切紙機壓紙器的設(shè)計
切紙機壓紙器的設(shè)計,切紙機壓紙器的設(shè)計,切紙機壓紙器,設(shè)計
畢業(yè)設(shè)計（論文）任務(wù)書
課題名稱： 切紙器壓紙器設(shè)計
專 業(yè)： 機電一體化
班 級： 機電0802
姓 名：
學 號： 0810201139
指導教師：
教師職稱：
2010年10月25日
畢業(yè)設(shè)計（論文）開題報告
課題名稱： 切紙機壓紙器
專 業(yè)： 機 電 一 體 化
班 級： 機電0802
姓 名：
學 號： 0810201139
指導教師：
教師職稱：
?
2010年11 月 20日
1、課題研究的現(xiàn)狀和意義
現(xiàn)狀：
現(xiàn)代的生活學習及各行各業(yè)中都離不開各種大小規(guī)格不同的紙張，如學習的書本用紙、合同用紙、發(fā)票用紙、圖畫用紙、印刷業(yè)更離不開各種大小的紙張。那么按一定規(guī)格生產(chǎn)出的紙張，必須要裁切才能得到各種大小不同的紙張。
人工切紙，勞動強度大，效率低，切出的紙張精度難以保證，因而有必要設(shè)計機器來裁切紙張。壓紙器的工作是由電機驅(qū)動，經(jīng)減速傳動系統(tǒng)蝸桿蝸輪，帶動螺母，再將動力傳給螺桿，由螺桿直接帶動壓紙器松開或緊壓紙張。 力太大，則會壓破紙張；力太小，則會使力不夠，而引起裁切達不到要求。
意義：通過本機器的設(shè)計，使自己得到很好的設(shè)計實踐的鍛煉，使自己更好的了解和掌握設(shè)計機器的全過程。
?2、課題要解決的問題或研究的基本內(nèi)容
?基本內(nèi)容：
切紙機上的壓紙傳動機構(gòu)，是保證切紙機切紙精度的重要環(huán)節(jié)，
本課題是為 QZ203C型壓紙器。
設(shè)計采用蝸桿、蝸輪傳動機構(gòu)來完成該壓紙傳動工作
因此，蝸輪、蝸輪傳動設(shè)計是本課題首要解決的問題。
1.根據(jù)任務(wù)要求，進行壓紙器機械系統(tǒng)總體方案設(shè)計，確定減速傳動系統(tǒng)、執(zhí)行系統(tǒng)的組成
2. 根據(jù)設(shè)計參數(shù)和設(shè)計要求，使執(zhí)行系統(tǒng)具有較好的傳力性能。
3. 選擇電動機型號，分配減速傳動系統(tǒng)中各級傳動的傳動比，并進行傳動機構(gòu)的工作能力設(shè)計計算：1.電動機的選擇 2.軸的設(shè)計 3.渦輪設(shè)計 4.蝸桿設(shè)計 5.螺桿螺母設(shè)計6.聯(lián)軸器的選用 7.減速器的選用 8.軸承的選用 9.鍵的選用。
4. 對壓紙系統(tǒng)進行結(jié)構(gòu)設(shè)計，繪制裝配圖及關(guān)鍵零件工作圖。
3、課題研究擬采用的手段和工作路線
手段：
1、 利用《機械設(shè)計手冊》第三卷 設(shè)計蝸桿與蝸輪極其傳動方式（在此之前應(yīng)先設(shè)計液壓傳動）。
2、 參考機械設(shè)計手冊及網(wǎng)上查詢法，設(shè)計推紙器及其構(gòu)件。
3、 蝸輪箱體、連接零件及傳動機構(gòu)需要的圖紙（采用電腦設(shè)計立體圖、平面圖）。
4、 用CAD 設(shè)計裝配圖
工作路線：
1) 設(shè)計準備
了解設(shè)計任務(wù)書，明確設(shè)計要求、工作條件、設(shè)計內(nèi)容的步驟；通過查閱有關(guān)設(shè)計資料了解設(shè)計對象的性能、結(jié)構(gòu)及工藝性；準備好設(shè)計需要資料、繪圖工具；擬定設(shè)計計劃等。
2) 壓紙傳動機構(gòu)草圖設(shè)計
繪制裝配草圖；進行結(jié)構(gòu)構(gòu)件設(shè)計。
3) 繪制裝配工作圖
標注尺寸、技術(shù)要求、零件序號；編寫標題欄、零件明細表等。
4) 繪制零件工作圖
繪出零件的必要視圖；標注尺寸、公差及表面粗糙度；編寫技術(shù)要求和標題欄等
5) 編寫設(shè)計說明書
4、課題研究進程計劃
2010年11月8日～2010年11月14日 開題報告
