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同時(shí)優(yōu)化的注塑模具設(shè)計(jì)和加工條件
毛里西奧·卡布雷拉Ríos3 ， 2布萊恩Lilly1 ，和何塞· Castro2的Carlos E. Castro1的
1 ，焊接工業(yè)及系統(tǒng)工程學(xué)的機(jī)械工程及Departement的
美國(guó)俄亥俄州立大學(xué)
美國(guó)俄亥俄州哥倫布市， 43210
3系統(tǒng)工程研究生課程
新萊昂州自治大學(xué)
圣·尼古拉斯·德洛斯加爾薩，墨西哥新萊昂州， 66450
抽象
注塑（IM）是最突出的方法，大量生產(chǎn)的塑料制品。注塑機(jī)如今所面臨的最大挑戰(zhàn)之一是確定IM過(guò)程變量的正確設(shè)置。選擇適當(dāng)?shù)脑O(shè)置為IM過(guò)程是至關(guān)重要的，因?yàn)樵诔尚途酆喜牧系男袨槭歉叨仁艿倪^(guò)程變量。因此，過(guò)程變量支配產(chǎn)生的部分的質(zhì)量。優(yōu)化IM過(guò)程的困難是，績(jī)效指標(biāo)（PMS），如表面質(zhì)量或周期時(shí)間，通常表現(xiàn)出的特點(diǎn)為目的的過(guò)程中，是否有足夠的一部分，或機(jī)器沖突的行為。因此，必須找到一種妥協(xié)，所有的PMS利益之間。在本文中，我們提出了一個(gè)方法，包括計(jì)算機(jī)輔助工程，人工神經(jīng)網(wǎng)絡(luò)，數(shù)據(jù)包絡(luò)分析（DEA） ，可以用來(lái)找到最好的幾個(gè)性能之間的妥協(xié)措施。這里討論的方法也可以用于識(shí)別強(qiáng)勁的變量設(shè)置，這可能有助于定義多個(gè)決策者之間的談判的出發(fā)點(diǎn)。
介紹
注射成型（ IM）是最突出的過(guò)程中，大量生產(chǎn)的塑料零件。據(jù)塑料工業(yè)協(xié)會(huì)，超過(guò)75％的所有塑料加工機(jī)器IM機(jī)器，接近60％的塑料加工設(shè)備注塑機(jī)[1] 。選擇適當(dāng)?shù)腎M過(guò)程設(shè)置是至關(guān)重要的，因?yàn)樵诔尚途酆喜牧系男袨槭歉叨仁艿倪^(guò)程變量。因此，過(guò)程變量支配產(chǎn)生的部分的質(zhì)量。甲大量的研究已朝向確定IM的方法，以及射出門(mén)的最優(yōu)位置的過(guò)程設(shè)置。
優(yōu)化IM過(guò)程所面臨的挑戰(zhàn)是，性能的措施往往表現(xiàn)出相互矛盾的行為時(shí)，他們是共同的工藝或設(shè)計(jì)變量的函數(shù)。例如，循環(huán)時(shí)間和部件翹曲都將可以由噴射溫度的影響。提高噴射溫度將有利于最大限度地減少周期時(shí)間。然而，讓冷卻至前脫模脫模溫度較低的部分將減少零件翹曲。因此，必須找到妥協(xié)這兩者之間性能的措施，設(shè)置脫模溫度。出于這個(gè)原因，優(yōu)化IM過(guò)程時(shí)，它幾乎是不可能找到一個(gè)最佳的解決方案。然而，它是可行的，以確定一組多個(gè)預(yù)防性維護(hù)之間的最佳折衷。
在的問(wèn)題，同時(shí)考慮幾個(gè)預(yù)防性維護(hù)，即尋找最佳的妥協(xié)，被稱為作為多個(gè)條件優(yōu)化。多準(zhǔn)則優(yōu)化的常規(guī)方法包括個(gè)別加權(quán)計(jì)算項(xiàng)目經(jīng)理相結(jié)合的一個(gè)目標(biāo)函數(shù)和優(yōu)化該功能。這些方法將收斂到一個(gè)解決方案，但它可能被證明是一個(gè)挑戰(zhàn)，以確定是否該解決方案是高效的前沿，尤其是在的情況下，在那里PMS非線性行為。此外，這種解決方案是依賴于用戶定義的權(quán)重的偏壓。在工程實(shí)踐中往往是不可能確定一個(gè)最佳的解決方案的所有標(biāo)準(zhǔn)。相反，它是可行的和有吸引力的，以確定最佳的妥協(xié)之間的PM ：是不能得到改善的預(yù)防性維護(hù)，在一個(gè)單一的尺寸而不損害另一個(gè)組合。數(shù)據(jù)包絡(luò)分析（DEA）提供一個(gè)公正的方式找到這些高效的妥協(xié)。
本文的目的是在IM情況下，通過(guò)一系列的案例研究，包括一些潛在的工業(yè)應(yīng)用中表現(xiàn)出的決心高效的解決方案（最好的妥協(xié)） 。這些解決方案處方IM工藝和設(shè)計(jì)變量的設(shè)置。此外，識(shí)別的強(qiáng)大的解決方案進(jìn)行了討論。
優(yōu)化策略
卡布雷拉 - 里奧斯，建議等人[ 2，3 ]的總體戰(zhàn)略，以找到最好的幾個(gè)項(xiàng)目經(jīng)理之間的妥協(xié)包括五個(gè)步驟：
步驟1）中定義的物理系統(tǒng)。確定利益的現(xiàn)象，的性能指標(biāo)，可控的和不可控的變量，實(shí)驗(yàn)區(qū)，將包括在研究的反應(yīng)。
步驟2）生成的基于物理的模型來(lái)表示系統(tǒng)中的感興趣的現(xiàn)象。定義模型，得到的答復(fù)利益相關(guān)的可控變量。如果這是不可行的，請(qǐng)?zhí)^(guò)此步驟。
步驟3）運(yùn)行的實(shí)驗(yàn)設(shè)計(jì)。無(wú)論是系統(tǒng)運(yùn)行的模型在上一步中創(chuàng)建數(shù)據(jù)集，或進(jìn)行實(shí)際試驗(yàn)時(shí)，在物理系統(tǒng)的數(shù)學(xué)模型是不可能的。
步驟4 ）適合的元模型的實(shí)驗(yàn)結(jié)果。創(chuàng)建經(jīng)驗(yàn)公式（元模型）來(lái)模擬數(shù)據(jù)集的功能。
步驟5）優(yōu)化的物理系統(tǒng)。使用元模型獲得的利益的現(xiàn)象的預(yù)測(cè)，并找到最好的妥協(xié)之間的PMS對(duì)原系統(tǒng)。通過(guò)DEA在這里發(fā)現(xiàn)最好的妥協(xié)。
在此處描述的方法中，元模型的功能之間的可控變量（獨(dú)立） ，和反應(yīng)（因變量）是經(jīng)驗(yàn)性的逼近。這些元模型的使用，也可以方便或必要性。由于要求作出許多響應(yīng)預(yù)測(cè)DEA的，因?yàn)樗怯迷谶@里，它是更方便地獲得這些預(yù)測(cè)從元模型，而不是更復(fù)雜的物理模型。此外，當(dāng)基于物理學(xué)的模型沒(méi)有可用來(lái)表示感興趣的現(xiàn)象，利用元模型變得至關(guān)重要。
數(shù)據(jù)包絡(luò)分析（DEA）
卡布雷拉 - 里奧斯等[ 2,3 ]在聚合物加工過(guò)程中使用DEA解決多個(gè)條件的優(yōu)化問(wèn)題。DEA ，技術(shù)由Charnes ，庫(kù)珀創(chuàng)建的，和羅得島[4]，提供了一種方法來(lái)衡量效率的項(xiàng)目經(jīng)理相類(lèi)似性質(zhì)的組合在一組給定的組合。每個(gè)組合的效率，通過(guò)使用兩個(gè)線性化版本的以下的數(shù)學(xué)編程問(wèn)題的比例形式計(jì)算：
其中，Y和X含有這些預(yù)防性維護(hù)的值要分別最大化和最小化，目前正在分析的載體，μ是一個(gè)向量，以最大化的預(yù)防性維護(hù)的乘法器，ν是一個(gè)矢量乘法器，用于以最小化的預(yù)防性維護(hù)，u0是一個(gè)標(biāo)量變量， n是在集總的組合的數(shù)量，和ε是一個(gè)非常小的常數(shù)通常被設(shè)置為一個(gè)值，為1×10 -6 。由上面所示的模型的兩個(gè)線性化版本當(dāng)作高效的解決方案代表的PM的組合設(shè)置中的（有限的）的最佳折衷。以及線性化過(guò)程的完整描述可以發(fā)現(xiàn)在該模型的應(yīng)用程序的任何參考文獻(xiàn)1至5 。
過(guò)程變量的設(shè)置和注射點(diǎn)的測(cè)定
考慮在圖1中示出的部分。的這一部分，我們介紹，在以前的工作[5,6,7] ，表示的焊接線的位置是至關(guān)重要的情況下，以及部分的平坦度起著重要的作用。部分是注射成型使用住友IM機(jī)使用PET 9CC / s的具有固定流量。9個(gè)項(xiàng)目經(jīng)理被包括在此研究： （1）最大噴射壓力， PI， （2）凍結(jié)的時(shí)間，轉(zhuǎn)鐵蛋白，（3）最大剪切應(yīng)力在墻， SW ，（4）的撓度范圍的z方向，RZ ，（5）中的時(shí)間流動(dòng)前沿接觸孔A ，TA， （6）的時(shí)間的流動(dòng)前沿接觸孔B，的tB ，（7）在該時(shí)間的流動(dòng)前沿接觸的部分的外邊緣，腳趾，（8）的垂直距離從邊緣1的焊接線中，d1 ，和（9）的水平距離，從邊緣2的焊接線， d2的。生產(chǎn)的目的是希望減少PI ，TF ，SW ，和RZ ： PI保持機(jī)的能力受到挑戰(zhàn)，TF ，以減少總周期時(shí)間，SW ，以盡量減少塑料降解，和RZ控制部分的尺寸。這是可取的最大化TA， TB，腳趾， D1，和D2： TA， TB，腳趾，以盡量減少泄漏的可能性，和d1和d2的焊接線保持距離邊角被假定為區(qū)域的壓力濃度。
