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造紙廠中的卷筒和平板紙切割
摘要：
這篇文章描述了發(fā)生在葡萄牙造紙廠，在設(shè)計和最優(yōu)化的切割卷筒和平板紙的生產(chǎn)過程中所出現(xiàn)的一個現(xiàn)實性的工業(yè)難題。全球性的工業(yè)難題主要是寬度的設(shè)定，這些寬度的設(shè)定在生產(chǎn)的過程中一定要符合相應(yīng)的條款，主要的目標(biāo)就是要能夠使一系列已經(jīng)定好的卷筒和平板紙與主要的卷筒相分離。寬度結(jié)合的這個過程將決定即將被生產(chǎn)加工的主要卷筒的質(zhì)量和重量、切割的樣式以使損失達到最小，從而滿足生產(chǎn)的需要。
進程中的技術(shù)進程對這個擁有兩個加工階段的進程起著決定性的作用。本文還介紹了模型的細節(jié)和解決的方法。還包括一些解說性的計算方面的結(jié)果。
2003 Elsevier Ltd. 版權(quán)所有
關(guān)鍵詞：結(jié)合的最優(yōu)化；成品紙庫存；自發(fā)式
1、引言
在造紙廠設(shè)計紙張的生產(chǎn)過程中采用了許多具有實質(zhì)效果的特殊方式，其中的每個方式又有自己獨特的特點，同時這種方式還要求必須有8個精確無誤的數(shù)學(xué)公式和解決方法[1-3]。然而，把所耗費的損失降到最小卻是由客觀性的物理作用而決定的。其余的部分則是由運行整個過程的時間、數(shù)字和切割型號的特性等因素列表而組成的。另外還有一些是常見的強制力、相關(guān)客戶的說明書、戰(zhàn)略決定和生產(chǎn)過程中的一些技術(shù)特性。
這篇文章主要闡述了應(yīng)葡萄牙造紙廠CPP的要求而設(shè)計的一套流程，主要用以產(chǎn)品的設(shè)計和切割紙張的卷筒。這套系統(tǒng)被命名為[COOL]（[COOL]代表葡萄牙的單詞，意思是能使寬度的聯(lián)合達到最佳化），是復(fù)雜系統(tǒng)的一部分，用來支撐生產(chǎn)紙張和操縱托盤的工具。
在這篇論文里所解決的就是制定切割的型號以及生產(chǎn)依照型號和質(zhì)量而進行分類的紙張來滿足預(yù)定規(guī)格的卷筒和平板紙。這個系統(tǒng)基本上就是處理在符合定單的前提下設(shè)計紙張，切割主要卷筒的過程中所出現(xiàn)的問題。切割預(yù)計要與主軸相連，并且要考慮把損失降到最小化同時還要滿足定單的數(shù)量要求。一個多樣化的技術(shù)上、操作上出現(xiàn)的強制力包含在設(shè)計的進程之中，并引發(fā)了一個奇怪的難關(guān)。
從這種觀點來看，這項難題存在于成品紙庫存這個問題之中[4-6]。這個問題忽視在卷筒末端將會出現(xiàn)損失而公式化的被采用，這樣1D進程就被設(shè)計出來。在切割過程中的技術(shù)特性決定了擁有兩個工業(yè)階段的進程的方法論的必要性。另一個1D的切割問題可以在已經(jīng)出版的文學(xué)上得到證實。不僅是造紙業(yè)，這項進程還可以應(yīng)用到其他工業(yè)，例如鋼鐵行業(yè)[7、8]和塑料行業(yè)[9]。
