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1、系統(tǒng)動(dòng)力學(xué)簡(jiǎn)稱(chēng)SD(System Dynamics),系統(tǒng)動(dòng)力學(xué)原理 建模基本步驟 VensimPLE軟件,系統(tǒng)動(dòng)力學(xué)模型(System Dynamics)被譽(yù)為實(shí)際系統(tǒng)的實(shí)驗(yàn)室,是美國(guó)麻省理工學(xué)院(MIT)福瑞斯特Forrester (Jay W .Forrester)教授于1956年首創(chuàng)的一種運(yùn)用結(jié)構(gòu)、功能、歷史相結(jié)合的方法,借助于計(jì)算機(jī)仿真而定量地研究非線(xiàn)性(Non-linearity)、多重反饋(Information feedback)、復(fù)雜時(shí)變(Dynamic complexity)系統(tǒng)的系統(tǒng)分析技術(shù)??捎糜谘芯刻幚砩鐣?huì)經(jīng)濟(jì)生態(tài)和生物等復(fù)雜系統(tǒng)問(wèn)題,它可在宏觀(guān)層次和微觀(guān)層次上對(duì)復(fù)雜

2、、多層次、多部門(mén)、非線(xiàn)性的大規(guī)模系統(tǒng)進(jìn)行綜合研究。,反饋(feedback)是控制系統(tǒng)的一種方法。它是把系統(tǒng)輸出去的一部分信息(給定信息)作用于被控對(duì)象后產(chǎn)生的結(jié)果(真實(shí)信息)再返回給輸入,并對(duì)系統(tǒng)的再輸出產(chǎn)生影響的過(guò)程。系統(tǒng)所具有的這種功能和過(guò)程稱(chēng)之為系統(tǒng)反饋。系統(tǒng)動(dòng)力學(xué)認(rèn)為幾乎所有人工的系統(tǒng)都是反饋系統(tǒng) 。,動(dòng)態(tài)( Dynamic)即系統(tǒng)動(dòng)力學(xué)所包含的量是隨時(shí)間變化的,能以時(shí)間為坐標(biāo)的圖形表示。譬如,人口的增長(zhǎng),就業(yè)人數(shù)的增減,城鎮(zhèn)與農(nóng)村的生活質(zhì)量和物價(jià)的漲落等都是動(dòng)態(tài)問(wèn)題。學(xué)習(xí)定義動(dòng)態(tài)問(wèn)題的技巧是學(xué)習(xí)系統(tǒng)動(dòng)力學(xué)的第一步。,1系統(tǒng)的流,系統(tǒng)動(dòng)力學(xué)主要利用四種流來(lái)構(gòu)成模型 物流 訂貨流 資金

3、流 信息流,2水平(積累、狀態(tài))level變量,水平(積累)是系統(tǒng)的流的積累。例如,庫(kù)存量、存款、人口、資源等都可作為水平變量。一個(gè)水平方程相當(dāng)于個(gè)容器,它積累變化的流速率。其流速有輸入流速和輸出流速,容器內(nèi)的水平正是其輸入流速與輸出流速的差量的積累。,怎樣計(jì)算水平變量?,用DT近似表示dt ,上式寫(xiě)成,一階差分方程,符號(hào)DT表示時(shí)間的差分,即兩次計(jì)算之間時(shí)間間隔的長(zhǎng)度。,因此,一個(gè)水平變量的新值等于它前一時(shí)刻的值加上或者減去時(shí)間間隔所產(chǎn)生的變化值。 這就是系統(tǒng)動(dòng)力學(xué)的水平方程(Level方程)。 在一個(gè)水平方程中,可以有一個(gè)或者幾個(gè)流入速率,也可以同時(shí)有一個(gè)或者幾個(gè)流出速率。 水平方程實(shí)際是

4、積分運(yùn)算,用微積分符號(hào)可將上式寫(xiě)成:,3速率(Rate)變量,速率(流速) (Rate)表示系統(tǒng)中水平變量變化的強(qiáng)度,具有瞬時(shí)性的特征,反映單位時(shí)間內(nèi)水平變量增加或減少的量。 水平變量是系統(tǒng)活動(dòng)結(jié)果的狀態(tài)變量,而速率則是對(duì)水平變量變化過(guò)程及其控制的描述。 速率的基本形式有兩種,流入速率和流出速率。,4延遲(Delay),在復(fù)雜的社會(huì)經(jīng)濟(jì)系統(tǒng)中存在廣泛的延遲(Delay)現(xiàn)象即系統(tǒng)中的物流或信息流從它們的輸入到它們的輸出響應(yīng),總不可避免地有一段時(shí)間的延遲,這段時(shí)間就是延遲時(shí)間或延遲。 例如,從訂貨到收貨,固定資產(chǎn)的投資到發(fā)揮經(jīng)濟(jì)效益,從下種到莊稼收獲,從投入教育經(jīng)費(fèi)到人才的產(chǎn)出,從污染物散入環(huán)境