圖1：有缺口部分的厚度不變
五可控變量完全因子設(shè)計(jì)中的表1中所示的水平變化。這些可控變量包括：（a）的熔融溫度，Tm ， （b）在模具的溫度， Tw的， （c）該噴射溫度，碲， （d）在注入點(diǎn)中，x，和（e）注射的垂直坐標(biāo)的水平坐標(biāo)點(diǎn)，y坐標(biāo)。Y。德，只在兩個(gè)層次上變化，因?yàn)橐粋€(gè)初步的研究表明，三分之一的水平并沒(méi)有添加任何有意義的變化。注入點(diǎn)的位置被限制為在圖1中所示的區(qū)域中，由于限制的IM機(jī)器。這一點(diǎn)將在一個(gè)直角坐標(biāo)系，其原點(diǎn)在左下角的部分，其特征在于變量x和y 。
表1：每個(gè)可控變量的初始數(shù)據(jù)集水平
創(chuàng)建于MoldflowTM有限元網(wǎng)格的一部分，以獲得性能的措施的估計(jì)。一個(gè)初始的數(shù)據(jù)集，得到從全因子設(shè)計(jì)。與一般的優(yōu)化策略，這初始數(shù)據(jù)集被用來(lái)創(chuàng)建元模型模仿的行為，每個(gè)性能指標(biāo)。在一般情況下，這是有利的，以適應(yīng)一個(gè)簡(jiǎn)單的模型的數(shù)據(jù)。在這項(xiàng)研究中，二階線性回歸最初被認(rèn)為模型的性能措施。當(dāng)簡(jiǎn)單的模型不足夠了，然后更復(fù)雜的模型，在這種情況下，人工神經(jīng)網(wǎng)絡(luò)，成為必要。在一般的人工神經(jīng)網(wǎng)絡(luò)的近似質(zhì)量和預(yù)測(cè)能力，每一個(gè)績(jī)效指標(biāo)的表現(xiàn)優(yōu)于二階線性回歸，因此，在以前從未嘗試過(guò)的預(yù)測(cè)，每個(gè)PM可控變量的組合?；貧w模型和人工神經(jīng)網(wǎng)絡(luò)的性能，可以發(fā)現(xiàn)結(jié)果見(jiàn)表2：
表2：性能和殘余的回歸元模型的分析結(jié)果的結(jié)果
完整的多標(biāo)準(zhǔn)優(yōu)化問(wèn)題最初這種情況下造成的，包含了所有9個(gè)績(jī)效指標(biāo)。要解決的優(yōu)化問(wèn)題，它是必要的，以產(chǎn)生大量的可控變量的可行的水平組合。這是通過(guò)不同的Tm和Tw的五個(gè)級(jí)別，其余的變量在三個(gè)層次內(nèi)的實(shí)驗(yàn)的感興趣區(qū)域的（見(jiàn)表1 ） ，在一個(gè)完整的階乘枚舉。本實(shí)驗(yàn)設(shè)計(jì)共675組合。運(yùn)用數(shù)據(jù)包絡(luò)分析后的結(jié)果是，超過(guò)400 675組合被認(rèn)為是有效的。如此大量的有效的組合，可以通過(guò)檢查在回歸形式每個(gè)PM方差分析的結(jié)果總結(jié)于表3中，解釋。請(qǐng)注意最后的5個(gè)項(xiàng)目經(jīng)理是只依賴于注射點(diǎn)的位置由變量x和y 。值（ X *，Y *）的任何特定組合給出相同的結(jié)果在所有這五個(gè)預(yù)防性維護(hù)的其余部分的其他可控變量的Tm ， Tw的，和Te的值無(wú)關(guān)。在三個(gè)層面上進(jìn)行了全階乘列舉的x和y ，如下，我們可以得到只有九個(gè)不同的值這些5的PM ，但每個(gè)九個(gè)特定組合（的x，y ）有實(shí)際上的可控變量的其余部分75的組合。在高維的問(wèn)題，此升高量高效的解決方案中有大量的重復(fù)結(jié)果。為了提高識(shí)別能力，即獲得較少的高效的解決方案，可以解決在方程的DEA模型。 1?5中，通過(guò)設(shè)置μ0等于零。由此產(chǎn)生的模型是類(lèi)似的Charnes - 庫(kù)珀 - 羅德（CCR ）DEA模型[8]。
表3的重要來(lái)源變化（線性，二階和二階相互作用的線性回歸元模型） ，以績(jī)效考核指標(biāo)
使用上述的簡(jiǎn)單的修改，高效的組合數(shù)歸結(jié)到149 。它可以表明，這些組合是400的一個(gè)子集的那些加以前找到。這些高效的組合示于圖2中的預(yù)防性維護(hù)方面。 圖2：水平的PMS ，在所有9時(shí)，對(duì)應(yīng)的有效解決方案
重要的是要注意到，我們可以利用我們的方法給我們的信息的PMS的功能，以量身定制的優(yōu)化問(wèn)題。為了說(shuō)明，五個(gè)子情況下被定義為實(shí)際應(yīng)用的概念在圖1中所示的部分：（?。┻^(guò)量的容量注入機(jī)中的應(yīng)用，（ⅱ）的尺寸關(guān)鍵的應(yīng)用程序的質(zhì)量和經(jīng)濟(jì)性，（ⅲ）的結(jié)構(gòu)的一部分的應(yīng)用程序，（四）部分質(zhì)量關(guān)鍵應(yīng)用，及（v）情況，包括項(xiàng)目經(jīng)理，只依賴于注射部位[7] 。
產(chǎn)能過(guò)剩射出成型機(jī)：
有關(guān)的情況下，其中的注塑成形機(jī)，具有多余的容量，這將有可能不考慮在優(yōu)化問(wèn)題的最大噴射壓力。為了簡(jiǎn)單起見(jiàn)，在這種情況下， SW， TA ，TB ，腳趾也下降了優(yōu)化，留下四個(gè)性能指標(biāo)。的DEA模型，再通過(guò)設(shè)置恒定的μ0等于0，以提高識(shí)別能力的DEA解決。于表3中所示的功能要求包含所有的變量，并使用與675組合的階乘枚舉。在這種情況下， 14的組合被發(fā)現(xiàn)是有效的。圖3示出了水平的有效的解決方案的預(yù)防性維護(hù)。的焊接線的位置之間的折衷是顯而易見(jiàn)的。一個(gè)明顯的妥協(xié)之間也出現(xiàn)TF和Rz 。這是可以理解的妥協(xié)，因?yàn)檫@兩個(gè)在噴射溫度相對(duì)依賴。
圖3：多余的機(jī)器容量的應(yīng)用水平的項(xiàng)目經(jīng)理認(rèn)為有效的解決方案。
圖4示出的有效的解決方案的射出門(mén)的位置。在這種情況下的位置有助于確定“吸引力”的區(qū)域定位進(jìn)樣口，因?yàn)樗麄兺??集群中的特定部分。在這種情況下，高效的噴射位置群集沿右邊緣和底部邊緣。受到影響的射出門(mén)的位置的三個(gè)預(yù)防性維護(hù)的焊接線的位置，在z-方向上的偏轉(zhuǎn)。這里附加的PM是凍結(jié)的時(shí)間，這是沒(méi)有由注射位置的影響，根據(jù)方差分析。
圖4 ：注射劑的高效解決方案，多余的機(jī)器容量的應(yīng)用轉(zhuǎn)化為介于-1和1的位置。
表4示出了在有效的解決方案可用于所有的可控變量的值。請(qǐng)注意，在攝氏120度和260度的有效的解決方案可用于所有分別Tw的和Tm 。在工業(yè)實(shí)踐中，如果在這種情況下，所涉及的PMS是唯一的興趣，這將是一個(gè)很好的跡象表明， Tm和總重量應(yīng)設(shè)置在這些溫度下。另請(qǐng)注意，噴射溫度值的高效解決方案，在整個(gè)范圍內(nèi)變化。根據(jù)方差分析，不依賴于噴射溫度d1和d2 ，所以必須是由于這個(gè)事實(shí)，前面提到的Rz和TF之間的折衷。
表4：多余的機(jī)器容量應(yīng)用的高效解決方案
尺寸質(zhì)量和經(jīng)濟(jì)性的關(guān)鍵應(yīng)用：
在這種情況下，它的經(jīng)濟(jì)問(wèn)題，包括減少周期時(shí)間和保持機(jī)器容量的未經(jīng)檢驗(yàn)的，機(jī)械壽命長(zhǎng)，功耗和更小的假設(shè)。這兩個(gè)問(wèn)題的定義的TF和PI分別。 RZ定義的尺寸質(zhì)量。方差節(jié)目的分析，所有的可控變量影響這些預(yù)防性維護(hù)中的至少一個(gè)，所以與675組合的枚舉再次施加。二十五年有效的解決方案。因?yàn)檫@個(gè)問(wèn)題是三維有效前沿可以可視化。高效的點(diǎn)是在圖5中所示，相對(duì)于其余的數(shù)據(jù)集。
圖5：一個(gè)重要的經(jīng)濟(jì)和三維應(yīng)用的有效前沿的可視化
圖6示出了在水平的預(yù)防性維護(hù)方面的有效的解決方案。在這里被確認(rèn)的凍結(jié)和偏轉(zhuǎn)的時(shí)間之間的直接的妥協(xié)。請(qǐng)注意，他們按照相反的趨勢(shì)，而這是有利于最大限度地減小了。
