Multivariate statistical process control of platinum: a case of mining company in Shabani, Zimbabwe

Transcription

1 Naional Universiy of Science and Technolgy NuSpace Insiuional Reposiory Applied Mahemaics hp://ir.nus.ac.zw Applied Mahemaics Publicaions Mulivariae saisical process conrol of plainum: a case of mining company in Shabani, Zimbabwe Mawonike, Romeo Inernaional Journal of Research in Engineering & Applied Sciences Mawonike, R., Geza, E. & Mwembe, D., 213. Mulivariae Saisical Process Conrol Of Plainum: A Case Sudy Of Mining Company In Shabani, Zimbabwe. Inernaional Journal of Research in Engineering & Applied Sciences, 3(1), pp hp://ir.nus.ac.zw/xmlui/handle/ /413 Downloaded from he Naional Universiy of Science and Technology (NUST), Zimbabwe

2 IJREAS Volume 3, Issue 1 (Ocober 213) (ISSN ) MULTIVARIATE STATISTICAL PROCESS CONTROL OF PLATINUM: A CASE STUDY OF MINING COMPANY IN SHABANI, ZIMBABWE Romeo Mawonike * Ephifania Geza ** Desmond Mwembe *** ABSTRACT Applicaion of saisical mehods in monioring and conrol of indusrial processes are generally known as saisical process conrol (SPC). Since mos of he modern day indusrial processes are mulivariae in naure, mulivariae saisical process conrol (MSPC), supplaned univariae SPC echniques. MSPC echniques are no only significan for scholasic pursui; i has been addressing indusrial problems in recen pas. Monioring and conrolling a chemical process is a challenging ask because of heir mulivariae, highly correlaed and non-linear naure. In his paper, a series of echniques were applied. Time series plo was implemened o deermine he saionariy of he daa. The Box-Jenkins mehodology of model idenificaion, esimaion and validaion; was used o generae ARIMA models based on muliple non sequenial daa. As a resul, he residuals from ARIMA models have shown four aribues: normally disribued, uncorrelaed, independen and no auocorrelaion beween successive ime poins. Two MSPC echniques; Mulivariae Cumulaive Sum (MCUSUM) and Mulivariae Exponenially Weighed Moving Average (MEWMA) were implemened as conrol chars for monioring residuals. All he chars indicae he ou of conrol signals in he process, which were believed o be from one or more variables combined ogeher. The problem of which variable is causing he ou-of-conrol process and when is ha ou-of-conrol happening was alleviaed hrough he consrucion of individual Cumulaive Sum (CUSUM) conrol chars. Eliminaion of ou-of-conrol signals resuled in a successful in conrol process shown in boh MCUSUM and MEWMA chars. Comparison beween hese wo mulivariae chars shows ha MCUSUM is more powerful in deecing smaller shifs han he MEWMA char. Therefore, monioring of residuals provided a valuable proof-of-concep ha validaed he use of ime series analysis in conjuncion wih MSPC ools in modeling and monioring he behaviour of indusrial processes. Inernaional Journal of Research in Engineering & Applied Sciences hp:// 22

3 IJREAS Volume 3, Issue 1 (Ocober 213) (ISSN ) Keywords: Mulivariae Saisical Process Conrol, Auoregressive Moving Average, indusrial processes, mean shif. 1. INTRODUCTION As saed by Mongomery (), Saisical Process Conrol (SPC) is a collecion of ools ha help achieving process sabiliy and improving capabiliy hrough reducion of variance. The objecive is mainly obained by quick deecion and eliminaion of unusual disrupions and fauls. Formally speaking, a fauly is defined as a deparure of a calculaed saisic from an accepable range. Usually his goal is achieved hrough he use of conrol chars. As defined by Sapenhurs (), a conrol char is a plo of process characerisics usually hrough ime wih saisically deermined limis. In order o ensure a high level of process monioring performance, researchers and praciioners frequenly use many saisical mehods called SPC. For many decades, he primary use of SPC mehods was focused on indusrial qualiy conrol applicaion. However, wih recen increase of informaion echnology sysems, new areas of SPC have emerged, for example applicaion range from video surveillance sysems (Elbasi e al ) o nework inrusion deecion (Park ) and healhy safey managemen. New echnology has also enhanced he divergence from univariae analysis o mulivariae analysis of conrol chars, due o complexiies and he increase of conrol improvemen requiremens, has led o he necessiy of enhancing advanced SPC known as he Mulivariae Saisical Process Conrol (MSPC) mehods. Nowadays, in indusry, here are many siuaions in which he simulaneous monioring or conrol of wo or more relaed qualiy process characerisics is necessary. Monioring hese qualiy characerisics independenly can be very misleading. Process monioring of problems in which several relaed variables of ineres are collecively known as mulivariae saisical process conrol. The mos useful ool of mulivariae saisical process conrol is he qualiy conrol char. Indeed, in order o apply MSPC mehods, very resricive assumpions ha are usually difficuly o be saisfied in several process conrol applicaions are imposed. * Deparmen of Mahemaics and Compuer Science, Grea Zimbabwe Universiy, Zimbabwe. ** Deparmen of Applied Mahemaics, Naional Universiy of Science and Technology, Zimbabwe. **** Deparmen of Applied Mahemaics, Naional Universiy of Science and Technology, Zimbabwe Inernaional Journal of Research in Engineering & Applied Sciences hp:// 23

