卷筒纸胶印机印刷滚筒设计【三维SW】【4张CAD图纸和毕业论文全套】
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【温馨提示】====设计包含CAD图纸 和 DOC文档，均可以在线预览，所见即所得，，dwg后缀的文件为CAD图，超高清，可编辑，无任何水印，，充值下载得到【资源目录】里展示的所有文件======课题带三维，则表示文件里包含三维源文件，由于三维组成零件数量较多，为保证预览的简洁性，店家将三维文件夹进行了打包。三维预览图，均为店主电脑打开软件进行截图的，保证能够打开，下载后解压即可。======详情可咨询QQ：414951605
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造纸厂中的卷筒和平板纸切割摘要：这篇文章描述了发生在葡萄牙造纸厂，在设计和最优化的切割卷筒和平板纸的生产过程中所出现的一个现实性的工业难题。全球性的工业难题主要是宽度的设定，这些宽度的设定在生产的过程中一定要符合相应的条款，主要的目标就是要能够使一系列已经定好的卷筒和平板纸与主要的卷筒相分离。宽度结合的这个过程将决定即将被生产加工的主要卷筒的质量和重量、切割的样式以使损失达到最小，从而满足生产的需要。进程中的技术进程对这个拥有两个加工阶段的进程起着决定性的作用。本文还介绍了模型的细节和解决的方法。还包括一些解说性的计算方面的结果。 2003 Elsevier Ltd. 版权所有关键词：结合的最优化；成品纸库存；自发式1、引言在造纸厂设计纸张的生产过程中采用了许多具有实质效果的特殊方式，其中的每个方式又有自己独特的特点，同时这种方式还要求必须有8个精确无误的数学公式和解决方法1-3。然而，把所耗费的损失降到最小却是由客观性的物理作用而决定的。其余的部分则是由运行整个过程的时间、数字和切割型号的特性等因素列表而组成的。另外还有一些是常见的强制力、相关客户的说明书、战略决定和生产过程中的一些技术特性。这篇文章主要阐述了应葡萄牙造纸厂CPP的要求而设计的一套流程，主要用以产品的设计和切割纸张的卷筒。这套系统被命名为COOL（COOL代表葡萄牙的单词，意思是能使宽度的联合达到最佳化），是复杂系统的一部分，用来支撑生产纸张和操纵托盘的工具。在这篇论文里所解决的就是制定切割的型号以及生产依照型号和质量而进行分类的纸张来满足预定规格的卷筒和平板纸。这个系统基本上就是处理在符合定单的前提下设计纸张，切割主要卷筒的过程中所出现的问题。切割预计要与主轴相连，并且要考虑把损失降到最小化同时还要满足定单的数量要求。一个多样化的技术上、操作上出现的强制力包含在设计的进程之中，并引发了一个奇怪的难关。从这种观点来看，这项难题存在于成品纸库存这个问题之中4-6。这个问题忽视在卷筒末端将会出现损失而公式化的被采用，这样1D进程就被设计出来。在切割过程中的技术特性决定了拥有两个工业阶段的进程的方法论的必要性。另一个1D的切割问题可以在已经出版的文学上得到证实。不仅是造纸业，这项进程还可以应用到其他工业，例如钢铁行业7、8和塑料行业9。我们设想一个原始的解决方法来解决前面所出现的问题，与仅用人工就能获得解决方法相比，这种方法就正如造纸厂所证实的那样用节省纸张来达到重视改良的效果。这种方法是基于两个独特的呈直线型的计划模型，而这个模型则是由单一的阿拉伯算法计算而得的。那么，为了满足以往被忽略的全部限制所得到的解决方法就必须类似于后选择步骤。这个解决方法所获得的有效性是通过工业难题的整合应用模型的改革运动而获得的，这种改革运动是用微软CPLEX0的商业最优化来解决的。这篇论文的整体结构如下：第2段介绍了生产的难题和它的工业背景。特殊强调了工业环境的特殊作用以及相关的解决办法。第3段和第4段将描述出现的问题和解决问题相应的方法论。第4段列举了一个小例子来说明解决问题的进程。第5段现实性的讨论了一些可能出现的结果。2、工业环境这一研究案例是在一家葡萄牙造纸厂中进行的，因为是从纸浆开始生产纸产品，所以此案例可视为一个垂直工业流程，产品包含卷筒纸和平板纸。工厂在两类市场中运做：一类是标准纸，另一类是按要求预定纸。整个生产周期一共六周，并且由于技术因素，纸生产中存在一个或加速或减速的提前预警产品的生产连续性问题。图1 制造流程图1显示了该厂通过生产线的成品纸从纸浆中生产出再按固定宽度绕在主卷筒机上,然后主卷筒机随卷线机将卷筒纸切成小筒纸,这些卷筒纸或被直接提供给客户或送入中间仓库。在卷线机和切割机上都有少量的固定宽度的切割碎片留在纸上。这主要由采用的生产流程所决定。图2显示了计划和生产流程的相干方面。重点是主产品和副产品计划和产量的不同。计划是基于客户同属的产品定货。同一纸种和等级的卷筒纸和辅助卷筒纸一起构成了预订产品的规格。这种助卷筒纸只包含卷筒纸和平板纸中的一种。因此两种辅助卷筒纸是有区别的：一种是平板辅助卷筒纸，一种是卷筒辅助卷筒纸。辅助卷筒纸只按照有关的主卷筒纸构成其切割方式的。介绍了辅助切割，以便于对于产品生产过程和采取的解决方法有一个更好地了解。这与有关的技术过程密切相关。且当应用到切纸机时，就要求对于额外的纸宽进行考虑。在主切割方式的副切割方式的定义由相应的两项解决方案决定。在副卷筒和切割方法的终结中必须进行讨论。