CG2-150型仿型切割機(jī)設(shè)計(jì)【說明書+CAD】

CG2-150型仿型切割機(jī)設(shè)計(jì)【說明書+CAD】,說明書+CAD,CG2-150型仿型切割機(jī)設(shè)計(jì)【說明書+CAD】,cg2,型仿型,切割機(jī),設(shè)計(jì),說明書,仿單,cad 目錄1 前言2 課題簡(jiǎn)介3 機(jī)架的設(shè)計(jì)4 割炬的設(shè)計(jì)5 基臂的鑄造工藝6 參考文獻(xiàn)7 體會(huì)畢業(yè)設(shè)計(jì)的目的畢業(yè)設(shè)計(jì)是學(xué)生完成教學(xué)計(jì)劃的最后一個(gè)極為重要的實(shí)踐性環(huán)節(jié)，是使學(xué)生綜合運(yùn)用所學(xué)過的基本理論，基礎(chǔ)知識(shí)與基本技能去解決專業(yè)范圍內(nèi)的工程技術(shù)問題而進(jìn)行的一次基本訓(xùn)練，這對(duì)學(xué)生即將從事有關(guān)的技術(shù)工作和未來事業(yè)的開拓有一定的意義。設(shè)計(jì)說明的概述-機(jī)械設(shè)計(jì)的任務(wù)是從社會(huì)需求出發(fā)；創(chuàng)造性的設(shè)計(jì)出具有特定功能的新機(jī)械或改進(jìn)原有機(jī)械設(shè)計(jì)的性能，以滿足人們?nèi)找嬖龃蟮纳詈蜕a(chǎn)的需要；機(jī)械設(shè)計(jì)使機(jī)械產(chǎn)品開發(fā)和技術(shù)改造的關(guān)鍵環(huán)節(jié)，使機(jī)械產(chǎn)品的第一步，機(jī)械的功能決定于設(shè)計(jì)機(jī)械的質(zhì)量；性能和成本；也只要是在設(shè)計(jì)階段決定的；而制造過程的本質(zhì)就在于實(shí)現(xiàn)設(shè)計(jì)時(shí)所規(guī)定的質(zhì)量。本人的畢業(yè)設(shè)計(jì)的課程是CG2-150型仿型切割機(jī)，是一種靠模生產(chǎn)的高效率半自動(dòng)氣割低碳合金鋼扳的機(jī)器，由于采用靠模仿型；可以很方便的切割出各種形狀，適用于批量生產(chǎn)。本機(jī)體積較小，自重輕，切割范圍大，特別適合于野外工作；它廣泛地被大中小型工廠使用。CG2-150型切割機(jī)的最顯著的特點(diǎn)是每分鐘能割完厚度為50-750mm地切割面，在相同工作量下與使用氣割 相比較，縮短工時(shí)近2倍，并且克服了以前用氣割下后鋼板，鋼管時(shí)地切割面積步光滑，尺寸誤差較大，而且有時(shí)因氧氣回火出現(xiàn)傷人現(xiàn)象等弊端，給電氣焊工帶來了很大地方便。本人只要側(cè)重于割炬（動(dòng)力部分），基壁，機(jī)架的設(shè)計(jì)。本機(jī)器有轉(zhuǎn)動(dòng)系統(tǒng)，樣板安裝在型臂前端地模板架上，磁滾輪面靠平樣板側(cè)面，表面吸力大于16N，磁滾輪由直流伺服電機(jī)帶動(dòng)三級(jí)減速機(jī)構(gòu)轉(zhuǎn)動(dòng)，三級(jí)減速由前級(jí)一對(duì)螺旋齒輪；中間一對(duì)平齒輪；后級(jí)一對(duì)蝸輪幅組成速度比為1/175，磁滾輪使用高級(jí)永磁合金直徑為10mm，當(dāng)磁滾輪沿樣板周邊移動(dòng)；帶動(dòng)主壁下面地割炬，割炬正確地割出與樣板相同形狀地工件。主軸固定在機(jī)座上，作為基壁地支點(diǎn)；設(shè)有平衡裝置，與主壁上安裝地磁滾輪，割炬保持平衡運(yùn)動(dòng)，底座下部裝有蝸桿可以在垂直方向調(diào)節(jié)割炬地升降位置，型壁套裝在聯(lián)結(jié)管上端可以作360度地回轉(zhuǎn)，360度范圍內(nèi)可作垂直升降和水平位置地調(diào)正。技術(shù)參數(shù)名稱150型150A型氣割鋼板厚度范圍5-100mm5-100mm氣割最大圓周直徑600mm1800mm氣割最大正（長(zhǎng)方形）500*500mm,400*900mm,450-750mm1270*1270mm氣割最大長(zhǎng)度1200mm1750mm氣割速度50-750mm/min電動(dòng)機(jī)Z3-11型直流電動(dòng)機(jī)，功率25W，電壓110V，電流0.5A，轉(zhuǎn)速3000-4000轉(zhuǎn)/min輸入電源50HZ,220V外形尺寸與重量型號(hào)150型150A型機(jī)器外型尺寸190*335*80mm149*335*800mm凈重43kg52kg1 機(jī)架機(jī)架時(shí)各種機(jī)械地基本部件，它主要起支承作用，機(jī)械的其他部件一般固定在機(jī)架上，有些部件是在機(jī)架的導(dǎo)軌面上運(yùn)動(dòng)。機(jī)架起基準(zhǔn)作用，以保證各部件間正確的相對(duì)位置，并且使整個(gè)機(jī)器組成一個(gè)整體。在其他部件及工件本身的重量與工作過程中的載荷（包括各種沖擊力）作用下，機(jī)架要有足夠的強(qiáng)度，而且變形部超過允許值。此外還應(yīng)考慮機(jī)架的動(dòng)剛度，阻尼，熱變形性，尺寸穩(wěn)定性，疲勞強(qiáng)度等。對(duì)于移動(dòng)式機(jī)器的機(jī)架則要求重量輕；對(duì)于在寒冷、酷熱、潮濕等環(huán)境下工作的機(jī)架分別要求耐高溫、耐低溫、抗銹燭等性能機(jī)架在機(jī)器中的尺寸一般較大，它往往與機(jī)器的整體布局，機(jī)器的造型美觀，操作方便，加工工藝性好。機(jī)架的分類 為了便于比較，下面機(jī)架的幾種分類：按機(jī)型求分，可分為：臥式、立式。臥式則分：橫梁式機(jī)架、平板式機(jī)架。立式則分：?jiǎn)瘟⒅?