4 IJREAS Volume 3, Issue 1 (Ocober 213) (ISSN ) 1.2 BACKGROUND OF THE STUDY Economy Plainum Group (PGE) - mineralizaion occurs wih he Main Sulphide Zone (MSZ) which is lihologically coninuous, generally 1 m o m from he op of he ulramafic sequence and ypically 2 m o 3 m hick. Alhough plainum was firs discovered in he Grea Dyke in 192, is exploiaion was overshadowed by ha of he Bushveld Complex in Souh Africa. Focus has reurned o he Grea Dyke following he increase in demand for plainum. Peak values for he Plainum Group Meals (PGM) and base meals are commonly offse, while he proporions beween plainum and palladium also vary verically. Anoher source of PGM which is becoming imporan, especially in he U.S., is he secondary source; namely, scrap of ceramics/glass, elecrical componens, and spen caalyss. Plainum is obained as a by - produc from Nickel, Copper and Palladium mining and processing. Noble meals like silver, gold and plainum sele o he boom of he cell as anode mud during elecro-refining. Pure plainum is isolaed from deposis and oher ores hrough differen levels of subracing impuriies. The feed maerial comes from a jaw crusher sysem designed o crush he mined ore o minus 1 mm. Furher comminuing is performed via a closed circui SAG mill, pebble mill and ball milling circui. This produces a cyclone sized ore produc of p8 14 microns ha repors o rougher floaion and flash floaion based on size. The Nye uses four major floaion reagens. These are poassium amyl xanhenes promoer, cye 3477 promoer, cmc for alc suppression and sulfuric acid for ph adjusmen. In essence he operaing philosophy is focused on maximum recovery of all sulfides ha bear he PGM s. The rougher concenrae is subjec o wo sages of cleaning followed by final column cell floaion. The ails from he rougher circui are reground and subjeced o milding and scavenger floaion. Simply, impuriies like iridium and Se are subraced hrough mechanical processing like floaion and filraion. 1.3 MAIN OBJECTIVES To idenify variables causing an ou of conrol siuaion. To fi an ARIMA model for each variable and hen obain he residuals. To deermine he in-conrol and ou-of-conrol signals by using MCUSUM and MEWMA conrol chars. 1.4 RESEARCH HYPOTHESIS : The process of refining plainum is in conrol. Inernaional Journal of Research in Engineering & Applied Sciences hp:// 24

5 IJREAS Volume 3, Issue 1 (Ocober 213) (ISSN ) : The process of refining plainum is ou of conrol. 1. ASSUMPTIONS Variables are independenly disribued. Process daa is saionary and linear. There is no mulicollineariy beween variables. The variables are random and normally disribued. 2. LITERATURE REVIEW Hawkins (1993), suggesed a MSPC conrol char using as a charing saisic, and showed ha his char was more effecive han one based on when he poenial shif occurs in only one measuremen componen. Hawkins (1991, also suggesed ha he char based on can perform poorly, as when he shif occurs in several highly correlaed componens and when he shif is proporional o wo leading principal componens of he covariance marix. Li e al. (), proposed an Adapive Principal Componens Analysis (APCA) algorihm for an adapive process monioring of processes ha run under muliple operaing modes by inroducing an efficien approach o updae he correlaion marix, he number of principal componens and he confidence limis recursively. The proposed adapive PCA mehod was successfully applied o a sequencing bach reacor (Lee e al. ). Choi e al. (), inroduced recursive updaed PCA along wih wo monioring merics, Hoelling s and he Q saisic, for monioring imevarying processes. However, PCA assumes ha he relaionship beween variables is linear and herefore is applicaion o nonlinear processes provided poor resuls. Thus, many researchers have proposed he use of Kernel Principal Componens Analysis (KPCA) mehod in order o monior such processes. Lee e al. (4), applied KPCA echnique as a new nonlinear process monioring echnique for faul deecion in wo mulivariae processes. Many auhors showed ha he proposed approach is effecive in capuring he nonlinear relaionship in he process variables and ha i has superior process monioring performance compared o linear PCA. Bersimis e al (7) discussed he basic procedures for he implemenaion of mulivariae saisical process conrol via conrol charing. Furhermore, hey reviewed mulivariae exensions for all kinds of univariae conrol chars, such as mulivariae Shewhar ype conrol chars, mulivariae CUSUM conrol chars Inernaional Journal of Research in Engineering & Applied Sciences hp:// 2

