米粉切割機(jī)的設(shè)計
米粉切割機(jī)的設(shè)計,米粉切割機(jī)的設(shè)計,米粉,切割機(jī),設(shè)計
米粉切割機(jī)米粉切割機(jī)一、設(shè)計目的及意義n 這次畢業(yè)設(shè)計目的是為了鞏固、擴(kuò)大和學(xué)會綜合地使用所學(xué)過各門課程知識。n 該米粉切割機(jī)的主要作用是把已經(jīng)成型的條狀米粉按照要求切割成一定的長度。不僅要保證了加工的效率，也要盡量減少米粉的斷頭損失，同時又能按需求能切割不同長度的米粉。二、米粉切割機(jī)的米粉切割機(jī)的總體設(shè)計方案總體設(shè)計方案 一般米粉切割機(jī)都是一個很簡單的系統(tǒng)。我們設(shè)計的米粉切割機(jī)由電動機(jī)提供動力，通過V帶和帶輪傳動給主軸，然后通過安裝在主軸上的鋸片切斷米粉。這樣就涉及到電動機(jī)的選擇，V帶、帶輪、以及主軸的設(shè)計。同時還包括機(jī)架、進(jìn)料裝置等等以及一些小零件的設(shè)計。還有各個部件如何裝配，安裝，直至最后整個機(jī)器安全正常工作。其中我主要負(fù)責(zé)機(jī)器的整體設(shè)計。三、三、米粉切割機(jī)參數(shù)米粉切割機(jī)參數(shù) n一、用途：用于切割捆狀的米粉。n二、主要規(guī)格參數(shù):1、配套功率：2.2千瓦單相（220V）2、主軸轉(zhuǎn)速：940轉(zhuǎn)/分 3、生產(chǎn)效率：單相機(jī)7080斤干米粉/小時 7、外形尺寸：11201100800（單位mm）總裝圖總裝圖四、四、機(jī)架設(shè)計機(jī)架設(shè)計機(jī)架設(shè)計主要包括機(jī)架的整體框架設(shè)計，以及各部分的截面形狀選材等等。如下圖所示四、四、主軸以及鋸片的工位主軸以及鋸片的工位n主軸是用來傳動的最重要零件，鋸片是直接接觸米粉并完成切斷。二者的安裝以及配合決定了米粉切割的效率以及穩(wěn)定性。五、五、進(jìn)料裝置的設(shè)計進(jìn)料裝置的設(shè)計n進(jìn)料裝置的作用及其要求：1、要求進(jìn)料裝置能固定好要切斷的米粉，并穩(wěn)定的前后平移。2、要求能輔助鋸片很好的完成切斷工作。六、六、皮帶輪的設(shè)計皮帶輪的設(shè)計 皮帶輪是傳動中最關(guān)鍵的部位，由它帶動鋸片工作。七七、電機(jī)安裝板的設(shè)計電機(jī)安裝板的設(shè)計n電動機(jī)要固定在一個合理的位置，才能穩(wěn)定安全工作，我把電機(jī)固定在一個電機(jī)安裝板上。這個電機(jī)安裝板則固定在底架上。八八、整體安裝設(shè)計整體安裝設(shè)計n至此，基本的每個部分都設(shè)計完畢，現(xiàn)在就要求合理把各部分安裝到機(jī)架上。九、總結(jié)九、總結(jié) 在設(shè)計過程中遇到不少問題，感謝各位老師的細(xì)心指導(dǎo)。但是由于我缺乏經(jīng)驗(yàn)、學(xué)識和水平有限，因此此設(shè)計缺點(diǎn)、錯誤在所難免，衷心期待各位老師、同學(xué)們批評指正。米粉切割機(jī)
題目： 米粉切割機(jī)
二OO九年 五 月
23
米粉切割機(jī)
摘 要
米粉在江西，乃至整個中國，都是一種傳統(tǒng)美食。隨著米粉銷量的增加，對米粉的制作工藝以及機(jī)器的要求越來越高?，F(xiàn)在國內(nèi)也沒有一個專門針對米粉制造而生產(chǎn)米粉加工機(jī)器的大型企業(yè)，大多數(shù)都是一些小型機(jī)械廠生產(chǎn)的米粉加工設(shè)備。國家也沒有一個標(biāo)準(zhǔn)化的規(guī)定。
基于提高生產(chǎn)效率，我和同學(xué)共同完成了這個米粉切割機(jī)的設(shè)計。其中包括米粉切割機(jī)的機(jī)架設(shè)計，進(jìn)料裝置及其自動化設(shè)計，還有傳動裝置的詳細(xì)構(gòu)造機(jī)器安全可靠性，以及切割米粉選用何種鋸片。還包括各部件之間如何配合、安裝等。
關(guān)鍵詞：米粉 米粉切割機(jī)
Abstract
Rice in Jiangxi, China as a whole, are a traditional food. With the increase in rice sales of rice production technology and machines have become increasingly demanding. Now there is no domestic rice manufactured specifically for the production of large-scale rice processing machinery enterprises, the majority are small-scale production of rice flour machinery factory processing equipment. Countries do not have a standardized requirement.
Based on increasing productivity, I and students completed the rice co-cutter design. Rice cutting machine which includes the design of the rack, feed devices and automation design, as well as the detailed structure gear machine security and reliability, as well as the selection of the saw blade cutting rice. Also include how to co-ordinate between the various components and installation.