2010年11月15日～2010年11月21日 電動機的選擇
2010年11月22日～2010年11月28日 軸的設(shè)計
2010年11月29日～2010年12月5日 渦輪設(shè)計
2010年12月6日～2010年12月10日 蝸桿設(shè)計
2010年12月11日～2010年12月16日 螺桿螺母設(shè)計
2010年12月17日～2010年12月21日 減速器的選用
2010年12月22日～2010年12月23日 裝配圖的繪制
2010年12月24日～2010年12月28日 零件圖的繪制
2010年12月29日～2011年1月1日 交電子文檔 PPT
5、課題成果
論文 圖紙 產(chǎn)品或作品□ 應(yīng)用程序□
其它：
指導教師意見：
指導教師（簽名）：
年 月 日
教研室主任意見：
教研室主任（簽名）：
年 月 日
4
畢業(yè)設(shè)計（論文）
課題名稱： 切紙機壓紙器的設(shè)計
目 錄
1、設(shè)計任務(wù)………………………………………………………………1
2、傳動方案…………………………………………………………2
3、設(shè)計內(nèi)容…………………………………………………………3
4、切紙機壓紙器傳動設(shè)計……………………………………………4
4.1電機選擇…………………………………………………………4
4.2計算傳動裝置的運動和動力參數(shù)…………………………………5
5、蝸桿蝸輪設(shè)計………………………………………………6
6、螺桿螺母傳動設(shè)計………………………………………………11
7、設(shè)計小結(jié)…………………………………………………………16
參考文獻…………………………………………………………17
切紙機壓紙器的設(shè)計
摘要：壓紙器的工作原理是由電機驅(qū)動，經(jīng)減速傳動系統(tǒng)蝸桿蝸輪，帶動螺母，再將動力傳給螺桿，由螺桿直接帶動壓紙器松開或緊壓紙張。 力太大，則會壓破紙張；力太小，則會使力不夠，而引起裁切達不到要求。
根據(jù)任務(wù)要求，進行壓紙器機械系統(tǒng)總體方案設(shè)計，確定減速傳動系統(tǒng)、執(zhí)行系統(tǒng)的組成； 根據(jù)設(shè)計參數(shù)和設(shè)計要求，使執(zhí)行系統(tǒng)具有較好的傳力性能；選擇電動機型號，分配減速傳動系統(tǒng)中各級傳動的傳動比，并進行傳動機構(gòu)的工作能力設(shè)計計算； 對壓紙系統(tǒng)進行結(jié)構(gòu)設(shè)計，用CAD繪制裝配圖及關(guān)鍵零件工作圖。參考機械設(shè)計手冊及網(wǎng)上查詢法，設(shè)計壓紙器及其構(gòu)件。蝸輪箱體、連接零件及傳動機構(gòu)需要的圖紙（采用電腦設(shè)計立體圖、平面圖）。
關(guān)鍵詞：蝸輪蝸桿； 螺母； 螺桿； 蝸桿軸
現(xiàn)代的生活學習及各行各業(yè)中都離不開各種大小規(guī)格不同的紙張，如學習的書本用紙、合同用紙、發(fā)票用紙、圖畫用紙、印刷業(yè)更離不開各種大小的紙張。那么按一定規(guī)格生產(chǎn)出的紙張，必須要裁切才能得到各種大小不同的紙張。
人工切紙，勞動強度大，效率低，切出的紙張精度難以保證，因而有必要設(shè)計機器來裁切紙張。壓紙器的工作是由電機驅(qū)動，經(jīng)減速傳動系統(tǒng)蝸桿蝸輪，帶動螺母，再將動力傳給螺桿，由螺桿直接帶動壓紙器松開或緊壓紙張。 力太大，則會壓破紙張；力太小，則會使力不夠，而引起裁切達不到要求。
1、設(shè)計任務(wù)：切紙機壓紙器
壓紙傳動是將壓紙器牢牢的壓在紙上，然后刀切紙，從而提高
紙張的裁切精度。它可分為三部分：
1.驅(qū)動設(shè)備即電動機；
2.傳動系統(tǒng)即蝸桿蝸輪傳動和螺桿螺母傳動；
3.壓紙執(zhí)行主體即壓紙器。
蝸輪壓紙，壓紙力平衡可靠，壓紙系統(tǒng)使用壽命長，這種類型和其他類型相比很經(jīng)濟。