5、到危害人類(lèi)健康,等等都存在著一段或長(zhǎng)或短的延遲時(shí)間,這些是物流的延遲。 同樣信息流也存在延遲。如,商品供求關(guān)系的變化要經(jīng)過(guò)一段時(shí)間才會(huì)引起商品價(jià)格的變動(dòng);產(chǎn)品質(zhì)量影響工廠(chǎng)的聲譽(yù)也需要一段時(shí)間。,延遲實(shí)際上是將系統(tǒng)中流入速率變?yōu)榱鞒鏊俾实姆N轉(zhuǎn)換過(guò)程。一個(gè)動(dòng)態(tài)的流體系統(tǒng)通常是同一瞬間流入速度不等于流出速率,這就是說(shuō)從這一輸入到輸出的過(guò)程中會(huì)有一種延遲傳送的流量:當(dāng)流入速率大于流出速率時(shí),延遲傳送的流量就增多;當(dāng)流出速率大于流入速率時(shí),延遲傳送的流量就減少。 因此,延遲是一種特殊的水平變量,特殊的積累,延遲是一個(gè)“積累容器”、但它又不同于一般的水平變量。因?yàn)檠舆t的流出速率只受延遲時(shí)間的影響,與外界因

6、素?zé)o關(guān);而一般水平變量的流出速率除受本身的特性決定以外,還受外界因素的影響。,一個(gè)簡(jiǎn)單的疾病蔓延模型,三個(gè)水平變量:未患病者;患病者;康復(fù)者; 為分析需要,增加新的水平變量:INC處于潛伏期者,其輸入速率為感染率INF,輸出速率為疾病顯現(xiàn)率SYMP。,5信息反饋系統(tǒng),控制論系統(tǒng)可以劃分為開(kāi)環(huán)系統(tǒng)和閉環(huán)系統(tǒng)。閉環(huán)系統(tǒng)又叫信息反饋系統(tǒng)。 如果系統(tǒng)的輸出量對(duì)系統(tǒng)的輸入量沒(méi)有影響,即對(duì)系統(tǒng)的控制作用沒(méi)有影響,則叫做開(kāi)環(huán)系統(tǒng)。例如,洗滌機(jī)就是這種系統(tǒng)在洗滌機(jī)中依次進(jìn)行著浸濕、洗滌和漂清的過(guò)程。在洗滌機(jī)中不需要對(duì)輸出信號(hào),即衣服的清潔度進(jìn)行測(cè)量。 開(kāi)環(huán)系統(tǒng),不需對(duì)輸出量進(jìn)行測(cè)量,也不需將輸出量反饋到輸入端

7、與輸入量進(jìn)行比較。這種系統(tǒng)不是反饋控制系統(tǒng),當(dāng)出現(xiàn)擾動(dòng)時(shí),開(kāi)環(huán)控制系統(tǒng)就不能完成既定任務(wù)。,閉環(huán)系統(tǒng)是指系統(tǒng)的輸出對(duì)系統(tǒng)的輸入有影響的系統(tǒng),也就是系統(tǒng)輸出信息對(duì)控制作用有直接影響的系統(tǒng)。因此,閉環(huán)系統(tǒng)又叫信息反饋系統(tǒng)。輸入信號(hào)和反饋信號(hào)之差,叫誤差信號(hào)。誤差信號(hào)送到控制器上,以減少系統(tǒng)的誤差,使系統(tǒng)的輸出量趨于所希望的目標(biāo)值。 例如,商業(yè)中,訂貨庫(kù)存系統(tǒng)就是反饋系統(tǒng)。訂貨與庫(kù)存水平產(chǎn)生生產(chǎn)決策,生產(chǎn)決策又引起供貨、調(diào)整庫(kù)存,最后再引起生產(chǎn)決策,具有市場(chǎng)信息反饋的生產(chǎn)系統(tǒng)是一個(gè)反饋系統(tǒng)。該系統(tǒng)根據(jù)市場(chǎng)信息控制產(chǎn)品生產(chǎn)。,信息反饋系統(tǒng)又可以劃分為兩種系統(tǒng):正反饋系統(tǒng)Positive (Reinfo