我們設(shè)想一個原始的解決方法來解決前面所出現(xiàn)的問題，與僅用人工就能獲得解決方法相比，這種方法就正如造紙廠所證實的那樣用節(jié)省紙張來達到重視改良的效果。這種方法是基于兩個獨特的呈直線型的計劃模型，而這個模型則是由單一的阿拉伯算法計算而得的。那么，為了滿足以往被忽略的全部限制所得到的解決方法就必須類似于后選擇步驟。這個解決方法所獲得的有效性是通過工業(yè)難題的整合應(yīng)用模型的改革運動而獲得的，這種改革運動是用微軟CPLEX⒍0的商業(yè)最優(yōu)化來解決的。
這篇論文的整體結(jié)構(gòu)如下：第2段介紹了生產(chǎn)的難題和它的工業(yè)背景。特殊強調(diào)了工業(yè)環(huán)境的特殊作用以及相關(guān)的解決辦法。第3段和第4段將描述出現(xiàn)的問題和解決問題相應(yīng)的方法論。第4段列舉了一個小例子來說明解決問題的進程。第5段現(xiàn)實性的討論了一些可能出現(xiàn)的結(jié)果。
2、工業(yè)環(huán)境
這一研究案例是在一家葡萄牙造紙廠中進行的，因為是從紙漿開始生產(chǎn)紙產(chǎn)品，所以此案例可視為一個垂直工業(yè)流程，產(chǎn)品包含卷筒紙和平板紙。工廠在兩類市場中運做：一類是標(biāo)準(zhǔn)紙，另一類是按要求預(yù)定紙。整個生產(chǎn)周期一共六周，并且由于技術(shù)因素，紙生產(chǎn)中存在一個或加速或減速的提前預(yù)警產(chǎn)品的生產(chǎn)連續(xù)性問題。
圖1 制造流程
圖1顯示了該廠通過生產(chǎn)線的成品紙從紙漿中生產(chǎn)出再按固定寬度繞在主卷筒機上,然后主卷筒機隨卷線機將卷筒紙切成小筒紙,這些卷筒紙或被直接提供給客戶或送入中間倉庫。
在卷線機和切割機上都有少量的固定寬度的切割碎片留在紙上。這主要由采用的生產(chǎn)流程所決定。
圖2顯示了計劃和生產(chǎn)流程的相干方面。重點是主產(chǎn)品和副產(chǎn)品計劃和產(chǎn)量的不同。計劃是基于客戶同屬的產(chǎn)品定貨。同一紙種和等級的卷筒紙和輔助卷筒紙一起構(gòu)成了預(yù)訂產(chǎn)品的規(guī)格。這種助卷筒紙只包含卷筒紙和平板紙中的一種。因此兩種輔助卷筒紙是有區(qū)別的：一種是平板輔助卷筒紙，一種是卷筒輔助卷筒紙。輔助卷筒紙只按照有關(guān)的主卷筒紙構(gòu)成其切割方式的。
介紹了輔助切割，以便于對于產(chǎn)品生產(chǎn)過程和采取的解決方法有一個更好地了解。這與有關(guān)的技術(shù)過程密切相關(guān)。且當(dāng)應(yīng)用到切紙機時，就要求對于額外的紙寬進行考慮。在主切割方式的副切割方式的定義由相應(yīng)的兩項解決方案決定。在副卷筒和切割方法的終結(jié)中必須進行討論。這些限制決定了方法的可行性。
訂購系統(tǒng)如圖3所示，可在國內(nèi)外市場訂購（因為此公司也在葡萄牙以外運作）由市場營銷部負責(zé)。如果認為合適的話，與外部訂購相同。可源于這些定單產(chǎn)品要求或是切割定單或是預(yù)期定單。當(dāng)一位客戶的卷筒紙的定單可由現(xiàn)存（存在半成品倉庫）的卷筒紙滿足其要求時就生成一份切割定單，當(dāng)客戶的平板紙定單可由現(xiàn)存（存于標(biāo)準(zhǔn)倉庫）的平板紙滿足時，就生成一份預(yù)期定單。
圖2計劃和生產(chǎn)流程