蝸桿蝸輪的傳動的特點：（1） 結(jié)構(gòu)緊湊、傳動比大。
（2） 傳動平穩(wěn)，噪音小。
（3） 可制成具有自鎖性的蝸桿。
因此，選擇蝸桿蝸輪傳動。但也存在著缺點：嚙合處有較大的相對滑動，因而發(fā)熱量大，效率低，傳動效率一般在70%~80%之間。還有一個缺點就是造價高。
螺桿螺母的傳動，如圖可知：它是以傳遞動力為主。
這種傳動的特點： 要求較小的轉(zhuǎn)矩產(chǎn)生較大的軸向力。這種傳動為間歇性工作，工作速度不高。
蝸輪箱固定在機身上起了支撐作用。
電機作勻速轉(zhuǎn)動1420r/min，要求壓紙器下壓速度達到0.104m/s，再通過螺桿螺母傳動，使較小的轉(zhuǎn)矩產(chǎn)生較大的軸向力。因而，受力較大的軸承應(yīng)用推力軸承。壓緊力大小在1000Ｎ左右。下壓距離在125mm內(nèi)。要求較小的轉(zhuǎn)矩產(chǎn)生較大的軸向力。這種傳動為間歇性工作，工作速度不高，且要求可自鎖。
2、傳動方案
方案一：齒輪傳動：
優(yōu)點:瞬間傳動比恒定且穩(wěn)定性高,結(jié)構(gòu)穩(wěn)定可靠,噪音低,傳動功率大效率高.
缺點:應(yīng)用環(huán)境要求高,潤滑條件要好,不適合灰塵較多以及距離較遠的兩軸之間的傳動,制造和安裝精度高.
方案二：鏈條傳動：
缺點:磨損變形影響使用,使用過程有一定的聲響.
優(yōu)點:對應(yīng)用環(huán)境要求一般吧不是很高,可用在高溫、重載、低速、塵埃較大的環(huán)境并且適合較遠兩軸間的傳動.其實自行車并不只是能用鏈條傳動,也可以用齒輪傳動的,只不過用齒輪傳動結(jié)構(gòu)會更復(fù)雜些.可能會費力些
方案三：蝸桿蝸輪的傳動：
優(yōu)點： 結(jié)構(gòu)緊湊、傳動比大。傳動平穩(wěn)，噪音小。 可制成具有自鎖性的蝸桿。
缺點：嚙合處有較大的相對滑動，因而發(fā)熱量大，效率低，傳動效率一般在70%~80%之間。還有一個缺點就是造價高。
壓紙器由電機驅(qū)動，經(jīng)減速傳動系統(tǒng)蝸桿蝸輪，帶動螺母，再將動力傳給螺桿，由螺桿直接帶動壓紙器松開或緊壓紙張。如圖1：
圖 1
電機驅(qū)動，經(jīng)減速傳動系統(tǒng)蝸桿蝸輪：
圖 2
蝸輪帶動螺母，再將動力傳給螺桿：
圖 3
3、設(shè)計內(nèi)容
1.根據(jù)任務(wù)要求，進行壓紙器機械系統(tǒng)總體方案設(shè)計，確定減速傳動系統(tǒng)、執(zhí)行系統(tǒng)的組成
2. 根據(jù)設(shè)計參數(shù)和設(shè)計要求，使執(zhí)行系統(tǒng)具有較好的傳力性能。
3. 選擇電動機型號，分配減速傳動系統(tǒng)中各級傳動的傳動比，并進行傳動機構(gòu)的工作能力設(shè)計計算。
4. 對壓紙系統(tǒng)進行結(jié)構(gòu)設(shè)計，繪制裝配圖及關(guān)鍵零件工作圖。
5. 編寫機械設(shè)計畢業(yè)設(shè)計報告。
4、切紙機的壓紙傳動設(shè)計
4.1電機選擇
（1） 選擇電機的類型：
按工作要求和條件，選用三相異步電動機，封閉式結(jié)構(gòu)，電壓380v，y型。
（2） 選擇電機的容量：
電機所需功率按式
由式
因此
由電機至壓紙器的傳動總效率為：
：軸承傳動效率： 0.98
：蝸桿蝸輪傳動效率： 0.78
：螺桿螺母傳動效率： 0.37
所以
（3）確定電動機的轉(zhuǎn)速
螺桿工作轉(zhuǎn)速為：
按表推薦的傳動比合理范圍，取蝸桿蝸輪的傳動比為10~40，故電動機轉(zhuǎn)速的可選范圍為：
根據(jù)容量和經(jīng)濟情況，并符合這一范圍的轉(zhuǎn)速，由有關(guān)手冊查出并比較出，電動機型號為YS7124較適合。其主要性能如下表：
型號
額定功率
Kw
滿載時
轉(zhuǎn)速
r/min
電流
（380v時）
A
效率
%
功率因數(shù)
YS7124
0.37
1420
1.12
84
0.77
1
2
?