、雙立柱、多立柱。按材料及制造方法分：金屬機(jī)架、非金屬機(jī)架。金屬機(jī)架分：鑄造機(jī)架、焊接機(jī)架、組合機(jī)架非金屬機(jī)架分：花崗巖機(jī)架、混凝土機(jī)架、塑料機(jī)架。本設(shè)備的機(jī)架屬于金屬機(jī)架中的鑄造機(jī)架，它采用了高強(qiáng)度的鋁錠合金精密壓鑄制成，重量輕，強(qiáng)度高。 選擇單立柱機(jī)架是因?yàn)椋簡(jiǎn)瘟⒅鶛C(jī)架結(jié)構(gòu)簡(jiǎn)單，操作零件所受彎矩大，受力不能太大，如圖： 1、主壁 2、基壁 3、立柱 4、底座力的基本類型機(jī)架上承受著多種多樣的力和和力矩（彎矩及轉(zhuǎn)矩），但大體上可以分為以下幾種類型：1、工藝力：這是最重要、最基本的力，即機(jī)器在完成相應(yīng)的主要工藝過程中所產(chǎn)生的力量，如各種切削機(jī)床中的切削力；各種扎制設(shè)備中的扎制力與扎制力矩，各種鍛壓設(shè)備中的鍛壓件塑性成型力等等。對(duì)于同一臺(tái)機(jī)器；由于完成的工藝不同，工藝力也相應(yīng)的有所變化。在同一道工序中，變形力隨工作行程的增加而上升，且在鍛件厚度越薄時(shí)；變形力上升越快。如圖所示：在許多機(jī)械中，工藝力隨眾多因素變化而變化。因此，在一般機(jī)械中，機(jī)架的受力分析按額定負(fù)載（公稱負(fù)載）來進(jìn)行。額定負(fù)載示該機(jī)械所允許的最大負(fù)荷。超過此負(fù)荷，過負(fù)荷保險(xiǎn)機(jī)構(gòu)應(yīng)起作用，以防事故發(fā)生。還應(yīng)注意，工藝力的作用位置或方式不同，機(jī)架的受力情況也會(huì)改變。2 預(yù)緊力對(duì)于預(yù)緊力機(jī)架，一直處于預(yù)緊力作用下；因此在受力分析時(shí)，除了工作載荷外，還要考慮預(yù)緊力。預(yù)緊力與工作載荷不是簡(jiǎn)單的疊加關(guān)系，即在工作載荷（軸向拉力）作用下，預(yù)緊力會(huì)響應(yīng)減少。3 輔助工藝力這是指在主要工藝完成過程中所需的主要輔助動(dòng)作產(chǎn)生的力量。4 重力指工件和機(jī)器部件的重量。當(dāng)與工藝力相比；所占比例很小時(shí)，往往可以忽略不計(jì)。7 慣性力。機(jī)械的運(yùn)動(dòng)部分加速或減速時(shí)，均會(huì)產(chǎn)生相應(yīng)的慣性力，并作用于機(jī)架上。8 沖擊或振動(dòng)干擾力。外來的撞擊，旋轉(zhuǎn)部件的離心力，往復(fù)運(yùn)動(dòng)的慣性沖擊，等都會(huì)產(chǎn)生干擾力。 機(jī)架的材料及制造方法機(jī)架的材料蝸選擇碳素鋼，以下就對(duì)鋼材此材料進(jìn)行分析尺寸系數(shù)是由于實(shí)際構(gòu)件壁疲勞實(shí)驗(yàn)用的試件尺寸要大恩多，因而降低了疲勞極限；表面系數(shù)數(shù)時(shí)考慮表面加工質(zhì)量對(duì)疲勞強(qiáng)度的影響；兩者均可以在有關(guān)機(jī)械工程手冊(cè)中查到，對(duì)于鋼，值可參考下圖：表面系數(shù)1 拋光 2 磨削 3 精車 4 粗車 5 軋制加工表面 6 淡水腐蝕表面 7 海水腐蝕表面本設(shè)備的材料使用的時(shí)鋼材，鋼材的穩(wěn)定系數(shù)為1.5-3.0之間。所以我在確定機(jī)架最大高度時(shí)，必須要首先計(jì)算出穩(wěn)定安全數(shù)： 查表得立柱穩(wěn)定系數(shù)為=7.87 立柱穩(wěn)定安全系數(shù)公式=P0/P=0 (材料穩(wěn)定系數(shù))所以：=0 即7.87=1.5-3.0平臺(tái)臨界載荷P0=El/L2=100kgFi=Pl3/3EL=42mmFd=(1+ 1+2L/Fd )Fi=64mm平臺(tái)得最大高度 hmax=Pl/W=fd/fi=800mm 割炬 割炬部分由乙炔調(diào)節(jié)閥，預(yù)熱氧調(diào)節(jié)閥，切割氧調(diào)節(jié)閥，壓力開關(guān)，夾遲器，管子，火口螺母和割嘴組成， 當(dāng)氧氣和乙炔通過管子來到火口螺母而產(chǎn)生一定的壓力當(dāng)壓力一定程度時(shí)可調(diào)節(jié)壓力開關(guān)釋放混合氣體，點(diǎn)燃，產(chǎn)生氣鋒，切割鋼板?；?、主壁的氣割范圍：基壁的使用范圍為：400mm之內(nèi)可實(shí)現(xiàn)360度旋轉(zhuǎn)。主壁的使用范圍為：300mm之內(nèi)可以實(shí)現(xiàn)360度旋轉(zhuǎn)。割炬部分的運(yùn)動(dòng)：1、將220V的電壓源插頭插入電源插座。2、打開電源開關(guān)。3、本機(jī)可作逆向運(yùn)動(dòng)切換開關(guān)可以改變磁滾輪旋轉(zhuǎn)方向，必須在電動(dòng)機(jī)停止后進(jìn)行。4、打開自動(dòng)控制閥 打開乙烯閥1/3轉(zhuǎn) 點(diǎn)燃 打開預(yù)熱氧閥調(diào)整火焰，對(duì)鋼板預(yù)熱 打開切割氧閥 切割樣板鋼 根據(jù)需要調(diào)整速度。管道的設(shè)計(jì)：管子：在液壓氣閥及潤(rùn)滑油系統(tǒng)的管道中常用的管子有鋼管、銅管、膠管、尼龍管和塑料管等，由于本管子的設(shè)計(jì)不計(jì)壓力和彎曲情況，按價(jià)格來選用銅管。1、膠管：膠管的彎曲半徑不宜過小，一般不應(yīng)小于（表30）膠管內(nèi)徑46810131922膠管外徑13151719232932最小彎曲半徑90100110130190260320膠管與管接頭的聯(lián)結(jié)外應(yīng)留有一段直的部分，比較長(zhǎng)度不應(yīng)小于管外徑的兩倍。