8、rcing) Feedback Loop 、負(fù)反饋系統(tǒng)Negative (Balancing) Feedback Loop 。 正反饋系統(tǒng)是指系統(tǒng)運(yùn)行發(fā)生一個(gè)持續(xù)增長(zhǎng)過(guò)程,因?yàn)樵谠撓到y(tǒng)中控制作用引起的結(jié)果會(huì)導(dǎo)致發(fā)生更強(qiáng)的控制作用。 負(fù)反饋系統(tǒng),是系統(tǒng)運(yùn)行時(shí)尋求目標(biāo),若沒(méi)有達(dá)到目標(biāo)就會(huì)不斷產(chǎn)生反應(yīng)的系統(tǒng)。 劃分一個(gè)特定反饋系統(tǒng)是正反饋系統(tǒng)還是負(fù)反饋系統(tǒng)或者開(kāi)環(huán)系統(tǒng),也是相對(duì)的。這取決于觀(guān)測(cè)者確定系統(tǒng)目標(biāo)時(shí)的特殊著眼點(diǎn)(觀(guān)點(diǎn))。,6因果關(guān)系,系統(tǒng)由相互依存、相關(guān)作用的要素組成。如果要素A的量的變化會(huì)引起要素B的量的變化,則稱(chēng)A與B之間存在著因果關(guān)系。反映系統(tǒng)各要素之間因果關(guān)系的圖就稱(chēng)為因果關(guān)系圖。

9、 系統(tǒng)動(dòng)力學(xué)用矢線(xiàn)表示系統(tǒng)中兩個(gè)要素(變量)之間的聯(lián)系,稱(chēng)為因果鏈或因果環(huán)(Causal Link)。如果有兩個(gè)因素(變量)A和B,它們之間存在因果聯(lián)系,如果A變化A,則引起B(yǎng)變化B或-B。這時(shí),可把A看作B變化的原因,B是A的結(jié)果。,正因果關(guān)系:設(shè)有兩個(gè)因素(變量)A和B。如果A變化A,則使B變化B,A和B變化是同號(hào)。也就是說(shuō),如果A增加引起B(yǎng)增加,或者A減少引起B(yǎng)減少,即A對(duì)B的關(guān)系是正因果關(guān)系,記作A+B,如人口總數(shù)和出生率的關(guān)系是正因果關(guān)系。 負(fù)因果關(guān)系:設(shè)有兩個(gè)因素(變量)A和B。如果A變化A,則使B變化-B,A和B變化是異號(hào)。也就是說(shuō),如果A增加引起B(yǎng)減少,或者A減少引起B(yǎng)增加,即

10、A對(duì)B的關(guān)系是負(fù)因果關(guān)系,記作A-B。如死亡率和人口總數(shù)的關(guān)系就是負(fù)因果關(guān)系。 A對(duì)B的因果關(guān)系無(wú)論是正因果關(guān)系還是負(fù)因果關(guān)系,反過(guò)來(lái)并不一定成立。例如,死亡率對(duì)人口總數(shù)是負(fù)因果關(guān)系,但人口總數(shù)對(duì)死亡率卻是正因果關(guān)系,7反饋回路,系統(tǒng)中有兩個(gè)或者兩個(gè)以上的變量(水平變量、關(guān)于水平變量的信息變量),按照它們的因果反饋關(guān)系,組成一個(gè)閉合的因果反饋回路(又稱(chēng)因果反饋環(huán))。,正反饋回路(self-reinforcement自我增強(qiáng)):一個(gè)反饋回路,假設(shè)其中某一變量有一個(gè)變化(增加),依次通過(guò)回路中其他變量的因果關(guān)系之后,結(jié)果是加強(qiáng)原來(lái)變量的變化(增加);或者,這個(gè)變量有一個(gè)變化(減少),經(jīng)過(guò)一系列因果