確定傳動裝置的總傳動比和分配傳動比
由于本裝置只有一個減速器，所以減速器傳動比為
4.2計算傳動裝置的運動和動力參數(shù)
將傳動裝置由高速至低速為相鄰的傳動比
相鄰兩軸間的傳動效率
為各傳動裝置的輸入功率（kW）
為各傳動裝置的輸入轉(zhuǎn)矩（kW）
為各軸的轉(zhuǎn)速（r/min）
則各軸轉(zhuǎn)速：
I蝸桿蝸輪 r/min
螺桿螺母
各傳動裝置輸入功率：
I 蝸桿蝸輪
II螺桿螺母
各傳動輸出功率分別為輸入功率乘軸承效率0.98，
則
各傳動裝置輸入轉(zhuǎn)矩：
電機輸出轉(zhuǎn)矩：
傳動裝置輸入轉(zhuǎn)矩 ：
蝸桿蝸輪輸入轉(zhuǎn)矩：
5、蝸桿蝸輪設(shè)計
（1）. 選擇材料并確定許用應(yīng)力
蝸桿：由于功率不大，采用45#鋼調(diào)質(zhì)處理，硬度:217~255HBS。
蝸輪：因轉(zhuǎn)速較高，采用抗膠合性能好的鑄錫青銅，正火處理，ZCuSn10P1金屬模鑄造。
查表得：蝸輪材料的基本許用接觸應(yīng)力：
查表得：蝸輪材料的基本許用彎曲應(yīng)力：
計算應(yīng)力循環(huán)次數(shù)Ｎ（蝸輪轉(zhuǎn)速＝1420/2=64.5r/min）
次
計算壽命系數(shù)?。?
計算許用應(yīng)力：
（２）確定蝸桿蝸輪頭數(shù)和渦輪齒數(shù)
根據(jù)傳動功率不大，并要求蝸桿有自鎖等情況，取。
（３）計算蝸輪轉(zhuǎn)矩
由于在電機的選擇中可知：
＝
（４）按齒面接觸疲勞強度計算
取載荷系數(shù)Ｋ＝１.08，由式得：
由表查得，按選取
得：
查表得：
由式得：
齒根的彎曲疲勞強度校核合格。
（5）驗算傳動效率
蝸桿分度圓速度為：
由表查得：
與原估計是相接近的。
（6） 熱平衡計算
取溫室 ，因通風散熱條件不好，取散熱系數(shù) 。
油溫 由式子得：
由于在范圍內(nèi)，符合要求。
（7） 中心距a及各部分尺寸
（8） 精度選擇
由選擇精度等級。精度等級選擇參考GB10085 –88。
由于<3m/s, 故選8級精度。
（9） 繪制蝸桿、蝸輪零件圖
蝸桿
蝸輪
（10） 支撐蝸桿的軸承選擇
從蝸桿圖可知，支撐右邊的軸承內(nèi)徑是20mm。
根據(jù)實際請況選軸承型號：GB278-82 型號 6004Z
軸承的校核：
因其為輕微沖擊，查表得沖擊負載系數(shù)
6、螺桿螺母設(shè)計
(1)選擇材料和許用應(yīng)力
螺桿材料選45鋼,熱處理T235, 。
螺母材料選HT200, 淬火處理，由表可得：
,取50 , 。
螺旋系不是高速,由表可得:[Ｐ]=4～7，取4。
(2)按時磨性計算螺紋中徑
取1.2,
根據(jù)實際情況，可設(shè)計d=38
梯形螺紋，中徑精度，螺旋副標記為：Tr3816(8) 7H/7e。
螺紋高度
則螺紋圈數(shù) 圈.