2、膠管的長(zhǎng)度應(yīng)考慮到膠管在通入氣體后長(zhǎng)度方向?qū)l(fā)生收束變形，一般收束量為管長(zhǎng)的3-4%。因此膠管安裝時(shí)應(yīng)避免處于拉緊狀態(tài)。3、膠管在安裝時(shí)應(yīng)保證不發(fā)生扭轉(zhuǎn)變形，為方便安裝可沿官長(zhǎng)涂以色紋，以便檢查。4、膠管的管接頭軸線，應(yīng)盡量放置在運(yùn)動(dòng)的平面內(nèi)，避免兩端互相運(yùn)動(dòng)時(shí)膠管受扭。5、膠管應(yīng)避免與機(jī)械上夾角部分相接觸和摩擦，以免管子損壞。管子內(nèi)徑的計(jì)算：管子的內(nèi)徑D，按流速選取d=1130 Q/V mm式中：Q-氣體流量 m3/s V-流速m/s薦用流速，一般選取V=2-3m/s d=1130 q/2 =30mm金屬管子壁厚=pd/2m式中工作壓力MPaD管子的內(nèi)徑mm許用壓力MPa，對(duì)于鋼管=b/s（0抗拉強(qiáng)度MPa，S安全系數(shù)，當(dāng)P7MPa時(shí)，S=8，當(dāng)P17.5MPa時(shí)，S=4。）對(duì)于銅管=Pd/2mm=p*30/2*25=1.5 （P根據(jù)壓力來選擇，見表。37-9-42接頭螺母的選擇由于接頭螺母是表準(zhǔn)件；根據(jù)螺紋直徑d=11.5mm 螺母M16*-6H GB3763.83所以根據(jù)表37-9-26選用割嘴的選擇等壓式割嘴采用整體式制造成形，以氧氣和乙烯在嘴內(nèi)混合為預(yù)熱火，具有壽命長(zhǎng)，安全可靠等優(yōu)點(diǎn)。30度配合面可與國(guó)內(nèi)外同類產(chǎn)品互換。由于割嘴式標(biāo)準(zhǔn)件，由下表選擇：割嘴號(hào)切割氧孔徑mm切割厚度mm切割速度mm/min氣體壓力Mpa氣體消耗量(參考）氧氣乙炔氧氣m/h乙炔l/h00.810204503800.20.3=0.31.31.83401120304003200.250.35=0.32.5347021.230503502800.250.353447031.450703002400.30.44.5665041.670902602000.30.4=0.045.5765051.8901202101700.40.58.510.5650表中數(shù)據(jù)切割條件1 氧氣純度不低于2 被切割鋼材含碳量=0.45。3 切割方式為垂直切割4 氧氣壓力是指割炬前的切割氧壓力閥門的選用 閥門是管道系統(tǒng)的重要組成部件，用于啟閉，節(jié)流和保證設(shè)備安全運(yùn)行。由于閥門式標(biāo)準(zhǔn)件，只要選用參考機(jī)械零件手冊(cè)選用。表37-9-3閥門的類型：1、閘閥：閘閥的關(guān)閉件主要式閘板；有楔形和平行兩種。2、截止閥：主要作用調(diào)節(jié)性比較好，流體以閥座下面向上流動(dòng)，所以安裝時(shí)應(yīng)順介質(zhì)流閥，不能裝反。3、止向閥：此閥又稱單向閥，主要有旋啟式與升降式兩種。4、旋塞閥：其關(guān)閉件為圓錐體塞子，繞閥件中心、旋轉(zhuǎn)來執(zhí)行開關(guān)的任務(wù)。5、球閥：其關(guān)閉件為球體。6、蝶閥：碟閥的閥座做在閥體上，圓盤狀閥瓣固定在閥桿上，將其旋轉(zhuǎn)90度，即可執(zhí)行開啟關(guān)閉的作用。根據(jù)設(shè)備的情況，所以選擇旋塞閥。參考文獻(xiàn) 機(jī)械零件手冊(cè) 中冊(cè) 機(jī)械設(shè)計(jì)手冊(cè) 機(jī)械零部件手冊(cè)畢業(yè)設(shè)計(jì)指導(dǎo)書機(jī)械設(shè)計(jì)書 上下冊(cè)心得體會(huì)這次的畢業(yè)設(shè)計(jì)是大學(xué)3年中的最后一個(gè)環(huán)節(jié)，是對(duì)3年的學(xué)習(xí)生活中所學(xué)的知識(shí)一個(gè)匯總和概括，使我們每個(gè)人都能了解自己學(xué)道了什么，理解多少，會(huì)運(yùn)用多少，還有多少知識(shí)不了解，需要進(jìn)一步了解，加深理解，所以，在以后的工作中，繼續(xù)學(xué)習(xí)和加深。在此我非常感謝常小章老師在設(shè)計(jì)過程中對(duì)我的指導(dǎo)和幫助。 由于是第一次做畢業(yè)設(shè)計(jì)難免有所不足之處，還望老師諒解。 *Correspondingauthor.Tel.:#30-31-498-143;fax:#30-31-498-180. E-mail address: georgiadcperi.certh.gr(M.C.Georgiadis). Computersreceivedinrevisedform1September2000;accepted1October2000 Abstract Thispaperpresentsanewmathematicalprogrammingformulationfortheproblemofdeterminingthe optimalmannerinwhichseveralproductrollsofgivensizesaretobecutoutofrawrollsofoneormore standardtypes.Theobjectiveistoperformthistasksoastomaximizetheprottakingaccountofthe revenuefromthesales,thecostsoftheoriginalrolls,thecostsofchangingthecuttingpatternandthecostsof