6 IJREAS Volume 3, Issue 1 (Ocober 213) (ISSN ) and mulivariae EWMA conrol chars and also reviewed unique procedures for he consrucion of mulivariae conrol chars, based on mulivariae saisical echniques such as principal componens analysis (PCA) and parial leas squares (PLS). In general, heir paper reviews ha MEWMA conrol chars perform beer han classical Shewhar chars. Likewise, MCUSUM chars perform beer han Shewhar chars, while having a performance similar o ha of he MEWMA. Bagshaw and Johnson (197, saed ha, for Auoregressive AR (1) or a Moving Average MA (1) processes, incorrec conclusions can be drawn by using convenional CUSUM schemes. Harris and Ross (1991), discussed he impac of auocorrelaion on he performance of CUSUM and EWMA chars, and showed ha i affecs he average and median run lengh. Noorossana and Vaghe (6) showed ha breaking he independence assumpion affecs he average run lengh (ARL) of he conrol chars and makes hem unreliable. Psarakis and Papaleonida (7) assered ha even small levels of auocorrelaion can have big effecs on he saisical properies of convenional conrol chars and may cause subsanial increase in he average false alarm rae and a decrease in he abiliy of deecing changes on he process. Callao and Rius (3), used AR (1) and MA (1) based conrol chars o monior a sequenial injecion analysis, and showed ha residuals conrol chars provide beer undersanding of he sysem behaviour over ime and efficien deecion abiliies. Longnecker and Ryan (1992) invesigaed AR (1), AR (2) and ARMA (1,1) models for a residuals X-char and showed ou ha he X - char may have poor capabiliy o deec he process mean shif. Recenly, Jamal e al. (7), have inroduced an Arificial Neural Nework (ANN) based model o consruc residuals Mulivariae CUSUM char for mulivariae AR processes and showed ha i performs beer han MCUSUM and also showed ha despie heir advanages, mos ANN mehods have some drawbacks ha can affec heir performance such as he problem of opimal parameers selecion. Pacella and Semeraro (7, proposed a simple Neural Nework (NN) model o conrol auo correlaed processes and showed ha i performs well for several mean shifs. Sudjiano and Wasserman (1996) have successfully applied principal componen analysis o exrac feaures from large daases. Huang e al () used PCA o cluser mulivariae ime series daa by spliing large clusers ino small clusers based on he percenage of variance explained by principal componen analysis. They discovered his mehod is resricive if he Inernaional Journal of Research in Engineering & Applied Sciences hp:// 26

7 IJREAS Volume 3, Issue 1 (Ocober 213) (ISSN ) number of principal componens is no known a priori and because predeermined principal componens are inadequae for some operaing condiions. Seshu (211) in he hesis, applied MVSPC mehod based on clusering ime series daa and moving window based paern maching echnique for successful process monioring. Biochemical reacor, Drum-boiler, coninuous sirred ank wih cooling jacke and Tennessee Easman challenge processes were aken up o implemen he proposed monioring echniques. Insead of using firs hand pla daa, he above processes were modeled using firs principles and processes were perurbed a indusrially relevan operaing condiions including fauly ones o creae vecor ime series daabases. Weherill and Brown (1991) among ohers, poin ou ha many mulivariae mehods give no indicaion of which variable or variables are causing he problem. Few mulivariae chars canno deermine which variable(s) are OOC unless furher analysis is done. However, recen work has been done o deermine he variable or variables ha led o he mulivariae ou of conrol (OOC) signal. A common mehod used in a pos - signal analysis is o run muliple univariae chars and see variables signal on he individual char. In boh classical univariae and mulivariae SPC problems, successive observaions or samples of observaions are assumed o be saisically independen. This however, may no be he case in many of oday s qualiy conrol applicaions. Hwarng and Wang (8) used all he chars, Hoelling s, MEWMA and MCUSUM in comparing heir neural-nework-based idenifier char. The char no only deecs he mulivariae signal bu deermines which variable(s) are a faul, bu i requires a large amoun of daa and ime o use properly. However, heir conrol limis were calculaed under he assumpion of no auocorrelaion and independence and his may have led o erroneous performance resuls and a misinerpreaion of hese resuls since he in-conrol ARL was no se in each case of auocorrelaion and correlaion. Adams and Tseng (1998) sudied he effec of parameer esimaion on forecas-based monioring schemes. They showed ha conrol chars applied o forecas residuals are very sensiive o errors in esimaion and ha he direcion of he error has an effec on he chars using esimaed parameers. Kramer and Schmid () sudied he effec on he ARL of univariae Shewharype chars including hose using he residuals of an auoregressive model. They showed ha Inernaional Journal of Research in Engineering & Applied Sciences hp:// 27

8 IJREAS Volume 3, Issue 1 (Ocober 213) (ISSN ) when parameers are esimaed, he modified Shewhar char, which uses modified conrol limis, is preferred since he radiional chars are affeced by known parameers. Champ e al. (), sudied effecs of esimaion wih Hoelling s conrol char. They showed ha when parameers are esimaed, he conrol limis for he known case are no appropriae. The in-conrol ARL is much higher wih esimaed parameers and hus he char is canno deec shifs, especially small shifs, as quickly as when he parameers are known. They also provide sample size recommendaions o obain a resul ha is similar o he case of known parameers. Champ and Jones-Farmer (7) also reviewed some papers peraining o he mulivariae EWMA conrol char and mulivariae CUSUM conrol chars. These works all show ha parameer esimaion has a serious effec on he performance of conrol chars. Soumbos and Sullivan (2) sudied he effecs of non-normaliy on he MEWMA and he char. They also showed ha he charing parameer of he MEWMA may be adjused o handle non-normal siuaions. However, his is bound o be edious and inefficien. In many poenial applicaions, he sample size is assumed o be large enough o assume ha he sample mean vecor is approximaely (univariae/mulivariae) normal by he cenral limi heorem. However, his assumpion is ofen quesionable wih smaller sample sizes, especially for individual chars. Chou, Mason, and Young (1), sudied he effec of mulivariae nonnormaliy on he char. They noed ha he phase II, UCL for he char based on he F disribuion may be very inaccurae. They used a kernel smoohing echnique o esimae he disribuion of he saisic so ha a more accurae UCL could be creaed. Smih (1987), developed a MCUSUM procedure based on he likelihood raio es, which is used o sudy shifs in he mean vecor of a mulivariae normal process. The procedure was adaped o sudy shifs in he covariance marix of a mulivariae normal process and o sudy in he probabiliies of a muli-nominal process, because of is cumulaive naure, his mehod is much beer a deecing small shifs in he covariance marix. Haisheng (21) applied mulivariae saisical analysis mehods such as principal componen analysis (PCA) and independen componen analysis (ICA) o exrac informaion regarding a pharmaceuical able. ICA was found o ouperform PCA and was able o idenify he presence of five differen maerials and heir spaial disribuion around he able. Inernaional Journal of Research in Engineering & Applied Sciences hp:// 28