圖6 ：高效的解決方案的尺寸質(zhì)量和水平的項(xiàng)目經(jīng)理認(rèn)為在經(jīng)濟(jì)方面的應(yīng)用
圖7示出的有效的解決方案的射出門(mén)的位置。這種情況違背了第一種情況。射出門(mén)'吸引力'的領(lǐng)域以大型機(jī)器容量的情況下，發(fā)現(xiàn)在可行區(qū)域的底部和右側(cè)邊緣，但在這種情況下，頂邊和左下角被證明是高效的位置。這是由于這樣的事實(shí)的焊接線的位置，在這種情況下，不考慮。從這些結(jié)果中，我們可以得出這樣的結(jié)論： d1和d2的主要驅(qū)動(dòng)因素保持注射部位的右側(cè)或底部邊緣。它們是由x和y的影響，在這種情況下，被包含在第一種情況下，而不是唯一的預(yù)防性維護(hù)。
圖7：注射位置的高效解決方案的尺寸質(zhì)量和經(jīng)濟(jì)性的關(guān)鍵應(yīng)用程序轉(zhuǎn)化為介于-1和1 。
表5示出的可控變量，證明是有效的尺寸質(zhì)量和經(jīng)濟(jì)關(guān)鍵的應(yīng)用程序25的組合。十八滿分25高效的解決方案有可行的區(qū)域，這是接近的部分的中心位于左上角的射出門(mén)。這是最可靠的注射此應(yīng)用程序的位置。根據(jù)方差分析，PI的射出門(mén)的位置的影響。定位注塑門(mén)向中心將有利降低PI。由于沒(méi)有包括在這種情況下， d1和d2被有向中心移動(dòng)的噴射柵沒(méi)有負(fù)面影響的。
表5 ：高效的解決方案的尺寸質(zhì)量和經(jīng)濟(jì)性的關(guān)鍵應(yīng)用
一個(gè)結(jié)構(gòu)的應(yīng)用程序
在此應(yīng)用中，包括這些PM的焊接線中，d1和的邊緣2焊縫線之間的水平距離， d2的邊緣1之間的垂直距離。焊接線的位置被認(rèn)為是至關(guān)重要的，設(shè)計(jì)一個(gè)結(jié)構(gòu)合理的部分。從方差分析，這是眾所周知的，這些預(yù)防性維護(hù)只取決于的射出門(mén)的位置，其特征在于，由變量x和y 。為了避免重復(fù)全套中描述，創(chuàng)建一個(gè)新的數(shù)據(jù)集，通過(guò)改變x和y在九個(gè)水平注射位置創(chuàng)建更精細(xì)的采樣網(wǎng)格對(duì)齊。其余的變量中的一個(gè)值被設(shè)置為中間的各自的范圍內(nèi)。用于該數(shù)據(jù)集的可控變量的水平示于表6。在這個(gè)數(shù)據(jù)集的可控變量的組合的總數(shù)是81。
表6：用于數(shù)據(jù)集的x，y依賴于項(xiàng)目經(jīng)理的可控變量的水平
這里在圖8中示出的這個(gè)兩維的情況下的有效前沿。
圖8：在結(jié)構(gòu)應(yīng)用程序的可視化的有效邊界
七個(gè)高效的解決方案在圖9中示出的結(jié)構(gòu)的一部分的條款的增大的順序?yàn)閐1的兩個(gè)水平的預(yù)防性維護(hù)。確認(rèn)的焊接線的位置之間的折衷。我們想要最大限度地提高焊接線的位置，但在其中一個(gè)處于最大值的，另一種是在最低限度。
圖9：有效解的結(jié)構(gòu)應(yīng)用中的焊接線的位置d1 D D2 。
圖10示出的位置對(duì)應(yīng)于七個(gè)最佳的射出門(mén)妥協(xié)。整個(gè)區(qū)域是可行的注射區(qū)域。在這種情況下會(huì)發(fā)生“吸引力”集群在可行的注射區(qū)域的右下角，并沿可行的注入?yún)^(qū)域的右邊緣。這些結(jié)果傾向于同意與大機(jī)臺(tái)產(chǎn)能的情況下。由于其他可控變量的焊接線的位置是獨(dú)立的，任何這些x ， y對(duì)將獲得相同的結(jié)果為d1和d2的溫度水平無(wú)關(guān)。在這種情況下，有效的解決方案被定義由注射位置，所以未示出的溫度水平。換句話說(shuō)，溫度被留在其各自的可行范圍的最大值或最小值，我們會(huì)到達(dá)的焊接線的位置相同的結(jié)果。
圖10：噴射位置的7個(gè)有效的解決方案的的結(jié)構(gòu)APPLICATI ransformed -1和1之間
強(qiáng)大的解決方案分析
在上一節(jié)的討論，不同的情況下會(huì)導(dǎo)致額外的分析：尋找穩(wěn)健高效的解決方案。在個(gè)別情況下，可以找到可靠的解決方案，和其中的一些進(jìn)行了討論?？梢酝茢?，一個(gè)強(qiáng)大高效的解決方案相結(jié)合的可控變量設(shè)置，維持高效率的性能指標(biāo)進(jìn)行分析時(shí)，不同的子集。這也有利于以確定哪些解決方案，強(qiáng)大的大規(guī)模， iewhich過(guò)程變量的組合被視為有效的幾個(gè)子集的優(yōu)化。事實(shí)上，這種情況下，它是可能的，以確定組合（ ，銩Tm， Tw的，碲的x，y ）=（ 20厘米， 10厘米， 120℃ ， 260度，149 ℃）是一個(gè)強(qiáng)大的有效的解決方案。確定一個(gè)合適的澆口位置是一個(gè)關(guān)鍵的決定。來(lái)運(yùn)行該過(guò)程的溫度時(shí)，可以容易地調(diào)整。另一方面，那里是只有一次的機(jī)會(huì)，決定將位于澆口。選擇適當(dāng)?shù)奈恢脧囊婚_(kāi)始就可以節(jié)省時(shí)間和金錢(qián)。在這項(xiàng)研究中，它被確定，在右上角的可行的注入?yún)^(qū)（ = 20厘米時(shí)，y = 25厘米）的注射澆口位置是一個(gè)解決方案。
這種分析可能會(huì)幫助建立了多個(gè)決策者之間的“共同點(diǎn)” ，然后再移動(dòng)到什么樣的妥協(xié)時(shí)，可以采取與其他的有效的解決方案[7] 。
結(jié)論與未來(lái)工作
在注塑成型的工藝和設(shè)計(jì)變量的設(shè)置是一個(gè)活躍的研究領(lǐng)域。在這項(xiàng)工作中，協(xié)調(diào)利用CAE ，統(tǒng)計(jì)，神經(jīng)網(wǎng)絡(luò)，數(shù)據(jù)包絡(luò)分析法已被證明在多目標(biāo)優(yōu)化的情況下找到這些設(shè)置。優(yōu)化的虛擬部分進(jìn)行討論，并提出了幾個(gè)分的情況下被定義為在行業(yè)中的實(shí)際應(yīng)用進(jìn)一步的細(xì)節(jié)。本文提出的面向分析的幾個(gè)性能指標(biāo)上的妥協(xié)，作出明智的決定。這些分析也可以用于識(shí)別強(qiáng)勁的變量設(shè)置，這可能有助于定義多個(gè)決策者之間的談判的出發(fā)點(diǎn)。今后的工作將包括增加信息的變異PMS的DEA分析和確定的過(guò)程窗口與效率的考慮。
參考文獻(xiàn)
[1] 塑料工業(yè)協(xié)會(huì)（2003年） 。 SPI：塑料的數(shù)據(jù)源，SPI ，可在www.plasticsindustry.org [ 2004年7月]
[2 ]卡布雷拉 - 里奧斯， M.，祖耶夫， K.，陳， X.，卡斯特羅， JM ，斯特勞斯， EJ高分子復(fù)合材料，優(yōu)化澆口位置和周期時(shí)間的模內(nèi)涂層（IMC）過(guò)程， 23:5 （2002） 723-738
[3 ]卡布雷拉 - 里奧斯， M.，安裝坎貝爾， CA，卡斯特羅JM ，高分子工程，多個(gè)質(zhì)量標(biāo)準(zhǔn)的優(yōu)化與數(shù)據(jù)包絡(luò)分析方法，在模內(nèi)涂層（IMC） 22時(shí)05分（2002年） 305-340
[4 ] Charnes ， WW，和羅得島，E.歐洲運(yùn)籌學(xué)雜志，測(cè)量效率的決策單位， 2:6 （1978） 429-444
[5 ]卡斯特羅， CE，卡布雷拉 - 里奧斯， M.，禮來(lái)公司， B.，卡斯特羅， JM ，和山 - 坎貝爾，CA [綜合設(shè)計(jì)與流程科學(xué)，確定最佳的妥協(xié)之間的多個(gè)性能指標(biāo)在注射成型（ IM）的使用數(shù)據(jù)包絡(luò)分析（DEA）， 7:1（ 2003年） 77-87
[6 ]卡斯特羅， CE， Bhagavatula的，N. ，卡布雷拉 - 里奧斯，M. ，禮來(lái)公司，B.，和卡斯特羅， JM ， 2003 ANTEC訴訟，確定最佳的多性能之間的妥協(xié)措施在使用數(shù)據(jù)包絡(luò)注射成型（ IM）分析， （2003） 377-381
[7 ]卡斯特羅，CE ，卡布雷拉 - 里奧斯，M. ，禮來(lái)公司，B.，和卡斯特羅， JM ， ANTEC訴訟選擇的多個(gè)性能指標(biāo)在注射成型的工藝和設(shè)計(jì)變量設(shè)置下， 2004年， （2004年）卷946 （即將出版）