这些限制决定了方法的可行性。订购系统如图3所示，可在国内外市场订购（因为此公司也在葡萄牙以外运作）由市场营销部负责。如果认为合适的话，与外部订购相同。可源于这些定单产品要求或是切割定单或是预期定单。当一位客户的卷筒纸的定单可由现存（存在半成品仓库）的卷筒纸满足其要求时就生成一份切割定单，当客户的平板纸定单可由现存（存于标准仓库）的平板纸满足时，就生成一份预期定单。图2计划和生产流程图3 订购系统3问题描述为了使完成生产定单时的浪费最少，造纸中存在的工作主要是切纸方法的整合过程。它决定了主卷筒纸的重量和质量。生产系统的发展将支持产品定单的切割计划。不会干扰相关定单的完成与每个生产循环的成品纸的生产。这些是市场营销部所做的决定，最终在使用cool系统的模拟中得到了支持。在与主卷筒机有关的切割方法的定义中必须考虑到一些限制条件。这些限制条件可分为以下两方面：操作限制（包括管理和客户要求）*只有每个宽度单位的等重的卷筒纸才可以结合在一起*只有内外直径相同才可以结合*客户的内外径规格的要求必须得到满足*必须考虑到辅助卷筒机的任务，因为切纸机有不同的特点。对于切割方法提出了最小宽度的要求，以便使可用的机器得以利用。技术限制（主要归因于机器的特点）* 在输入时主卷筒机的最大和最小宽度；* 旋切刀的限制数；* 切割机最大，最小板纸宽度；* 切割机的最大切割卷筒纸直径；* 在切割机和卷线机中的边料；在造纸工业中还必须考虑一系列的欧洲耗材标准。当在完成订单的过程中（见表1）在这个范围内，客户有义务接受订货数量的不足。当产量大于最大定购数量时，市场部会努力劝说客户接受这些额外数量的产品。由于产品固有的损耗，在计划阶段是决不会考虑负损耗的。4.解决步骤所采取的解决方法在生产中已经清楚表明。主要可分三步。如图4所示。第一部包括的内容见表1。基于主卷筒的固定宽度和订单的固定宽度来选择辅助卷筒和切割方法。之后一系列的切割方法要通过排除不完善的辅助卷筒机方法或切割方法来进行筛选。所有剩下的切割方法都必须排除生产过程中存在的技术操作限制而具有可行性。第二步，在第一步中被选择和接受的切割方法在解决问题的线性规划模式的应用。发展两个真实问题的可行方法。按照循环步骤的线性解决方案，要满足在前面步骤中被忽略的各种相关变化的限制条件。在随后部分都会得到详细的阐述。介绍一个小的真实的工业例子来说明解决步骤，它与主卷筒纸长度不定情况下的生产要求。内直径或外直径没有被确定的含义是在平板纸购货或卷筒纸定购中客户并未指定直径的具体数值。宽2520mm纸的等级为250g/mm,厚度为345mm。相关的产品要求见表2。 自发循环循环过程来解决cp模式和在前面被忽略的整合属性的采用。限制例如：(1)客户确定的卷筒纸直径必须得满足,这就意味包括卷筒必须总是按要求的直径采取多样性，为了使这个过程的影响最小，定购数量如表3。解决方案按照固定直径卷筒的在建立zp模型前要选择卷筒长度最多样的一个。(2)板纸的结合重量最少,相当于纸最小长度,以便避免切纸机的无效率使用 (3)与以前的几项相似切割方法限制的重量最小以便卷线机避免无效率使用,同时使用每种切割方法来切割最小量的纸.循环模式用Lp模式的最终解决方案来开始并努力调整这些方式长度来满足以上提到的限制.新的方法虽可能与Lp1保持相似,但必须满足定购数量,第一,循环过程中尽量排除那些没有最低重量条件的方式(以上限制2和3).必须提前注明不要排除订单的独特形式.然后,剩下的形式要足足包括使用,以补偿被破坏的方法的效果.这个过程基本由连续的选择在每个方法中不能被满足的项目数量的切割方式.然后讨论用第一种切割方法切割的数量,最后,使没有被满足的项目得以满足.这个过程不断重复,直到在所有一切切割方式中没有满足的项目都得以满足为止. 即使当使用模式1时,这个循环过程也能导致标准耗材以上的过量生产.表3所示的解决方案中只有与板纸结合的最小重量相公的限制没有被fp16的长度所满足。因为它由此方法决定的板纸结合的最小重量比限制条件小，为（2730：00mm）.因为在此种方法中只有PR1002而且在FP21（x14）中也存在。所以FP16方法可被排除。FP21的数量最终的解决方法见表4。图5显示了表2中COOL系统产出的数据。图5大规模例证的计算结果5计算结果计算测试的主要目的在于确认所采取的解决步骤的有效性和在发展出的两个线形程序模式（模式1和模式2）中建立一个对比分析。在第一组计算中所甬道的数据是由市场部提供的，它与造纸长中要解决的实际问题一致。有关的定单数从3到16，定单的最大和最小宽度分别为1392mm和238mm,平均宽度为690mm。尽管这些只有相对较小的例子，但通过这些例子，公司希望使系统的应用者能够容易的对于COOL系统在使用的初始阶段的表现给以评价。计算所用的数据可在www.apdio/sicuo中找到。计算法则由c语言完成。计算结果由450赫兹的奔腾3处理器完成。为了对用以上的描述的线形模式和自发循环得到的解决方法的质量进行评价，使用了IP模式。这种IP模式能使生产的纸的数量最小同时又能严格满足定单的数量。为了考虑上述提到的全部限制，包括几个不同的变化：平板纸结合的最小重量（最小重量平板纸），应用复合整合程序模式CPLEXV。6。0版软件来解决IP模式。见图6。每个发展得到解决方法的表现（基于两个LP模式。