Pattern Generation for Two Dimensional. cutting stock problem.

Transcription

1 Internton Journ of Memtcs Trends nd Technoogy- Voume3 Issue- Pttern Generton for Two Dmenson Cuttng Stock Probem W N P Rodrgo, W B Dundseker nd A A I Perer 3 Deprtment of Memtcs, Fty of Scence, Unversty of Perdeny Sr Lnk Abstrct Seecton of fesbe ttng ptterns n order to mnmze e rw mter wstge whch s known s ttng stock probem hs become key fctor of e success n tody s compettve mnufcturng ndustres In s pper, sovng two-dmenson ttng stock probem s dsssed Our study s restrcted to rw mter (mn sheet) n rectngur shpe, nd ttng tems re so consdered s rectngur shpe w known dmensons The Brnch nd Bound pproch n sovng nteger progrmmng probems s used to sove e probem Keywords Two-Dmenson ttng stock probem, Cuttng ptterns, Brnch nd Bound gorm I INTRODUCTION Mnmzng wstge s key fctor n mprovng productvty of mnufcturng pnt Wstge cn ocr n mny wys nd ttng stock probem cn be descrbed under e rw mter wstge An optmum ttng stock probem cn be defned s ttng mn sheet nto smer peces whe mnmzng tot wstge of e rw mter or mxmzng over proft obtned by ttng smer peces from e mn sheet Mny reserchers hve worked on e ttng stock probem nd deveoped dfferent gorms to sove e probem Among em, Hf et () nd Coromoto et (7) hve mde n pproch to t rge rectngur stock of known dmensons to n types of smer rectnges of known dmensons Hf hs mde ssumptons t peces hve fxed orentton (e pece of eng nd wd w s dfferent from pece of eng w nd wd, for w, nd pped ts re of guotne type ( t from one edge of e rectnge to e opposte edge whch s pre to e two remnng edges) Hf deveoped memtc mode to mxmze over proft by ttng smer rectngur peces from e rge rectngur stock Aso, Coromoto hs used Pre Agorm nd Sequent Agorm to sove e memtc mode whch mxmzes e tot proft nrred by ttng n number of rectngur tems from rge rectngur mn sheet Coromoto hs mde n observton t ttng ptterns cn be obtned by mens of horzont nd vertc buds of met-rectnges nd used Vswnn nd Bgch Agorm to produce best horzont nd vertc buds In ddton to bove two studes, mny reserchers hve ntroduced dfferent pproches to mxmze e utzton re of e mn sheet or to mnmze e wste re of e mn sheet, nd hve ssumed bo mn sheet nd smer peces re n rectngur shpe w known dmensons 3,4,5 There re dfferent rrngements to t requred peces from e exstng rw mter to mxmze e used re Ech rrngement s defned s ttng pttern In s study, modfed Brnch nd Bound Agorm s presented nd computer progrm usng Mtb softwre pckge s deveoped to generte fesbe ttng ptterns for twodmenson ttng stock probem ISSN: Pge 54

2 Internton Journ of Memtcs Trends nd Technoogy- Voume3 Issue- II MATERIALS AND METHODS Pror to fndng mnmum rw mter wstge of twodmenson ttng stock probem, rectngur shped mn sheet w known dmensons nd requred tems re seected Accordng to e seecton, memtc mode to mnmze e wstge s formuted s foows: Foowng nottons re ntroduced to descrbe e mode: m = Number of tems, n = Number of ptterns, p j = Number of ocrrences of e j pttern, tem n e x j = Number of mn sheets beng t ccordng to e j pttern, m Here, p j A L W for j,,, n, where A, L nd W re e re of e mn sheet respectvey A Modfed Brnch nd Bound Agorm tem, eng nd wd of e Step : Arrnge requred engs,, =,,, m decresng order, e > > > m, where m = number of tems Arrnge requred wds, w, =,,, m ccordng to e correspondng eng, =,,, m Step : For =,,, m nd j = do Steps 3 to 5 n c j = Cuttng oss for ech d = Demnd for e Memtc Mode: tem j pttern, Step 3: Set L L - z j z z j (), Mnmze z Subject to n j n j c j x j Tot Cuttng Loss p j x j d for,,, m Demnd Constrnts where L s e eng of e mn sheet Here, j s e number of peces of e tem n e j pttern ong e eng of e mn sheet nd y nteger ess n or equ to y s e gretest x j, p j nd nteger for, j, Step 4: If j >, en set b W j w () The number of ocrrences of e pece n e j pttern (p j ) needs to be determned to fnd e optmum souton (mnmum-wste rrngement) for e gven memtc mode Therefore, modfed Brnch nd Bound Agorm s used to generte fesbe ttng ptterns ese set b j =, where W s e wd of e mn sheet Here, b j s e number of peces of e e tem n j pttern ong e wd of e mn sheet Step 5: Set p b, j j j ISSN: Pge 55

