Column Generation. Teo Chung-Piaw (NUS) 25 th February 2003, Singapore

Transcription

1 Colun Generaton Teo Chung-Paw (NUS) 25 th February 2003, Sngapore

2 1 Lecture 1.1 Outlne Cuttng Stoc Proble Slde 1 Classcal Integer Prograng Forulaton Set Coverng Forulaton Colun Generaton Approach Connecton wth Lagrangan Relaxaton Coputatonal ssues 2 Cuttng Stoc Proble 2.1 Introducton Exaple A paper copany has a supply of large rolls of paper, each of wdth W. Slde 2 Slde 3 Custoers deand n rolls of wdth w (,...,). (w W ) Exaple: Quantty Ordered n Order Wdth (nches) w The deand can be et by slcng a large roll n a certan way, called a pattern. Slde 4 For exaple, a large roll of wdth 100 can be cut nto 1

3 4 rolls each of wdth 25, or 2 rolls each of wdth 35, wth a waste of Soluton Approach I 3.1 L. V. Kantorovch Forulaton (1939 Russan, 1960 Englsh) Matheatcal Methods of Plannng and Organsng Producton Manageent Scence, 6, Slde 5 K: Set of avalable rolls. y : 1 f roll s cut, 0 otherwse. x : nuber of tes te s cut on roll. Objectve: To nze the nuber of rolls used to eet all the deand n y Constrants K Slde 6 Total nuber of tes te s cut s not less than the deand. x n K The wdth of a roll s at ost W w x Wy 2

4 n K y K x n for a fxed te, w x Wy for fxed roll, x 0, 0 y 1 Integralty constrants on all varabes Slde Qualty of Soluton Scenaro I: N1 Slde 8 n : unfor, between 1 and 100 (rand(100)+1); w : unfor, between 1 and 30 (rand(30)+1); Wdth of Roll, W = 3000; Rolls Ites constr varables CPU (s) Note 1 The OPL code for the proble: Code 3

5 Slde 9 Scenaro II: Change wdth of the roll fro 3000 to 150. Rolls Ites constr varables CPU (s) never never Out of eory The perforance has deterorated. Why? Slde 10 How good s the LP relaxaton? w n Observaton: ZLP = W. N2 Ths bound s trval: The objectve s to n y the optal soluton wll satsfy: K Choose y as sall as possble. Therefore w x = Wy for all Choose x as sall as possble. Therefore x = n 4 Slde 11

6 for all. y = w x W = w W x K K K w = W x K = w n W Note 2 We have the constrants w x Wy. Proof In the LP, snce the objectve s to nze y, the optal LP soluton wll be such that w x = y. W y wll be sall f the x values are sall. At the sae te, because x n, K to ae x sall, at the optal soluton, we ust have x = n. K So the objectve functon, for the optal LP soluton, reduces to = w x y W K K w = x W K w = x W K w n = W Another Exaple: W = 273 Quantty Ordered Order Wdth (nches) Slde 12 5

7 LP: solved n 0.27s. Soluton IP: halted after 12 hours of coputatonal te! 4 Approach II 4.1 Glore and Goory Set Coverng P. C. GILMORE AND R. E. GOMORY, A lnear prograng approach to the cuttng-stoc proble, Oper. Res., 8 (1961), pp x j = nuber of tes pattern j s used a j = nuber of tes te s cut n pattern j For exaple, a large roll of wdth 100 can be cut nto 4 rolls each of wdth w = 25 (pattern j, a j =4) 2 rolls each of wdth w = 35 (pattern l, a l =2) n n j=1 x j n j=1 a j x j n, = 1,...,, x j Z +, j =1,...,n Exaple: An nstance: W = 100, = 3. pattern w n Another way to forulate the cuttng stoc proble: Slde 13 Slde 14 Slde 15 Slde 16 nze 6 j=1 x j x 1 x 2 x 3 x 4 x 5 x 6 RHS 4x 1 + 2x 2 + 2x 3 + 1x 4 + 0x 5 + 0x x 1 + 1x 2 + 0x 3 + 2x 4 + 1x 5 + 0x x 1 + 0x 2 + 1x 3 + 0x 4 + 1x 5 + 2x x j = nuber of rolls to be cut usng pattern j Coputatonal Issues Lnear relaxaton: (LP) n n j=1 x j n j=1 a j x j n, =1,..., x j 0, j = 1,...,n Slde 17 6

