Chapter 1: Introduction

Transcription

1 Chapter 1: Introducton Materals requrements plannng (MRP) s a wdely used method for producton plannng and schedulng. Planned lead-tme (PLT) and lot sze are two of the nput parameters for MRP systems, whch determne planned order release dates. The followng secton descrbes the background of the problem under study Background Presently, planned lead-tme and lot sze are estmated usng ndeendent methodologes. Marln (1986) surveyed and lsted the dfferent methods prescrbed to determne PLT. Mohan and Rtzman (1998) also lsted the dfferent methods prescrbed by prevous researchers to determne PLT. Spper and Bulfn (1998) dscussed some of the wdely used or recommended lot szng rules. There are some dsadvantages assocated wth both PLT estmaton methods lsted n Mohan and Rtzman (1998) and lot szng rules lsted n Spper and Bulfn (1998). No exstng PLT estmaton methods consder factors such as machne breakdown, scrap-rate, etc. Moreover, they dd not consder the capacty of a shop, whch changes dynamcally, because the avalable capacty at any gven tme s determned by the loadng of the shop at that tme. All exstng PLT estmaton methods use hstorcal data for estmaton. However, manufacturng envronment changes randomly over tme. Factors such as capacty and shop congeston vary dynamcally. Therefore, the estmaton methods based on hstorcal data leads to a huge lead-tme dfference between the actual lead-tme and PLT,.e., lead-tme error 1. Moreover, the processng tme for a product vares due to product varablty. 1 Lead-tme error s defned as the dfference between actual lead-tme and PLT. 1

2 Chapter 1: Introducton Consderng the average value of processng tme for PLT estmaton msleads the purpose. A large PLT (.e., a PLT that s more than the actual lead-tme) leads to a hgh nventory, because lots are released early. However, t may also cause customer dssatsfacton because of the excessvely long customer lead-tme. A small PLT (.e., a PLT that s less than the actual lead-tme) leads to low nventory, because lots are released later. Ths also causes customer dssatsfacton, however, because of mssed due dates. It s up to planners to decde the rght PLT that compromses both nventory and customer satsfacton. Snce no standard-method s prescrbed so far, planners choose PLT based on hstorcal data and prevous experence. Lot-szng rules, lsted n Spper and Bulfn (1998), are appled at each level of the bll of materal separately, hopng to mnmze the holdng and setup costs. In recent years, companes are tryng to reduce workng captal. However, an MRP system has no control over WIP. Ths s partly because that none of the conventonal lot-szng methods ncorporates WIP cost n ther objectve functon whle they attempt to mnmze t,.e., they do not mnmze the total cost assocated wth producton. It s mportant to note that WIP cost determnes the cost assocated wth queues at each workstaton and nventory holdng cost determnes the cost assocated wth end tem nventory for satsfyng future demand. Lot-szng rules, lsted n Spper and Bulfn (1998), can be effectvely appled to determne the order sze for an tem n order to satsfy the future demands exstng n certan perods by mnmzng the holdng and setup costs. Nevertheless, they do not mnmze the actual cost assocated wth producton. It can only be acheved when lot-szng rules mnmze the combnaton of WIP cost, setup cost, and holdng cost. However, lot sze and other manufacturng system parameters, such as shop congeston, dynamc shop capacty, machne falures, product varablty, processng tme, setup tme, watng tme, and scrap rate, nteract dynamcally and determne PLT. 2

3 Chapter 1: Introducton Work-n-process cost s a functon of PLT. A small lot sze leads to small PLT and hence small work-n-process cost. A bg lot sze leads to large PLT and hence large work-n-process cost. Therefore, only the best combnaton of PLT and lot sze determnes the lowest cost functon. So far, there s no prescrbed method avalable to determne PLT by consderng the dynamc nteractons among the varous parameters mentoned earler. Therefore, lot sze and PLT are estmated usng dfferent ndependent methods separately. By dong so, the advantage of the dependency nature of PLT on lot sze s omtted. The followng secton descrbes the problem statement Problem statement Based on the dscusson presented n the prevous secton, the research problem under study s defned as how to set both planned lead-tme and lot sze n such a way that the total cost,.e., the combnaton of setup cost, holdng cost, and work-n-process cost, s mnmzed by consderng the dynamc nteractons among varous manufacturng system parameters. In order to estmate PLT wth less lead-tme error, the estmaton procedure should consder parameters such as, lot sze, shop congeston, dynamc shop capacty, machne falures, product varablty, processng tme, setup tme, watng tme and scrap rate. The complex problem set by these parameters s stochastc n nature because of the varablty exstng n the above-mentoned parameters, and t s dynamc n nature as well because these parameters can nteract dynamcally. Therefore, the estmaton procedure should capture both the dynamc and stochastc nature of the parameters. Moreover, lot sze and lead-tme are dependent on each other. Bg lot sze leads to bg queue tme and hence, results n bg lead-tme. Moreover, changng the sze of one lot wll have an effect on the lead-tme of other lots. Conventonal lot-szng models are formulated based on setup and holdng costs. Ther objectve s to determne lot sze n such a way that the combnaton of setup and 3

4 Chapter 1: Introducton holdng costs are mnmzed. These, conventonal models do not capture the nterdependent nature of both lot sze and lead-tme. Such nterdependence can only be acheved by combnng the WIP cost wth the cost functon of lot-szng models. It s mportant to note that the WIP cost of a lot s dependent on the average lead-tme of that lot and ts sze. Therefore, the cost functon of lot-szng model s modfed n such a way that the combnaton of setup cost, holdng cost, and work-n-process cost, s mnmzed. Total cost of lot (Y) = setup cost + holdng cost + WIP cost WIP cost = average WIP of lot (Y) * lead-tme of lot (Y) * unt WIP cost Karmarkar, Kekre, and Kekre (1985) argue that the setup cost n the cost functon denotes the opportunty cost of lost producton tme, and they are ncurred when producton tme s a bndng constrant. However, the setup cost s not only the opportunty cost of the lost producton tme but so s the cost assocated wth the utlzaton of resources, such as labor for setups. Therefore, the term setup cost n the cost functon represents the combnaton of opportunty cost of lost producton tme and the cost assocated wth the utlzaton of resources for setups. WIP cost represents the cost that occurs due to parts watng n queues. If lot sze s too bg, the holdng cost and WIP cost wll be hgh and the setup cost wll be low. However, f lot sze s too small, the holdng cost and WIP cost wll be low and the setup cost wll be hgh. Moreover, changng the sze of a lot wll have an effect on the WIP and lead-tme of other lots. Hence, changng the sze of a lot wll have an effect on the cost functon of other lots also. The WIP and lead-tme of a partcular lot sze are dependent on factors such as the sze of remanng lots, machne falures, the varablty of processng tme, scrap rate and setup tme. WIP and lead-tme should be estmated by a method that captures both the stochastc and dynamc aspects of the 4

5 Chapter 1: Introducton above-mentoned parameters. Only then can the realstc estmaton of WIP and leadtme be acheved. Thereby, lead-tme error can be reduced, and lot sze based on the realstc producton cost can be estmated. The followng secton descrbes the dscusson on the research objectve Research objectve Based on the dscusson presented n the prevous secton, t s found that the soluton procedure should tackle two dfferent aspects of the problem smultaneously, n order to fnd the mproved soluton. These two aspects are: 1. Fndng lot sze dynamcally for gven demand values, and 2. Estmatng the WIP and lead-tme of a lot realstcally by consderng both stochastc and dynamc aspects of the manufacturng parameters The objectve of ths research s to develop a feasble methodology to determne the soluton for the problem under study by tacklng both aspects of the problem Thess outlne Chapter 1 ntroduces the motvaton for ths research and defnes the problem statement. Chapter 2 revews prevous research work done n felds such as queung network based manufacturng model, lot szng, and lead-tme effect on MRP models and planned lead-tme estmaton procedures. Chapter 3 descrbes the general approach to the problem under study and presents the fundamental theory behnd the proposed approach. Chapter 4 descrbes the detals of the proposed methodology. Chapter 5 descrbes the detals of the manufacturng system modelng n MPX. Chapter 6 descrbes the detals of smulaton modelng and estmaton procedure. Chapter 7 descrbes the detals of the expermental desgn. Chapter 8 contans experment results and dscusson. Chapter 9 descrbes the concludng remarks. Appendx A provdes tabulated results and descrbes the step-by-step executon of the proposed approach. 5

6 Chapter 2: Lterature revew The lterature revew s dvded nto three parts. They are Revew of queung network based manufacturng model, Revew of lot-szng and planned lead-tme effect on MRP systems, and Revew of planned lead-tme estmaton methods Revew of queung network based manufacturng model Karmarkar, Kekre, and Kekre (1985) proposed calculatng the relatonshp between lot sze and shop performance usng an open queung network model, whch s then embedded n an optmzaton routne that searches for optmal lot szes. They descrbed the mportance of WIP nventory cost and they modfed the objectve functon of the conventonal lot-szng models by addng the WIP nventory cost. They suggested that a manufacturng system model, whch s based on open queung network theory, could be used to estmate average flow tme more accurately than the conventonal PLT estmaton methods. However, they dd not consder parameters such as, machne falures and scrap rate n ther model. Because of the complex computatons, the method whch they proposed cannot be mplemented easly. They dd not descrbe any method whch explans how an optmzaton routne can be embedded on an open queung network model. Karmarkar (1987) nvestgated a mult-tem open queung network model. He ndcated that because of the mult varable nature of the model, t could not be solved easly. He stated queung delays are externaltes n that changng one tem s lot sze potentally has an mpact on delays of all other tems. Sur and Dehl (1985) proposed a software tool called MANUPLAN for assstng n the plannng and desgnng of manufacturng systems. They embedded a dynamc model, whch was based on queung network theory, on statc the allocaton model n order to 6