圖3 訂購系統(tǒng)
3問題描述
為了使完成生產(chǎn)定單時的浪費最少，造紙中存在的工作主要是切紙方法的整合過程。它決定了主卷筒紙的重量和質(zhì)量。生產(chǎn)系統(tǒng)的發(fā)展將支持產(chǎn)品定單的切割計劃。不會干擾相關(guān)定單的完成與每個生產(chǎn)循環(huán)的成品紙的生產(chǎn)。這些是市場營銷部所做的決定，最終在使用cool系統(tǒng)的模擬中得到了支持。
在與主卷筒機有關(guān)的切割方法的定義中必須考慮到一些限制條件。這些限制條件可分為以下兩方面：
操作限制（包括管理和客戶要求）
*只有每個寬度單位的等重的卷筒紙才可以結(jié)合在一起
*只有內(nèi)外直徑相同才可以結(jié)合
*客戶的內(nèi)外徑規(guī)格的要求必須得到滿足
*必須考慮到輔助卷筒機的任務(wù)，因為切紙機有不同的特點。對于切割方法提出了最小寬度的要求，以便使可用的機器得以利用。
技術(shù)限制（主要歸因于機器的特點）
* 在輸入時主卷筒機的最大和最小寬度；
* 旋切刀的限制數(shù)；
* 切割機最大，最小板紙寬度；
* 切割機的最大切割卷筒紙直徑；
* 在切割機和卷線機中的邊料；
在造紙工業(yè)中還必須考慮一系列的歐洲耗材標(biāo)準(zhǔn)。當(dāng)在完成訂單的過程中（見表1）在這個范圍內(nèi)，客戶有義務(wù)接受訂貨數(shù)量的不足。當(dāng)產(chǎn)量大于最大定購數(shù)量時，市場部會努力勸說客戶接受這些額外數(shù)量的產(chǎn)品。由于產(chǎn)品固有的損耗，在計劃階段是決不會考慮負損耗的。
4.解決步驟
所采取的解決方法在生產(chǎn)中已經(jīng)清楚表明。主要可分三步。如圖4所示。第一部包括的內(nèi)容見表1。基于主卷筒的固定寬度和訂單的固定寬度來選擇輔助卷筒和切割方法。之后一系列的切割方法要通過排除不完善的輔助卷筒機方法或切割方法來進行篩選。所有剩下的切割方法都必須排除生產(chǎn)過程中存在的技術(shù)操作限制而具有可行性。
第二步，在第一步中被選擇和接受的切割方法在解決問題的線性規(guī)劃模式的應(yīng)用。發(fā)展兩個真實問題的可行方法。按照循環(huán)步驟的線性解決方案，要滿足在前面步驟中被忽略的各種相關(guān)變化的限制條件。
在隨后部分都會得到詳細的闡述。
介紹一個小的真實的工業(yè)例子來說明解決步驟，它與主卷筒紙長度不定情況下的生產(chǎn)要求。內(nèi)直徑或外直徑?jīng)]有被確定的含義是在平板紙購貨或卷筒紙定購中客戶并未指定直徑的具體數(shù)值。寬2520mm紙的等級為250g/mm,厚度為345mm。相關(guān)的產(chǎn)品要求見表2。
自發(fā)循環(huán)
循環(huán)過程來解決cp模式和在前面被忽略的整合屬性的采用。限制例如：
(1)客戶確定的卷筒紙直徑必須得滿足,這就意味包括卷筒必須總是按要求的直徑采取多樣性，為了使這個過程的影響最小，定購數(shù)量如表3。解決方案按照固定直徑卷筒的在建立zp模型前要選擇卷筒長度最多樣的一個。
(2)板紙的結(jié)合重量最少,相當(dāng)于紙最小長度,以便避免切紙機的無效率使用
(3)與以前的幾項相似切割方法限制的重量最小以便卷線機避免無效率使用,同時使用每種切割方法來切割最小量的紙.