[8 ] Charnes ， A.，庫(kù)珀， WW，盧因， AY ，并Seiford ，LM波士頓， Kluwer學(xué)術(shù)出版社，數(shù)據(jù)包絡(luò)分析法：理論，方法與應(yīng)用，（1993） 36-42
UNIVERSIDAD AUTNOMA DE NUEVO LEN FACULTAD DE INGENIERA MECNICA Y ELCTRICA Divisin de Posgrado en Ingeniera de Sistemas Serie de Reportes Tcnicos Reporte Tcnico PISIS-2004-04 Simultaneous Optimization of Mold Design and Processing Conditions in Injection Molding Carlos E. Castro 1 Mauricio Cabrera Ros 2 Blaine Lilly 1,3 Jos M. Castro 3 (1) Department of Mechanical Engineering Ohio State University Columbus, Ohio, EUA (2) Programa de Posgrado en Ingeniera de Sistemas FIME, UANL E-mail: mcabrerauanl.mx (3) Department of Industrial, Welding, and Systems Enginering Ohio State University Columbus, Ohio, EUA 08 / Septiembre / 2004 2004 by Divisin de Posgrado en Ingeniera de Sistemas Facultad de Ingeniera Mecnica y Elctrica Universidad Autnoma de Nuevo Len Pedro de Alba S/N, Cd. Universitaria San Nicols de los Garza, NL 66450 Mxico Tel/fax: +52 (81) 1052-3321 E-mail: pisisyalma.fime.uanl.mx Pgina: http:/yalma.fime.uanl.mx/pisis/ Simultaneous Optimization of Mold Design and Processing Conditions in Injection Molding Carlos E. Castro 1 , Mauricio Cabrera Ros 3 , Blaine Lilly 1,2 , and Jos M. Castro 2 1 Department of Mechanical Engineering and 2 Departement of Industrial, Welding however it might prove a challenge to determine if this solution lies in the efficient frontier, especially in the case where the PMs show nonlinear behavior. In addition, this solution is dependent on the bias of the user defining the weights. In engineering practice it is often times impossible to define one optimal solution to all criteria. Instead, it is both feasible and attractive to determine the best compromises between PMs: that is the combinations of PMs that cannot be improved in one single dimension without harming another. Data Envelopment Analysis (DEA) provides an unbiased way to find these efficient compromises. It is the purpose of this paper to demonstrate the determination of efficient solutions (best compromises) in an IM context through a series of case studies comprising several potential industrial applications. These solutions prescribe the settings for IM process and design variables. Additionally, the identification of robust solutions is discussed. The Optimization Strategy Proposed by Cabrera-Rios, et al 2, 3 the general strategy to find the best compromises between several PMs consists of five steps: Step 1) Define the physical system. Determine the phenomena of interest, the performance measures, the controllable and non-controllable variables, the experimental region, and the responses that will be included in the study. Step 2) Build physics-based models to represent the phenomena of interest in the system. Define models that relate the controllable variables to the responses of interest. If this is not feasible, skip this step. 3 Step 3) Run experimental designs. Create data sets by either systematically running the models from the previous step, or by performing an actual experiment in the physical system when a mathematical model is not possible. Step 4) Fit metamodels to the results of the experiments. Create empirical expressions (metamodels) to mimic the functionality in the data sets. Step 5) Optimize the physical system. Use the metamodels to obtain predictions of the phenomena of interest, and to find the best compromises among the PMs for the original system. The best compromises are identified here through DEA. In the method outlined here, the metamodels are empirical approximations of the functionality between the controllable (independent) variables, and the responses (dependent variables). These metamodels are used either for convenience or for necessity. Because DEA as it is used here requires that many response predictions be made, it is more convenient to obtain these predictions from metamodels rather than more complicated physics-based models. In addition, when physics-based models are not available to represent the phenomena of interest, the