The Optim ization A lgor ithm s for Solv ing Resource-con stra ined Project Schedul ing Problem: A Rev iew

Transcript

1 16 V o l. 16 Supp l. CON T ROL A N D D EC IS ION N ov : (2001) 0S ,, (, ) :,,,,, : ; ; ; : O 224 : A The Optim ization A lgor ithm s for Solv ing Resource-con stra ined Project Schedul ing Problem: A Rev iew L IU S h i2x in, W A N G M eng 2g uang, TA N G J ia2f u (Schoo l of Info rm ation Science and Engineering, N o rtheastern U niversity, Shenyang , Ch ina) Abstract: Because of the im po rtant theo retical values and w ide p ractical app lications of the resource2 constrained p ro ject scheduling p roblem, there have been fast developm ents in the op tim azation algo2 rithm s fo r so lving th is p roblem. Som e of the recent developm ents on branch and bound algo rithm s, p ri2 o rity2based heuristics, sam p ling algo rithm s and intelligent algo rithm s fo r so lving th is p roblem are re2 view ed, and the develop ing tendency is discussed. Key words: p ro ject scheduling; resource2constrained; branch2and2bound; heuristics 1, (P ro ject2 o rien ted),,, ;,,,, N P2hard,,,, : ; : : ( ) : (1968 ),,,,, ; (1936 ),,,,,

2 (RCPSP) :, J,, j h, h P j (P j j ),, P j j 1, J, ( 0),, j (j = 1, 2,, J ) k rjk, d jk R k (k = 1, 2,, K ), j S j,a t (t - 1, t ], ( [ 1 ]) m in S J (1) s. t. S j - S i d i, i P j (2) j A t rjk R k, t = 1, 2,, f J k = 1, 2,, K (3), (1), ; (2) ; (3) ( ), ( ), RCPSP g PSg tg, PSg,, A g PSg, PSg A g, tg = m in{s j + d j, j A g}, tg E g = {j g j PSg, Π i P j, S i + d i tg }, tg E g,,, PSg ( ), D em eu lem eester [2 ] tg E g,, DA g g,,, [ 3 ] W indow s N T, M ingozzi [4 ], Patterson [5 ], g, PSg, B rucker [6 ], ( ),,,, () (), : RCPSP, [7, 8 ]

3 16 : 649 :,, J, n (n = 1, 2,, J ) PSn E n: = {j g j PSn, P j Α PSn} E n (, ), J, n tn, tn C n, tn A n, tn E n, E n: = {j g j {C n A n}, P j Α C n, rjk ΠR k, k = 1, 2,, K } n n + 1 : 1) tn+ 1, C n+ 1 A n+ 1, ΠR k E n+ 1; 2) E n+ 1 j, j, j, E n+ 1, 2) E n+ 1 30, RCPSP, [9 11 ] 1 RCPSP N C, R F R S [14 ],,,, ,, v (j ) Ω(j ), Ω(j ) E n Ω(j ), (R S) (BR S) (RBR S) R S, E n, Ω(j ) = 1 ge ng BR S Ω(j ),, j [15 ] Ω(j ) = v (j ) v (i) (4) i E n RBR S D rex l [16 ], Ω(j ) j v (j ),, Ω(j ) 1 RCPSP Θj: = v (j ) - m in i E n v (i) (5) Ω(j ) : = (Θj + Ε) Α (Θi + Ε) Α (6) i E n Θj j, Ε(> 0) 0, GR PW M ax [ 10 ] d j + i SA j d i M T S M ax [ 10 ] gsa T j g RSM M ax [ 10 ] [ 10 ] W CS M ax [ 10 ] [ 10 ] L ST M in [ 10 ] L Sj M IN SL K M in [ 12 ] L Sj - tn M INL FT M in [ 12 ] L Sj + d j GRD M ax [ 12 ] d j k rjk GRU M ax [ 12 ] k rjk L FS M in [ 12 ] (L Sj - ESj) gsa T j g R &M gbgl T I M ax [ 12 ] [ 12 ] WACRU W RU P M ax [ 13 ] [ 13 ] :, SA j j, SA T j j ( ),

4 Α> 0 1) RCPSP, RCPSP, ;, RCPSP,,,, RCPSP [ 22 ],, : 2) 1) ,,,,,,, RCPSP,,,, [ 17, 18 ],, 2) , J, j j [23 ],,, [ 19, 20 ] 3) 3), J ( ), [24 ], ( : j, j ), g,,, g () 4),, J RCPSP, [ 21 ],,,,, RCPSP,,,,,,,, 4 5 RCPSP RCPSP,