循環(huán)模式用Lp模式的最終解決方案來開始并努力調(diào)整這些方式長度來滿足以上提到的限制.新的方法雖可能與Lp1保持相似,但必須滿足定購數(shù)量,第一,循環(huán)過程中盡量排除那些沒有最低重量條件的方式(以上限制2和3).必須提前注明不要排除訂單的獨特形式.然后,剩下的形式要足足包括使用,以補償被破壞的方法的效果.
這個過程基本由連續(xù)的選擇在每個方法中不能被滿足的項目數(shù)量的切割方式.然后討論用第一種切割方法切割的數(shù)量,最后,使沒有被滿足的項目得以滿足.這個過程不斷重復(fù),直到在所有一切切割方式中沒有滿足的項目都得以滿足為止.
即使當(dāng)使用模式1時,這個循環(huán)過程也能導(dǎo)致標(biāo)準(zhǔn)耗材以上的過量生產(chǎn).
表3所示的解決方案中只有與板紙結(jié)合的最小重量相公的限制沒有被fp16的長度所滿足。因為它由此方法決定的板紙結(jié)合的最小重量比限制條件小，為（2730：00mm）.因為在此種方法中只有PR1002而且在FP21（x14）中也存在。所以FP16方法可被排除。FP21的數(shù)量最終的解決方法見表4。
圖5顯示了表2中COOL系統(tǒng)產(chǎn)出的數(shù)據(jù)。
圖5大規(guī)模例證的計算結(jié)果
5計算結(jié)果
計算測試的主要目的在于確認所采取的解決步驟的有效性和在發(fā)展出的兩個線形程序模式（模式1和模式2）中建立一個對比分析。在第一組計算中所甬道的數(shù)據(jù)是由市場部提供的，它與造紙長中要解決的實際問題一致。有關(guān)的定單數(shù)從3到16，定單的最大和最小寬度分別為1392mm和238mm,平均寬度為690mm。盡管這些只有相對較小的例子，但通過這些例子，公司希望使系統(tǒng)的應(yīng)用者能夠容易的對于COOL系統(tǒng)在使用的初始階段的表現(xiàn)給以評價。
計算所用的數(shù)據(jù)可在www.apdio/sicuo中找到。
計算法則由c語言完成。計算結(jié)果由450赫茲的奔騰3處理器完成。
為了對用以上的描述的線形模式和自發(fā)循環(huán)得到的解決方法的質(zhì)量進行評價，使用了IP模式。這種IP模式能使生產(chǎn)的紙的數(shù)量最小同時又能嚴格滿足定單的數(shù)量。為了考慮上述提到的全部限制，包括幾個不同的變化：平板紙結(jié)合的最小重量（最小重量平板紙），應(yīng)用復(fù)合整合程序模式CPLEXV。6。0版軟件來解決IP模式。
見圖6。每個發(fā)展得到解決方法的表現(xiàn)（基于兩個LP模式。模式1，模式2）都得到了客觀的評價圖6（A），用IP模式得到的結(jié)果的速率和用線形步驟得到的IP模式在每個測試實驗中得到表現(xiàn)：Y軸的數(shù)值為1。00與IP模式結(jié)果相同。從此章中可知基于過程的線形結(jié)果大部分是與用IP模式得到的數(shù)據(jù)一致的：模式1得到的數(shù)據(jù)的與測試的%70相同，而從模式二得到的數(shù)據(jù)中有50%與之相同。但有一個例外，IP結(jié)果決不超過22%。
本章中的圖6（B）用于證明所采用的線形方法的充分性。在循環(huán)過程前后的結(jié)果速率都以計算。在循環(huán)過程之前在Y軸上的數(shù)值1。00就與LP模式的結(jié)果相同。在絕大多數(shù)情況下，LP路徑的結(jié)果都與最終結(jié)果一致，這就意味著在循環(huán)過程中考慮的整個屬性限制都不會改變線形程序的結(jié)果。
這兩章都表明了用模式1（使生產(chǎn)的紙長度最小而且不允許超出耗材標(biāo)準(zhǔn)過量生產(chǎn)）得到的結(jié)果都比模式2（不產(chǎn)生中間庫存）的結(jié)果好。而且，這些說明了有必要改進模式2的循環(huán)過程。
表5對比了由兩個線形程序得到的結(jié)果，包括3個組成部分：生產(chǎn)出的中間庫存的數(shù)量，在標(biāo)準(zhǔn)上超出生產(chǎn)的產(chǎn)品的數(shù)量以及不可再利用的紙的數(shù)量（廢品）。所有的數(shù)量都按照整體數(shù)量的百分數(shù)來表示不考慮采取的每一個模式的客觀作用：模式2盡量不產(chǎn)生中間庫存而模式1盡量不超量生產(chǎn)產(chǎn)品。盡管如此有時這些超出部分是循環(huán)過程的必然結(jié)果。但與模式2比較它的數(shù)量就遠小于模式2所產(chǎn)生的庫存量。
因為只有廢品是不可再利用的部分所以圖6對基于此過程的兩種LP模式所得到的價值進行了一個對比。最終結(jié)果是在產(chǎn)生的廢物最小化方面，用模式1得到的價值比用模式2得到的價值略微小一些。