7 Chapter 2: Lterature revew capture the dynamcs, nteractons, and uncertantes n the system. They mentoned that these models estmate steady state average values. They ntutvely descrbed that these estmates can be used to analyze the performance of the system over the perod of 3 months whle the processng tme of an tem s on the order of 15 mnutes to 4 hours. They also mentoned some of the lmtatons of the software. They argued that they developed ths software tool n order to overcome the requrement of an enormous amount of detaled nformaton, labor and computer tme for developng a smulaton model. MANUPLAN overcame the computaton dffcultes assocated wth mult-tem open queung network model, as mentoned n Karmarkar (1987). However, they dd not explore the possblty of usng ths tool to estmate lot sze and lead-tme smultaneously by embeddng an optmzaton routne on ths tool, as argued n Karmarkar, Kekre, and Kekre (1985). Zjm and Butenhek (1996) developed a method that determnes the earlest possble completon tme of any arrvng job, wthout sacrfcng the delvery performance of any other job n a shop. They used an advanced queung model n ther approach, argung that treatng each workstaton as M/G/1 queue, as proposed by Karmarkar (1987), s not exact. They reasoned t s because that the arrval streams of a work staton are the super-postons of departure streams from other work statons and an external arrval stream and snce M/G/1 work staton s departure streams are not Posson, the arrval stream can not be Posson. Zjm and Butenhek (1996) suggested that lead-tme estmaton could be refned usng more sophstcated approxmates of queung network models. They posted that these technques are mplemented n a software tool called MPX 2. However, they dd not nvestgate how ths sophstcated queung network model could be used to set lead-tme and lot sze optmally. 2 MPX s a regstered trademark of Network Dynamcs, Inc. Framngham, MA 01701, USA. 7

8 Chapter 2: Lterature revew 2.2. Revew of lot szng and lead-tme effect on MRP systems Karmarkar, Kekre, and Kekre (1985) llustrated the mpact of lot sze on queues. They ntutvely found that bg lot sze causes large queue buld up. They also found that ntally lot sze reducton causes queue reducton but eventually the queue started to buld up because of an ncreased number of setups. They also suggested the conventonal objectve functon modfcaton for fndng the optmal lot sze. They argued that an nvestment assocated wth work-n-process (WIP) s the opportunty cost and lot sze models should ncorporate the WIP cost n ther objectve functon n order to capture the mplct effect of lot sze on lead-tme. Melnyk and Pper (1985) nvestgated the effect of dfferent lot szng rules on leadtme error. They examned the nteracton between lot szng rules and lead-tme estmaton methods. They beleved that lot sze and lead-tme are two nter dependent functons. They found that PLT nflaton nfluences lot sze effectveness and vce-versa. Bllngton, Mcclan, and Thomas (1983) proposed lnear and nteger programmng approaches for estmatng lot sze and lead-tme n capacty-constraned MRP systems. They dd not consder factors such as processng tme varablty, machne breakdowns, and dynamc nteractons among lots n ther mathematcal modelng. Molnder (1997) proposed a methodology for optmzng lot sze, safety stock and safety lead-tme jontly by applyng a smulated annealng based search algorthm. He estmated planned lead-tme and demand at dfferent levels based on ther varance. He optmzed lot sze and safety stock for each constant value of planned lead-tme and demand by applyng the smulated annealng technque. Thereafter, he optmzed safety lead-tme and lot-sze. However, he dd not consder factors such as processng tme varablty, machne breakdowns, and dynamc nteractons among lots n hs model. 8

9 Chapter 2: Lterature revew Gude and Srvastava (2000) revewed dfferent bufferng technques used for tacklng the uncertanty n MRP systems. Ther study report ndcates that only a few research efforts have been made n the area of lead-tme uncertanty n MRP systems. Most of the research has tackled lead-tme uncertanty usng the safety lead-tme factor and they have all used ndependent approach for estmatng lot-sze and planned leadtme. Yeung, Wong, and Ma (1998) revewed dfferent parameters whch affect the effectveness of MRP systems. They found that capacty constrant s not consdered n most of the research. They reasoned that t s due to complcated nteractons among varous envronmental factors. They recommended further study on varous uncertantes such as processng tme, machne falures, etc. Koh, Saad, and Jones (2002) revewed and categorzed uncertantes assocated wth MRP systems. They concluded that MRP, MRPII, or ERP works only as a planner rather than an optmzer and that these tools would be effectve only under the absence of uncertantes. They suggested that a planner should use some other technques to cope wth uncertantes. They found that safety lead-tme s a wdely used tool to cope wth lead-tme uncertanty. Most of the research dd not consder safety lead-tme as a functon of uncertanty assocated wth lead-tme. Instead, they all measured safety leadtme based on the varance of lead-tme dstrbuton functon. By dong so, the nter dependency nature of both lot sze and lead-tme was neglected. Marln (1986) nvestgated the effect of planned lead-tme accuracy on MRP systems. He estmated PLT usng some of the wdely used methods and then nvestgated the effect of planned lead-tme on MRP systems usng a smulaton model. He concluded that the estmated value of PLT does affect MRP system performances and care should be exercsed n selectng a procedure for determnng planned lead-tme. He found that the smple hstorcal average method s preferred n many cases. 9

10 Chapter 2: Lterature revew Mohan and Rtzman (1998) nvestgated the mpact of planned lead-tme on MRP system performances. They used four dfferent levels of planned lead-tme. At each level, they used dfferent magntudes of nflaton. They concluded that planned lead-tme does affect customer servce, but t has a lesser effect on WIP than that of lot sze. They dd not consder the nterdependent nature of both lot sze and planned lead-tme. Nagendra and Das (2001) proposed a method called Progressve Capacty Analyzer (PCA) for determnng lot sze subjected to capacty restrctons, for mult-level bll of materal. They observed that no commercally avalable MRP and Enterprse Resource Plannng (ERP) software tools dd actually resolved the fnte capacty problem effcently. PCA procedure conssts of two lnear programs and a lot-aggregatng heurstc, and t s embedded on the tradtonal MRP exploson. The authors estmated planned lead-tme usng an ndependent methodology. They dd not consder the nterdependent nature of both lot sze and planned lead-tme. Pandey, Psal, and Sunsa (2000) proposed a fnte capacty-materal-requrement algorthm for obtanng capacty-based producton plans. They used an ndependent methodology to estmate planned lead-tme. They also dd not consder the nterdependent nature of both lot sze and planned lead-tme Revew of planned lead-tme estmaton methods Boeder and Gurnee (1982) proposed a heurstc method for estmatng planned leadtme. They used the followng set of data to estmate planned lead-tme: Routngs Transt tme table Structured blls of materal Standard run tme Standard setup tme Queue tme 10

11 Chapter 2: Lterature revew Planned lead-tme s estmated by summng setup tme, run tme, transfer tme, and queue tme, for a gven lot sze. Ths method does not consder the dynamc nteractons among varous parameters. Orlcky (1975) derved a lnear equaton for estmatng planned lead-tme. The lnear equaton s of the form PLT = a+ Nb. The parameters, a and b are derved emprcally and N denotes the total number of operatons to be performed on a part. Marln (1986) nvestgated the above-mentoned equaton n hs research. He sad, Unless the work content of and backlogs at each operaton are dentcal, ths method performs poorly. Hoyt (1978) proposed a technque called QUOAT (QUeue/Output Analyss Technque). QUOAT s developed based on the prncple that the lead-tme through a gven work center s the most recent week s average queue dvded by ts average output. However, ths method dd not consder the dynamc nteractons among varous parameters completely. Pandey, Psal, and Sunsa (2000) used a conventonal method to estmate planned lead-tme. PLT = K *(setup tme + processng tme), where K= two and eght. Ths heurstc method also dd not consder the dynamc nteractons among varous parameters completely. Based on the exhaustve lterature revew, t s evdent that there s no research has been done on calculatng both planned lead-tme and lot sze smultaneously n such a way that the combnaton of setup cost, holdng cost and work-n-process cost s mnmzed. Moreover, t can be reasoned from the prevous dscusson that the PLT estmaton methods cannot produce a better soluton f they do not consder the dynamc nteractons among varous factors such as lot sze, processng tme varablty, etc. Therefore, the need exsts for developng a methodology to fnd an mproved soluton. 11

12 Chapter 3: Approach The core objectve of the proposed approach s to fnd a soluton by tacklng two dfferent aspects of the problem under study. The core dea s to apply dynamc programmng concept to estmate lot sze for lumpy demand and apply advanced queung network theory to estmate realstcally the WIP and lead-tme of a lot. Ths proposed approach embeds an optmzaton routne, whch s based on dynamc programmng of a manufacturng system model, whch s based on open queung network theory. Then, t optmzes lot sze usng realstc estmates of WIP and leadtme of dfferent lots smultaneously for sngle-product, sngle-level bll of materal. Fgure 1 shows the schematc representaton of the proposed approach. Demand Dynamc programmng routne Advanced queung network theory based manufacturng model Lot sze and Planned lead tme Fgure 1: Schematc dagram of the proposed approach 12