5 16 : 651,, state2of2the2art heuristics fo r the resource2constrained, p ro ject scheduling p roblem [J ]. 2000, 127: Eur J of Op s R es,, [ 12 ] L aw rence S R, M o rton T E. R esource2constrained m ulti2p ro ject scheduling w ith tardy co sts: Com paring : [ 1 ] H erroelen W, R eyck B D e, D em eulem eester E. R e2 source2constrained p ro ject scheduling: A survey of re2 cent developm ents[j ]. Computers & Op s R es, 1998, 25 (4): [ 2 ] D em eulem eester E, H erroelen W. A branch2and2bound p rocedure fo r the m ultip le resource2constrained p ro ject scheduling p roblem [J ]. M anagem ent Science, 1992, 38 (12): [ 3 ] D em eulem eester E, H erroelen W. N ew benchm ark re2 sults fo r the resource2constrained p ro ject scheduling p roblem [J ]. M anagem ent Science, 1997, 43 (11): [4 ] M ingozzi A, M aniezzo V, R icciardelli S et al. A n exact algo rithm fo r the resource2constrained p ro ject schedul2 ing p roblem based on a new m athem atical fo rm ulation [J ]. M anagem ent Science, 1998, 44 (5): [5 ] Patterson J H, Slow insk i R, T albo t F B et al. A n algo2 rithm fo r a general class of p recedence and resource con2 strained scheduling p roblem [ A ]. Slow insk i R and W eglarz J (Eds). A dvances in P ro ject Scheduling [M ]. Am sterdam : E lsevier Publisher, [6 ] B rucker P, Knust S, Schoo A et al. A branch and bound algo rithm fo r the resource2constrained p ro ject scheduling p roblem [J ]. Eur J of Op s R es, 1998, 107: [ 7 ] K lein R, Scho ll A. Computing low er bounds by destruc2 tive im p rovem ent: A n app lication to resource2con2 strained p ro ject scheduling[j ]. Eur J of Op s R es, 1999, 112: [ 8 ] B rucker P, Knust S. A linear p rogramm ing and con2 strained p ropagation2based low er bound fo r the RCPSP [J ]. Eur J of Op s R es, 2000, 127: [ 9 ] Ko lisch R. Serial and parallel resource2constrained p ro2 ject scheduling m ethods revisited: T heo ry and com puta2 tion [J ]. Eur J of Op s R es, 1996, 90: [10 ] Ko lisch R, H artm ann S. H euristic algo rithm s fo r so lv2 ing the resource2constrained p ro ject scheduling p rob2 lem : C lassification and com putational analysis [ J ]. M anusk rip te aus den Instituten fur Betriebsw irtschafts2 leh re der U niversit a t K iel, 1998, 469. [ 11 ] H artm ann S, Ko lisch R. Experim ental evaluation of m yop ic, bo ttleneck, and resource p ricing heuristics [J ]. Eur J of Op s R es, 1993, 64: [ 13 ] T hom as P R, Salh i S. A n investigation into the rela2 tionsh ip of heuristic perfo rm ance w ith netw o rk2re2 source characteristics [ J ]. search Society, 1997, 48: J of the Operational R e2 [14 ] Ko lisch R, Sp recher A, D rexl A. Characterization and generation of a general class of resource2constrained p ro ject scheduling p roblem [J ]. M anagem ent Science, 1995, 41 (10): [ 15 ] A lvarez2v aldes R J, T am arit M. A lgo ritmo s heuristi2 co s determ inistas y aleato rio s en secuenciacion de p royecto s con recurso s lim itado s [J ]. 13: Q estiio, 1989, [16 ] D rexl A. Scheduling of p ro ject netw o rk s by job assign2 m ent [J ]. M anagem ent Science, 1991, 30: [ 17 ] H artm ann S. A competitive genetic algo rithm fo r re2 source2constrained p ro ject scheduling [ J ]. search L ogistics, 1998, 45: N aval R e2 [ 18 ],. [J ]., 2002, 17. [19 ] L ee J K, K im Y D. Search heuristics fo r resource con2 strained p ro ject scheduling [J ]. 1996, 47 (3): J of the Op s R es Soc, [20 ] Cho J H, K im Y D. A sim ulated annealing algo rithm fo r resource constrained p ro ject [J ]. J of the Op s R es Soc, 1997, 48: scheduling p roblem [21 ] O zdam ar L. A genetic algo rithm app roach to a general catego ry p ro ject scheduling p roblem [J ]. on Syst, M an and Cyber, 1999, 29 (1): IEEE T rans [ 22 ] L iu Sh ixin, W ang M engguang. A n object2o riented m ethodo logy fo r so lving the RCPSP s w ith heuristics and m etaheuristics [ J ]. tro l, 2000, 11 (5): P roduction P lanning & Con2 [ 23 ],. SA gga [J ]., 2001, 16 (5). [24 ],,. [J ]., 1999, 14 (3):