use of metamodels becomes essential. Data Envelopment Analysis (DEA) Cabrera-Rios et al 2,3 have demonstrated the use of DEA to solve multiple criteria optimization problems in polymer processing. DEA, a technique created by Charnes, Cooper, and Rhodes 4, provides a way to measure the efficiency of a given combination of PMs relative to a finite set of combinations of similar nature. The efficiency of each combination is computed through the use of two linearized versions of the following mathematical programming problem in ratio form: 4 free nj T T T T j T j T T T 0 min 0 min 0 min 0 max min 0 0 max 0 0 ,.,11 , 1 Y 1 Y Y Y s.t. Y Y Maximize toFind = + + (1) (2) (3) (4) (5) where, and are vectors containing the values of those PMs currently under analysis to be maximized and minimized respectively, is a vector of multipliers for the PMs to be maximized, is a vector of multipliers for the PMs to be minimized, max 0 Y min 0 Y 0 is a scalar variable, n is the number of total combinations in the set, and is a very small constant usually set to a value of 1x10 -6 . The solutions deemed efficient by the two linearized versions of the model shown above represent the best compromises in the (finite) set of combinations of PMs. A complete description of the linearization procedure as well as the application of this model can be found in any of the references 1 through 5. Determination of settings of process variables and injection point Consider the part shown in Figure 1. This part, which we introduced in previous works 5,6,7, represents a case where the location of the weld lines is critical, and the part flatness plays a major role. The part is to be injection molded using a Sumitomo IM machine using PET with a fixed flow rate of 9cc/s. Nine PMs were included in this 5 study: (1) maximum injection pressure, P I , (2) time to freeze, t f , (3) maximum shear stress at the wall, S W , (4) deflection range in the z-direction, R Z , (5) time at which the flow front touches hole A, t A , (6) time at which the flow front touches hole B, t B , (7) time at which the flow front touches the outer edge of the part, t oe , (8) the vertical distance from edge 1 to the weld line, d 1 , and (9) the horizontal distance from edge 2 to the weld line, d 2 . For production purposes it is desirable to minimize P I , t f , S W , and R Z : P I to keep the machine capacity unchallenged, t f to reduce the total cycle time, S W to minimize plastic degradation, and R Z to control the part dimensions. It is desirable to maximize t A , t B , t oe , d 1 , and d 2 : t A , t B , t oe in order to minimize the potential for leakage, and d 1 and d 2 to keep the weld lines away from corners which were assumed to be areas of stress concentration. Figure 1: Part of constant thickness with cutouts. 6 Five controllable variables were varied at the levels shown in Table 1 in a full factorial design. These controllable variables include: (a) the melt temperature, T m , (b) the mold temperature, T w , (c) the ejection temperature, T e , (d) the horizontal coordinate of the injection point, x, and (e) the vertical coordinate of the injection point, y. T e was only varied at two levels because a preliminary study showed that a third level did not add any meaningful variation. The injection point location is constrained to be in the region shown in Figure 1, due to limitation of the IM machine. This point will be characterized by the variables x and y in a Cartesian coordinate system with its origin at the lower left corner of the part. Table 1: Levels of each of the controllable variables for the initial dataset T m T w T e x y Label C C C cm cm -1 260 120 149 15 10 0 275 130 159 20 17.5 1 290 140 25 25 A finite element mesh of the part was created in Moldflow TM in order to obtain estimates for the performance measures. An initial dataset was obtained from the full factorial design. Following with the general optimization strategy, this initial dataset was used to create