11、關(guān)系后,結(jié)果是減弱這個(gè)變量的變化(減少)。那么,這個(gè)回路叫正反饋回路,記作“(+)”。 負(fù)反饋回路(goal-seeking向目標(biāo)追尋):一個(gè)反饋回路,假設(shè)其中某一變量有一個(gè)變化(增加),依次通過(guò)回路中其他變量的因果關(guān)系之后,結(jié)果減弱了原來(lái)變量的變化(減少);或者,這個(gè)變量有一個(gè)變化(減少),經(jīng)過(guò)一系列因果關(guān)系后,結(jié)果是增強(qiáng)這個(gè)變量的變化(增加)。那么,這個(gè)回路叫負(fù)反饋回路,記作“(-)” 判斷一個(gè)反饋回路是正還是負(fù)有一條簡(jiǎn)明的規(guī)則:把反饋回路中所有的負(fù)號(hào)(負(fù)因果鏈)的個(gè)數(shù)都加起來(lái),如果負(fù)號(hào)鋒的個(gè)數(shù)之和為零或偶數(shù)時(shí)則為正反饋回路,如果是奇數(shù),則為負(fù)反饋回路,8因果反饋回路圖,一個(gè)復(fù)雜的系統(tǒng)由幾

12、個(gè)或多個(gè)子系統(tǒng)組成系統(tǒng)動(dòng)力學(xué)用數(shù)學(xué)模型來(lái)模擬系統(tǒng)并利用信息把系統(tǒng)定量化。一個(gè)系統(tǒng)就是一個(gè)模型。一個(gè)復(fù)雜模型可以分解為若干個(gè)子模型(模塊)。一個(gè)子模型中又有若干個(gè)反饋回路,包括正、負(fù)相結(jié)合的反饋回路,把這些反饋回路按其內(nèi)在因果關(guān)系聯(lián)結(jié)起來(lái),就構(gòu)成整個(gè)模型的因果反饋圖。,一階正反饋回路,一階是指一個(gè)水平變量 舉例:已知人口的年增加值和總?cè)丝跇?gòu)成一階正反饋回路,現(xiàn)假設(shè)人口的年增長(zhǎng)率為2%,期初人口為100萬(wàn)。試構(gòu)造因果反饋結(jié)構(gòu)并進(jìn)行計(jì)算機(jī)仿真模擬。,手工仿真計(jì)算的前幾步:,(1)birth=birth rate* population Units: Year (2)birth rate=0.02 U

13、nits: *undefined* (3)FINAL TIME = 100 Units: Year The final time for the simulation. (4)INITIAL TIME = 0 Units: Year The initial time for the simulation. (5)population= INTEG (birth,100) Units: *undefined* (6)SAVEPER = TIME STEP Units: Year 0,? The frequency with which output is stored. (7)TIME STEP

14、 = 1 Units: Year 0,? The time step for the simulation.,注釋?zhuān)?INTEGRAL:積分 即population是birth的積分,初始值為100。 SAVERPER:the frequency with which values are saved for latter display TME STEP:the integration solution interval.,Time (Year)population Runs: 0 100 1102 2104.04 3106.121 4108.243 5110.408 6112.616 71

15、14.869 8117.166 9119.509 10121.899 11124.337 12126.824 13129.361 14131.948 15134.587 16137.279 17140.024 18142.825 19145.681,一階負(fù)反饋回路,舉例:一個(gè)基本的庫(kù)存控制系統(tǒng)。 現(xiàn)假設(shè)初始庫(kù)存量為 1000噸,期望庫(kù)存量為6000噸,由當(dāng)前庫(kù)存量調(diào)整到期望庫(kù)存量所需要的時(shí)間AT為5周,試構(gòu)造因果反饋結(jié)構(gòu)并進(jìn)行計(jì)算機(jī)仿真模擬。,手工仿真計(jì)算的前幾步:,水平變量是庫(kù)存量,記為I。 訂貨速率為OR,OR可以為正,也可為負(fù)。正的表示訂貨增加庫(kù)存量,負(fù)的表示將現(xiàn)有的庫(kù)存量退回供應(yīng)部門(mén)以