9 IJREAS Volume 3, Issue 1 (Ocober 213) (ISSN ) Garland e al. (1997) compared he scanner s buil-in qualiy assurance sysem, visual inspecion, a muli-rule Shewhar char, and CUSUM conrol chars in esablishing he bes procedure for he qualiy assurance of DXA scanners. They uilized hree crieria for analysis, he number of fauls deeced ou of eigh non-mechanical fauls, he rue posiive fracion, and he Type I error rae. Their analysis was based on simulaed (phanom) image daa. Based on heir crieria, hey found ha visual inspecion, he muli-rule Shewhar char, and he CUSUM conrol char performed much beer han he scanner s buil-in qualiy assurance sysem. Psarakis and Papaleonida, (7), suggesed anoher mehod ha would be fi an auoregressive inegraed moving average (ARIMA) model o he daa and monior he residuals of he process wih a common univariae char. However, his assumes ha he chosen ARIMA model is appropriae for he daa o yield residuals ha are independenly and idenically disribued. Harris and Ross (1991) have shown ha using a Shewhar conrol char on he residuals may no be efficien in deecing small shifs in he process. Choi e al. (2), A general mulivariae exponenially weighed moving average conrol char, proposed a general MEWMA char in which he smoohing marix is full insead of one having only diagonal. The performance of his char appears o be beer han ha of he MEWMA proposed by Lowry e al. (1992). Runger e al. (1999) showed how he shif deecion capabiliy of he MEWMA can be significanly improved by ransforming he original process variables o a lower-dimensional subspace hrough he use of he U ransformaion. Wilksrom e al (1998) applied Auoregressive Moving Average (ARMA) models in principal componens. The use of radiional mulivariae Shewhar chars or Mulivariae Cumulaive Sum (MCUSUM) and Mulivariae Exponenially Weighed Moving Average (MEWMA) schemes may be impracical for high - dimensional sysems wih collineariies. Rigdom (199) gave an inegral and a double - inegral equaion for he calculaion of in-conrol and ou of conrol Average Run Lengh (ARLs) respecively. However, in his paper, ARIMA models are going o be applied on each variable of he five variables under sudy. s are o be moniored using he mulivariae Cumulaive Sum (MCUSUM) chars and mulivariae Exponenial Weighed Moving Average (MEWMA) chars and comparison of he mehods will be done based on heir robusness o deec he mean shif. 3. METHODOLOGY 3.1 Deerminaion of variables Inernaional Journal of Research in Engineering & Applied Sciences hp:// 29

10 IJREAS Volume 3, Issue 1 (Ocober 213) (ISSN ) Several processes done during he purificaion of plainum have similar qualiy characerisics which need o be moniored a he same ime. For insance, he qualiy of ore feeds grade should have cerain percenages per on of he mineral o be processed, he impuriy grade should have prescribed percenages which mus remain in he sysem as i is very expensive and ime consuming o remove all he impuriies such as iridium when refining plainum. On he oher hand if impuriies remaining have a high percenage, hen he qualiy of he final produc (plainum) will be lowered resuling in poor performance of he produc and also rework on he processes. The concenrae (lower grade and high grade) mus also have cerain percenages of he mineral before i is ranspored o Souh Africa for furher purificaion. In his paper, five qualiy characerisics were moniored for qualiy. Daily daa was colleced for six and half monhs ( days). The percenage of plainum per on of ore fed ino he sysem (feed grade). The percenage of rougher ails (RT) remaining in he sysem (impuriies). The percenage of final ails (FT) remaining in he sysem (impuriies). The percenage of Medium Grade Concenrae (MGC) in he sysem. The percenage of High Grade Concenrae (HGC) in he sysem. Table 3.1: Required percenage per each variable Variable Lower limi (%) Mean (%) Upper limi (%) Feeds FT RT MGC HGC Box Jenkins Mehodology Box Jenkins approach was used o come up wih he bes ARIMA model for each qualiy characerisic. The residuals obained were moniored hrough he use of conrol chars o saisfy he second objecive of he paper. The Box-Jenkins mehodology is an economeric framework, named afer saisicians George Box (1933) and Gwilym Jenkins (1919), which applies auoregressive moving average (ARMA) models or auoregressive inegraed moving average (ARIMA) models o ime-series daa o faciliae forecasing. Inernaional Journal of Research in Engineering & Applied Sciences hp:// 3