Key words: Ground rice Rice cutting machine
目 錄
1 緒 論 4
1.1 引言 4
1.2 精制直條米粉生產(chǎn)工藝 5
1.3 米粉切割機(jī)的主要規(guī)格參數(shù) 5
2 機(jī)架設(shè)計部分： 6
2.1 機(jī)架設(shè)計的一般要求 6
2.2 米粉切割機(jī)機(jī)架結(jié)構(gòu)選擇： 6
2.2.1 機(jī)架的一般分類： 6
2.2.2 機(jī)架的截面形狀： 7
3 主軸及鋸片安裝位置： 9
3.1 主軸概述 9
3.1.1 軸的常用材料 9
3.1.2 主軸的工作位及其安裝 10
3.2 切斷及鋸片工作位 14
3.2.1 切割破碎原理 14
3.2.2 切割器的類型 16
4 切割位及切割器選擇 17
5 進(jìn)料裝置設(shè)計 19
5.1 托料架設(shè)計 19
5.2 軸承的選擇 19
5.2.1 滾動軸承和滑動軸承 19
5.2.2 滾動軸承的主要類型、性能與特點(diǎn) 19
參考文獻(xiàn) 21
致 謝 22
1 緒 論
1.1 引言
從20世紀(jì)90年代起，江西的精制直條米粉由于采用了不同于傳統(tǒng)作坊式的生產(chǎn)工藝和設(shè)備，產(chǎn)品的質(zhì)量比傳統(tǒng)米粉有了顯著的提高，取得了米粉生產(chǎn)歷史上的重大技術(shù)創(chuàng)新。產(chǎn)品外觀潔白光亮、晶瑩透亮、條形均勻挺直，久煮不糊燙、吐漿度很小、不斷條，食用口感柔嫩爽滑、有咬勁，是米粉中的上品。外形有各種粗細(xì)的圓、扁條狀，分別稱為江西米粉、銀絲米粉、上等沙河粉等。現(xiàn)已開發(fā)出如糙米米粉、香菇米粉、淮山米粉、蓮子米粉、芥麥米粉、螺旋藻米粉等系列產(chǎn)品。目前，在我國港澳地區(qū)和東南亞、美國、歐洲、加拿大等地的海外華人聚居地有很好的銷量，并在穩(wěn)步增長，成為出口米粉市場的一大名牌，已在江西省形成了大規(guī)模工業(yè)生產(chǎn)，取得了良好的經(jīng)濟(jì)效益。該米粉生產(chǎn)，原料全部采用大米，沒有任何添加劑，而且主要以市場上銷量較差、糧食部門往往庫存較多的早稻米為主，原材料十分豐富；同時，在我國這樣一個農(nóng)業(yè)大國。對促進(jìn)農(nóng)產(chǎn)品加工轉(zhuǎn)化，提高附加值具有很積極的意義。
近幾年期間，精制直條米粉成套設(shè)備已從江西省推廣到福建、浙江、湖南、廣西、四川、山東、大連等省市，并出口到菲律賓、越南等東南亞國家和委內(nèi)瑞拉等南美國家。
1.2 精制直條米粉生產(chǎn)工藝
精制直條米粉的生產(chǎn)工藝如下：大米配比→洗米→浸泡→脫水→粉碎→混合→榨粉→第一次時效處理→復(fù)蒸→第二次時效處理→梳條整理→低溫烘干→切割→包裝→成品。
1.3 米粉切割機(jī)的主要規(guī)格參數(shù)
1．配套功率：2.2千瓦單相（220V）。
2．主軸轉(zhuǎn)速：940轉(zhuǎn)/分
3．生產(chǎn)效率：單相機(jī)70~80斤干米粉/小時。
4．主機(jī)凈重：81公斤
5．加工原料：大米，玉米，綠豆等雜糧
6．米粉絲形狀：圓形、扁形、寬帶形
7. 外形尺寸：臺式1120×1100×800（單位mm）
針對米粉的切割，在保證效率和產(chǎn)量的前提條件下，我對米粉切割機(jī)有了一定構(gòu)想。
2 機(jī)架設(shè)計部分：
2.1 機(jī)架設(shè)計的一般要求
1.在滿足強(qiáng)度和剛度的前提條件下，機(jī)架的重量應(yīng)要求輕、成本低。
2.抗振性好。
3.由于內(nèi)應(yīng)力及溫度變化引起的結(jié)構(gòu)變形應(yīng)力最小。
4.結(jié)構(gòu)設(shè)計合理，工藝性良好，便于鑄造、焊接和機(jī)械加工。
5.結(jié)構(gòu)力求便于安裝與調(diào)整，方便修理和更換零部件。
6.有導(dǎo)軌的機(jī)架要求導(dǎo)軌面受力合理、耐磨性良好
根據(jù)機(jī)架的不同用途對各項又有所偏重。
2.2 米粉切割機(jī)機(jī)架結(jié)構(gòu)選擇：
2.2.1 機(jī)架的一般分類：
1.梁柱式機(jī)架：如大多數(shù)金屬切削機(jī)床的床身、立柱及橫梁等。
2.框架式機(jī)架：a)閉框式機(jī)架：如軋鋼機(jī)機(jī)架鍛壓機(jī)機(jī)身、汽車車架（臥式閉框）等。b)開框式（C形）機(jī)架：如開式壓力機(jī)機(jī)身等。
3.平板式機(jī)架：如水壓機(jī)的基礎(chǔ)平臺、機(jī)器的底座、金屬切削機(jī)床的工作臺等。
4.箱殼式機(jī)架：如齒輪傳動箱箱體、泵體及動力機(jī)械的機(jī)身（如柴油機(jī)機(jī)體）等。
米粉切割機(jī)的工作震動小，支撐在機(jī)架上的傳動裝置和托料架總體重量大概也只有60KG。各項指標(biāo)對機(jī)架的強(qiáng)度以及剛度要求不高，初步選定機(jī)架結(jié)構(gòu)為框架式機(jī)架中的開框式機(jī)架即可。
米粉切割機(jī)要正常工作，其機(jī)架穩(wěn)定性是基本條件，綜合成本方面考慮設(shè)計出架的外形初步訂為如下圖所示：
2.2.2 機(jī)架的截面形狀：
由于零件的抗彎、抗扭強(qiáng)度和剛度除了與其截面面積有關(guān)外，還取決于截面的形狀。因此合理的確定機(jī)架的截面形狀是機(jī)架設(shè)計中的一個重要問題。
機(jī)械設(shè)計手冊中列出了幾種常用的截面形狀如下：
以圖1的抗彎慣性矩和抗扭慣性矩為標(biāo)準(zhǔn)建立如下表格：
截面形狀
（面積相等）
抗彎慣性矩相對值
抗扭慣性矩相對值
圖1
1
1
圖2
3.03
2.89
圖3
5.04
5.37
圖4
1.04
0.88
圖5
7.35
0.82
圖6
3.45
1.27
圖7
6.90
3.98
圖8
19
0.09
圖9
16.70
0.49
由慣性矩的相對值可以看出：
圓形截面有較高的抗扭剛度，但抗彎強(qiáng)度較差，故宜用于受扭為主的機(jī)架。工字形截面的抗彎強(qiáng)度大，但是抗扭很低故宜用于承受純彎的機(jī)架。方形截面抗彎，抗扭分別低于工字形和圓形截面，有一定的綜合性能。
另外，截面面積不變，加大外形輪廓尺寸，減小壁厚，亦即使材料遠(yuǎn)離中性軸的位置，可提高截面的抗彎，抗扭剛度。從結(jié)構(gòu)上來看，由于空心矩形內(nèi)腔容易安設(shè)其他零件，故許多機(jī)架的截面常采用方形或矩形截面。