8 LP soluton provdes a lower bound to IP. How to solve (LP)? Feasble patterns: all nonnegatve nteger vectors (z 1,...,z ) satsfyng w z W Slde 18 Not all feasble patterns are needed n the above forulaton. Issue: In the constrant atrx A, the nuber of decson varables (.e., nuber of feasble patterns) s large! 5 Cuttng Stoc Proble 5.1 Colun Generaton Algorth 1. Start wth a basc feasble soluton B. For exaple, use the sple pattern to cut a roll nto W/w rolls of wdth w. (The bass atrx s a dagonal atrx.) 2. For any pattern j, reduced cost s 1 π a j, where (π 1,...,π ) (=c B B 1 ) s the splex ultplers vector assocated wth the current bass. 3. Identfy a pattern wth negatve reduced cost, or prove that none exsts. Update bass and repeat. For nstance, we ay want to fnd the colun wth ost negatve reduced cost: Z (π) = n 1 π x subject to w x W, x ntegral. If Z (π) 0, all coluns have negatve reduced cost! Otherwse, the soluton gves rse to a colun wth negatve reduced cost! Identfyng Coluns n 1 π x subject to w x W, x ntegral. s equvalent to solvng Z (π) = ax π x subject to w x W, x ntegral. Slde 19 Slde 20 Slde 21 7

9 5.1.3 Knapsac proble Z (π) =ax πx subject to wx W, x ntegral. Slde 22 The colun generaton ethod depends crtcally on how fast we can solve the napsac proble. How dffcult s t to solve the napssac proble? Coputatonal Result on rando nstances usng the MIP solver fro CPLEX: Slde 23 n CPU (s) 1, , , More specalzed algorth can be used to solve the Knapsac proble effcently n practce How Good s the bound? Nuber of tes = 5. Tested on several nstances: Optal Col Gen (LP) Slde 24 Round Up Conjecture: Unfortunately, ths s not true: W = 273 w 1 =18 n 1 = 233 w 2 =91 n 2 = 310 w 3 =21 n 3 = 122 w 4 = 136 n 4 = 157 w 5 =51 n 5 = 120 Z IP Z LP? Slde 25 8

10 Z LP (CG) = Z IP = 230. N3 Note 3 Round Up Conjecture In 1985 the theoretcal result of Marcotte (The cuttng stoc proble and nteger roundng, Math. Prograng, 33 (1985), pp ) shed lght on the relatonshp between solutons of the LP relaxaton and the cuttng stoc proble tself. She proved that for soe practcal nstances of the proble a so-called round-up property s vald. It eans that to fnd the optal value of the cuttng stoc proble, t s suffcent just to solve the LP relaxaton and to round up the value of the objectve functon. Unfortunately, ths conjecture s not true for all nstances of the cuttng stoc proble. Feldhouse (The dualty gap n tr probles, SICUP-Bulletn No. 5, 1990) presents an exaple of the cuttng stoc proble wth a gap of Ths gves rse to the odfed round up conjecture Modfed Round Up Conjecture Z IP Z LP +1? Slde 26 Ths conjecture has not been answered. Can you dsprove t? Gettng Intergal Soluton The soluton obtaned fro solvng the colun generaton proble ay fractonal. How to obtan ntegral soluton? Slde 27 - round up the fractonal soluton. (e.g., change 18.3 to 19). - round down the fractonal soluton, and resolve the proble wth saller set of deand. - branch and bound to obtan the optal ntegral soluton Roundng Up Let x j be the (fractonal) LP soluton obtaned fro the colun generaton ethod. Slde 28 Let x j = x j. x j ntegral. j a,j x j n ples j a,j xj n. So feasble ntegral soluton. x j defned n ths way s a How good s ths heurstc? 9

11 Slde 29 W = 273 w 1 =18 n 1 = 233 w 2 =91 n 2 = 310 w 3 =21 n 3 = 122 w 4 = 136 n 4 = 157 w 5 =51 n 5 = 120 Z LP (CG) = Round-Up produces a soluton of 231 Round-Up Fractonal cut cut cut cut cut cut cut Slde 30 Dsadvantages of the round-up heurstc? 6 Colun Generaton 6.1 Dual Perspectve Lagrangan Relaxaton How would you use LR to solve the cuttng stoc proble? n K K =1 y =1 x n = 1, 2,...,, x w Wy = 1,...,K, y {0, 1}, x 0,x ntegral ( ) L(u) = n K =1 y + u n K =1 x x w Wy = 1,...,K, Slde 31 Whch constrants would you relax? Slde 32 Suppose you relax the frst class of constrants: y {0, 1}, x 0,x ntegral 10