3 Internton Journ of Memtcs Trends nd Technoogy- Voume3 Issue- Step 6: Cuttng Loss where p j s e number of peces of e tem n e j pttern n e mn sheet () Cut oss ong e eng of e mn sheet: m L j W For,,, m m If L j w nd W, en (Consderng 9 o rotton for e gven ttng tems) m L j set A j w Bj W,, oerwse pj pj Aj Bj f Aj where, A j nd B j re e number of peces of e tem n e j pttern ong e eng nd wd of e c u rectnge respectvey nd C u nd C v re e tot t oss re ong e eng nd wd of e mn sheet respectvey () Cut oss ong e wd of e mn sheet: c v Here, k If where k pttern j j j If b j set w k k j j W b j w s e remnng wd of For z set,, en nd k w, en j z j Azj z kj w, f z Bzj, oerwse j z ech tem n ech Azj ese set A j = pzj pzj Azj Bzj If Aj set B j = P j = P j, en m Cu L m Cv L j Aj w j W Bj m ese Cu L j W, Bj ese set A j = B j = P j = P j If Azj, en set Cu j Azj z Bzj wz Cv j k j Bzjz ese Cv j kj, ISSN: Pge 56

4 Internton Journ of Memtcs Trends nd Technoogy- Voume3 Issue- where, A zj nd B zj re e number of peces of e tem n e j pttern ong e eng nd wd of e c v rectnge respectvey For z r set zj z j bzj bz j Step 7: Set r = m Whe r >, do Step 8 Step 8: Whe rj > set j = j + nd do Step 9 For z = r +,, m cte z j nd b z j usng Equtons () nd () Go to Step 5 Step : Set r = r Step 9: If, en generte new pttern rj brj ccordng to e foowng condtons: Step : STOP B Iustrtve Exmpe For z,,, r set zj z j bzj bz j For z r set z j z j f z j, en set bz j W wz Foowng exmpe w ustrte how to generte fesbe ttng ptterns by mnmzng tot ttng wste: A foor te mnufcturng pnt uses rectngur shped mrbe sheets of eng 3 mm nd wd 4 mm s rw mter to t tes ccordng to e gven specfctons The compny hs receved n order for broom tes ccordng to e dmensons gven n Tbe I: ese set b z j For z = r +,, m cte z j nd b z j usng Equtons () nd () Go to Step 5 TABLE I Requred tem dmensons nd demnd Item No Requred Dmensons (mm) Demnd ese generte new pttern ccordng to e foowng condtons: For z,,, r set zj z j bzj bz j Beow ustrtes e meod descrbed n e reserch pper to t e mn sheet ccordng to e dmensons so t e tot rw mter wstge s mnmzed ISSN: Pge 57

5 Internton Journ of Memtcs Trends nd Technoogy- Voume3 Issue- III RESULT, en b Modfed Brnch nd Bound Agorm s pped to e bove exmpe to generte fesbe ttng ptterns s gven beow: Step : For =,, 3, 4, 5, 6 engs =, 6, 4, 4,, wds w = 6, 8, 8, 6, 8, 6 Leng (L) nd wd (W) of e rw mter re 3 mm nd 4 mm respectvey Dmensons of ech tem: Item no () Leng (mm) Wd w (mm) Step : For =,,, 6 nd j = do Steps 3 to 5 Step 3: Set L L , en b3 4, en b4 5, en b5 6, en b6 Step 5: Set Pttern Step 6: Cut oss () Cuttng oss ong e eng of e mn sheet: L ,, mm For =, set A j = B j = (Condtons re not stsfed gven n Step 6 prt ()) W L 3 3 L L L Step 4:, en set b W w For = 3, dmensons of Item 3 re of eng ( 3 ) 4 mm nd wd (w 3 ) 8 mm nd condtons re stsfed gven n Step 6 prt () set 8 A3 w3 4 B 3 3 A 3 >, en set 8 A w B 4 mm C u C v B 8 mm 3 3 ISSN: Pge 58