11 IJREAS Volume 3, Issue 1 (Ocober 213) (ISSN ) ARIMA (p, q) Processes Auoregressive inegraed moving average ARIMA was used o come up wih he bes model ha gives qualiy residuals. ARIMA ( ) has an auoregressive componen of order and a moving average componen of order. For each qualiy process o be saionary, i mus saisfy he condiion below: } is saionary and if for every,. (3.1) Where, The polynomials (3.2) (3.3) have no common facors. The process { } is said o be an ARMA (p, q) process wih mean μ if { μ} is an ARIMA ( ) process. I is more convenien o express in he form, (3.4) Where and are he pah and degree polynomials Model Idenificaion The purpose of model idenificaion is o undersand he paern in he ime series daa in erms of saionariy. If he daa of qualiy characerisics menioned above is no saionary, ha is violaing he second assumpion. I is made saionary wih respec o mean (mean equal o zero) and variance hrough ransformaions. In he firs sage of he Box-Jenkins approach, charing echniques consising of run sequence plos, auocorrelaion plos and parial auocorrelaion plos will be used o check saionariy, amongs oher characerisics of he ime series. The model idenificaion sage helps decide wheher he ime series may be modeled by ARIMA or no, and if so, wha order of auoregressive and moving average erms should be chosen for he validaion sage Auocorrelaion Plo Inernaional Journal of Research in Engineering & Applied Sciences hp:// 31

12 IJREAS Volume 3, Issue 1 (Ocober 213) (ISSN ) Auocorrelaion plo was used o idenify an ARIMA model o be fied o he daa by checking he behaviour of he spikes as illusraed in able 3.2. The auocorrelaion funcion (ACF) measures he degree of correlaion beween lagged values of he imes series. The auocorrelaion value is bounded by he inerval [ 1, 1] where a value close o 1 indicaes srong, posiive correlaion; a value close o 1 indicaes srong negaive correlaion; and a value close o indicaes weak or no correlaion. A char ha plos he sample auocorrelaion coefficien agains he associaed lag k is known as an ACF plo. The lagged process auo covariance and sample auo-covariance are: (3.) (3.6) Where is he covariance beween and, μ is process mean, is he sample mean and is he observaion Parial Auocorrelaion Plo Parial auocorrelaion plo was used o idenify an ARIMA model o be fied o he daa by checking he behaviour of he spikes as illusraed in able 3.2 below. The parial auocorrelaion coefficien a lag k is he auocorrelaion beween observaions and ha is no explained by lag k = 1 hrough o lag. Similar o he ACF, he F is bounded on he [ 1, 1] inerval; he numerical inerpreaion of he F wih respec o correlaive behaviour and srengh are also similar o he ACF. Meanwhile, he - lagged process auocorrelaion and sample auocorrelaion are: (3.7) (3.8) Table 3.2: Behaviour of ACF and F plos and recommended Models Shape Indicaed Model Inernaional Journal of Research in Engineering & Applied Sciences hp:// 32

13 IJREAS Volume 3, Issue 1 (Ocober 213) (ISSN ) Exponenial, decaying o zero Alernaing posiive and negaive, decaying o zero One or more spikes, res are essenially zero Decay, saring afer a few lags All zero or close o zero High values a fixed inervals No decay o zero Auoregressive model use F plo o idenify he order. Auoregressive model use F plo o idenify he order. Moving average model order idenified by where plo becomes zero. Mixed Auoregressive and Moving average (ARMA) model. Daa is essenially random Include seasonal auoregressive erm. Series is no saionary Parameer Esimaion The Yule-Walker se of linear equaions were used o esimae he parameers of he model. Since he Yule-Walker equaions are useful in deermining he F, his sudy focuses only on he Yule - Walker equaions among ohers. Having idenified a enaive model, he nex sep is o esimae he parameers idenified in he model. This is done by an ieraive process. An iniial se of values for he parameers were assumed. Ieraively, an opimal se was obained. The crierion for comparison of differen ses of parameer values is he mean square error Model Validaion Validaion of he bes ARIMA model on each of he five variables under sudy was carried ou. If he esimaion is deermined o be inadequae, he Box- Jenkins mehodology espouses reurning o he model idenificaion sage o reexamine he appropriaeness of he ARIMA model o he daa series. Validaing he model was o examine he residuals using chars and es saisics, which separaely, form he qualiaive and quaniaive bases of analysis respecively, and ogeher consiue he graphical residual analysis oolki. A powerful echnique o perform residual analysis proposed was o develop and examine 4-plos. 4-plos consis of a collecion of exploraory graphical analysis (EDA) graphs which are especially suiable for he graphical residual analysis of univariae models. A 4-plo ypically consiss of a run sequence plo, a lag plo (residuals versus fied values), a hisogram, and a normal probabiliy plo. For he model o be adequae: he run sequence plo of residuals mus have a mean zero and all poins should be inside he conrol limis, he hisogram mus approximae a normal disribuion curve, lag plo Inernaional Journal of Research in Engineering & Applied Sciences hp:// 33

14 IJREAS Volume 3, Issue 1 (Ocober 213) (ISSN ) (residuals agains fied values) mus show no sign of a paern and poins mus no be concenraed a one poin and finally, he normal probabiliy plo should exhibi a sraigh line and he value mus be greaer han zero in mos cases. 3.3 Mulivariae Cumulaive Sum (MCUSUM) conrol chars Mulivariae cumulaive sum (MCUSUM) conrol chars were used o monior he five ses of residuals obained from he ARIMA models simulaneous. The conrol char was used o deermine he in-conrol and ou of conrol signals. MCUSUM char was chosen by he researcher because of is abiliy o deec smaller shifs in he process mean vecor. In order o develop a MCUSUM, we look a a series of a sequenial probabiliy raio es of CUSUM. Le be he observaion, which derives from wih an in-conrol mean vecor and a known common variance-covariance marix. Le be he ou-of-conrol vecor means. The MCUSUM is based on he saisic:. (3.9) Where if if is half he magniude of he shif, and. This MCUSUM char signals when (3.1) 3.4. Mulivariae Exponenially Weighed Moving Average (MEWMA) conrol chars Mulivariae exponenially weighed moving average (MEWMA) chars are he second caegory of chars examined. The char also funcions as he MCUSUM char menioned above. These wo chars were used for comparison purposes in erms of performance. MEWMA char also has abiliy o deec smaller shifs in he process mean vecor. The formula is as follows: Le be he, -dimensional observaion. Also assume ha follows a wih a known variance-covariance marix and a known p-dimensional mean vecor. Inernaional Journal of Research in Engineering & Applied Sciences hp:// 34

