,., ;, :, ( ), ,., TFF j, ( ij) ( ), : FF 3 ij = FF ij + TFF j (1. 30) FF 3 ij ( ij) ; FF ij ( ij) ; TFF j j, : S 3 j. TFF j = min { FF 3 jk } j ( jk) S 3 j 0 j ( ) (1131) j, ( jk), FF jk 51212,,., TSS i, ( ij) ( ), : SS 3 ij = SS ij + TSS i (1. 32) SS 3 ij ( ij) ; SS ij ( ij) ; TSS i i, : P 3 i. TSS i = min { SS 3 hi } i ( hi) P 3 i 0 i ( ) (1133) i, ( hi), SS hi

8 , 1. 2,., B, 0, 2 ; F, 0, A B C D E F G H I A A,B B,C C D,E E,F,G D,E E,F F,G H H,I I I 2 ES EF LS L F TF FF SS TF FF SS A B C D E F G H I ,. 2, : : :, 7 4. : 7. ( ) : 4. 1) 4. FF 45 = E 5 - E 4 - d 45 = = 4 ; FF 46 = E 6 - E 4 - d 46 = = 2 2) , TFF 5 = 0 ; TFF 6 = 0, : FF 3 45 = FF 45 + TFF 5 = = 4 ; FF 3 46 = FF 46 + TFF 6 = = 2 3) 4, : TFF 4 = 2 ; 4) 4 : FF 3 14 = FF 14 + TFF 4 = = 2. : :, 5 6. : 5. ( )

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