槽形截面鋼和工字形截面比較，抗彎強(qiáng)度相差不大，抗扭強(qiáng)度又高于工字形截面。其綜合性能最適合用于米粉切割機(jī)的機(jī)架，能承受比較高的彎曲應(yīng)力，也能滿足抗扭要求。故整體機(jī)架的四個支撐柱截面和四個橫梁均選用槽形截面。但下面的布肋主要受力是在機(jī)架正面的兩根，側(cè)邊的布肋受力很小，其功用主要是保持機(jī)架的整體穩(wěn)定性，考慮到成本問題選用角鋼即可。
因此機(jī)架所用材料為：槽鋼[503731100 數(shù)量2；槽鋼[50373800 數(shù)量2；槽鋼[50373646 數(shù)量4；角鋼L404041026 數(shù)量2；角鋼L40404700 數(shù)量2。
3 主軸及鋸片安裝位置：
3.1 主軸概述
3.1.1 軸的常用材料
軸的材料首先應(yīng)有足夠的強(qiáng)度，對應(yīng)力集中敏感性低；還應(yīng)滿足剛度、耐磨性、耐腐蝕性及其良好的加工性，以及價格低廉、易于獲得的要求。
軸常用的材料主要有碳鋼、合金鋼、球墨鑄鐵和高強(qiáng)度鑄鐵。
碳鋼有足夠高的強(qiáng)度，對應(yīng)力集中的敏感性較低，便于進(jìn)行各種熱處理及機(jī)械加工，價格低，供應(yīng)充足，故應(yīng)用最廣。一般機(jī)器中的軸，可用30、40、45、50等牌號的優(yōu)質(zhì)中碳鋼制造，尤以45號鋼經(jīng)調(diào)質(zhì)處理最常用。對低速輕載或不重要的軸，可用A3、A4、A5等普通碳素鋼制造。
因此軸主要起傳動作用，連接皮帶輪和刀片，對其他性能無特殊要求故采用最為常見的45號優(yōu)質(zhì)中碳鋼，正火處理。
3.1.2 主軸的工作位及其安裝
主軸一段與電動機(jī)的帶輪相連，連接方式為螺紋加鍵連接。
1.螺紋選擇
螺紋有外螺紋和內(nèi)螺紋之分，他們共同組成螺旋副。起連接作用的螺紋稱為連接螺紋；起傳動作用的螺紋稱為傳動螺紋。根據(jù)其母體形狀可分為圓柱螺紋和圓錐螺紋兩類，圓錐螺紋主要用于管連接，圓柱螺紋用于一般連接和傳動。螺紋又有米制和英制（螺距以每英寸牙數(shù)表示）之分，我國除管螺紋保留英制外、都采用米制螺紋。按照牙型的不用螺紋又分為普通螺紋、管螺紋、梯形螺紋、矩形螺紋和鋸齒形螺紋等。前兩種主要用于連接，后三種主要用于傳動。其中除矩形螺紋外，都已標(biāo)準(zhǔn)化。
主軸主要是傳動，因此從《機(jī)械設(shè)計》上截取幾種常用傳動螺紋如下表：
螺紋類型
牙型圖
特點(diǎn)和應(yīng)用
矩
形
螺
紋
牙型為正方形，牙型角。其傳動效率教其他螺紋高，但牙根強(qiáng)度弱，螺旋副磨損后，間隙難以修復(fù)和補(bǔ)償，傳動精度降低。為了便于銑、磨削加工，可制成10°的牙型角
矩形螺紋尚未標(biāo)準(zhǔn)化，推薦尺寸：，。且目前已逐漸被梯形螺紋所代替。
梯
形
螺
紋
牙型為等腰梯形，牙型角。內(nèi)外螺紋以錐面貼緊不易松動。與矩形螺紋相比，傳動效率略低，但工藝性好，牙根強(qiáng)度高，對中性好。如用剖分螺母，還可以調(diào)整間隙。梯形螺紋是最常用的傳動螺紋。
鋸
齒
形
螺
紋
牙型為不等腰梯形，工作面的牙側(cè)角為3°，非工作面的牙側(cè)角為30°。外螺紋牙根有較大的圓角，以減小應(yīng)力集中。內(nèi)、外螺紋旋合后，大徑處無間隙，便于對中。這種螺紋兼有矩形螺紋傳動效率高、梯形螺紋牙根強(qiáng)度高的特點(diǎn)，但只能用于單向受力的螺紋連接或螺旋傳動中，如螺旋壓力機(jī)。
米粉切割機(jī)的電動機(jī)轉(zhuǎn)速為940轉(zhuǎn)/分、功率2.2千瓦，屬于低速的電動機(jī)。而且切割米粉對傳動效率要求不是很高。只要齒根有足夠強(qiáng)度即可。
牙型角、牙型為等腰梯形的梯形螺紋。內(nèi)外螺紋貼緊不易松動，雖然傳動效率略低，但工藝性好，牙根強(qiáng)度高，對中性好。最適合主軸兩段的傳動螺紋連接，故選用梯形螺紋。
2.螺紋連接的強(qiáng)度計算
由主軸兩端螺紋受力分析可知，螺紋所受的載荷包括軸向載荷、橫向載荷、彎矩和轉(zhuǎn)矩等。但對其中每一個具體的螺紋而言，其受載的形式不外乎是受軸向力或受橫向力。在軸向（包括預(yù)緊力）的作用下，螺栓桿和螺紋部分都有可能發(fā)生塑性變形和斷裂；而在橫向力的作用下，當(dāng)采用鉸制孔用螺栓時，螺栓桿和孔壁的貼合面上可能發(fā)生壓潰或螺栓桿被剪斷等。根據(jù)統(tǒng)計分析，在靜載荷下螺栓連接是很少發(fā)生破壞的，只有在嚴(yán)重過載的情況下才會發(fā)生。就破壞性質(zhì)而言，約90%的螺栓屬于疲勞破壞 。而且疲勞斷裂常發(fā)生在螺紋根部，即截面面積較小并有缺口應(yīng)力集中的部位（約占其中的85%），有時也發(fā)生在螺栓頭與光桿的交接處（約占其中的15%）。
綜上所述，對于受拉螺栓，其主要破壞形式是螺栓桿螺紋部分發(fā)生斷裂，因而其設(shè)計準(zhǔn)則是保證螺栓的靜力或疲勞拉伸強(qiáng)度；對于受剪螺栓，其主要破壞形式是螺栓桿和孔壁的貼合面上出現(xiàn)壓潰或螺栓桿被剪斷，其設(shè)計準(zhǔn)則是保證連接的擠壓強(qiáng)度和螺栓的剪切強(qiáng)度，其中連接的擠壓強(qiáng)度對連接的可靠性起決定性作用。
對于米粉切割機(jī)的主軸，兩端的螺紋主要受到的是螺栓連接的預(yù)緊力和工作剪力。受到的工作拉力可以忽略不計。
切割米粉所受的工作剪力由試驗(yàn)測得=N，螺栓連接的預(yù)緊力=N，，。
緊螺栓連接強(qiáng)度校核
緊螺栓連接裝備時，螺母需要擰緊，在擰緊力矩的作用下，螺栓除受預(yù)緊力的拉伸而產(chǎn)生的拉伸應(yīng)力外，還受螺紋摩擦力矩的扭轉(zhuǎn)而產(chǎn)生扭轉(zhuǎn)切應(yīng)力，使螺栓處于拉伸與扭轉(zhuǎn)的復(fù)合應(yīng)力狀態(tài)下。因此，進(jìn)行僅承受預(yù)緊力的緊螺栓強(qiáng)度計算時，應(yīng)綜合考慮拉伸應(yīng)力和扭轉(zhuǎn)切應(yīng)力的作用。
螺旋副間的摩擦力矩為
(2 - 1)
螺栓危險截面的拉伸應(yīng)力為
(2 - 2)
螺栓危險截面的扭轉(zhuǎn)切應(yīng)力應(yīng)為
(2 -3)
對于普通螺紋的鋼制螺栓，取，，，由此可得
(2 - 4)