13 Chapter 3: Approach 3.1. General assumptons 1. End tem demand s dynamc n the sense that t may vary across perods and they are determnstc n the sense that the expected values of demand are known a pror. 2. Setup tme s determnstc. Processng tme and machne falures are stochastc. 3. Backorderng s not permtted. 4. Setup cost denotes the combnaton of opportunty cost of the lost producton tme and the cost assocated wth the utlzaton of resources for setup. Snce the lost producton tme and the utlzaton of resources for setup are very dffcult to quantfy, the value of setup cost s chosen by applyng a rule of thumb. In ths model, the value of setup cost s ten tmes hgher than that of holdng cost. 5. Producton cost remans constant for the entre perod of the plannng horzon. 6. Queue space s nfnte at each workstaton. 7. Materal handlng and labor resources are not consdered n the modelng. 8. The release date of each lot s found by offsettng ts PLT from ts due date. At each workstaton, lots are processed based on the FCFS rule Dynamc programmng based optmzaton routne One of the challenges of the problem under study s to fnd lot sze dynamcally. A dynamc lot-szng model can be represented ether as dynamc programmng problem or as mxed nteger-programmng model. Detals of the varous dynamc lot-szng models can be found n Shapro (1993). However, the representaton of the dynamc lot-szng model as mxed nteger-programmng model s only possble when work-nprocess cost s not present n the objectve functon of the dynamc lot-szng model. Ths s because work-n-process cost s a functon of average WIP and lead-tme. Moreover, the average WIP and lead-tme of a lot s dependent on the dynamc and stochastc nature of the varous manufacturng system parameters, whch were 13

14 Chapter 3: Approach dscussed prevously. Therefore, the formulaton of mxed nteger-programmng model for the problem under study wll be very dffcult and tme consumng. As a result, the problem under study s modeled as a dynamc programmng model. The followng secton descrbes the general approach of dynamc programmng General approach Dynamc programmng (DP) s a mathematcal approach desgned to solve a problem by breakng t nto smaller sub-problems. Dynamc programmng solves the problem n stages. At each stage, exactly one varable s optmzed. Computatons at dfferent stages are lnked through recursve computatons n a manner that yelds a feasble optmal soluton to the entre problem. The dynamc programmng approach solves a mult-stage problem by solvng a seres of sngle-stage problems. Ths s acheved by tandem projecton onto the space of each varable. In other words, t projects frst onto a subset of varables, then onto a subset of these, and so on. DP facltates the easy formulaton of the problem under study. In each stage of the DP model, lot szes are chosen from a set of values whch are determned by combnng dfferent demand values exstng over the perod of plannng horzon. Wagner and Whtn (1958) proposed that lot szes assgned by combnng dfferent demand values exstng over the perod of the plannng horzon actually produce the better soluton. DP converts a sequental or multstage decson process contanng many nterdependent varables nto a seres of sngle-stage problems and each sngle-stage problem contans only a few varables. Each demand perod can be represented as a stage n dynamc programmng because t has to be decded at each perod whether to order some quanttes for producton. Actually, the decson made at each perod decdes the quantty due n that perod. Inventory at the begnnng of each perod 14

15 Chapter 3: Approach represents the ntal state of the stage. Inventory at the end of each perod represents the fnal state of the stage. Decson Stage 1 Begnnng state Stage t Endng state Stage T Return Fgure 2: General schematc dagram of dynamc programmng * F = mn ( + K( l )) * t F l 1 t < l t 0 (3.1) Where, t, * F t = Optmal cost ncurred to satsfy the demand exstng from perod 1 to perod ( l t) K = Cost ncurred to satsfy demand exstng from perod l to perod t at perodl. 15

16 Chapter 3: Approach The above-mentoned equaton wll be evaluated recursvely from perod 1 to perod T and lot sze wll be estmated dynamcally n such a way that the shortest path from stage T to stage 1 s obtaned. By applyng the dynamc programmng concept, t s evdent that the lot sze can be determned dynamcally for lumpy demand. Lot sze Perod 1 Inventory at the begnnng of perod t Perod t Inventory at the end of perod t Perod T Cost functon value Fgure 3: Schematc dagram of modfed dynamc programmng for lot-szng model Prncple of optmalty Bellman s prncple of optmalty s quoted n Nemhauser (1966) An optmal polcy has the property that whatever the ntal state and decson are, the remanng decsons must consttute an optmal polcy wth regard to the state resultng from the frst decson. The proof of the prncple of optmalty (by contradcton) can be found n Nemhauser (1966). It smply states, If the remanng decsons were not optmal then 16

17 Chapter 3: Approach the whole polcy could not be optmal. It can be proved that equaton 3.1 actually satsfes the prncple of optmalty. Proof: Consder a lot-szng problem wth 3 perods. The lot-szng problem can be represented by nodes. Node 1 n Fgure 4 represents the begnnng of perod 1 whle node 2 represents the end of perod 1. A path n Fgure 4 from node 1 to node 4 corresponds to settng up for producton only n perods that correspond to nodes n the path. The path dstance s determned by the total producton cost. All possble combnatons of setups n perods1 through 4 are represented by some path from node 1 to node 4 on ths graph. Therefore, fndng the shortest path on ths graph corresponds to the optmal producton lot-szng schedule wth mnmum total cost. The objectve of the lot-szng problem s to satsfy the demand of all perods at mnmal (optmal) cost,.e., fnd the shortest path from node 1 to node 4. It can be found by movng backwardly from node 4 to node1. K1-4 K1-3 K2-4 K1-2 K2-3 K Fgure 4: Node representaton of the mult perod lot-szng problem 17

18 Chapter 3: Approach Let, K -j represents the cost of an arc emanatng from node to j,.e., the cost ncurred to satsfy demand exstng between perod and j. Let, * Ft represents the optmal cost of reachng node t,.e., the cost ncurred to satsfy demand exstng untl perod t. Then, the optmal cost of reachng node 4 can be wrtten as follows: F = mn * 4 The optmal value of F F F * 4 * 1 * 2 * 3 + K + K + K (3.2) F s based on the value of F, F or F,.e., the decson made at node1, node2, or node3. It can also be sad that whatever the ntal state and decson are, the remanng decson consttutes an optmal polcy wth regard to the state resultng from the frst decson. Smlarly, the optmal cost of reachng nodes 3, 2 and 1 can be wrtten as: * 1 * 2 * 3 * * F 1 + K1 3 F 3 = mn * (3.3) F2 + K 2 3 * F = mn{ F + K } (3.4) * * F 1 = 0. (3.5) In general, t can be sad that, * F = mn ( + K( l )) * t F l 1 t < l t 0. (3.6) 18

19 Chapter 3: Approach 3.3. Queung network theory based manufacturng system model The need for ncorporatng a manufacturng system model n the proposed approach s to estmate realstcally the WIP and lead-tme of a lot by consderng the stochastc and dynamc nature of the parameters, whch were dscussed earler. Smulaton technques are commonly used for stochastc modelng of manufacturng systems. However, such models are dffcult and extremely tme consumng to develop and valdate. Sur and Dehl (1985) dscussed the dffcultes assocated wth usng smulaton based manufacturng models for short-perod analyss. Substantal research has been done n the feld of queung theory based manufacturng system modelng. It has been shown that queung theory based manufacturng system models can effectvely be used for a quck analyss of a manufacturng system Basc elements of queung model for manufacturng system As a part arrves at a faclty, t jons a queue. A workstaton chooses from the queue lne to begn servce part based on the servce dscplne rule. Upon completon of the servce, the process of choosng a new part s repeated. It s assumed that no tme s lost between the release of a servced part from the workstaton and the admsson of a new one from the queue. The prncpal actors n a queung model are parts and workstatons. In queung models, the nteracton between parts and workstatons s of nterest only n as far as t relates to the perod of tme a part needs to complete ts servce. Thus, from the standpont of parts arrvals, the tme nterval that separates successve arrvals needs to be measured. In the case of the workstaton, t s the servce tme per part that counts n the analyss. In queung models, parts arrvals and servce tmes are summarzed n terms of probablty dstrbutons, normally referred to as arrvals and servce tme dstrbutons. Although the patterns of arrvals and departures are the man factors n the analyss of queues, other factors, such as queue 19

20 Chapter 3: Approach sze, servce dscplne rule and number of workstatons are mportant n the development of a model. The queung process s usually descrbed usng a shorthand notaton. It conssts of a seres of symbols and slashes such as A/B/X/Y/Z, where A ndcates the nter-arrvaltme dstrbuton, B ndcates the servce tme dstrbuton, X ndcates the number of parallel servce channels, Y ndcates the restrcton on system capacty, and Z ndcates queue dscplne. Taha (1987) can be referred to addtonal nformaton on queung theory. A manufacturng system can be modeled usng networks of queues. Networks of queues can be descrbed as a group of nodes where each node represents a workstaton. In general, parts arrve at any node from nsde or outsde of the system and depart when servce s completed. Thus, parts may enter the system at some nodes, traverse from node to node n the system, and depart from some node, not all the parts necessarly enterng and leavng at the same nodes, or takng the same path once havng entered the system. An open network of queues suts the modelng of a manufacturng system where the arrval rate (dstrbuton) s a controlled parameter. As ths occurs n most manufacturng systems (job shops, flow shops, GT systems), the open network approach wll be used n ths research Analyss of approxmate GI/G/m queues Zjm and Butenhek (1996) argued that the arrval stream at a workstaton, whch s modeled as M/G/1, s determned by combnng a departure stream from other workstatons and an external arrval stream. Addtonally, t can be sad that the departure stream from a workstaton, whch s modeled as M/G/1, cannot be Posson. Therefore, the arrval stream at a workstaton, whch s modeled as M/G/1, cannot be Posson. However, the M/G/1 queung model assumes that the nter arrval tme of parts follows the exponental dstrbuton. Snce, t s evdent that the nter arrval tme 20