16、減少庫(kù)存量。 系統(tǒng)的目標(biāo)是要求的庫(kù)存量,記為DI。 由當(dāng)前庫(kù)存量調(diào)整到期望庫(kù)存量所需時(shí)間,記為AT,一階負(fù)反饋回路:庫(kù)存量模擬結(jié)果,Current表示調(diào)整時(shí)間為2天run2表示調(diào)整時(shí)間為1天,總結(jié):兩種反饋回路,一階正反饋回路具有自我加強(qiáng)的作用,對(duì)系統(tǒng)起強(qiáng)化作用,經(jīng)過(guò)若干年后總?cè)丝跀?shù)會(huì)越來(lái)越多,每年增加的人口數(shù)也相應(yīng)地越來(lái)越多。 一階負(fù)反饋回路具有自我平衡的作用,經(jīng)過(guò)若干時(shí)間(周、天)后,庫(kù)存量會(huì)越來(lái)越接近于期望庫(kù)存量,定貨速率也相應(yīng)地越來(lái)越小,庫(kù)存差額相應(yīng)地減小。,畫(huà)出系統(tǒng)的因果反饋圖是建立系統(tǒng)動(dòng)力學(xué)的數(shù)學(xué)模型的基礎(chǔ)。 在因果反饋圖中采用名詞,不用動(dòng)詞。變量之間的影響和作用用以帶箭頭的因果鏈

17、表示。 明確變量增減的涵義。 盡可能確定變量的量綱,這有助于突出因果反饋圖中文字?jǐn)⑹龅暮x。 反饋結(jié)構(gòu)一定形成閉合回路。 畫(huà)好各子模型中的反饋回路,區(qū)別正反饋還是負(fù)反饋,不能產(chǎn)生差錯(cuò)。,系統(tǒng)動(dòng)力學(xué)仿真的基本步驟,Vensim軟件介紹,界面及工具 建立因果反饋圖( causal loop diagrams )和流圖( Stock and Flow Diagrams ) 模擬過(guò)程 函數(shù):LOOKUP、TIME、STEP,What is Vensim?,Vensim is a visual modeling tool that allows you to conceptualize, documen

18、t, simulate, analyze, and optimize models of dynamic systems. Vensim provides a simple and flexible way of building simulation models from causal loop or stock and flow diagrams. By connecting words with arrows, relationships among system variables are entered and recorded as causal connections. Thi

19、s information is used by the Equation Editor to help you form a complete simulation model. You can analyze your model throughout the building process, looking at the causes and uses of a variable, and also at the loops involving the variable. When you have built a model that can be simulated, Vensim

20、 lets you thoroughly explore the behavior of the model. -Vensim Users Guide Version 5,界面及工具,標(biāo)題Title Bar,菜單Menu,Toolbar,繪圖工具Sketch Tools,狀態(tài)條Status Bar,Structural Analysis Tools Causes Tree creates a tree-type graphical representation showing the causes of the Workbench Variable.因果樹(shù)圖 Uses Tree create

21、a tree-type graphical representation showing the uses of the Workbench Variable.使用樹(shù)圖 Loops displays a list of all feedback loops passing through the Workbench Variable.反饋圖 Document reviews equations, definitions, units of measure, and selected values for the Workbench Variable. Dataset Analysis Tool

22、s Causes Strip Graph displays simple graphs in a strip, allowing you to trace causality by showing the direct causes (as shown) of the Workbench Variable.顯示與所選擇變量有直接關(guān)系的變量的模擬結(jié)果 Graph displays behavior in a larger graph than the Strip Graph, and contains different options for output than the Strip Gra

23、ph. Table generates a table of values for the Workbench Variable. Runs Compare compares all Lookups and Constants in the first loaded dataset to those in the second loaded dataset.,分析工具,Analysis Tool Output,控制板The Control PanelThe Control Panel allows you to change internal settings that govern the

24、operation of Vensim, such as which Workbench Variable is selected or what Datasets are loaded.,2. 建立因果反饋圖( Causal loop diagram)和建立流圖(Stock and flow diagram),1)建立因果反饋圖(causal loop diagrams):以Savings-Interest為例,演示,3種分析工具:Cause Tree ; Uses Tree;Loops,分析因果反饋圖的邏輯結(jié)構(gòu),兩個(gè)圖經(jīng)由變量savings的因果關(guān)系而連接,剛好分別前后對(duì)調(diào);當(dāng)反饋環(huán)向后回

25、到該變量,則用括號(hào)括起來(lái)。,確定savings為變量后,點(diǎn)擊Loops,列舉全部有關(guān)該變量的因果反饋圖,Loop Number 1 of length 1 saving interest Loop Number 2 of length 2 saving work effort income,因果反饋圖只能描述反饋結(jié)構(gòu)的基本方面,不能表示不同性質(zhì)的變量的區(qū)別,這是其根本弱點(diǎn)。如狀態(tài)變量的積累概念,是系統(tǒng)動(dòng)力學(xué)中最重要的量,然而因果反饋圖全然忽視了這一點(diǎn)。因此,需要建立流圖。,2)建立流圖(Stock and Flow Diagrams ) -以workforce-inventory為例,cons