6 Internton Journ of Memtcs Trends nd Technoogy- Voume3 Issue- Pttern Pttern Step 8: =, en go to Step Step : Set r = = > () Cuttng oss ong e wd of e mn sheet: c v 4 b w 44, mm For z,, 6 set A j = C v Pttern = B j = (Condtons re not stsfed gven n Step 6 prt ()) 44, nd mm tot ttng oss = 44, mm Step 7: Set r = 6 = 5 > Step 8: 5 =, en go to Step Step 8: >, en set j = j + = nd go to Step 9 Step 9: < b, en generte new pttern j (= ) ccordng to e foowng condtons: set = = b = b = L b L 3 3 b 3 L b L b L b 6 Step : Set r = 5 = 4 > Step 8: 4 =, en go to Step Step : Set r = 4 = 3 > Step 8: 3 =, en go to Step Step : Set r = 3 = > Step 5: Set Pttern = ISSN: Pge 59

7 Internton Journ of Memtcs Trends nd Technoogy- Voume3 Issue- Step 6: Cut oss () Cuttng oss ong e eng of e mn sheet: L ,, mm For =, set A j = W c v 4 b w 8, 76, mm For z =, dmensons of Item re of eng ( ) 6 mm nd wd (w ) 8 mm nd condtons re stsfed gven n Step 6 prt () set A 8 B w B j = A >, en For = 3, dmensons of Item 3 re of eng ( 3 ) 4 mm nd wd (w 3 ) 8 mm nd condtons re stsfed gven n Step 6 prt () w set A A 3 >, en 4 B 3 3 Set C u 8 A3 w3 B3 3 4 mm C v 8 4 B33 8 mm Pttern C u A w B , mm, C v 8 B mm Pttern Pttern Pttern Pttern nd tot ttng oss = 48, mm () Cuttng oss ong e wd of e mn sheet: ISSN: Pge 6

8 Internton Journ of Memtcs Trends nd Technoogy- Voume3 Issue- The gorm proceeds n e sme mnner to generte e ttng ptterns shown n Tbe II for e 3 mm 4 mm stndrd dmenson The Tbe II exhbts e generted ttng ptterns for optmum wste: TABLE II Generted ttng ptterns Requred Dmensons Cuttng ptterns mm 6 8 mm 4 8 mm 4 6 mm mm 6 mm Cut Loss ( x 4 ) mm Requred Dmensons Cuttng ptterns mm 6 8 mm 4 8 mm 4 6 mm 8 mm 3 6 mm 4 6 Cut Loss ( x 4 ) mm There re 5 fesbe ttng ptterns vbe to t rw mter w e dmensons 3 mm 4 mm nto requred rectngur shped tems The memtc mode s deveoped to desgn generted ttng ptterns so t wste (t oss) w be mnmzed nd e optmum souton to e mode s gven n Tbe III: Requred Dmensons TABLE III Optmum Souton Optm Ptterns 3 Demnd 6 mm 6 8 mm 4 8 mm 4 6 mm 8 mm 6 mm # of sheets from ISSN: Pge 6

9 Internton Journ of Memtcs Trends nd Technoogy- Voume3 Issue- ech pttern Z mn = 94, cm (Tot t oss = 94, mm ) [4] Hssn Jvnshr, Shghyegh Reze, Sed Shekhzdeh Njr nd Gng SS, Two Dmenson Cuttng Stock Mngement n Fbrc Industres nd Optmzng e Lrge Object s Leng, IJRRAS, August [5] Chen Chen, Chu Km Hut, Optmum Shpyrd Stee Pte Cuttng Pn Bsed On Genetc Agorm, EPPM, Sngpore, - September IV CONCLUSION In s study, ttng stock probem s formuted s memtc mode bsed on e concept of ttng ptterns As gven n Tbe II, twenty fve ttng ptterns re generted nd ony two ttng ptterns re seected s gven n Tbe III to t e mn sheet ccordng to e requrements In s cse study, e pnt ssumes t e extr peces from ech tem s wstge Aso, dmensons of ttng tems re rge nd e tot t oss cn be decresed f ere re smer rectngur shped ttng tems Twenty ree fesbe ttng ptterns cn be generted by ppyng 9 o rotton to e mn sheet In s cse study, t s better to use ttng ptterns wout permttng rotton to e mn sheet bese Item ( mm 6 mm) nd Item (6 mm 8 mm) cnnot be t, f 9 rotton s pped to e mn sheet REFERENCES [] Vn-Dt Cung, Mhnd Hf, Bertrnd Le Cun, Constrned twodmenson ttng Stock probems, best-frst brnch-nd-bound gorm, Internton Trnsctons In Operton Reserch, 7 () [] Coromoto Leon, Gr Mrnd, Csno Rodrguez, & Cros Segur, D Cuttng Stock Probem: A New Pre Agorm nd Bounds, (wwwsprngernkcom/ndex/96t3v338q7u64pdf), 5 November [3] Sd MA Sumn, Pttern genertng procedure for e ttng stock probem, Internton Journ of Producton Economcs 74 () 93-3 ISSN: Pge 6