21 Chapter 3: Approach of parts can not follow an exponental dstrbuton, the M/G/1 queung model can not exactly represent a realstc manufacturng work staton. Sur, Sanders, and Kamath (1993) stated that when parts do not assure the condton of Posson arrval streams and workstatons do not assure the condton of exponental servce tme n networks of queues, t would be effectve to mplement approxmate network decomposton. Ths decomposton conssts of a set of workstatons, whch are approxmately represented as GI/G/m models lnked together usng a set of equatons smlar to those requred for Jackson networks. In general GI/G/1 queues, both nter arrval-tme and servce tme of a part follow a general probablty dstrbuton. The symbol GI s often used to specfy that nter arrval-tme of a part s general and ndependently dstrbuted. Kuehn (1979) proposed an approxmate method for the analyss of general queung networks. Accordng to hs method, a network s splt nto subsystems whch are analyzed n solaton Basc GI/G/1 open queue network and decomposton analyss GI/G/1 queue network conssts of varous elements such as, servers, queues, transton path, feedback loops, decomposton ponts (splttng of processes), and composton ponts (super poston of processes). The GI/G/1 queue network s formed by lnkng a seres of GI/G/1 queue statons together. An elementary queung staton conssts of a sngle server, a sngle queue wth unlmted capacty, an nput termnal (composton pont C ), and an output termnal (decomposton pont D ). It s assumed that external parts arrve at a composton pont and depart from the network at a decomposton pont. Snce t s an open network of queues, there exsts at least one external arrval process and at least one staton from whch customers can leave the network. Parts arrve from the outsde of a queung network based on GI arrval process and they are served at the varous statons based on G servce processes. 21

22 Chapter 3: Approach Composton pont Queue Work staton Decomposton pont q 0 q 1 C G ( µ, c ) H D q N Fgure 5: Elementary GI/G/1 queue staton The basc prncple of decomposton analyss s defned n the followng steps: Queung networks are decomposed nto sngle GI/G/m queue models; Each GI/G/m model s analyzed ndependently related to ts network surroundngs by arrval and departure process; All non-renewal processes are approxmated usng statonary renewal processes (A renewal process s defned as a process arses from any sequence of random varables whch are non-negatve, dentcal, and ndependently dstrbuted); and Two moments, the mean and the coeffcent of varaton are consdered for all renewal processes consstently. Some basc operatons necessary for mplementng the decomposton analyss are dscussed n the followng secton. They were all developed by Kuehn (1979). Before presentng the basc operatons, the followng notatons have to be defned. 22

23 Chapter 3: Approach Notatons λ The mean arrval rate of workstaton or node λ o The external arrval rate of parts at workstaton or node λ o The external arrval rate of parts c o Vector of coeffcents of varaton of the external arrval processes G Vector of servce processes GI Vector of external arrval processes µ Servce rate at node c H Vector of coeffcents of varaton of the servce processes c A Vector of coeffcents of varaton of the arrval processes q j Routng probablty for customers leavng staton and changng to staton j where =1, 2...N, j =0, 1.N, N ρ Total number of queung statons Server utlzaton of staton Mean arrval rates The mean arrval rate at workstaton s developed based on the statonarty assumpton. It s obtaned from the followng set of lnear equatons representng the conservaton of flow: N λ = λ o + λ j, = 1, 2 N. (3.7) j= 1 j q In the statonary case, for all statons t must hold that = λ µ, (3.8) ρ / ρ < 1, for all = 1, 2 N. 23

24 Chapter 3: Approach The transton rate λ of the path from staton to staton j follows from j λ j = λqj, for all = 1, 2 N & j =0, 2.N. (3.9) Mean values of GI/G/1 queung system The arrval process s a renewal process, whch follows the general probablty dstrbuton functon, GI, wth a mean arrval rate at node ( λ ), and a coeffcent of varaton of the arrval processes at node ( c A ). The servce process follows the general probablty dstrbuton functon of G wth a mean servce tme at node h = µ ) and a coeffcent of varaton of the servce processes at node ( c H ). Then, ( 1/ the closed form equaton for mean watng tme at node ( w ), whch was developed by Kuehn (1979), who appled a new approxmaton formula developed by Kraemer and Langenbach-Belz for GI/G/1 queung statons whch rests on the frst two moments of the arrval and servce processes, s used. w = h * ( ρ ) 2 1 ρ * ( c + c )* g( ρ, c, c ) A H A H (3.10) Where, g 2 2 ( ρ, c, c ) A H 2 1 exp 3ρ c A < 1 = exp c A 1. 2 ( ρ ) ( 1 c ) ( 1 ρ ) * c c * 2 c A 2 A 2 A + c 1 + 4c 2 A 2 H 2 H,, (3.11) 24

25 Chapter 3: Approach Flow tme f can be calculated usng the followng relatonshp. f = w + h. (3.12) Output process of GI/G/1 queue system The output process s characterzed by the dstrbuton functon of the nter departure tme at node (T D ). The frst moment of T D s the recprocal of the arrval rate λ, and the second moment of T D s calculated from the followng set of equatons: w c A + 2ρ ch 2ρ ( 1 ρ )*, h 2 c = ρ < 1 D (3.13) c + ρ ( ), A ch c A ρ Traffc and traffc varablty equatons Sur, Sanders, and Kamath (1993) calculated the arrval rate at staton ( λ ) and the vector of coeffcents of varaton of the arrval processes at staton ( c A ) by solvng two sets of smultaneous lnear equatons, whch are developed based on the abovementoned relatonshps. The followng set of lnear equatons, taken from Sur, Sanders, and Kamath (1993). λ, s calculated by solvng the followng equaton: 0 ( I Q ) 1 λ (3.14) = λ O 25

26 Chapter 3: Approach where ( Q ) 1 I s known as the fundamental matrx and the elements of ths matrx, O say n j, are the expected number of vsts to staton j by a part whch enters the system at staton before that part departs the system. Smlarly, the coeffcent of varaton of a composte arrval process at each staton s obtaned from the followng set of equatons: c 2 aj = a j + M = 1 c 2 a b j, (3.15) Where aj M ( q c 1) + q [( p ) + p x ] = 1+ w j 0 j 0 j j 1 j j ρ, (3.16) = 1 b j j j 2 ( ρ ) j 1 = w p q, (3.17) where q j ( λ / λ j ) p j =, (3.18) 2 ( max[ c,0.2] 1) 0.5 x = 1+ m H, (3.19) 1 2 = 0 = M j q j v, (3.20) [ ( ) ( )] v 1 1 w ρ. (3.21) j = j j p j represents the probablty wth whch a part receved servce at staton and wll go to staton j next. 26

27 Chapter 3: Approach GI/G/m queung model wth nterference The above-mentoned GI/G/1 open queung network model assumes that there s no stoppage at the workstaton. Therefore, ths model cannot represent workstatons wth falures. Ths type of problem s generally called a machne-operator nterference problem. Sur, Sanders, and Kamath (1993) revewed a wde range of lterature, whch addressed the machne-operator nterference problem. Practcal solutons to the machne-operator nterference problem have been appled and used n a few software tools. Sur, Sanders, and Kamath stated, The MPX package models multple classes of operators tendng multple classes of machnes. MPX a powerful analytcal tool developed by Network Dynamcs Inc., Framngham, MA, solves the machne-operator nterference problem n two stages. In the frst stage, the problem s analyzed as a multple-class-closed network and the performance measures are generated. In the second stage, the performance measures calculated, whch are calculated n the frst stage, wll be used to modfy the machne servce tme n open network of queues. Zjm and Butenhek (1996) mentoned, All of the approxmaton procedures for GI/G/1 open queung network models are mplemented n MPX Summary of queung model dscusson Based on the above dscusson, t can be reasoned that the followng characterstcs are requred for a queung model so that t can be used for calculatng the WIP and lead-tme of a lot by capturng the dynamc and stochastc nature of the varous parameters, whch are mentoned earler. The characterstcs are: Each node n the queung model should be modeled as a GI/G/m queue Advanced approxmate procedures should be appled to evaluate the performances of the GI/G/m queue Machne-operator problem can be addressed and solved Machne falures can be captured n the model 27

28 Chapter 3: Approach MPX MPX s a powerful analytcal tool used for modelng manufacturng systems va advanced queung theory. MPX can be used to estmate WIP and lead-tme of lots by capturng the dynamc and stochastc nature of the system. Detals of MPX are dscussed n the followng subsectons Theory behnd MPX MPX s based on open network model wth multple classes of customers. It s solved usng a node decomposton approach. Each node s analyzed as GI/G/m queue, wth an estmate for the mean watng tme based on the frst two moments of the arrval and servce dstrbutons. Then, MPX soluton consders the nterconnecton of nodes as well as the mpact of falures on servce tme and departure dstrbutons for further calculatons. The machne-operator problem s solved by applyng the method proposed by Sur, Sanders, and Kamath (1993). The exact algorthm, whch runs MPX, s not publshed yet. However, based on the nvestgaton performed by Sur, Sanders, and Kamath (1993) as well as Zjm and Butenhek (1996), the approxmate algorthm, whch runs MPX, s developed. The approxmate algorthm conssts of two stages. They are: 1. In the frst stage, the machne operator nterference problem s solved by treatng t as a multple class closed queung network and the requred performance measures are calculated. Detals of a multple class closed queue network are llustrated n Sur, Sanders, and Kamath (1993). 2. In the second stage, each node s treated as GI/G/m queue and the performance measures, whch are calculated n the stage 1 as well as the factor corresponds to the effect of machne falures are used to modfy the servce tme of GI/G/m 28