模式1，模式2）都得到了客观的评价图6（A），用IP模式得到的结果的速率和用线形步骤得到的IP模式在每个测试实验中得到表现：Y轴的数值为1。00与IP模式结果相同。从此章中可知基于过程的线形结果大部分是与用IP模式得到的数据一致的：模式1得到的数据的与测试的%70相同，而从模式二得到的数据中有50%与之相同。但有一个例外，IP结果决不超过22%。本章中的图6（B）用于证明所采用的线形方法的充分性。在循环过程前后的结果速率都以计算。在循环过程之前在Y轴上的数值1。00就与LP模式的结果相同。在绝大多数情况下，LP路径的结果都与最终结果一致，这就意味着在循环过程中考虑的整个属性限制都不会改变线形程序的结果。这两章都表明了用模式1（使生产的纸长度最小而且不允许超出耗材标准过量生产）得到的结果都比模式2（不产生中间库存）的结果好。而且，这些说明了有必要改进模式2的循环过程。表5对比了由两个线形程序得到的结果，包括3个组成部分：生产出的中间库存的数量，在标准上超出生产的产品的数量以及不可再利用的纸的数量（废品）。所有的数量都按照整体数量的百分数来表示不考虑采取的每一个模式的客观作用：模式2尽量不产生中间库存而模式1尽量不超量生产产品。尽管如此有时这些超出部分是循环过程的必然结果。但与模式2比较它的数量就远小于模式2所产生的库存量。因为只有废品是不可再利用的部分所以图6对基于此过程的两种LP模式所得到的价值进行了一个对比。最终结果是在产生的废物最小化方面，用模式1得到的价值比用模式2得到的价值略微小一些。按照这一系列的对比实验，模式1在所有方面的表现均优于模式2。但模式2仍可在COOL系统的最终版本中使用。因为每种模式都有可能使得到的解决方案都更或甚至要求不同的工业条件：当允许或建议产生中间库存的模式1可被利用。当要求生产足够多的中间库存以满足市场的模式2就会被使用。就效率而言，LP 方法可以只使用采用IP方法生产时的时间的75%尽管对于测试的例子中的IP方法的平均解决时间为18小时，当在实际生产过程中需要时也会使用。进行搜集和测试了更大规模的一系列例子以便评价当面对大规模定货时基于IP而发展的方法的效率的表现。所有这些例子包括30个不同面的例子和在以上提到的真实定单中随机抽取的例子。主要的目的是为了对于我们的方法在特殊条件下的效率进行评估。这些测试采用模式1。结果速率和LP+ROUND-UP/IP见图5正如我们所看到的，我们用我们的方法和用基于CPLEX的IP方法在客观作用上并无大的异同。采用两种方法，用于解决10个例子所使用的时间见表6。正如我们预测的，在整个程序的时间中选择所使用的时间总是很长。但这一缺点并未经常限制整合程序的使用，例如在例子第5，7，10中便是如此。在这些例子中操作效率的不同也许更大一些。COOL系统已在造纸厂中证明了其有效性并正在广泛使用。在经济和环境上的巨大利益得到认可。根据报道转换消耗已经降低了3%。这意味每年多余1000的节约。而且在能源上也得到了巨大的节约。况且，与纸不同，能源不可重复利用。6 结论这篇论文介绍了COOL系统，此系统是在葡萄牙造纸长中解决特定的切割储存问题时发展出来的。使得在生产和切割主卷筒纸时的边缘废料最少是发展此解决过程的主要目的。由于技术原因，主卷筒纸分成两个部分，同时满足一系列的技术和操作限制。两项切割的特点对于采取的解决过程是至关重要的。由于此问题的结合属性，基于切割方法计算的解决过程得以发明。为了满足大部分的限制条件，这些方法是要进行选择的。并且这些方法在决定每种纸的生产的重量和数量的问题的线性程序计算中被用作选择列。以往被忽略的整合属性的限制通过线形程序解决方案在之后的选择中也被包括进去了。基于模式的两个线形程序得以发展和得到测试。尽管使用两个模式得到的结果非常令人满意，但是在它们中的对比分析和在每一个中的对比分析以及用整合程序模式的到的方法的分析表明循环程序仍有必要改进。尽管如此，却应该摒弃发展自发解决问题系统的想法。自动化切纸机在工业上有很大优势：可以减少产品循环和可以完成即时的定单，还可以提高客服质量。由于巨大的经济和环境利益以及操作优势，COOL系统已经在造纸厂中得以应用，并得到了积极的反馈。参考文献：1 Haessler RW. A heuristic programming solution to a nonlinear cutting stock problem. Management Science1971; 17(12):B793802.2 Johnson MP, Rennick C, Zak E. Skiving addition to the cutting stock problem in the paper industry. SIAM Review1997; 39(3):47283.3 Johnston RE. OR in the paper industry. OMEGA the International Journal of Management Science 1981; 9(1):4350.4 Dowsland KA, Dowsland WB. Packing problems. European Journal of Operational Research 1992; 56:214.5 