metamodels to mimic the behavior of each the performance measures. In general, it is favorable to fit a simple model to the data. In this study, second order linear regressions were initially considered as models for the performance measures. When simple models do not suffice, then more complicated models, in this case ANNs, become necessary. In general the ANNs outperformed the second order linear regression for every performance measure in terms of approximation quality and prediction capability, and were therefore used to obtain predictions for each PM at previously untried 7 combinations of controllable variables. The results for the performance of the regression models and the ANNs obtained can be found Table 2. Table 2: Summary of performance and results from residual analysis results for the regression metamodels The complete multiple criteria optimization problem originally posed for this case contained all nine performance measures. To solve the optimization problem, it was necessary to generate a large number of feasible level combinations of the controllable variables. This was achieved by varying T m and T w at five levels, and the rest of the variables at three levels within the experimental region of interest (see Table 1) in a full factorial enumeration. This experimental design resulted in a total of 675 combinations. The results after applying DEA were that over 400 of the 675 combinations were found to be efficient. Such a large number of efficient combinations can be explained by examining Table 3, which summarizes the results of the analysis of variance of each PM in regression form. Notice that the last five PMs are only dependent on the injection point position determined by variables x and y. Any specific combination of values (x * ,y * ) will 8 give the same result on all of these five PMs regardless of the values that the rest of the other controllable variables T m , T w , and T e take. Having used a full factorial enumeration with x and y at three levels, it follows that we can obtain only nine different values for these five PMs, but each of the nine specific combinations (x,y) have in fact 75 combinations of the rest of the controllable variables. In the high dimensionality of the problem, this elevated amount of repetition results in a large number of efficient solutions. In order to increase the discrimination power i.e. obtain fewer efficient solutions, one can solve the DEA model shown in Eqs. 1 through 5 by setting 0 equal to zero. The resulting model is similar to the Charnes-Cooper-Rhodes (CCR) DEA model 8. Table 3: The significant sources of variation (linear, quadratic and second order interaction terms in the linear regression metamodel) to each performance measure. 9 Using the simple modification described above, the number of efficient combinations comes down to 149. It can be shown that these combinations are a subset of those 400 plus found previously. These efficient combinations are shown in terms of the PMs in Figure 2. Figure 2: Levels of the PMs that corresponded to the efficient solutions when all nine were included It is important to notice that we can exploit the information our methods gave us about the functionality of the PMs in order to tailor the optimization problem. To illustrate, five sub cases were defined for practical applications of the conceptual part shown in Figure 1: (i) an excess capacity injection machine application, (ii) a dimensional quality and economics critical application, (iii) a structural part application, 10 (iv) a part quality critical application, and (v) a case including PMs that are only dependent on the injection location 7. Excess Capacity Injection Molding Machine: For a case in which the injection-molding machine has excess capacity, it would be possible to not consider the maximum injection pressure in the optimization problem. For simplicity, in this case S W , t A , t B , and t oe were also dropped from the optimization, leaving