29 Chapter 3: Approach queue. Then, the performance measures are calculated for GI/G/m queue by applyng the decomposton technque that s dscussed earler Input parameters of a MPX model MPX model captures the dynamc and stochastc nature of the followng nput parameters. The nput parameters are: Scrap rate ( percentage of parts rejected at work staton ), Mean tme to falure of work staton, Mean tme to repar of work staton, Labor avalablty, Work staton utlzaton lmt, Number of labors for each knd of group, Number of work statons for each knd, Processng tme, Setup tme, Lot sze. The Varablty factor of each of the above-mentoned parameters can be altered Lmtatons of modelng n MPX Some lmtatons exst n modelng a manufacturng system usng MPX. They are as follows: 1. MPX models a manufacturng system based on the assumpton that the buffer capacty or queue length at each workstaton s unlmted. Therefore, MPX cannot model manufacturng systems whch have fnte queue capacty at each workstaton. 2. MPX apples the FCFS (Frst Come Frst Serve) servce dscplne rule for selectng parts from a queue for servce at each workstaton. 29

30 Chapter 3: Approach 3. The MPX model cannot be drectly used for MRP calculatons. For example, the lumpy demand values that exst at dfferent perods cannot be loaded drectly. Moreover, MPX cannot apply any dynamc lot szng rules. 4. MPX can only be used drectly for a unform demand pattern,.e., the EOQ model. 5. MPX output values, such as WIP, flow tmes, etc., are steady-state average values. 6. The MPX model executes the producton of dfferent lots at the same tme. Tme phasng among dfferent lots cannot be acheved drectly. 30

31 Chapter 4: Methodology Lot sze and planned lead-tme are estmated smultaneously by embeddng a dynamc programmng based optmzaton routne on MPX. The core mechancs of ths method can be descrbed n three steps as follows: 1. At each stage of the optmzaton routne, dfferent lot szes are evaluated to satsfy demand exstng across a plannng horzon. 2. For each lot sze determned n the frst step, the average WIP and planned lead-tme are estmated by runnng the MPX based manufacturng model. The estmaton of average WIP for each lot s requred n order to calculate the WIP cost. 3. The total cost of each lot s calculated by executng the modfed cost functon descrbed n Secton 4.4. All the above-mentoned steps wll be executed teratvely at each stage of the dynamc programmng based optmzaton routne. At the end of the executon, the optmal path s chosen by movng backward from the destnaton node (perod) to the startng node as explaned n Secton Fgure 6 descrbes the smultaneous lot sze and lead-tme settng methodology. However, developng a MPX model for estmatng WIP and planned lead-tme s not straghtforward. MPX lmtatons are dscussed n Secton Some of the lmtatons can be overcome by crcumventng MPX. Dscusson on overcomng the MPX lmtatons s n Secton

32 Chapter 4: Methodology Dynamc programmng based optmzaton routne Tme-phased demand values Step 1 Determne lot szes Step 3 Calculate cost for each lot Step 2 Calculate WIP & PLT for each lot by usng MPX model Start from the last perod Move to the new perod whose path to the current perod has mnmal cost Lot sze= sum of demand satsfed durng transton. Estmated values of Lot sze & PLT Fgure 6: Descrpton of the smultaneous lot sze and lead-tme settng (SLLS) methodology 32

33 Chapter 4: Methodology 4.1. Overcomng the MPX lmtatons Varous MPX lmtatons are dscussed n the prevous secton. The followng sectons descrbe how these lmtatons are overcome n the proposed algorthm (SLLS) Unlmted buffer capacty MPX assumes that the buffer capacty and queue length at each workstaton s unlmted. Therefore, MPX cannot model manufacturng systems whch have fnte queue capacty at each workstaton. MPX (2001) nvestgated and found, n general, most dscrete manufacturng systems do not have sgnfcant blockng effects and typcally only hghly automated, and ntegrated manufacturng lnes are affected by blockng. However, MPX can calculate the watng tme (queue tme) of parts n queues. In general, fnte queue capacty s not commonly encountered n manufacturng systems. It s qute frequently encountered n hghly automated manufacturng systems such as FMS. Hence, ths lmtaton does not have a huge mpact on the applcaton of SLLS algorthm n lowlevel automated manufacturng systems lke a job shop. Therefore, the proposed algorthm (SLLS) consders manufacturng systems wth unlmted buffer capacty. The followng secton descrbes the dscusson on ntroducng tme phasng among lots Introducng tme phasng among lots The frst step of the optmzaton routne determnes lot sze at each stage. Each lot wll have a dfferent due date. Its startng date s determned by offsettng ts lead-tme from ts due date. Therefore, dfferent lots may have dfferent startng dates. Hence, tme phasng among lots s mportant and has to be captured n the analytcal model. Unfortunately, MPX does not permt the tme phasng among dfferent lots drectly. 33

34 Chapter 4: Methodology Dfferent lots of a product wll be treated as dfferent products n the MPX model. All these products wll vst the same set of STANDARD type equpment. MPX provdes an opton of treatng equpment as DELAY type equpment. When a lot vsts a pece of equpment of the DELAY type, the lot wll smply be delayed for a specfed amount of tme. Tme phasng among lots s acheved by ntroducng a pece of dummy equpment of the DELAY type n ther routngs. By allowng a lot to vst the equpment of the DELAY type frst, the requred tme phasng effect for that lot s acheved. The length of the delay tme for lot Y s determned usng ether the completon tme of the frst operaton of all the prevous lots or the tme dfference between the startng dates of lot 1, whch s started frst for processng, and lot Y. It s mportant to note that MPX can estmate the tme requred for processng the operaton of each lot separately. Fgure 7 dsplays the mplementaton of the tme phasng concept n MPX. For example, n Fgure 7, lot 1 of sze 60 wll be released to a manufacturng shop at tme zero. Lot 2 of sze 60 wll vst the frst workstaton after 2600 mnutes from tme zero. Ths s shown n Fgure 8 and Fgure 9. MPX actually releases lot 2 at tme zero. However, lot 2 wll be held for 2600 mnutes at the dummy workstaton, delay1, as shown n Fgure 8 before t s allowed to start ts actual routng. Thereby, lot 2 s made to start ts orgnal routng after 2600 mnutes. 34

35 Chapter 4: Methodology Fgure 7: Dsplay of tme phasng concept n MPX Fgure 8: Dsplay of DELAY concept n MPX for lot2 35

36 Chapter 4: Methodology Fgure 9: Dsplay of DELAY concept n MPX for lot1 The estmaton of average WIP and lead-tme for a tme-phased lot s not exact. Ths s because MPX calculates WIP and lead-tme for the tme-phased lot by consderng all the workstatons n the routng ncludng the dummy workstaton. Therefore, the effect of the dummy workstaton on WIP and lead-tme has to be truncated. It s mportant to note that the delay tme, whch s set n the dummy staton, s a constant. λ L Mean arrval rate of a tme-phased lot Average number of parts n the system L a Average number of parts n all the workstatons except the dummy staton, d L d Average number of parts n the dummy staton, d W Average tme a part spends n the system W d Average tme a part spends at the dummy staton, d W a Average tme a part spends n all workstatons except the dummy staton, d 36

37 Chapter 4: Methodology MPX calculates L and W. However, t s desred to estmate La and W a. From Lttle s law, L = L a + L d (4.1) W = W a + W d (4.2) ( ) L = λ *W L = λ * W a + W d (4.3) ( λ * W ) L ( λ * ) = (4.4) a W d By usng Lttle s law, equaton 4.4 can be wrtten as, Where, ( λ ) L = L * (4.5) a W d La W a= (4.6) λ λ = lotsze n prod perod W d s equvalent to the delay tme set n the dummy equpment and L s found from the MPX output result. From equaton 4.5 and 4.6, the actual WIP and lead-tme of a tmephased lot can be calculated Dscusson on usng steady state average values As mentoned earler, the estmates of WIP and lead-tme calculated by MPX are steady state average values. Obvously, achevng steady state n a dscrete manufacturng system s not always possble. Ths s because the manufacturng faclty has to start from an dle system n most cases. Therefore, the transton state wll have an effect on the performance of the manufacturng system. There are some ssues assocated wth usng 37

38 Chapter 4: Methodology the steady state average values of WIP and PLT. These ssues are dscussed n the followng subsectons Determnng the arrval rate of a lot As dscussed n Secton , the mean arrval rate s one of the crtcal factors for estmatng parameters usng an open queung network model. In MPX, the arrval rate of lot λ s determned usng the followng equaton, D λ = (4.7) PP In equaton 4.7, D denotes end demand and PP denotes producton perod. Snce, end demand and lot sze are the same n ths research, equaton 4.7 can be wrtten as lotsze λ = (4.8) PP From equaton 4.8, t s evdent that the arrval rate of lot λ s dependent on lot sze and producton perod. Snce lot sze s determned by applyng the optmzaton routne, the selecton of producton perod value s qute crtcal. It s mportant to note that the value of the producton perod has to be chosen n such a way that all lots, whch are determned at dfferent stages of dynamc programmng, can be released nto the shop and the producton can be completed durng the producton perod. Therefore, the producton perod should represent the perod measured between the earlest release date and the latest due date of all lots. As a result, t s very dffcult to determne the exact producton perod value. That s because the producton perod s determned by usng the sze of the plannng horzon and the earlest release date of all lots whch are quantfed at each stage of dynamc programmng. Nevertheless, lot sze vares over dfferent stages of dynamc programmng. It adds a further complcaton to the problem. Generally, the producton perod can be found by usng the followng relatonshp: 38