disposalofthetrim.Amixedintegerlinearprogramming(MILP)modelisproposedwhichissolvedto globaloptimalityusingstandardtechniques.Anumberofexampleproblems,includinganindustrialcase study,arepresentedtoillustratethee$ciencyandapplicabilityoftheproposedmodel. Scope and purpose One-dimensional cutting stock (trim loss) problems arise when production items must be physically dividedintopieceswithadiversityofsizesinonedimension(e.g.whenslittingmasterrollsofpaperinto narrower width rolls). Such problems occur when there are no economies of scale associated with the productionofthelargerraw(master)rolls.Ingeneral,theobjectivesinsolvingsuchproblemsareto5: p69 minimizetrimloss; p69 avoidproductionover-runsand/or; p69 avoidunnecessaryslittersetups. Theaboveproblemisparticularlyimportantinthepaperconvertingindustrywhenasetofpaperrollsneed tobecutfromrawpaperrolls.Sincethewidthofaproductisfullyindependentofthewidthoftherawpaper ahighlycombinatorialproblemarises.Ingeneral,thecuttingprocessalwaysproducesinevitabletrim-loss whichhastobeburnedorprocessedinsomewastetreatmentplant.Trim-lossproblemsinthepaperindustry have, in recent years, mainly been solved using heuristic rules. The practical problem formulationhas, therefore,inmostcasesbeenrestrictedbythefactthatthesolutionmethodsoughttobeabletohandlethe entireproblem.Consequently,onlyasuboptimalsolutiontotheoriginalproblemhasbeenobtainedand 0305-0548/02/$-seefrontmatter p7 2002ElsevierScienceLtd.Allrightsreserved. PII: S0305-0548(00)00102-7 veryoftenthisrathersignicanteconomicproblemhasbeenlefttoamanualstage.Thisworkpresents a novel algorithm for e$ciently determining optimal cutting patterns in the paper converting process. Amixed-integerlinearprogrammingmodelisproposedwhichissolvedtoglobaloptimalityusingavailable computertools.Anumberofexampleproblemsincludinganindustrialcasestudyarepresentedtoillustrate theapplicabilityoftheproposedalgorithm. p7 2002ElsevierScienceLtd.Allrightsreserved. Keywords: Integerprogramming;Optimization;Trim-lossproblems;Paperconvertingindustry 1. Introduction Animportantproblemwhichisfrequentlyencounteredinindustriessuchaspaperisrelatedwith themosteconomicmannerinwhichseveralproductrollofgivensizesaretobeproducedby cuttingoneormorewiderrawrollsavailableinoneormorestandardwidths.Thesolutionofthis probleminvolvesseveralinteractingdecisions: p69 Thenumberofproductrollsofeachsizetobeproduced. Thismaybeallowedtovarybetweengivenlowerandupperbounds.Theformernormallyre#ect the rmordersthatarecurrentlyoutstanding,whilethelattercorrespondtothemaximum capacityofthemarket.However,certaindiscountsmayhavetobeo!eredtosellsheetsoverand