由于螺栓材料是塑性的，故可根據(jù)第四強(qiáng)度理論，求出螺栓預(yù)緊狀態(tài)下的計算應(yīng)力為
(2 - 5)
由此可見，對于普通螺紋的鋼制緊螺栓連接，在擰緊時雖是同時承受拉伸和扭轉(zhuǎn)的聯(lián)合作用，但在計算時可以只按拉伸強(qiáng)度計算，并將所受的拉力（預(yù)緊力）增大30%來考慮扭轉(zhuǎn)的影響。
當(dāng)普通螺栓連接承受橫向載荷時，由于預(yù)緊力的作用，將在接合面產(chǎn)生摩擦力來抵抗工作載荷。這時，螺栓僅承受預(yù)緊力的作用，而且預(yù)緊力不受工作載荷的影響，在連接承受工作載荷后仍保持不變。預(yù)緊力的大小，根據(jù)接合面不產(chǎn)生滑移的條件確定。
假設(shè)各螺栓所需的預(yù)緊力均為，螺栓數(shù)目為，則其平衡條件為
由此得預(yù)緊力為
(2 - 6 )
式中：—接合面的摩擦系數(shù)，查表的；
—接合面數(shù)，式中
—防滑系數(shù)，。
螺栓危險截面的拉伸強(qiáng)度條件根據(jù)式(2 - 2)、(2 - 5)及 (2 - 6 )可寫為
(2 - 7)
同樣可得螺絲桿的剪切強(qiáng)度條件為
(2 - 8)
由(2 - 7)得螺栓危險截面的拉伸應(yīng)力
螺栓危險截面的剪切應(yīng)力
而螺紋連接的許用切應(yīng)力和許用擠壓應(yīng)力分別按下式確定
(2 - 8)
(2 -10)
軸的材料選用45號鋼，查表（詳見GB/T 3098.1-2000和GB/T 3098.2-2000 ）得，，。
由此可得螺紋連接的許用切應(yīng)力，許用壓應(yīng)力。
由于，且，故該主軸的螺紋連接滿足要求。
3.2 切斷及鋸片工作位
本產(chǎn)品是為了把從抽絲機(jī)里出來的長度不一的米粉切割成需要的長度，以達(dá)到包裝的要求。
一般精制米粉的規(guī)格：
米粉直徑：。
米粉長度：。
3.2.1 切割破碎原理
1.切割形式
在進(jìn)行切割時，在切割平面內(nèi)的切割方向上刀片與物料之間必須保持一定的相對運(yùn)動，才能完成切入直至切斷。動刀片刃口某點(diǎn)與物料間相對運(yùn)動在切割平面上的分速度與其在刃口于該點(diǎn)處法平面上的投影間的夾角稱為滑切角（如圖1所示），。而成為滑切系數(shù)，滑切系數(shù)取決于動刀片自身刃口形狀、動刀片安裝位置及切割過程中物料在切割平面上的運(yùn)動。
圖 1 滑切角的概念
（1） 砍切 當(dāng)時的切割形式為砍切，切割阻力大，切割過程中物料變形較大，物料流失較多。
（2） 斜切 當(dāng)（為刀片與物料間的摩擦角）時，，其中為刀片結(jié)構(gòu)刃角，即刃口法平面與兩平面形成的交線間的夾角，如圖 2 所示。而為刀片實(shí)際切割工作刃角，雖然，雖未形成滑切，任較為省力，有時為使切割過程阻力均勻，采用斜置刃口逐漸完成對于整個切割斷面的切割而形成斜切。
（3） 滑切 當(dāng)時，形成滑動切割，微觀上呈鋸切割，故省力，切割過程中圖 2 刀片刃角與斜角物料變形較小，所得片狀物料的厚度較為均勻，且失水較少。屬于運(yùn)動參數(shù)，它表明該切割器在切割過程中滑切作用的大小。滑切系數(shù)越大滑切作用越強(qiáng)，切割越省力。
切割米粉時主要從正面切，所受阻力不是太大，大約N。對米粉長度精度要求在5㎜內(nèi)即可，因此選用砍切足以滿足米粉的切斷要切，而且對刀片的要求也有所降低。
3.2.2 切割器的類型
切割器是指直接完成切割作業(yè)的部件，是切割機(jī)械的核心。切割器的類型及結(jié)構(gòu)直接影響著切割機(jī)械的功能及整體性能。切割器一般可按切割方式和結(jié)構(gòu)形式劃分。
1.按切割方式分
按切割方式分，切割器可分為有支撐切割器和無支持切割器兩種。
（1）有支撐切割器 （2）無支撐切割器
（1）有支撐切割器 即在切割點(diǎn)附近有支撐面，切割物料起阻止物料沿刀片刃口運(yùn)動方向移動的作用。這種切割器在結(jié)構(gòu)上表現(xiàn)為由動刀和定刀（或另一動刀）構(gòu)成切割副。為保證切割形式整齊穩(wěn)定的切割斷面質(zhì)量，要求動刀與定刀之間在切割斷面質(zhì)量，要求動刀與定刀只間在切割點(diǎn)初的刀片間隙盡可能小且均勻一致。這種切割器所需刀片切割速度較低，碎段尺寸均勻、穩(wěn)定，動力消耗少，多用于切片、段、絲等要求形狀尺寸穩(wěn)定一致的場合。
（2）無支撐切割器 指物料在被切割時，由物料自身的慣性和變形力阻止其沿切割方向移動。這種切割器僅包含有一個（組）動刀，而無定刀（或另一動刀）。所需刀片切割速度高，碎斷尺寸不均勻，動力消耗多，多用于碎塊、漿、糜等形狀及尺寸一致性要求不高的場合。
2.按結(jié)構(gòu)形式分
按結(jié)構(gòu)形式，切割器分為盤刀式、滾刀式和組合刀式三種。
（1）盤刀式切割器 動刀刃口工作時所形成的軌跡近似為圓盤形，即刃口所在平（曲）面近似垂直回轉(zhuǎn)軸線，所得到的產(chǎn)品斷面為平面，是應(yīng)用廣泛的一種切割器。這種切割器便于布置，切割性能好，易于切割出幾何形狀規(guī)則的片狀、塊狀產(chǎn)品。切割出產(chǎn)品的尺寸（如切片的厚度）：當(dāng)物料喂入進(jìn)給方向與動刀主軸方向垂直時，取決于相鄰刀片的間距；當(dāng)物料喂入進(jìn)給方向與動刀主軸方向平行時，取決于相鄰兩次切割過程中物料進(jìn)給量。
盤刀式刀片刃口基本類型有直刃口、凸刃口和凹刃口。
（a）直刃口：隨著切割點(diǎn)由近而遠(yuǎn)、滑切角減小，參與切割的刃口增長，因而切割阻力矩變化幅度大；同時近端鉗住角較大，但制造、刃磨容易。
（b）凸刃口：有偏心圓和螺線。隨切割點(diǎn)漸遠(yuǎn)?；薪窃龃螅懈钭枇剌^為穩(wěn)定，但遠(yuǎn)端鉗住角較大，將形成推料，使刀片刃口磨損不均勻；但不便于刃磨，常需要配置專用刃磨架，對于連續(xù)進(jìn)給場合的刀片間隙調(diào)整困難。常見的圓盤刀也屬于凸刃口，一般速度較高，滑切作用強(qiáng)烈，切割斷面質(zhì)量好，尤其適合于剛度較差的物料切片。
（c）凹刃口： 與凸刃口相比，其鉗住性能好，切割阻力矩均勻，但滑切現(xiàn)象不明顯，且制造、磨刃困難。
同時還有其他專用刃口（如鋸齒形、缺口形等）形狀均屬于上述基本類型的衍生，但運(yùn)用的比較少。
4 切割位及切割器選擇
而滾刀式切割器和組合刀式切割器，在米粉切割中應(yīng)用很少。為了達(dá)到既能完成切割要求又減低成本，用盤刀式的凸刃口切割器即可，同時需要用有支撐切割器。