12 where u 0 for all. Slde 33 where K L(u) = L (u)+ u n L (u) = n =1 y u x x w Wy y {0, 1} L (u) s the nu of the two values: zero (when y = 0), or 1-Z (when Slde 34 y =1), where Z s obtaned by solvng the napsac proble Z =ax x x w W x 0,x ntegral x 0,x ntegral The subproble n Lagragan Relaxaton reduces agan to a Knapsac proble! Proposton: N4 ax u 0 L(u) = Z LP (CG). The optal Lanagrangan ultplers s the LP dual ultplers to the constrants N j=1 a,j Kλ j n n the colun forulaton. Lagrangan relaxaton solves the dual of the colun forulaton! Slde 35 Note 4 Dervaton of proposton K L(u) = L (u)+ u n =1 The value L (u) does not depend on the ndex. where L(u) = KL(u)+ u n L(u) =n y u x x w Wy y {0, 1} x 0,x ntegral 11

13 Let z j =(a 1,j,...,a,j ), j = 1,...,N be the extree ponts of the polytope x w W, x 0, x ntegral. L(u) =n(0, 1 ax j=1,...,n a,j u )= n j=1,...,n n(0, 1 a,j u ) The Lagrangan Dual reduces to L(u) = KL(u)+ u n ax L(u) = ax n K n(0, 1 a,j u ) u 0 u 0 j=1,...,n + u n ) ( ax y y u n for the zero extree pont y K(1 u a,j )+ u n for the jth extree pont u 0 = 1,...,. Equvalently, ax y (λ 0 ) y u n 0 (λ j ) y + K( u a,j ) u n K u 0 = 1,...,. λ 0,λ j are the assocated dual varables. The dual of ths proble: N n j=1 Kλ j λ 0 + N j=1 λ j =1 N n (λ 0 + N j=1 λ j )+ j=1 a,j Kλ j 0. λ j 0 j = 1,...,N. Ths s just n ( ) N j=1 Kλ j 12

14 λ 0 + N j=1 λ j =1 N j=1 a,j Kλ j n λ j 0 j = 1,...,N. For K large, the constrant λ 0 + N j=1 λ j = 1 s redundant as long as there s a feasble soluton λ j wth N j=1 λ j 1 Let x j = Kλ j. We obtan the colun forulaton of the cuttng stoc proble! 6.2 Lagrangan Relaxaton Coparson The bounds obtaned by both ethods are dentcal. Whchethodsbetter? Colun Generaton Langrangean Relaxaton Pral, dual optal soluton Dual but not pral soluton (Pral) Bounds onotone (Dual) Bounds zg-zag Dual soluton zg-zag Dual soluton sutably selected LP solver needed Easy to pleent 6.3 Speedng Up Colun Selecton Prevent generaton { of redundant } coluns. Instead of n j c j π T N j,solve or n j:π T N j>0 π T, N j { } c j π T N j n j 1 T. N j { } { } n j c j π T c N j versus n j π T N j j 1 T N j c j Slde 36 Slde 37 Slde 38 Round-Up Fractonal cut cut cut cut cut cut cut

15 Mantan a colun pool. Chec for coluns wth negatve reduced cost n the colun pool before solvng the prcng subproble. Replensh the colun pool everyte you solve a prcng subproble Dual Selecton Restrct doan of dual prces Slde 39 Use propertes of optal dual prces to restrct the doan. In the cuttng stoc proble, suppose the orders are raned such that w 1 < w 2 <... < w, then t s easy to see that the dual prces satsfy: π 1 π 2... π. Translatng nto the pral, ths s equvalent to addng the followng coluns wth zero cost to the pral proble: , 0, Colun Generaton 7.1 Applcatons Exaples Other applcaton of colun generaton: Slde 40 Vehcle Routng wth te wndow or other types of constrant: Aset of custoers. Each custoer ust be served wthn certan te wndow. Fnd a set of routes to serve all the custoers, so that each custoer wll be vsted by a vehcle wthn the stpulated te wndow. Here each colun ay represent a feasble trp. One lely objectve s to fnd nal nuber of vehcles to cover all the custoers. Slde 41 14

16 Colun generaton phase now reduces to the followng: Gven a proft π for each deand pont, fnd a route that satsfes: Slde 42 N5 feasblty constrants (eet the te wndow constrants); total profts accrued by servng the deand ponts on the route s axu - a varant of the TSP proble. Note 5 Papers J. Desrosers, Y. Duas,F. Sous & M. Soloon. Te Constraned Routng and Schedulng, Handboos n OR & MS, 8 (1995) G. Desaulners et al. A Unfed Fraewor for Deternstc Vehcle Routng and Crew Schedulng Probles T. Cranc & G. Laporte (eds) Fleet Manageent & Logstcs (1998). 8 Conclusons Colun Generaton has been successfully used to solve any large scale nteger prograng proble arsng n the ndustry. Slde 43 Able to handle large scale odel that standard coercal MIP solver cannot handle. Ablty to solve the prcng subproble effcently s ey to the approach Connecton between Colun generaton and Lagragan Relaxaton Non-lneartes occurng n practcal probles taen care of n the subproble (next lecture) 15