15 IJREAS Volume 3, Issue 1 (Ocober 213) (ISSN ). (3.11) For where R = and for and I is he ideniy marix. The iniial is usually obained as equal o he in-conrol mean vecor of he process. I is obvious ha when R = I, hen he MEWMA conrol char is equivalen o he - char. The MEWMA gives an ou-of-conrol signal if, Where is he variance-covariance marix of. The value is calculaed by simulaion o achieve a specified in-conrol ARL. The MEWMA char is generally used in he phase II wih individual daa and uses he charing saisic, Where, =1, 2, 3 (3.12) The covariance marix is given by: (3.13) which he scalar charing consan λ, is he vecor of observaions a ime and. 3.. Average Run Lengh (ARL) The average run lengh is he number of samples before a process becomes ou of conrol. The average run lengh (ARL) of he mulivariae char when he process is in conrol and and are known can be calculaed as (3.14) Where α is he probabiliy ha exceeds Lu under he assumpion ha he process is in conrol, where, Furhermore, he ou-of-conrol ARL (ARL1) of he mulivariae Shewhar char depends on he mean vecor of variance-covariance marix only hrough he non - cenraliy parameer. Inernaional Journal of Research in Engineering & Applied Sciences hp:// 3

16 IJREAS Volume 3, Issue 1 (Ocober 213) (ISSN ) (3.1) Where is a specific ou of conrol mean vecor, bearing in mind ha he _ is sill in conrol. Hence i is possible o consider he ARL as a funcion of, he square roo of and o consruc curve by using he equaion, (3.16) Parameer β is he probabiliy ha he procedure fails o diagnose an ou of conrol siuaion CUMULATIVE SUM (CUSUM) CONTROL CHARTS The CUSUM chars were used o idenify he in-conrol and ou-of-conrol of individual residuals in he even ha he MCUSUM and or MEWMA char has deeced ou of conrol signal(s) a some poin(s) in he process. The CUSUM char helps he research o pinpoin he individual qualiy process or processes causing he in-conrol and ou-of-conrol of he processing of Plainum. In order o design he CUSUM conrol char, esimaes for he process sandard deviaion were compued, hese were obained from he I-MR chars. The CUSUM char plos wo one-sided series: upper one-sided values ( ) and lower one-sided value ( ). The upper one-sided values aggregae deviaions above he process arge mean of calculaed as:, and are (3.17) Similarly, he lower one-sided value aggregae deviaions below, and are calculaed as: (3.18) Where is he observaion of he process be he mean or arge value is he reference value. The criical size of he shif δ mus be deermined, he shif from he in-conrol mean o he ou of conrol upper and lower means and respecively. Again, resricing he sample size o one for simpliciy: (3.19) (3.2) Inernaional Journal of Research in Engineering & Applied Sciences hp:// 36

17 IJREAS Volume 3, Issue 1 (Ocober 213) (ISSN ) Where be he mean or arge value be he upper shif mean be he lower shif mean be he upper criical shif be he lower criical shif _ be he sandard deviaion of he process. The reference value K is half he magniude of he shif, hus a value half way beween he arge and he upper CUSUM mean, or he lower CUSUM mean. The reference value is a line, which if crossed, provides an early warning of a shif in he process mean. The upper and lower reference values and are calculaed as follows: (3.21) (3.22) A decision variable, which acs as a conrol limis, mus be creaed o deermine he sae of he process. The decision variable, if exceeded eiher from above (by he one-sided upper CUSUM) or below (by he one-sided lower CUSUM), deermines a shif in he process mean. The upper and lower decision variables and are calculaed as follows: (3.23) (3.24) However, o calculae H, he average run lengh (ARL) parameer h for boh he one sided upper ( ) and lower ( ) mus be deermined. 4. DATA ANALYSIS AND RESULTS 4.2. Model Idenificaion for each variable Time series plo Time series plo 2... feeds 1.9 FT Index 1 Index 1 Inernaional Journal of Research in Engineering & Applied Sciences hp:// 37

18 IJREAS Volume 3, Issue 1 (Ocober 213) (ISSN ) Time series plo Time series plo.2 RT.42 MGC Index 1 Index 1 Time series plo HGC Index 1 Fig 4.1: Time series plos Figure 4.1, shows ime series plos of all five variables and hey all exhibi non saionary behaviour in he sense ha he mean is varying over ime, ha is he mean is no zero. Therefore, he daa on all variables is no saionary. There is no need o plo he ACF and F for hese daa bu only requires ransformaions such as logarihms and differencing. Time series plo Time series plo 1. C3 C Index 1 Index 1 Time series plo Time series plo C4. C Index 1 Index 1 Inernaional Journal of Research in Engineering & Applied Sciences hp:// 38