abovethequantitiesforwhichrmordersareavailable. p69 Thenumberofrawrollsofeachstandardwidthtobecut. Rollsmaybeavailableinoneormorestandardwidths,eachofadi!erentunitprice. p69 Thecuttingpatternforeachrawroll. Cuttingtakesplaceonamachineemployinganumberofknivesoperatinginparallelonarollof standardwidth.Whilethepositionoftheknivesmaybechangedfromonerolltothenext,such changesmayincurcertaincosts.Furthermore,theremaybecertaintechnologicallimitationson theknifepositionsthatmayberealizedbyanygivencuttingmachine. Theoptimalsolutionoftheaboveproblemisoftenassociatedwiththeminimizationofthe trima waste that is generally unavoidable since rolls of standard widths are used. However, trim-lossminimizationdoesnotnecessarilyimplyminimizationofthecostoftherawmaterials (rolls)beingusedespeciallyifseveralstandardrollsizesareavailable.Amoredirecteconomic criterionisthemaximizationoftheprotoftheoperationtakingaccountof: p69 therevenuefromproductrollssales,includingthee!ectsofanybulkdiscounts; p69 thecostoftherollsthatareactuallyused; p69 thecosts,ifany,ofchangingtheknifepositionsonthecuttingmachine; p69 thecostofdisposingoftrimwaste. Theaboveconstitutesahighlycombinatorialproblemanditisnotsurprisingthattraditionally itssolutionhasoftenbeencarriedoutmanuallybasedonhumanexpertise.Thesimpliedversion ofthisproblemissimilartothecuttingstockproblemknownintheoperation-researchliterature, whereanumberoforderedpiecesneedtobecuto!biggerstoredpiecesinthemosteconomic fashion.Inthe1960sandthe1970s,severalscienticarticleswerepublishedontheproblemof 1042 G. Schilling, M.C. Georgiadis/Computers that raw rolls of the type t that permits the smallest minimum 1044 G. Schilling, M.C. Georgiadis/Computers andthateachrawrollwillbeusedtoproduceproductrollsof asingletypeonly.Overall,thisleadstothefollowingupperboundonthenumberofrawrollsthat mayberequired: Jp13p0p24 p39 p9 p71p14p16 Np13p0p24 p71 p87min p82 Bp13p9p14 p82 /B p71 p88 . (1) We can also calculate a lower bound Jp13p9p14 on the minimum number of raw rolls that are necessarytosatisfytheminimumdemandfortheexistingorders.Wedothisbyassumingthatrolls of the type t allowing the maximum possible engagement Bp13p0p24 p82 are used, and that no trim is produced.However,wemustalsotakeaccountofpossiblelimitationsonthenumberofavailable knives. Overall, this leads to the following lower bound on the number of rolls that may berequired: Jp13p9p14max p7 p9p39 p71p14p16 Np13p9p14 p71 B p71 max p82 Bp13p0p24 p82 , p9p39 p71p14p16 Np13p9p14 p71 max p82 Np13p0p24 p82 p8 . (2) 3. Mathematical formulation Theaimofthemathematicalformulationistodeterminethetypetofeachrawrolljtobecut andthenumberofproductrollsofeachtype i tobeproducedfromit. 