four performance measures. The DEA model was again solved here by setting the constant 0 equal to zero in order to improve the discrimination power of DEA. The functionality shown in Table 3 called for inclusion of all variables, and the factorial enumeration with 675 combinations was used. In this case, fourteen combinations were found to be efficient. Figure 3 shows the levels of the PMs for the efficient solutions. The compromise between the locations of the weld lines is evident. A noticeable compromise also arises between t f and R z . This is an understandable compromise, because the two depend oppositely on the ejection temperature. 11 Figure 3: Efficient solutions for the excess machine capacity application in terms of the levels of the PMs considered. Figure 4 shows the locations of the injection gate for the efficient solution. The positions in this case help to define attractive areas to locate the injection port, since they tend to cluster in specific sections. In this case the efficient injection locations clustered along right and bottom edges. The three PMs that are affected by the location of the injection gate are the weld line positions and the deflection in the z-direction. The additional PM here is the time to freeze, which is not affected by the injection location according to the analysis of variance. 12 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 X Y Figure 4: Injection Locations of the efficient solutions to the excess machine capacity application transformed to fall between -1 and 1. Table 4 shows the values for all of the controllable variables at the efficient solutions. Notice that T w and T m were at 120 and 260 degrees Celsius respectively for all of the efficient solutions. In industrial practice, if the PMs involved in this case were the only ones of interest, this would be a good indication that T m and T w should be set at these temperatures. Also notice that the ejection temperature values of the efficient solutions vary over the entire range. According to the analysis of variance, d 1 and d 2 do not depend on the ejection temperature, so this fact must be due to the compromise between R z and t f previously mentioned. 13 Table 4: Efficient Solutions for the excess machine capacity application xyT w T m T e t f R z d 1 d 2 cm cm C C C s mm mm mm 25 25 120 260 149 20.89 0.0005 37.9 130.9 25 25 120 260 154 18.99 0.002 37.9 130.9 25 25 120 260 159 17.27 0.007 37.9 130.9 25 25 120 260 149 22.71 0.000 37.9 131.4 25 17.5 120 260 149 20.90 0.001 94.9 107.8 25 17.5 120 260 154 19.00 0.005 94.9 107.8 25 17.5 120 260 159 17.28 0.010 94.9 107.8 25 17.5 120 260 149 22.72 0.001 94.9 108.9 15 10 120 260 149 20.92 0.001 124.6 67.4 15 10 120 260 154 19.02 0.005 124.6 67.4 15 10 120 260 159 17.30 0.011 124.6 67.4 25 10 120 260 149 20.92 0.001 124.7 82.1 25 10 120 260 154 19.02 0.006 124.7 82.1 25 10 120 260 159 17.30 0.012 124.7 82.1 Controllable Variables Performance Measures Dimensional Quality and Economics Critical Application: In this case it was assumed that the economic concerns included minimizing the cycle time and keeping the machine capacity untested in order to have long machine life and smaller power consumption. These two concerns are defined by t f and P I respectively. R z defines the dimensional quality. The analysis of variance shows that all of the controllable variables affect at least one of these PMs, so the enumeration with 675 combinations again was applied. Twenty-five efficient solutions were found. Since the problem is three-dimensional the efficient frontier can be visualized. The efficient points are shown in Figure 5 with respect to the rest of the data set. 14 Figure 5: A Visualization of the efficient frontier of the economics critical and dimensional Application Figure 6 shows the efficient solutions in terms of the levels of the PMs. The direct compromise between the time to freeze and deflection is confirmed here. Notice that they follow opposite trends while it