(3)自鎖性驗算
由于系雙頭螺紋,導程S=16，P=8mm.故螺紋升角為:
查表得：鋼對鑄鐵的,取0.15
可得：
，故可自鎖。
(4)螺桿強度驗算
由公式得，螺紋摩擦力矩：
，故螺桿強度可靠。
(5)螺母螺紋強度驗算
螺母材料強度低于螺桿故只需算螺母羅紋強度即可。
由公式可得出：
牙根寬度： b=0.65p=0.65*8=5.2
基本牙高：
代入公式得：
所以，螺母強度可靠。
(6)螺桿穩(wěn)定性驗算
從實際工作情況測得，壓紙器的最大工作高度125mm。
由表查得：
由公式得：
由表查得：
臨界載荷：
故穩(wěn)定性條件滿足。
(7)效率計算
由公式得：
(8)繪圖
螺母
螺桿
(9)支撐螺母的軸承選擇
根據(jù)實際請況選軸承型號：7511E和7211AC
軸承的校核：查表得沖擊系數(shù)為
當量動載荷：
額定動載荷：
????
7、設(shè)計小結(jié)
在指導老師劉老師和同學的細心指導和幫助下，這次畢業(yè)設(shè)計終于完成了。在畢業(yè)設(shè)計中自己遇到了很多的困難和問題，但經(jīng)過自己的努力和老師的指導，克服這些難題。從這次畢業(yè)設(shè)計中自己收獲了很多。不僅系統(tǒng)的復(fù)習了大一大二所學的專業(yè)知識，還更好的把所學的知識和實際聯(lián)系到了一起
從開始直到設(shè)計基本完成，我有許多感想。這是我們比較獨立的在自己的努力下做一個與畢業(yè)相關(guān)的設(shè)計。首先要多謝老師給我們的這個機會，還要感謝諸多工程師的幫助。我深切的感覺到，在這次設(shè)計中也暴露出我們的許多薄弱環(huán)節(jié)，很多學過的知識不能靈活應(yīng)用，在這次作業(yè)后才漸漸掌握，以前學過的東西自己并不是都掌握了，很多知識都已很模糊，經(jīng)過這次設(shè)計又回憶起來了。
做設(shè)計的期間用到的AutoCAD畫圖軟件也在不斷練習中進一步深入，學會了如何去應(yīng)用工程手冊，我體會到老師的良苦用心。
總的說來，我感覺這次畢業(yè)設(shè)計中學到了很多東西，是很有意義的。
?
?
參考文獻
[1]陳立德主編.機械設(shè)計基礎(chǔ).北京：高等教育出版社，2004
[2]龔桂義主編.機械設(shè)計畢業(yè)設(shè)計指導書.北京：高等教育出版，1990
[3]成大先主編.機械設(shè)計手冊.第一卷，第二卷，第三卷.北京：化學工業(yè)出版社，2008
[4]李澄，吳天生，聞百橋主編.機械制圖.北京：高等教育出版社，2008
[5]陳于平，高曉康.互換性與測量技術(shù).北京：高等教育出版社，2005年
[6] 東北工學院《機械零件設(shè)計手冊》編寫組主編.機械零件設(shè)計手冊.北京：冶金工業(yè)出版社，1991
[7]張志禮等.機械設(shè)計.北京：科學出版社，2008
[8]鞏云鵬等.機械設(shè)計課程設(shè)計.北京：冶金工業(yè)出版社，1999
[9]石永剛、吳央芳主編.凸輪機構(gòu)設(shè)計與應(yīng)用創(chuàng)新.北京：機械工業(yè)出版社，2007
[10]王金，張錫安.機械設(shè)計程序設(shè)計.沈陽：東北工學院出版社，1991
17
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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