19 IJREAS Volume 3, Issue 1 (Ocober 213) (ISSN ) Time series plo C3. - Index 1 Fig 4.2: Time Series Plos afer 1 s difference and aking logs Time series plos in figure 4.2 shows ha he daa in all variables is now saionary. This is indicaed by he zero mean and a consan variance. Since daa is saionary, idenificaion of he ARMA models is done hrough ploing ACF and F for each variable. acf plo Auocorrelaion Plo 1: feeds acf plo Auocorrelaion Plo 2: final ail Inernaional Journal of Research in Engineering & Applied Sciences hp:// 39

20 IJREAS Volume 3, Issue 1 (Ocober 213) (ISSN ) Inernaional Journal of Research in Engineering & Applied Sciences hp:// Auocorrelaion acf plo Plo 3: rougher ail E-1-3.7E-1 -.8E-1 Auocorrelaion acf plo Plo 4: medium grade concenrae E-1-3.6E-1 -.6E-1 Auocorrelaion acf plo Plo : high grade concenrae Fig 4.3: ACF Plos of he saionary daa

21 IJREAS Volume 3, Issue 1 (Ocober 213) (ISSN ) Inernaional Journal of Research in Engineering & Applied Sciences hp:// Parial Auocorrelaion pacf plo Plo 6: feeds Parial Auocorrelaion pacf plo Plo 7: final ail Parial Auocorrelaion pacf plo Plo 8: rougher ail

22 IJREAS Volume 3, Issue 1 (Ocober 213) (ISSN ) pacf plo Parial Auocorrelaion Plo 9: medium grade concenrae pacf plo Parial Auocorrelaion Plo 1: high grade concenrae Fig 4.4: F Plos for saionary daa ACF and F plos in fig 4.3 and fig 4.4 respecively were used o idenify he bes ARMA model for each se of daa. The idenified models are summarized in able 4.1 below. Table 4.1: Model specificaion and model parameers Variable Model specificaion Model parameers Feeds ARIMA(1,1,2), FT RT MGC HGC ARIMA(,1,1) ARIMA(,1,1) ARIMA(,1,1) ARIMA(,1,1) Inernaional Journal of Research in Engineering & Applied Sciences hp:// 42

23 IJREAS Volume 3, Issue 1 (Ocober 213) (ISSN ) 4.3 Model Validaion plo Normal Plo of s I Char of s.6 1 UCL= X= Normal Score Observaion Number LCL= Hisogram of s s vs. Fis Frequency Fi.6 Fig 4.: residual plos for Feeds plo Normal Plo of s I Char of s Normal Score Observaion Number UCL=.488 X=2 LCL=-.13 Hisogram of s s vs. Fis Frequency Fi Fig 4.6: residual plos for FT Inernaional Journal of Research in Engineering & Applied Sciences hp:// 43

24 IJREAS Volume 3, Issue 1 (Ocober 213) (ISSN ) plo Normal Plo of s I Char of s UCL= X= LCL= Normal Score 2 3 Observaion Number Hisogram of s s vs. Fis 4.1 Frequency E-1-1.E-1-8.E-2 -.E-2-3.E-2 -.E Fi.1 Fig 4.7: residual plos for RT plo Normal Plo of s I Char of s.1.2 UCL= X= Normal Score Observaion Number LCL= Hisogram of s s vs. Fis 6.1 Frequency Fi.1 Fig 4.8: residual plos for MGC Inernaional Journal of Research in Engineering & Applied Sciences hp:// 44

25 IJREAS Volume 3, Issue 1 (Ocober 213) (ISSN ) Normal Plo of s Normal Score 2 plo I Char of s 6 Observaion Number UCL=.4446 X=-6.2E-4 LCL=-.47 Frequency Hisogram of s s vs. Fis Fi Fig 4.9: residual plos for HGC The individual chars of residuals in fig 4. up o fig 4.9 indicae ha he residuals have consan locaion and scale. Daa poins are in conrol excep for fig 4.9 where oher poins are ou of conrol; his could be he resul of he presence of whie noise in he original daa. The lag plo also indicae ha here is no auocorrelaion beween successive poins a lag1 and he probabiliy plos show ha he all residuals follow a normal disribuion wih a p-value of.,.38,.392,.37 and.874 respecively. All models are echnically accepable a he % significance level since he p - values are above he criical value of. excep for he feeds which is cenered a zero. Plo of residuals agains fied values show no sign of paern and show randomness which is expeced from a good model. As a resul, hese residuals are adequae o moniored for qualiy by he echnique of conrol chars. 4.. Mulivariae Chars Inernaional Journal of Research in Engineering & Applied Sciences hp:// 4

26 IJREAS Volume 3, Issue 1 (Ocober 213) (ISSN ) Fig 4.1: Mulivariae Cumulaive Sum char of residuals (λ=.2) The MCUSUM char of residuals in figure 4.1 shows he ou of conrol process failing a several poins. The process is failing a poin 1 12 hen becomes normal from 13 24, again failing a poin 2 and 3 afer which i goes back o normal up o poin 73 and hen goes ou of conrol from and again fails a poins 12, 14, 141, 161, 162, 163, 18, 182, 183, 18, 187 and. Since five variables were being involved simulaneously, i could be differen variables failing a differen poin during he process. I is difficul o idenify which variable is causing he ou of conrol signal a a paricular ime. Therefore, i is necessary o char each of he residuals separaely using he CUSUM chars and poin ou which is characerisic(s) is causing he whole process o fail a some poins. Fig 4.11: Mulivariae Exponenially Weighed Moving Average char of residuals. Inernaional Journal of Research in Engineering & Applied Sciences hp:// 46