3.1. Key variables Thefollowingintegervariablesareintroduced: n p71p72 :numberofproductrollsoftype i tobecutoutofrawroll j afii9773 p71 : numberofproductrollsoftype i producedoverandabovetheminimumnumberordered. Wenotethat n p71p72 cannotexceed: p69 themaximumnumber Np13p0p24 p71 ofproductrollsoftype i thatcanbesold; p69 themaximumnumberofproductrollsofwidth B p71 thatcanbeaccommodatedwithinamax- imumengagementBp13p0p24 p82 forarawrolloftype t; p69 themaximumnumber Np13p0p24 p82 ofknivesthatcanbeappliedtoarawrolloftype t. Thisleadstothefollowingboundsfor n p71p72 : 0)n p71p72 )min p1 Np13p0p24 p71 ,max p16p87p82p87p50 Bp13p0p24 p82 B p71 ,max p16p87p82p87p50 Np13p0p24 p82 p2 i1, 2 ,I, j1, 2 ,Jp13p0p24. (3) Also 0)afii9773 p71 )Np13p0p24 p71 !Np13p9p14 p71 , i1, 2 ,I. (4) G. Schilling, M.C. Georgiadis/Computers thissimplyimpliesthatitisnot necessarytocutroll j. Furthermore,thelimitedavailabilityofrawrollsofagiventypetmaybeexpressedintermsof theconstraint p40 p13p0p24 p9 p72p14p16 y p82p72 )JH p82 , t1, 2 ,. (6) 3.3. Cutting constraints Weneedtoensurethat,ifarolljistobecut,thenthelimitationsontheminimumandmaximum engagementareobserved.Thisisachievedviatheconstraints p50 p9 p82p14p16 Bp13p9p14 p82 y p82p72 ) p39 p9 p71p14p16 B p71 n p71p72 ) p50 p9 p82p14p16 Bp13p0p24 p82 y p82p72 , j1, 2 ,Jp13p0p24. (7) Wenotethatthequantityp9p39 p71p14p16 B p71 n p71p72 representsthetotalwidthofallproductrollstobecutoutof rawroll j.Ify p82p72 1forsomerolltype t,thenconstraint(7)ensuresthat Bp13p9p14 p82 ) p39 p9 p71p14p16 B p71 n p71p72 )Bp13p0p24 p82 . 1046 G. Schilling, M.C. Georgiadis/Computers onceagain,atmostoneofthetermsinthissummation canbenon-zero(cf.constraints(5a)and(5b).Thelatterquantityisgivenbyp9p39 p71p14p16 B p71 n p71p72 .Overall, trimdisposalresultsinthefollowingcostterm cp3p9p19p16 p40 p13p0p24 p9 p72p14p16 p1 p50 p9 p82p14p16 Bp18p15p12p12 p82 y p82p72 ! p39 p9 p71p14p16 B p71 n p71p72 p2 . Theabovetermscannowbecollectedinthefollowingobjectivefunction: max p3 p39 p9 p71p14p16 (p p71 Np13p9p14 p71 #afii9773 p71 (p p71 !cp3p9p19p2 p71 )! p40 p13p0p24 p9 p72p14p16 p9 p82p14p16 cp18p15p12p12 p82 y p82p72 !cp2p8p0p14p6p4 p40 p13p0p24 p9 p72p14p17 z p72 !cp3p9p19p16 p40 p13p0p24 p9 p72p14p16 p1 p50 p9 p82p14p16 Bp18p15p12p12 p82 y p82p72 ! p39 p9 p71p14p16 B p71 n p71p72 p2p4 . (11) Notethatthersttermintheaboveobjectivefunction(i.e.p9p39 p71p14p16 p p71 Np13p9p14 p71 )isactuallyaconstantand doesnota!ecttheoptimalsolutionobtained. 