按照這一系列的對比實驗，模式1在所有方面的表現(xiàn)均優(yōu)于模式2。但模式2仍可在COOL系統(tǒng)的最終版本中使用。因為每種模式都有可能使得到的解決方案都更或甚至要求不同的工業(yè)條件：當(dāng)允許或建議產(chǎn)生中間庫存的模式1可被利用。當(dāng)要求生產(chǎn)足夠多的中間庫存以滿足市場的模式2就會被使用。就效率而言，LP 方法可以只使用采用IP方法生產(chǎn)時的時間的75%盡管對于測試的例子中的IP方法的平均解決時間為18小時，當(dāng)在實際生產(chǎn)過程中需要時也會使用。進行搜集和測試了更大規(guī)模的一系列例子以便評價當(dāng)面對大規(guī)模定貨時基于IP而發(fā)展的方法的效率的表現(xiàn)。
所有這些例子包括30個不同面的例子和在以上提到的真實定單中隨機抽取的例子。主要的目的是為了對于我們的方法在特殊條件下的效率進行評估。
這些測試采用模式1。結(jié)果速率和LP+ROUND-UP/IP見圖5正如我們所看到的，我們用我們的方法和用基于CPLEX的IP方法在客觀作用上并無大的異同。采用兩種方法，用于解決10個例子所使用的時間見表6。
正如我們預(yù)測的，在整個程序的時間中選擇所使用的時間總是很長。但這一缺點并未經(jīng)常限制整合程序的使用，例如在例子第5，7，10中便是如此。在這些例子中操作效率的不同也許更大一些。
COOL系統(tǒng)已在造紙廠中證明了其有效性并正在廣泛使用。在經(jīng)濟和環(huán)境上的巨大利益得到認可。根據(jù)報道轉(zhuǎn)換消耗已經(jīng)降低了3%。這意味每年多余1000的節(jié)約。而且在能源上也得到了巨大的節(jié)約。況且，與紙不同，能源不可重復(fù)利用。
6 結(jié)論
這篇論文介紹了COOL系統(tǒng)，此系統(tǒng)是在葡萄牙造紙長中解決特定的切割儲存問題時發(fā)展出來的。使得在生產(chǎn)和切割主卷筒紙時的邊緣廢料最少是發(fā)展此解決過程的主要目的。由于技術(shù)原因，主卷筒紙分成兩個部分，同時滿足一系列的技術(shù)和操作限制。兩項切割的特點對于采取的解決過程是至關(guān)重要的。
由于此問題的結(jié)合屬性，基于切割方法計算的解決過程得以發(fā)明。為了滿足大部分的限制條件，這些方法是要進行選擇的。并且這些方法在決定每種紙的生產(chǎn)的重量和數(shù)量的問題的線性程序計算中被用作選擇列。以往被忽略的整合屬性的限制通過線形程序解決方案在之后的選擇中也被包括進去了。
基于模式的兩個線形程序得以發(fā)展和得到測試。盡管使用兩個模式得到的結(jié)果非常令人滿意，但是在它們中的對比分析和在每一個中的對比分析以及用整合程序模式的到的方法的分析表明循環(huán)程序仍有必要改進。盡管如此，卻應(yīng)該摒棄發(fā)展自發(fā)解決問題系統(tǒng)的想法。
自動化切紙機在工業(yè)上有很大優(yōu)勢：可以減少產(chǎn)品循環(huán)和可以完成即時的定單，還可以提高客服質(zhì)量。由于巨大的經(jīng)濟和環(huán)境利益以及操作優(yōu)勢，COOL系統(tǒng)已經(jīng)在造紙廠中得以應(yīng)用，并得到了積極的反饋。
參考文獻：
[1] Haessler RW. A heuristic programming solution to a nonlinear cutting stock problem. Management Science
1971; 17(12):B793–802.
[2] Johnson MP, Rennick C, Zak E. Skiving addition to the cutting stock problem in the paper industry. SIAM Review
1997; 39(3):472–83.
[3] Johnston RE. OR in the paper industry. OMEGA the International Journal of Management Science 1981; 9(1):43–50.
[4] Dowsland KA, Dowsland WB. Packing problems. European Journal of Operational Research 1992; 56:2–14.
[5] Golden BL. Approaches to the cutting stock problem. AIIE Transactions 1976; 8(2):265–74.