在設(shè)計手冊中查得幾種常用圓鋸床及其功用：
臥式 鋸刀箱（齒輪變速箱）與圓鋸片沿水平方向移動進(jìn)給，只能作與工件軸線成90°角的直角鋸斷。工件由液壓臺虎鉗沿水平方向和垂直方向夾緊。
立式 鋸刀箱與圓鋸片沿垂直方向移動進(jìn)給，能作與工件軸線成90°角的直角鋸斷，也能做斜角鋸斷。機(jī)床導(dǎo)軌具有間隙補(bǔ)償和防震動裝置。
擺式 圓鋸片繞一固定的支撐軸擺動，同時向下垂直進(jìn)給?？勺髋c工件成角的直角鋸斷，也可作與工件不成90°的斜角鋸削。
至于用到切割米粉，當(dāng)然是用圓鋸片對其進(jìn)行垂直切斷。故鋸片的位置與工件垂直固定即可?？沙醪蕉ǖ娩徠墓ぷ骱唸D如下
如圖所示，鋸片與主軸用螺紋連接。鋸片外部有直徑530㎜、寬60㎜的鋸片罩。鋸片轉(zhuǎn)速900r/min，在高速轉(zhuǎn)動切割米粉的過程中，肯定會有大量的斷頭隨著鋸片的轉(zhuǎn)動飛出，為了保證工人操作時候的安全性，鋸片外面必須有防止斷頭飛出的鋸片罩。綜合實(shí)際考慮，鋸片罩尺寸稍稍大于鋸片的尺寸即可。一般鋸片罩為圓形，在鋸片的工作位置應(yīng)留有一定的空間，所以鋸片罩用3/4圓盤即可。鋸片工作時，有大量的斷頭，所以鋸片罩下方開口，并作一圓弧，讓斷頭從下部流出。鋸片罩的外形如下圖所示：
5 進(jìn)料裝置設(shè)計
5.1 托料架設(shè)計
米粉要完成切割，必須把米粉先固定在一個裝置上，然后用鋸片切斷，每次切斷的都是一捆長度不一的米粉，固定裝置可用一個從中間斷開的U型槽，下端用螺母固定在進(jìn)料裝置的底架上，可以隨底架前后移動。因?yàn)橐苯咏佑|到米粉，為保證食用的安全性，材料應(yīng)該選用不銹鋼。
鋸片是固定在主軸上的，要想鋸斷米粉，只能是放在托料架上的米粉移動，移動沿垂直鋸片的方向，這樣才能按要求切斷米粉。
首先推動進(jìn)料裝置沿垂直鋸片的方向向著鋸片移動，直至切斷米粉，然后讓進(jìn)料裝置回到初始位置，取下切好的米粉，繼續(xù)上面的步驟，就可以連續(xù)的完成米粉切割。
這樣就要求進(jìn)料裝置能自由的在底架上前后移動，滑動摩擦的摩擦阻力太大不適合。所以進(jìn)料裝置的移動應(yīng)該是滾動形式的。至于進(jìn)料裝置的移動則可以用軸承完成。
5.2 軸承的選擇
5.2.1 滾動軸承和滑動軸承
根據(jù)軸承中摩擦性質(zhì)的不同，可把軸承分為滑動摩擦軸承（簡稱滑動軸承）和滾動摩擦軸承（簡稱滾動軸承）兩大類。滾動軸承由于摩擦系數(shù)小，起動阻力小，而且它已經(jīng)標(biāo)準(zhǔn)化，選用、潤滑、維護(hù)都很方便，因此在一般機(jī)器中應(yīng)用較廣。但由于滑動軸承本身具有的一些獨(dú)特優(yōu)點(diǎn)，使的它在某些不能、不便或使用滾動軸承軸承沒有優(yōu)勢的場合，如在工作轉(zhuǎn)速較高、特大沖擊與震動、徑向空間尺寸受到限制或必須剖分安裝（如曲軸的軸承）、以及需在水或腐蝕性介質(zhì)中工作等場合，仍占有重要地位。
進(jìn)料裝置的移動對各方面要求都不高，只要求移動平穩(wěn)?；跐L動軸承絕大多數(shù)已經(jīng)標(biāo)準(zhǔn)化，并由專業(yè)工人大量制造及供應(yīng)各種常用規(guī)格的軸承，且滾動軸承具有摩擦阻力小，功率消耗少，起動容易等優(yōu)點(diǎn)。故選用最常用的滾動軸承。
5.2.2 滾動軸承的主要類型、性能與特點(diǎn)
如果僅按軸承用于承受的外載荷不同來分類時，滾動軸承可以概括地分為向心軸承、推力軸承和向心推力軸承三大類。
滾動軸承的類型很多，現(xiàn)將常用的各類滾動軸承的性能和特點(diǎn)簡要介紹于下。
類型名稱
結(jié)構(gòu)代號
基本額定動載荷比
極限轉(zhuǎn)速比
軸向承載能力
軸向限位能力
性能和特點(diǎn)
調(diào)心球軸承
10000
0.60.9
中
少量
Ⅰ
圓外圈滾道表面是以軸承中點(diǎn)為中心的球表面，故能自動調(diào)心，允許內(nèi)圈相對外圈軸線便宜斜量。一般不宜承受純軸向載荷
調(diào)心滾子軸承
20000
1.84
低
少量
Ⅰ
性能、特點(diǎn)與調(diào)心球軸承相同，但具有較大的徑向承受能力，允許內(nèi)圈對外圈軸線偏斜量
圓錐滾子軸承
30000
1.52.5
中
較大
Ⅱ
可以同時承受徑向載荷及軸向載荷。外圈可分離，安裝時可調(diào)整軸承的游隙。一般成對使用
推力球軸承
51000
1
低
只能承受單向的軸向載荷
Ⅱ
只能承受軸向載荷。高速時離心力大，鋼球與保持架磨損，發(fā)熱嚴(yán)重，壽命降低，故極限轉(zhuǎn)速很低
深溝球軸承
60000
1
高
少量
Ⅰ
主要承受徑向載荷，也可同時承受小的軸向載荷。當(dāng)量摩擦系數(shù)最小。在高轉(zhuǎn)速時，可用來承受純軸向載荷。工件中允許內(nèi)、外圈軸線偏移量，大量生產(chǎn)，價格最低
注： 1基本額定動載荷比：指同一尺寸系列（直徑及寬度）各種類型和機(jī)構(gòu)形式的軸承的基本額定動
動載荷與單列深溝球軸承（推力軸承則與單向推力球軸承）的基本額定動載荷之比。
2極限轉(zhuǎn)速比：指同一尺寸系列0級公差的各類型軸承脂潤滑時的極限轉(zhuǎn)速比與單列深溝球軸承
脂潤滑時極限轉(zhuǎn)速之比。高、中、低的意義為：高為單列深溝球軸承極限轉(zhuǎn)速的90%100%；
中為單列深溝球軸承極限轉(zhuǎn)速的60%90%低為單列深溝球軸承極限轉(zhuǎn)速的60%以下
3軸向限位能力：Ⅰ 為軸的雙向軸向位移限制在軸承的軸向游隙范圍內(nèi)；Ⅱ 為限制軸的單向軸
向位移。
進(jìn)料裝置下方用于移動的軸承要承受徑向載荷，同時也要承受少量的軸向載荷。綜合考慮經(jīng)濟(jì)成本與適用性，選用深溝球軸承為最佳。在進(jìn)料裝置的兩邊需要四個滾動軸承，其作用是讓進(jìn)料裝置能前后移動。為了使進(jìn)料裝置在前后移動時，能與鋸片保持水平，在其下部安裝四個滾動軸承。示意圖如下：
參考GB/T276-93，選這控制前后移動的滾動軸承代號為6305，固定進(jìn)料裝置左右移動的滾動軸承代號為6304。
同時在托料架的一端安裝了一個可以左右移動的擋片，調(diào)節(jié)擋片就可以調(diào)節(jié)要切斷米粉的長度。到此整個機(jī)器的外形設(shè)計完畢。
參考文獻(xiàn)
[1]
徐澋主編.機(jī)械設(shè)計手冊.機(jī)械工業(yè)出版社,1991
[2]