3.7. Degeneracy reduction and constraint tightening Ingeneral,thebasicformulationpresentedaboveishighlydegenerate:givenanyfeasiblepoint, onecangeneratemanyotherssimplybyformingallpossibleorderingoftherollsselectedtobecut. Moreover,providedallrawrollsofthesametypearecutconsecutively,allthesefeasiblepointswill correspondtoexactlythesamevalueoftheobjectivefunction. Theabovepropertymayhaveadversee!ectsonthee$ciencyofthesearchprocedure.Therefore, in order to reduce the solution degeneracy without any loss of optimality, we introduce the followingorderingconstraints: p39 p9 p71p14p16 n p71p11p72p92p16 * p39 p9 p71p14p16 n p71p72 , j2, 2 ,Jp13p0p24. (12) Thisensuresthatthetotalnumberofproductrollscutoutofrawrollj!1isneverlowerthanthe correspondingnumberforroll j;allcompletelyunusedrawrollsareleftlastinthisordering. 1048 G. Schilling, M.C. Georgiadis/Computers Bp13p0p24 p82 !Bp13p9p14 p82 ,whichresultsinone lessconstraintforeachroll j. 4. Example problems In this section, we consider four example problems of increasing complexity in order to investigatethecomputationalbehaviorofourformulation.Furthermoreanindustrialcasestudyis alsopresented.Inallcases,weassumethatthemaximumrawrollengagementBp13p0p24 p82 isequaltothe correspondingrollwidthBp18p15p12p12 p82 .TheGAMS/CPLEXvs6.0solverhasbeenusedforthesolution15 andallcomputationswerecarriedoutonaAlphaServer4100.Anintegralitygapof0.1%was assumedforthesolutionofallproblems. 4.1. Example 1 OurrstexampleisbasedonthatgivenbyHarjunkoski9.Sometranslationofthevarious costcoe$cientswasnecessarytoaccountforslightdi!erencesintheobjectivefunctionsusedby thetwoformulations.Alsonotethattheobjectiveusedbythoseauthorsistheminimizationofcost asopposedtothemaximizationofprot;therefore,thesignoftheirobjectivefunctionisopposite tothatofours. G. Schilling, M.C. Georgiadis/Computers thus,withthegiveneconomicdatatheoperationincursaloss. Theoptimalsolution(withinamarginofoptimalityof0.1%)isfoundwithinlessthan1CPUs atnode49ofthebranch-and-boundalgorithmusingabreadthrstsearchstrategy.Itmustbe notedthattheintegralitygapofourformulationiscomparabletothatforoneoftheformulations presentedbyHarjunkoski9despitethefactthatitdoesnotemployanyapriorienumerationof thecuttingpatterns.Ourformulationalsoexaminesasmallnumberofnodesinordertodetectthe optimalpoint(Table3). 1050 G. Schilling, M.C. Georgiadis/Computers 9:84959. 2 GilmorePC,GomoryRE.Alinearprogrammingapproachtothecuttingstockproblem*partII.Operations Research1963;11:86388. 3 Hinxman AI. The trim-loss and assortment problems. a survey. European Journal of Operational Research 1980;5:818. G. Schilling, M.C. Georgiadis/Computers 44:17584. 