39 Chapter 4: Methodology Producton perod = Latest due date - earlest release date (4.9) In the above relaton, the latest due date s constant, for any gven dynamc lot-szng model. However, the earlest release date may vary over dfferent stages of dynamc programmng. In order to resolve ths complex ssue, the followng tral and error methodology s used: Step1. Set a bg value for the producton perod Step2. Execute the SLLS algorthm and determne the frst optmal lot. Step3. Reset the producton perod at a value determned by usng the followng relatonshp, Producton perod = Plannng horzon + lead-tme of the frst optmal lot sze. Step4. Repeat Step 2 and Step 3 untl the producton perod does not change. The above-mentoned methodology mplctly assumes that the frst optmal lot sze has the earlest release date. Though t happens often n many cases, t does not happen always n all the cases. If t s found that the earlest release due date has been changed, the producton perod value should be reset to a new value, and the executon of SLLS algorthm should be started agan. The followng secton descrbes the comparson between MPX steady state values and correspondng smulaton results. Fgure 10 descrbes a general data sub-menu bar n MPX. 39

40 Chapter 4: Methodology MPX steady state values vs. smulaton results As mentoned earler, MPX estmates steady state average values. It s sgnfcant to compare the values of WIP and PLT estmated by usng the MPX model wth values estmated by usng a smulaton based manufacturng model. In order to facltate ths, an experment s conducted. The objectve of the experment s to evaluate and compare both WIP and lead-tme for dfferent lot szes of a product whose structure and other manufacturng system parameters are mentoned n the next paragraph, by usng both MPX and smulaton models. The detals of the smulaton model are dscussed n Secton 6.1. Fgure 10: Dsplay of producton perod parameter n MPX 40

41 Chapter 4: Methodology Product structure, whch specfes the parent-component relatonshp, can be characterzed by usng parameters such as number of tems, number of levels, average number of components per parent, and average number of parents per component. Values of those parameters for ths experment are gven below: 1. Number of tems = 1 2. Number of levels = 1 3. Average number of components per parent = 1 4. Average number of parents per component = 1 A manufacturng faclty, workng n one eght-hour shft each day, 5 days per week, has two machnes organzed n to two departments. Each department has one machne. Labor s not consdered n the desgn. Machne utlzaton lmt s set at 95% for both machnes. Parts routng and operaton parameter values are descrbed n Fgure 11. The scrap rate s set at zero for both machnes. These parameter values are loaded n both MPX and smulaton models. Other parameters are set as follows. Setup cost = 50 unts per setup Holdng cost = 0.1 unt per day per tem WIP cost = 0.1 unt per day per tem Due date mssng cost = 8 unts per day per tem Plannng horzon = 15 days,.e., 3 perods Work staton 1 Work staton 2 Processng tme= 5 mnutes Setup tme = 145 mnutes MTTF = 4800 mnutes MTTR = 130 mnutes Processng tme = 3 mnutes setup tme = 130 mnutes MTTF = 4800 mnutes MTTR= 150 mnutes Fgure 11: Parts routng 1 41

42 Chapter 4: Methodology The experment results are shown n Table 1. The notatons used n the table are defned as follows: Q Sze of lot, SD DD DM PLT WIP Startng date of the lot, Delay of a lot measured from the startng date of the frst lot (n days), Delay of a lot measured from the startng date of the frst lot (n mnutes), Planned lead-tme of a lot (n days), and Work n process of a lot (n number of parts). Test No. Producton Lot no. Q SD DD DM MPX results Arena results perod PLT WIP PLT WIP 1 27 lot lot lot lot lot lot lot lot lot lot lot lot Table 1: Comparson between MPX and smulaton results From Table 1, t s apparent that the steady state values of WIP and PLT are hgher than those of the smulaton results. It s because that MPX dd not capture the effect of transton state on the performance of the manufacturng system. However, ths nference cannot be guaranteed snce t has been made based on the sample sze of four. Snce, the steady state values and the smulaton results are not the same, t s sgnfcant to clarfy 42

43 Chapter 4: Methodology whether the steady state values ntroduce any bas nto the soluton, whch s generated usng a dynamc programmng based optmzaton routne. It s dscussed n the next secton Unbased nature of steady state values In ths secton, t s shown that the qualty of the soluton generated by applyng the proposed optmzaton routne does not appear to be degraded by usng the steady state average values of WIP and PLT. The proposton s stated as, By usng the steady state average values, the objectve of fndng the best lot sze s not based. An experment s conducted to evaluate the proposton. The objectve of the experment s to compare the soluton evaluated by usng the steady state average values and smulaton results for a two-perod lot-szng problem. The product structure parameters used n ths experment, are defned as follows: 1. Number of tems = 1, 2. Number of levels = 1, 3. Average number of components per parent = 1, 4. Average number of parents per component = 1. A manufacturng faclty, workng n one eght-hour shft each day, 5 days per week, has two machnes organzed n to two departments. Each department has one machne. Labors are not consdered n the desgn. The machne utlzaton lmt s set at 95% for both machnes. Parts routng and operaton parameter values are descrbed n Fgure 12. Scrap rate s set at zero for both workstatons. These parameter values are loaded n both the MPX and smulaton models. Other parameters are set as follows: 43

44 Chapter 4: Methodology Setup cost = 50 unts per setup Holdng cost = 0.1 unt per day per tem WIP cost = 0.1 unt per day per tem Due date mssng cost = 8 unts per day per tem Plannng horzon = 10 days,.e. 2 perods Work staton 1 Work staton 2 Processng tme= 5 mnutes Setup tme = 145 mnutes MTTF = 4800 mnutes MTTR = 130 mnutes Processng tme = 3 mnutes setup tme = 130 mnutes MTTF = 4800 mnutes MTTR= 150 mnutes Fgure 12: Parts routng 2 In Fgure 13, nodes one, two and three represent current perod, perod one and perod two, respectvely. The dynamc programmng mechansm fnds the shortest path to reach node three. Path 2 Path 1 Path D1 D2 Fgure 13: Node representaton of the two-perod lot-szng problem In ths experment, dfferent demand values are generated, the optmal path for reachng node three s evaluated usng both steady state average values (usng MPX) and smulaton results, and the results are compared. Detals of the smulaton model are 44

45 Chapter 4: Methodology dscussed n Secton 6.1. The sets of demand values and lot szes are shown n Table 2. The experment results are shown n Table3. Notatons used n Table 3 are defned as follows: PP WIP PLT Producton perod n days Work-n-process of a lot (n unts) Planned lead-tme of a lot (n days) Sl.No Perod 1 2 Demand Path 1 lot sze Path 2 lot sze 950 Demand Path 1 lot sze Path 2 lot sze 560 Demand Path 1 lot sze Path 2 lot sze 1075 Demand Path 1 lot sze Path 2 lot sze 2055 Demand Path 1 lot sze Path 2 lot sze 2190 Table 2: Set of demand values and lot sze 45

46 Chapter 4: Methodology Sl.No PP Path Lot sze Steady state values Smulaton values days unts WIP PLT Total cost WIP PLT Total cost Table 3: Comparson of the decsons made by usng both the steady state average values and the smulaton results In Table 3, hghlghted cells represent the optmal path. It appears from Table 3 that the steady state average values, whch are estmated by usng MPX, dd not bas the soluton,.e., the decson made based on the steady state values turns out to be the same as the decson made based on the smulaton results. However, ths nference cannot be guaranteed snce t has been made based on the sample sze of fve. Therefore, the soluton generated by the SLLS algorthm may or may not be optmal, though t embeds dynamc programmng based optmzaton routne. Therefore, the qualty of the soluton generated by SLLS algorthm has to be tested. From ths experment, t can be concluded that the heurstc approach appled by the SLLS algorthm has to be valdated by 46

47 Chapter 4: Methodology conductng a seres of experments. These experments wll determne the qualty of the soluton generated by the SLLS algorthm. Detals of the experments are dscussed n Chapter Assumptons on lot-szng problem formulaton The followng assumptons concernng the condtons for the lot-szng problem are made: Unt processng cost, unt setup cost and unt holdng cost reman constant over the producton perod; Sze of a tme bucket (perod) s set at 5 days; Weekends and holdays are not accounted n holdng cost calculaton; Back orderng s not allowed; Demand values are known a pror; Negatve demand values are not allowed; Resource capacty, such as number of servers and number of labor, cannot be altered; Producton can only be acheved durng regular tme, 8 hours per day, and 5 days per week; Over-tme and work on weekends are not allowed; Buffer capacty between workstatons s nfnte; The due date of a perod s always the frst day of that perod; The varablty of processng tme s set at 30%; Processng tme follows the Trangular dstrbuton functon; and The machne utlzaton lmt s set at 95%. 47