27 IJREAS Volume 3, Issue 1 (Ocober 213) (ISSN ) The MEWMA chars of residuals in figure 4.11 shows an ou of conrol process which is failing a poin 89, 9, 16, 166 and 187. Again he process consiss of five variables being moniored a he same ime. These ou of conrol signals canno indicae which variable(s) may have caused he failing a some poin(s). Therefore furher invesigaion is required o idenify he behaviour of each variable I-MR and Cumulaive Sum (CUSUM) chars I and MR Char for: res feed I and MR Char for: res FT 1 UCL= UCL=.1 Individuals MU=. Individuals. MU=. -1 LCL= LCL=-.1 Observaion Observaion Moving Range UCL=1.19 R=.34 LCL=. Moving Range UCL=.618 R=.1884 LCL=. I and MR Char for: res RT I and MR Char for: res MGC Individuals UCL=.1436 MU=. LCL= Individuals UCL=.183 MU=. LCL=-.183 Observaion Observaion Moving Range.2.1. UCL=.1764 R=398 LCL=. Moving Range.2.1. UCL=.221 R=.6778 LCL=. I and MR Char for: res HGC UCL=.48 Individuals. - MU=. LCL=-.48 Observaion Moving Range UCL=4 R=.169 LCL=. Fig 4.12: Individual Moving Range chars of residuals Inernaional Journal of Research in Engineering & Applied Sciences hp:// 47

28 IJREAS Volume 3, Issue 1 (Ocober 213) (ISSN ) From he fig 4.12 above, we can esimae he sandard deviaions for each individual process and use hem o char CUSUM chars. The esimaed sandard deviaions and oher parameers are in able 4.3 below. Table 4.3: Esimaes of parameers and Sample Sandard Deviaions s σ Δ h K ARL Mean Feeds FT RT MGC HGC CUMULATIVE SUM CHARTS CUSUM Char for : res feed CUSUM Char for : res FT 1 Upper CUSUM Upper CUSUM.8711 Cumulaive Sum Cumulaive Sum. -1 Lower CUSUM Lower CUSUM -.9E-1 Subgroup Number Subgroup Number CUSUM Char for : res RT CUSUM Char for : res MGC Upper CUSUM Upper CUSUM Cumulaive Sum Cumulaive Sum Lower CUSUM -1.6E Lower CUSUM -2.E-1 Subgroup Number Subgroup Number Inernaional Journal of Research in Engineering & Applied Sciences hp:// 48

29 IJREAS Volume 3, Issue 1 (Ocober 213) (ISSN ) CUSUM Char for : res HGC Upper CUSUM 4.81E-2 Cumulaive Sum E-2 Lower CUSUM Subgroup Number Fig 4.13: Cumulaive Sum chars of residuals Fig 4.13, show ou of conrol signals on he Feeds a poin 183, and a poins 1, 2, 3 for FT, 81, 83, 98 for RT, 96, 97, 98, 99 for MGC and 1, and 168 for HGC. All he variables are conribuing o he failure of he process, looking a boh he MCUSUM and MEWMA chars. I could be he ou of conrol signals in he individual CUSUM chars which are also signaling in one or boh of he mulivariae chars. I is now possible o immediaely deermine which variable or variables conribued o he signals in he mulivariae chars. Therefore o improve he process, individual variables are o be moniored by removing all he ou of conrol poins appearing in each of he CUSUM chars above. The revised mulivariae conrol chars are shown in fig 4.14 and 4.1 below. Fig 4.14: Revised MCUSUM char Inernaional Journal of Research in Engineering & Applied Sciences hp:// 49

30 IJREAS Volume 3, Issue 1 (Ocober 213) (ISSN ) Fig 4.1: Revised MEWMA char MCUSUM char (fig 4.14) wih subgroup and MEWMA char (fig 4.1) wih show he in conrol process afer all he ou of conrol poins were removed from individual chars as indicaed by he CUSUM chars above. This indicaes ha all qualiy characerisics which deermine he qualiy of refined plainum are now wihin prese specificaions. As a resul, i could be all he variables (five of hem) or a few conribued o he failure of he process since hese variables (qualiy characerisics) joinly affec he qualiy of plainum.. CONCLUSION In his research, all objecives were me. The assumpions of normaliy, randomness, and saionary of variables have been implemened excep for lineariy which could no be shown clearly. Due o he lack of lineariy of variables, i is believed ha some of he ou of conrol signals could be false even hough all he fauly alarms were eliminaed from he process. Comparing wo MSPC echniques, MCUSUM seems o ouperform is couner echnique, MEWMA in deecing small shifs in he mean of he process. The process of refining plainum only became normal (in conrol) afer all he faul alarms were removed from individual processes, which means ha indeed chemical processes are mulivariae in naure. Several characerisics mus be moniored simulaneously o improve qualiy and reduce re-work and coss. The Average Run Lengh around 3 means ha he ou of conrol of he process occurs afer every 3 days. I suggess ha qualiy of plainum decreases owards monh end and become normal during oher days. I could be workers who will be working drunk because hey would be having money o spend from heir salaries and even abseneeism from work rying o fix heir bills and debs which cause poor qualiy of plainum owards and jus afer monh end. Inernaional Journal of Research in Engineering & Applied Sciences hp://