26、truct Inventory as a Level, then add a rate flowing in and a rate flowing out.,Now we need to figure out how production gets determined. add the level Workforce. net hire rate can either increase or decrease the workforce.,add the proportionality constant productivity. Also, net hire rate is depende

27、nt on the value of Workforce.,add in the variables target workforce and time to adjust workforce and connect them.,Add the concept of target production, and connect it to target workforce. We will set target production on the basis of sales.,introduce target inventory, inventory correction and two a

28、dditional Constants. The idea is simple . target inventory is the amount of stock that should be held based on expectations about sales. The inventory correction is the correction for a deviation of Inventory from its target.,3. 模擬:以workforce-inventory為例,The following steps are typical for building

29、and using Vensim models. Construct a model or open an existing model. Examine the structure using the structural Analysis tools (Tree Diagrams.). Simulate the model moving around model parameters to see how it responds. Examine interesting behavior in more detail using the dataset Analysis tools (Gr

30、aphs and Tables). Perform controlled simulation experiments and refine the model. Present the model and its behavior to your audience using SyntheSim results, Analysis tool output customer Graphs and Tables.,1) open an existing model:c:Program FilesVensimmodelsguidechap03WFINV.MDL,2)Examine the stru

31、cture using the structural Analysis tools (Tree Diagrams.).,Causes Tree Diagram (Workbench Variable, Inventory, is on the right and everything that causes it to change (up to 2 connections distant) is on the left.),the Workbench Variable on the left and where it is used in the model (what it causes

32、change, up to 2 connections distant) on the right.,Uses Tree Diagram,the Loops Analysis tool:variable net hire rate,Loop Number 1 of length 1 net hire rate Workforce Loop Number 2 of length 6 net hire rate Workforce production Inventory inventory correction target production target workforce,display

33、s all variables in all feedback loops (two) that pass through the Workbench Variable (net hire rate).,3)Simulating the Model:amount of Inventory over time.,Double click on the simulation Runname editing box on the Toolbar to highlight the default name Current (or click once and drag over the name Cu

34、rrent), then type in the name baserun. Click on the SyntheSim button,Double click on the simulation Runname editing box on the Toolbar again and replace the name baserun with experiment.,Using the mouse drag the slider below productivity back and forth. As you move the slider the model will simulate

35、 and the results will display in blue, with the results from baserun being shown in red.,4)Examining Behavior,Position the mouse over Workforce and leave it there. A graph should pop up just below workforce.,Click on Workforce to select it into the Workbench. Click on the Graph Analysis tool Two thi

36、ngs are worth noting about the above graph. First both runs show a pattern of behavior known as damped oscillation. Second, the two runs are identical except for scaling.,Click on the variable Inventory appearing in the sketch and then click on the Graph tool,We see a graph of Inventory with oscilla

37、ting behavior similar to workforce, although Inventory starts out by declining before increasing in value. More importantly, there is only one graph visible.,Click on the Table tool,Only one graph line is visible for Inventory because the values are identical for both runs. Changing productivity imp

38、acts only Workforce, target workforce and net hire rate. This is quite clear just by looking at the model diagram as you drag the slider and occurs because productivity really just scales the number of people required to produce one item.,Click on the Reset Slider button or press the Home key to ret

39、urn the value of productivity to its original setting. Repeat the above experimentation process with each of the three remaining constants. Drag the sliders observing behavior and then bring up more detailed graphs when you see something interesting.,The things to be looking for when you are evaluat

40、ing behavior are the period of oscillation(振動(dòng)), the extent by which variables change and the degree of damping(衰減). The period of oscillation is the time from one peak to the next on the time graphs. Damping is the decrease in amplitude that occurs from peak to peak, where amplitude is the distance

41、along the y axis from the eventual value the variable settles to. It should be easy to see that productivity and inventory coverage do not significantly change these while time to adjust workforce and time to correct inventory do.,time to adjust workforce,5)Causal Tracing,Just as you looked at the c

42、auses of Inventory by using the Causes Tree Diagram Analysis tool, you can also look at graphs of behavior of the variables that cause Inventory to change.,Click on the Reset All button or use the key combination Ctrl+Home. Click on Inventory to select it into the workbench. Click on the Causes Strip Analysis tool,A strip graph is generat

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