48 Chapter 4: Methodology 4.3. Notaton defntons C s C h Setup cost per lot Holdng cost per unt tem per unt tme C wp WIP cost per unt tem per unt tme d j Demand n perod j j k L, Lot sze, evaluated at stage (node), covers the demand exstnt M ( L ) between perod j and k, due at perod j. Set of lot szes, evaluated for the M th ncomng path of stage (node) and covers the demand exsts up to the perod. * L ( ) WIP L j k ( ) L j k Optmal set of lot szes, evaluated at stage, Average WIP of lot sze, L j, k LT, Average lead-tme of lot sze, TC ( * L ) TC{ L } L j, k Total cost of optmal set of lot szes, evaluated at stage j, k Total cost of lot sze, L j, k M TC ( L ) Total cost of set of lot szes, evaluated for M th ncomng path of stage (node) and covers the demand exsts between perod j N ( ) L j k and k. Total number of perods ES, Early start length of lot sze, L j, k M G Total number of lots n the set ( L ) 48

49 Chapter 4: Methodology M g L, g th lot of sze j k ( ) DT, Setup tme of L j k L n the set M ( ) L, j k j k L, for DELAY type equpment 4.4. Problem formulaton Equaton 4.10 s executed at each stage of dynamc programmng. Equaton 4.10 determnes the optmal way to satsfy the demand exstng up to perod. In other words, t determnes the crtcal path to node. Snce, TC ( * L ) = mn M ( L ) M ( L ) TC (4.10) TC = * l TC( L ) TC( L ) +, (4.11) l+ 1, TC ( * L * l ) = mn { TC( L ) TC( L )} + (4.12) l+ 1, where, 1 M, l + 1, 1 N. The stage =0, represents the tme at zero. Therefore, TC * ( L 0 ) = 0 (4.13) TC ( L ) l 1, + = setup cost + nventory cost + WIP cost + early startng cost WIP cost = C * WIP( L )* LT ( L ) wp l 1, l + 1, + (4.14) 49

50 Chapter 4: Methodology Inventory cost = * C * ( L d ) h 5 (4.15) k= l+ 1 l+ 1, k TC Early startng cost = ( ) 5 * (4.16) C h* ES L j, k * L j, k ( L ) C + * C * ( L d ) + ES( L )* L C * WIP( L ) LT ( L ) = l+ 1, s 5 h l+ 1, k j, k j, k + wp l 1, * + l+ 1, k= l+ 1 (4.17) Where, * L = M ( L ) M TC * L = TC L (4.18) : ( ) ( ) M ( L ) = * l { L L } + (4.19) l+ 1, Where, 1 M, l + 1, 1 N Determnaton of lot release dates Release dates of lots are determned by offsettng ther PLT from ther due dates. Then, order by whch lots are released nto the shop s determned by usng ther release dates. The notatons are defned as follows: ( L j k ) ( ) CT, Cumulatve tme of lot sze, DD, Due date of lot sze, L j k L j, k L j, k 50

51 Chapter 4: Methodology ( ) P, Order of lot sze, L j k M ( L ) L j, k M LD Perod n whch latest due date falls for lot set ( L ) a If ( ) DD < ( d ), L b c DD,, then cumulatve tme of L e f a L b, c and d L e, f are defned as, CT In general, a a ( L ) = LT ( L ) + 5 * ( e b) b, c b, c, (4.20) d d ( L ) LT ( L ) CT e f e, f CT, =. (4.21) M ( L ) = LT ( L ) + 5 LD ( L ) j j, k j, k * (4.22) If then CT (, ) < ( ) a L b c d L e f a ( ) P, < ( d ) L b c CT,, (4.23) P, (4.24) L e f The objectve of usng cumulatve tme s to fnd the datum lne, whch can then be used to fnd release dates for lots. Lot release dates cannot be determned smply by offsettng ther PLT from ther due dates. Ths s because dfferent lots wll have dfferent due dates. Moreover, the calendar system s not adopted n the algorthm. The earlest release date can be found by fndng a lot, whch has the longest cumulatve tme. Then, release dates for remanng lots are determned by keepng the earlest release date as the datum lne. Fgure 14 shows the concept of cumulatve tme. 51

52 Chapter 4: Methodology 5 days Lot2 startng date Lot1 startng date Lot 2 PLT ( cumulatve tme of lot 2) Lot1 PLT Lot 1 Due date Lot 2 Due date Cumulatve tme of lot 1 Fgure 14: Descrpton of cumulatve tme concept Consder the problem of fndng the release date for lots 1 and 2. Lead-tme of lot 1 s 11 days. Lead-tme of lot 2 s 13 days. Lot 2 s due 5 days later than lot1. In order to fnd the release date for lots 1 and 2 wth a common reference pont, cumulatve tme for both lots s calculated. Cumulatve tme of lots 1 and 2 are 16 and 13 day, respectvely. Therefore, lot 1 has the earlest release date. The followng secton descrbes estmatng lead-tme usng the MPX model Lead tme estmaton n the MPX model The MPX model apples FCFS prorty-queue-dscplne rule. Moreover, MPX can calculate the tme requred for each operaton of a lot. The planned lead-tme of a lot whch has the earlest due date s calculated drectly from MPX model. The lead-tme of a lot, whch does not have the earlest due date, s calculated by usng the followng algorthm. The logc of ths algorthm s that t uses ndvdual operaton tme of a lot along wth that of ts mmedate predecessor to calculate the average watng tme of the lot before t s taken up for producton. Before presentng detals of the algorthm, the followng notatons have to be defned. 52

53 Chapter 4: Methodology M ( g L ) P, Prorty of lot j k M ( g L j k ) a ( ) O, Operaton tme of xy M O z g L b c M ( g L j k ) ( L ) M g L j, k th y operaton, n workstaton x, of lot M a, Operaton tme of g L, n work staton z, whle b c sequence for processng ts ( y +1) th operaton. TW, Total watng tme of lot xy M g W, Watng tme of j k M g L, due to prortzaton j k M g L j, k M g L, s next n j k th y operaton, n work staton x, of lot M g L j, k M ( g L ) j k due to prortzaton TO, Total number of operatons requred for lot M g L j, k TM M ( g L ) j k Total number of machnes PLT, Planned lead tme of lot M g L j, k The Algorthm can now be descrbed n detal. Step 1. Defne M g L, and j k M a g 1L b. c M Step 2. Set y = 2, where y= 2, 3 TO( g L j k ) M Step 3. Set ( g L ) j k,. TW, =0. Fnd workstaton x correspondng to operaton y, where x = 1...TM. Step 4. Calculate ( L ) xy M g O, wth dynamc nteracton between j k M g L j, k and M a g 1 L b. c. Ths s acheved by runnng the MPX model wth only loads. Step 5. Fnd workstaton z, for operaton y +1. M g L, and j k M a g 1 L b. c as the 53

54 Chapter 4: Methodology M a Step 6. Calculate O zy ( L ) + 1 g 1 b, c. M M a O xy g j, k < O zy + 1( g 1L b, c ), M M a M W xy g j, k = O zy + 1( g 1L b, c ) - xy ( g L j k ) Step 7. Calculate the watng tme. If ( L ) M then ( L ) O, ; other wse xy ( g L j k ) M M Step 8. Calculate TW ( g L, ) = TW ( g L, ) + ( L ) M g L j, k j k Step 9. If = TO( ) 1 2. j k M W xy g j, k W, = 0. y, then go to the next step; otherwse y = y + 1 and go to step M Step 10. Calculate ( g L j k ) M M Step 11. Calculate PLT ( g L, ) = LT ( g L, ) + M ( g L ) LT, wth dynamc nteracton between j k j k j k M g L j, k TW, and STOP. and M a g 1 L b. c Concept of early startng As dscussed earler, the MPX model apples the FCFS prorty-queue-dscplne rule. Secton 4.6 descrbes the dscusson on estmatng the lead-tme of a lot by usng the MPX model. Based on the dscusson descrbed n prevous sectons, a lot whch has a late release date (lower order) has to be started after the completon of the frst operaton of a lot whch has an early release date (hgher order). In case of nsuffcent tme, the processng of a lower order lot may not be completed before ts due date. In order to avod tardness, t s reasonable to start the hgher order lot a lttle early so that the lower order lot can be completed before ts due date. Ths concept s explaned n the followng example. Consder a problem of orderng lots 1 and 2 based on ther release dates. Lead-tme of lot 1 s 11 days; lead-tme of lot 2 s 13 days. Lot 2 s due 5 days later than the due date of lot1. By applyng the technque dscussed n Secton 4.5, t s found that lot 1 has the earlest release date. Therefore, lot 2 starts 3 days after the startng date of lot 1. It s assumed that the frst operaton of lot 1 s completed 6 days after ts startng date. In 54

55 Chapter 4: Methodology realty, lot 2 has to wat 6 days before t s taken up for producton. Ths complex detal cannot be captured drectly n MPX. In order to capture the release date order, lot 2 has to be started 6 days after the startng date of lot 1. If so, ths leaves 8 days to complete lot 2, whch actually requres 13 days to complete. In order to avod tardness of lot 2, lot 1 s started 3 days earler than ts actual startng date so that ts frst operaton s completed just before the startng date of lot 2. Fgure 15 explans the concept of early startng. Lot1 1 st operaton completon date Lot2 startng date Lot 2 lead tme 5 days Lot1 lead tme Lot1 new startng date Lot1 startng date Early startng length Lot 1 Due date Lot 2 Due date Fgure 15: Concept of early startng The concept of early startng actually reduces shop congeston. Therefore, t mproves the shop performance. Moreover, t s economcally preferable. Ths s because the cost of mssng the due date s hgher than that of early completon. The early completon cost s part of the total cost defned n equaton 4.8, n Secton 4.4. Therefore, the cost of startng a lot early wll have an mpact on the decsons made at each stage. The concept of early startng may not be requred when the capacty of a shop s changeable. For example, f a planner sees the possblty of completng lot 2 n 8 days by ncreasng the servers, then lot 1 need not to be started early. Snce, t s assumed for ths study that resources are unchangeable, the concept of early startng s the only vable and economcal soluton for reducng the tardness of jobs. 55