[6] Hinxman A. The trim loss and assortment problems: a survey. European Journal of Operational Research 1980;5:8–18.
[7] Carvalho JVd, Rodrigues AG. An LP-based approach to a two-stage cutting stock problem. European Journal of
Operational Research 1995;84:580–9.
[8] Ferreira JS, Neves MA, Fonseca e Castro P. A two-phase roll cutting problem. European Journal of Operational
Research 1990;44:185–96.
[9] Haessler RW. Solving the two-stage cutting stock problem. OMEGA the International Journal of Management
Science 1979; 7(2):145–51.
[10] Oliveira JF, Ferreira JS. A faster variant of the Gilmore and gomory technique for cutting stock problems. JORBEL
1994; 34(1):23–38.
References
[1] Haessler RW. A heuristic programming solution to a nonlinear cutting stock problem. Management Science
1971;17(12):B793–802.
[2] Johnson MP, Rennick C, Zak E. Skiving addition to the cutting stock problem in the paper industry. SIAM Review
1997;39(3):472–83.
[3] Johnston RE. OR in the paper industry. OMEGA the International Journal of Management Science 1981;9(1):43–50.
[4] Dowsland KA, Dowsland WB. Packing problems. European Journal of Operational Research 1992;56:2–14.
[5] Golden BL. Approaches to the cutting stock problem. AIIE Transactions 1976;8(2):265–74.
[6] Hinxman A. The trim loss and assortment problems: a survey. European Journal of Operational Research 1980;5:8–18.
[7] Carvalho JVd, Rodrigues AG. An LP-based approach to a two-stage cutting stock problem. European Journal of
Operational Research 1995;84:580–9.
[8] Ferreira JS, Neves MA, Fonseca e Castro P. A two-phase roll cutting problem. European Journal of Operational
Research 1990;44:185–96.
[9] Haessler RW. Solving the two-stage cutting stock problem. OMEGA the International Journal of Management
Science 1979;7(2):145–51.
[10] Oliveira JF, Ferreira JS. A faster variant of the Gilmore and gomory technique for cutting stock problems. JORBEL
1994;34(1):23–38.
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 t
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