is favorable to minimize both. 15 Figure 6: Efficient solutions for the dimensional quality and economic application in terms of the levels of the PMs considered. Figure 7 shows the locations of the injection gate for the efficient solutions. This case contradicts the first case. In the large machine capacity case, the attractive areas for the injection gate were found at the bottom and right edges of the feasible area, but in this case, the top edge and bottom left corner proved to be the efficient locations. This is due to the fact that the positions of the weld lines were not considered in this case. From these results we can conclude that d 1 and d 2 are the main drivers for keeping the injection location on the right or bottom edge. They are the only PMs affected by x and y that were included in the first case and not in this case. 16 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 X Y Figure 7: Injection Locations of the efficient solutions to the dimensional quality and economics critical application transformed to fall between -1 and 1. Table 5 shows the twenty-five combinations of the controllable variables that proved to be efficient for the dimensional quality and economics critical application. Eighteen out of the twenty-five efficient solutions had the injection gate located at the upper left corner of the feasible region, which is close to the center of the part. This is the most robust injection location for this application. According to the analysis of variance, P I is affected by the location of the injection gate. Locating the injection gate towards the center would favorably decrease P I . Since d 1 and d 2 were not included in this case there were no negative effects of moving the injection gate towards the center. 17 Table 5: Efficient Solutions for the dimensional quality and economics critical application xyT w T m T e P I t f R z cm cm C C C MPa s mm 15 25 140 290 159 9.35 26.7 0.0098 15 25 140 290 149 9.35 37.9 0.0015 15 25 140 282.5 159 9.55 23.8 0.0099 15 25 140 282.5 149 9.55 32.0 0.0009 15 25 140 275 149 9.75 27.4 0.0006 15 25 140 275 159 9.75 21.9 0.0101 15 25 140 267.5 159 9.96 20.4 0.0102 15 25 140 267.5 149 9.96 24.5 0.0006 15 25 140 260 154 10.17 20.9 0.0045 15 25 140 260 159 10.17 19.1 0.0101 15 25 140 260 149 10.17 22.7 0.0006 15 25 130 275 149 12.25 26.5 0.0005 15 25 125 260 149 14.69 21.4 0.0005 15 25 125 260 159 14.69 17.8 0.0095 15 25 120 275 149 16.00 25.5 0.0004 15 25 120 260 154 16.61 19.0 0.0032 15 25 120 260 149 16.61 20.9 0.0005 15 25 120 260 159 16.61 17.3 0.0090 25 25 140 260 149 17.46 22.7 0.0004 25 25 135 260 149 19.43 22.4 0.0004 20 25 120 260 159 26.92 17.3 0.0071 25 25 120 260 159 27.16 17.3 0.0065 25 25 120 260 154 27.16 19.0 0.0020 15 10 120 282.5 149 28.50 29.8 0.0003 15 10 120 275 149 29.21 25.5 0.0003 Controllable Variables Performance Measures A Structural Application In this application, the PMs included were the vertical distance from edge 1 to the weld line, d 1 , and the horizontal distance from edge 2 to the weld line, d 2 . The location of weld lines is considered critical to design a structurally sound part. From the analysis of variance, it was known that these PMs depended only on the position of the injection gate, characterized by variables x and y. In order to avoid the repetition described in the full set, a new dataset was created by varying x and y at nine levels creating a finer sampling grid for the injection location. The rest of the variables were set to a value in the 18 middle of their respective ranges. The levels of the controllable variables for this dataset are shown in Table 6. The total number of combinations of controllable variables in this dataset was 81. Table 6: Levels of controllable variables used for the dataset for x,y dependent PMs he efficient frontier for this two-dimensional case is shown here in Figure 8. T m T w T e xy CCC cm cm 130 275 154 15 10 16.25 11.875 17.5 13.75 18.75 15.625 20 17.5 21.25 19.375 22.5 21.25 23