于永泗，齊明主編.機(jī)械工程材料.大連:大連理工大學(xué),2003
[3]
曾志新, 呂明編.機(jī)械制造技術(shù)基礎(chǔ).武漢理工大學(xué)出版社, 2001
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濮良貴,紀(jì)名鋼.機(jī)械設(shè)計.西北工業(yè)大學(xué)機(jī)械原理及機(jī)械零件教研室.2003,12(8)
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楊明忠,朱家誠.機(jī)械設(shè)計.武漢理工大學(xué)出版社,2006
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譚建榮，張樹有，陸國棟，施岳定編.圖學(xué)基礎(chǔ)教程.高等教育出版社,2004
[7]
劉鴻文編.材料力學(xué).高等教育出版社.2004
[8]
傅曉如編.米粉條生產(chǎn)技術(shù).金盾出版社.1999
致 謝
Reel and sheet cutting at a paper mill
M. Helena Correia, Jose F. Oliveira, J. Soeiro Ferreira
INESC Porto, Instituto de Engenharia de Sistemas e Computadores do Porto, 4200-465 Porto, Portugal
Faculdade de Economia e Gestao, Universidade Catolica Portuguesa, 4169-005 Porto, Portugal
Faculdade de Engenharia, Universidade do Porto, 4200-465 Porto, Portugal
Abstract
This work describes a real-world industrial problem of production planning and cutting optimization of reels and sheets, occurring at a Portuguese paper mill. It will focus on a particular module of the global problem which is concerned with the determination of the width combinations of the items involved in the planning process: the main goal consists in satisfying an order set of reels and sheets that must be cut from master reels. The width combination process will determine the quantity/weight of the master reels to be produced and their cutting patterns, in order to minimize waste, while satisfying production orders.
A two-phase approach has been devised, naturally dependent on the technological process involved.Details of the models and solution methods are presented. Moreover some illustrative computational results are included.
2003 Elsevier Ltd. All rights reserved.
Keywords: Combinatorial optimization; Cutting-stock; Heuristics
1. Introduction
Planning the paper production at a paper mill assumes several essentially distinct forms, each of which has its own particular characteristics, requiring different mathematical formulation and solution methods [1–3]. However, trim loss minimization is usually a component of the objective function. Other components take account of factors such as setup processing time, number and characteristics of cutting patterns. Additionally, there are usually several constraints involved, concerning customers specifications, strategic decisions and technological characteristics of the production process.
This paper describes a system developed by request of a Portuguese paper mill, Companhia dePapel do Prado (CPP), to support its production planning, focusing on the production and cutting of paper reels. This work is part of a broader system, named COOL (COOL stands for the Portuguese words meaning optimized combination of widths), which is intended to support the implementation of an optimizing policy for paper production and stock management.