5 Sweeney PE, Haessler RW. One-dimensional cutting stock decisions for rolls with multiple quality grades. EuropeanJournalofOperationalResearch1990;44:22431. 6 FerreiraJS,NevesMA,FonsecaeCastroP.Atwo-phaserollcuttingproblem.EuropeanJournalofOperational Research1990;44:18596. 7 GradisarM,JesenkoJ,ResinovicG.Optimizationofrollcuttinginclothingindustry.Computers10:S94553. 8 GradisarM,KljajicM,ResinovicG,JesenkoJ.Asequentialheuristicprocedureforone-dimensionalcutting. EuropeanJournalofOperationalResearch1999;114:55768. 9 HarjunkoskiI,WesterlundT,IsakssonJ,SkrifvarsH.Di!erentformulationsforsolvingtrimlossproblemsin apaper-convertingmillwithilp.ComputersandChemicalEngineering1996;20:S1216. 10 HarjunkoskiI,WesterlundT,PornR.Di!erenttransformationsforsolvingnon-convextrim-lossproblemsby minlp.EuropeanJournalofOperationalResearch1998;105:594603. 11 Westerlund T, Isaksson J, Harjunkoski I. Solving a two-dimensional trim-loss problem with milp. European JournalofOperationalResearch1998;104:57281. 12 WesterlundT,HarjunkoskiI,IsakssonJ.Solvingaproductionoptimisationprobleminapaper-convertingmill withmilp.ComputersandChemicalEngineering1998;22:56370. 13 WesterlundT,IsakssonJ.Somee$cientformulationsforthesimultaneoussolutionoftrim-lossandscheduling problems in the paper-converting industry. Transactions of the Institution of Chemical Engineering Part A1998;76:67784. 14 HarjunkoskiI,WesterlundT,PornR,SkrifvarsH.Numericalandenvironmentalconsiderationsonacomplex industrial mixed integer non-linear programming (minlp) problem. Computers and Chemical Engineering 1999;23:154561. 15 BrookeA,KendrickD,MeerausA,RamanR.GAMS.AUsersGuide.GAMSDevelopmentCorporation,1998. Michael C. Georgiadis, Ph.D., is a full time researcher at Chemical Process Engineering Research Institute in Thessaloniki,Greece.HeearnedhisDiplomaofChemicalEngineeringfromAristotleUniversityofThessalonikiand receivedhisM.Sc.andPh.D.inProcessSystemsEngineeringfromImperialCollege,London.Hisresearchinterestsliein theareasofmixedintegerandcomputer-aidedoptimizationtechniquesfor#exiblemanufacturinginprocessindustries, production scheduling, planning and dynamic modeling and simulation. He is the author of over 15 journal and conferencepublications. GordianSchilling,Ph.D.,isaprocessdevelopmentchemicalengineeratCibaSpecialtyChemicalsInc.inSwitzerland. HeearnedhisDiplomaofEngineeringfromETH,ZurichandreceivedhisPh.D.inProcessSystemsEngineeringfrom ImperialCollege,London. 1058 G. Schilling, M.C. Georgiadis/Computers & Operations Research 29 (2002) 10411058