Golden BL. Approaches to the cutting stock problem. AIIE Transactions 1976; 8(2):26574.6 Hinxman A. The trim loss and assortment problems: a survey. European Journal of Operational Research 1980;5:818.7 Carvalho JVd, Rodrigues AG. An LP-based approach to a two-stage cutting stock problem. European Journal ofOperational Research 1995;84:5809.8 Ferreira JS, Neves MA, Fonseca e Castro P. A two-phase roll cutting problem. European Journal of OperationalResearch 1990;44:18596.9 Haessler RW. Solving the two-stage cutting stock problem. OMEGA the International Journal of ManagementScience 1979; 7(2):14551.10 Oliveira JF, Ferreira JS. A faster variant of the Gilmore and gomory technique for cutting stock problems. JORBEL1994; 34(1):2338.References1 Haessler RW. A heuristic programming solution to a nonlinear cutting stock problem. Management Science1971;17(12):B793802.2 Johnson MP, Rennick C, Zak E. Skiving addition to the cutting stock problem in the paper industry. SIAM Review1997;39(3):47283.3 Johnston RE. OR in the paper industry. OMEGA the International Journal of Management Science 1981;9(1):4350.4 Dowsland KA, Dowsland WB. Packing problems. European Journal of Operational Research 1992;56:214.5 Golden BL. Approaches to the cutting stock problem. AIIE Transactions 1976;8(2):26574.6 Hinxman A. The trim loss and assortment problems: a survey. European Journal of Operational Research 1980;5:818.7 Carvalho JVd, Rodrigues AG. An LP-based approach to a two-stage cutting stock problem. European Journal ofOperational Research 1995;84:5809.8 Ferreira JS, Neves MA, Fonseca e Castro P. A two-phase roll cutting problem. European Journal of OperationalResearch 1990;44:18596.9 Haessler RW. Solving the two-stage cutting stock problem. OMEGA the International Journal of ManagementScience 1979;7(2):14551.10 Oliveira JF, Ferreira JS. A faster variant of the Gilmore and gomory technique for cutting stock problems. JORBEL1994;34(1):2338.Reel and sheet cutting at a paper millM. Helena Correia, Jose F. Oliveira, J. Soeiro FerreiraINESC Porto, Instituto de Engenharia de Sistemas e Computadores do Porto, 4200-465 Porto, PortugalFaculdade de Economia e Gestao, Universidade Catolica Portuguesa, 4169-005 Porto, PortugalFaculdade de Engenharia, Universidade do Porto, 4200-465 Porto, PortugalAbstractThis 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; Heuristics1. IntroductionPlanning 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 13. 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 46. 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 environmentThis 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. Plannings 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 descriptionThe 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 procedureThe 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 heuristicThe 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 4Fig.5.Computational results for large-scale instances5. Computational resultsThe 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 5Table 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 developed LP-based approach when a larger number of ordered items are considered. All of these instances include 30 items of various dimensions and were generated by randomly choosing among the items of the real instances considered above. The main purpose of these tests was to evaluate the efficiency of our approach under extreme conditions. These tests were performed using Model(1) and the resulting ratio LP+round-up/IP is depicted in Fig. 6. As can be observed, the solutions obtained with both our approach and the IP approach based on Cplex do not significantly different in terms of objective function. The computational time used by both approaches to solve the 10 instances tested is listed in Table 6. As it would be expected, the optimization time is almost always larger in integer programming. Although the magnitude of this difference does not usually restrain the use of integer programming, some cases occur (P5, P7 and P10) in which this di8erence may be significant in terms of operational efficiency. COOL has been validated by the paper mill and is currently in use. Considerable benefits, both in economic and environmental terms, are proclaimed. The transformation losses were reported as having decreased at least 3%, which corresponds to more than 1000 tons of paper a
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