The problem tackled in this paper concerns the definition of cutting patterns and quantity of paper to produce in order to satisfy a set of ordered reels and sheets, grouped by type of paper and grade.
It basically deals with the problem of planning the paper production and cutting of the master reels in order to satisfy a set of orders. The cutting plans to associate to the master reels must be defined considering minimization of waste while satisfying the ordered quantities. Varieties of technological and operational constraints are involved in the planning process, causing an interesting and dig cult trim problem.
From this perspective, this problem can be included in the broad family of Cutting-Stock Problems [4–6]. The problem formulation adopted disregards trim loss at the end of the reels (as it was considered irrelevant when compared with that occurring at the edges of the paper reels, which runs all along the paper length) and so, a 1D approach has been devised. The need of a two-phase methodology was determined by the technological characteristics of the cutting process. Other 1D two-phase cutting-stock problems can be found in published literature. Besides paper industry, similar approaches are also applied in other industries, such as the steel industry [7,8] and the plastic Flm industry [9].
We propose an original solution method for the problem described above, which leads to considerable improvements in terms of paper savings when compared with those solutions obtained manually, as confirmed by the paper mill. The procedure developed is based on two distinct linear programming models, which are solved by a Simplex algorithm. Then, the solutions obtained are rounded in a post-optimization procedure, in order to satisfy integer constraints previously ignored. The quality of the solutions obtained are also validated by the resolution of an integer programming model of the problem, solved using the commercial optimization software CPLEX v.6.0.
The paper is organized as follows. Section 2 introduces the production problem and its industrial background. Particular emphasis will be given to those features of the industrial environment, which were relevant for the solution approach developed. Sections 3 and 4 will describe the problem and the methodology developed to solve it, respectively. A small example is considered throughout Section 4 in order to illustrate the solution procedure. In Section 5 some results will be presented and discussed.
2. Industrial environment
This case study takes place at a Portuguese paper mill, which can be considered as a vertical industry, since it produces paper products from pulp. The products are supplied both in reels and sheets. This industry operates in two types of markets: one in which the paper products have standard dimensions and other where paper products have make-to-order dimensions. The production cycle is of 6 weeks and, for technological reasons, there is a pre-defend production sequence in which paper is produced in ascending or descending rates.
Fig. 1 shows the production Jow of the paper products through out the production line. The paper is produced at the paper machine from pulp and is wound into a master reel of fixed width. Then, the master reel follows to the winder where it is cut into smaller reels. These reels either go straight to the customer or to the Intermediate Stock, or are cut into sheets at the cutters. These cut-to-sizes sheets either go to the customer or to the Standard Stock.
Both at the winder and cutters there is a small shred of fixed width cut-o8 all along the paper length. This scrap has been quite determinant for the solution process adopted.
Fig. 2 illustrates the relative perspectives of planning and production processes, emphasizing the products and sub-products involved. Planning and Production follow opposite directions. Planning’s based on the customers specifications of ordered products. Ordered reels and sheets of the same type of paper and grade, and belonging to the same Production Order, are combined into auxiliary reels. These auxiliary reels may include either reels or sheets, but never both. So, two types of auxiliary reels will be distinguished: auxiliary reels of sheets and auxiliary reels of reels. Auxiliary reels are then combined into cutting patterns that are associated to master reels.
The concept of auxiliary reel has been introduced for a better understanding of both the production procedure and the solution approach adopted. It is strictly related to the technological process involved, which requires the consideration of additional scrap width whenever the cutters are used. The definition of sub-patterns inside the main cutting patterns to be cut from the master reels has determined the two-phase solution approach considered.
There is a set of constraints that must be considered in the generation of the auxiliary reels and cutting patterns and which will be described later in Section 3. These constraints determine pattern feasibility.
The order system is schematized in Fig. 3. An order can be placed by the national market or by the international market (as this company also operates outside Portugal) and is processed by the Marketing Department. The Marketing Department can also generate an internal order, similar to the external orders, if it is considered appropriated. These orders can originate a Production Requisition, a Cutting Order or an Expedition Order. A Production Requisition is grouped with other existing Production Requisitions of the same type of paper and grade, resulting in a Production Order, which then follows to production. A Cutting Order occurs when a customer order of reels can be satisfied by existing reels (stocked at the Intermediate Stock) and an Expedition Order occurs when a customer order of sheets can be satisfied by existing sheets (stocked at the Standard Stock).
3. Problem description
The work presented in this paper is mainly concerned with the cutting patterns generation process, which will determine the quantity/weight of the master reels to produce and the associated cutting patterns, in order to minimize waste while satisfying a production order. The system developed will support the cutting planning of a Production Order, not interfering with decisions related to the orders to satisfy and the type of paper to produce in each production cycle. These are previous decisions made by the Marketing Department, eventually supported by a simulation using the system COOL.
Some constraints must be considered during the definition of the cutting patterns to associate to a master reel. These constraints can be grouped in two sub-sets: ?Operational constraints (imposed by management and customers specifications):
? Only reels of identical weight per width unit (reels with the same length of paper) can be combined.
? Only reels of identical internal and external diameters can be combined.
? Customer specifications of internal and external diameters must be satisfied.
? Assignment of the auxiliary reels to the cutters must be considered, since cutters have different characteristics.
? Minimum width is imposed to cutting patterns, in order to optimize the use of the machinery available.
? Technological constraints (mainly due to machinery characteristics):
? Maximum and minimum widths of the master reel at the winder (input).
? Limited number of winder slitting knives.
? Maximum and minimum sheet lengths at the cutters.
? Maximum and minimum sheet widths at the cutters.
? Limited number of slitting knives at the cutters.
? Maximum diameter of input reels at the cutters.
? Edge trims loss both at the winder and cutters.
There are European Standard Tolerances in use at the paper industry, which must be taken into account when fulfilling order (see Table 1). The client is obliged to accept deviations of the quantity ordered in these ranges. When over-production above maximum tolerances occurs, the Marketing Department can try to negotiate the acceptance of this extra quantity with the client. Due to losses inherent to production, negative tolerances are never considered during the planning phase.
4. Solution procedure
The solution procedure adopted is clearly injected by the production Jow. It is divided into three main stages, which are represented in Fig. 4.
The First stage consists in enumerating all the auxiliary reels and cutting patterns, based on a fixed width for the master reel and on the widths of the ordered items. The resultant set of cutting patterns is then submitted to a selection process through which undesirable auxiliary reels/cutting patterns are eliminated. All the remaining cutting patterns must be feasible in terms of the technological and operational constraints imposed to the production process.
In the second stage, the cutting patterns generated and accepted during the First stage are used as columns in a linear programming model of the optimization problem. Two linear programming models were developed. These models are solved by a Simplex algorithm [10].
In the following sections each one of these stages will be presented in detail.
A small real industrial example is introduced to illustrate the solution procedure and will be followed through out its description. It concerns the production planning of paper in master reels of 2520 mm width. The paper grade is 250 g=m2 and its thickness is 345 _m. The Production Requisitions involved are described in Table 2.
Rounding heuristic
The rounding procedure is applied to the solution of both LP models and is intended to fulfill those constraints of integer nature previously ignored, such as:
(1) Fixed 7nished reels diameters imposed by the customer must be satisfied, meaning that the paper length of cutting patterns including such reels must always be multiple of the requested diameter. In order to minimize the impact of this heuristic procedure, the quantities ordered of reels of Fixed diameter are adjusted to the closest multiple of the length of one reel before building the LP model.
Table 3
(2) The minimum weight for combination of sheets constraint, equivalent to a minimum paper length, intends to avoid inefficient use of the cutters.
(3) Alike the previous item, the minimum weight for cutting pattern constraint is intended to prevent inefficient use of the winder, while establishing a minimum quantity of paper to cut with each cutting pattern used.
The rounding heuristic starts with the Final solution of the LP model (non-zero length patterns) and tries to adjust those pattern lengths in order to satisfy the referred constraints. The new solution is kept as close as possible to the LP one and must satisfy the ordered quantities. First, the rounding procedure tries to eliminate those patterns which do not respect the minimum weight conditions (constraints 2 and 3 above). Precaution must be taken not to eliminate the unique pattern containing some ordered item. Then, the remaining patterns must be rounded up in order to compensate the e8ect of the destroyed ones.
This procedure consists basically in successively sorting the cutting patterns by the number of items not satisfied in each pattern, and augmenting the quantity to be cut with the First cutting pattern of the list until, at least, one unsatisfied item becomes satisfied. This procedure is repeated until all the items in all cutting patterns are satisfied.
This rounding procedure can lead to over-production above standard tolerances, even when Model(1) is used.
In the solution presented in Table 3, only the constraint concerning the minimum weight for combination of sheets is not being satisfied by the length of FP 16(x12) since it is smaller than the minimum weight for combination of sheets determined for that pattern (2730:00 mm). As the only order in that pattern is PR 1002 and it also exists in FP 21 (x14), pattern FP 16 can be eliminated and the length of FP 21 must be adjusted to include the quantity of PR 1002 that was being cut from FP 16. The Final solution is presented in Table 4.
Fig.5.shows the output of COOL for the data in Table 2.
Table 4
Fig.5.Computational results for large-scale instances
5. Computational results
The main purpose of the computational tests was to validate the solution procedure adopted and to establish a comparative analysis between the two linear programming models developed (Model(1) and Model(2)). The data used in this First set of computational runs was provided by the Marketing Department of the company and corresponds to real problems solved at the paper mill. The number of ordered items involved range from 3 to 16 and the maximum and minimum width of the ordered items are 1392 and 238 mm, respectively, being the average width 690 mm, approximately. These are relative small instances but, by doing this, the company intends to allow the system user to easily evaluate the performance of COOL in the initial phase of usage.
Data used in the computational tests is available at www.apdio/sicup.
The algorithms were implemented using the C programming language. The computational results were obtained with a Pentium III at 450 MHz.
In order to evaluate the quality of the solutions obtained with the linear models and rounding heuristic described above, an IP model was implemented. This IP model minimizes the amount of paper produced while strictly satisfying the ordered quantities. In order to consider those integer constraints mentioned above, several integer variables are included: ? Minimum weight for combination of sheets (Min Weight Sheets): The IP model was solved using the Mixed Integer Programming module of the optimization software CPLEX v.6.0.
In Fig.6, the performance of each solution procedure developed (based on the two LP models, Model(1) and Model(2)) is evaluated in terms of objective function value. In Fig.6(a), for each model, the ratio of the results obtained with the IP model and those obtained with the linear procedure followed by the rounding heuristic are depicted for each test instance: the value of 1.00 in the y-axis corresponds to the IP model solution. From this chart it can be observed that the results of the linear based procedure are, in most cases, coincident with those obtained with the IP model: Model(1) attains the same objective function values of IP in 70% of the test instances while only approximately 50% of the results obtained with Model(2) are coincident with the IP results. Though, with only one exception, the IP results are never exceeded in more than 22%.
The chart in Fig.6(b) intends to prove the adequacy of the linear approach adopted and, so, the ratio of the results before and after the rounding procedure is computed. The value of 1.00 in the y-axis corresponds to the LP model solution before the rounding procedure. In most cases, the results of the LP routine are coincident with the Final result, which means that, in those cases, the constraints of integer nature considered in the rounding procedure do not change the linear programming result.
Both charts show that the results obtained with Model(1), which minimizes the paper length produced and does not allow over production above tolerances, are never worse than those obtained with Model(2), which does not produce to the Intermediate Stock. Moreover, these results suggest the need to improve the rounding procedure in case of Model(2).
Table 5
Table 5 compares the results obtained with the two linear programming models in terms of the three exceeding components: quantity produced to the Intermediate Stock (QuantStock), overproduction above standard tolerances (QuantTolExc) and quantity of paper that cannot be re-used in any way (Waste). All the values are expressed in terms of a percentage of the total weight of paper produced and reject the objective function adopted in each model: Model(2) does not produce to the Intermediate Stock while Model(1) tries not to exceed standard tolerances. The amounts in which, sometimes, these tolerances are exceeded in Model(1) are a consequence of the rounding procedure. However, they are quite small when compared to those obtained with Model(2).
Since waste is the only component which can not be re-used, Fig.7draws attention to the comparison between the values obtained with the two LP based procedures: Final solutions based on Model(1) are seldom significantly worse than those attained with Model(2), in terms of paper waste minimization.
According to the comparative tests performed with this set of instances, Model(1) seems to perform better than Model(2) in all of them. Nevertheless, Model(2) was kept available in the Final version of COOL, as each model may generate solutions more adequate to, or even required by, different industrial situations: when production to the Intermediate Stock is allowed or even recommended, Model(1) can be used; situations in which Intermediate Stock levels are high enough to forbid stock enlargement, Model(2) solutions may be required. In terms of efficiency, the LP approach lead to a reduction of the processing time of approximately 75% of the time used by the IP approach. Although the average resolution time of the IP approach for the instances tested was of 18 s, situations may occur which would preclude the use of the IP approach in practice.
A set of larger instances was generated and tested in order to evaluate the performance in terms of efficiency of the develo
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