56 Chapter 4: Methodology 4.8. Algorthm The algorthm for settng lot sze and planned lead-tme smultaneously s splt nto two sectons. Algorthm I s developed for lot-szng problems and algorthm II s developed for estmatng the WIP and planned lead-tme of a lot. Algorthm I has to be embedded n algorthm II. However, embeddng the optmzaton routne on MPX s techncally beyond the scope of ths research. Algorthm I can now be descrbed n detal n the followng steps. Notatons are already defned n Secton 4.3. Algorthm I Step 1. Set = 1, where = 1, 2, 3 N. Step 2. Set M =1 and l =0, where M = 1, 2, 3 and l =0, 1, 2 1. M Step 3. Fnd ( L ) and G, where ( ) M L ={ * l L, L } l + 1,. Note * L 0 = 0. M Step 4. Calculate the WIP and planned lead-tme for g L,, for all g = 1, 2 G, n the M set ( L ) by executng algorthm II. Step 5. Calculate M TC ( L ) by usng equaton 4.3, 4.4, and 4.9, descrbed n Secton 4.4. Step 6. Set M = M +1 and l = l +1. Step 7. If M and l 1, then go to step 3; otherwse go to the next step. Step 8. Calculate ( ) L * TC * and L. Step 9. Set = + 1. If N, then go to step 2; otherwse go to the next step. j k Step 10. Set * = N. Fnd L. Step 11. Fnd set of lots that consst n * L calculated planned lead-tme and startng date. from equaton 4.10, n Secton 4.4 and ther 56

57 Chapter 4: Methodology Step 12. Stop Algorthm II Step 1. If G=1, then go to the next step; otherwse go to step 4. Step 2. Determne M ( L ) M L j, k j k 1. Estmate the lead-tme of lot M L j, k M 1, ( L ) LT 1, and WIP 1, wthout the dynamc nteractons among other lots by executng the MPX model. Ths s acheved by executng the MPX model exclusvely for M 1 L j, k Step 3. Set ( L ) LT ( L ) M g j k M g PLT, =,. Then go to step 41. M Step 4. Determne each lot n ( L ). Set g =1, where g =1, 2, 3 G. Step 5. Set M LD ( L ) =. Step 6. Assgn g th lot n ( ) j k M L as M g L,. j k Step 7. Estmate the lead-tme of lot M M g L,, LT ( g L ) j k j k nteractons among other lots by executng the MPX model. Step 8. Calculate the cumulatve tme of mentoned n Secton 4.5. M M g L,, ( g L ) j k j k, wthout the dynamc CT, by usng equaton 4.14, Step 9. Set g = g + 1. If g G, then go to step 6; otherwse go to the next step. M Step 10. Arrange CT ( g L j, k ) n descendng order and fnd the rankngs of M M M P( g L, ) by assgnng P( g L, ) =1 for the largest value of ( g L ) so on. j k j k j k M g L,, j k j k CT, and 57

58 Chapter 4: Methodology Step 11. Set e = 1, where e = 1,2, 3 G. Step 12. Load M M g L, n the MPX model such that P( g L ) j k j k, = e. Step 13. Set the frst operaton of M g L, as delay at equpment of DELAY type and j k Step 14. If = Step 15. Set = 1 Step 16. Fnd set remanng operatons as per the routng. M e 1, then set ( g L j k ) M r and ( g L j k ) M a M a g L, : P( g L ) b c Step 17. Set setup tme of DT, =0 and go to step 19; otherwse go to the next step. DT, = 0. b c, = r. M M g L, at DELAY type equpment, ( g L ) j k M M a DT ( g L, ) = DT ( g L, ) + O x ( L ) j k j k M 1 g b, c. DT, where Step 18. Set r = r + 1. If r e 1, then go to step 15; otherwse go to the next step. Step 19. Set e = e + 1. If e G, then go to step 12; otherwse go to the next step. M Step 20. Run the MPX model and calculate ( g L ) Step 21. Set e = G, where e = G, G-1.3, 2 Step 22. Fnd Step 23. Set = 2 j k j k WIP, for all g = 1, 2 G. M M M a M a g L j, k : P( g L j, k ) = e and fnd g L b, c : P( g L b c ) M M y, where y= 2, 3 TO( g L, ) and set ( g L ) j k, = e -1 TW, =0. M Step 24. Fnd workstaton x correspondng to operaton y of g L,, where x = 1... M. Step 25. Calculate ( ) xy L j k j k j k O,. It can be found n the MPX results menu. Step 26. Fnd workstaton z correspondng to operaton y + 1 a Step 27. Calculate O zy ( L ) M a of g L,, where z = 1... M. b c M + 1 g b, c. It can be found n the MPX results menu. 58

59 Chapter 4: Methodology M Step 28. Calculate the watng tme W xy ( g L j, k ). M M a M M a If O xy ( g L j, k ) < O zy + 1( g L b, c ), then W xy ( g L j, k ) = O zy + 1( g L b, c ) - M M O xy ( g L j, k ); otherwse W xy ( g L j, k ) = 0. M M M Step 29. Calculate TW ( g L, ), where TW ( g L, ) = TW ( g L, ) + ( L ) M g L j, k j k Step 30. If = ( ) 1 step 24. j k j k xy M g W,. y TO, then go to the next step; otherwse y = y + 1 and go to M M M M Step 31. Recalculate LT ( g L j, k ), where LT ( g L j, k ) = LT ( g L j, k ) + ( g L j k ) M M Step 32. Recalculate CT ( g L, ) based on ( g L ) j k j k j k TW,. LT, obtaned n step 31 by usng equaton 4.14, mentoned n Secton 4.5. Step 33. If orderng of lots s changed, then go to step 10; otherwse go to the next step. Step 34. Set e = e 1. If e >1, then go to step 22; otherwse go to the next step. Step 35. Set e = G, where e = G, G-1.3, 2 M M M a M a Step 36. Fnd g L j, k : P( g L j, k ) = e and fnd g L b, c : P( g L b c ) M a Step 37. Calculate O x 1 ( g L b, c ). M M M a M a Step 38. If ES( g L j, k )+ CT ( g L j, k ) CT ( g L b, c ) - O x ( g L b, c ) M a ES ( g L ) = 0; otherwse b, c, = e -1 1, then M a M M M a a ES( g Lb, c ) = CT ( g L j, k ) + ES( g L j, k ) CT ( g Lb, c ) Ox1( Lb, c ). M Use CT ( g L, ) value obtaned n the step 32. j k M a Step 39. Calculate PLT ( g L ) M a b, c, where PLT ( g L b c )= M a M a, LT ( g L, ) + ES( g L ) b c b, c. 59

60 Chapter 4: Methodology a a a Step 40. Set LT ( L ) LT ( L ) + ES( L ) M g b, c M M = g b, c g b, c. Step 41. Set e = e 1. If e >1, then go to step 35; otherwse go to the next step. Step 42. STOP Algorthm executon The SLLS algorthm s executed by usng Mcrosoft Excel and MPX. Before executng the algorthm, manufacturng system parameters (held constant factors, explaned n Chapter 7) ncludng producton perod, processng tme, number of workstatons, parts routng, setup tme, MTTR, MTTF, and number of workng days are loaded n MPX. The executon of the algorthm s descrbed n the followng steps. (An example of SLLS algorthm executon s presented n Appendx A.) Step 1. The frst three steps of Algorthm I are executed manually n an Excel Worksheet Step 2. Lot sze values are loaded n MPX and the MPX model s executed manually Step 3. Algorthm II s executed and results are loaded manually n the Excel Work sheet Step 4. The remanng steps of algorthm I are executed manually untl computaton of all nodes s completed 3 Mcrosoft Excel 2000 s a regstered trademark of Mcrosoft Inc. USA. 60

61 Chapter 4: Methodology Approxmatng MPX results The followng procedures are appled for approxmatng the values obtaned from MPX results. 1. The MPX result descrbes lead-tme n fractons of days. Those fractonal values are approxmated to the next hgher nteger values.e., both 4.2 and 4.6 are approxmated to fve. 2. The MPX result descrbes the completon tme of an ndvdual operaton n mnutes. It s approxmated by usng the followng equaton. Approxmate ( L ) H = ( MPX result of ( L ) xy M g j k xy M g O, )/ 100 O, = ( ( approxmate H to the next hgher nteger value ) + 1 )*100 j k 3. Values represented n mnutes are approxmated to days by usng the followng equaton. R= ( target value V, n mnutes/ 480 ) Target value V, n days = approxmate R to the next hgher nteger value 61

62 Chapter 4: Methodology Lmtatons of the SLLS algorthm Generally, dynamc lot-szng problems are classfed nto three dfferent categores based on the product structure and product type. They are Sngle-level, sngle-product models Sngle-level, mult-product models Mult-level, mult-product models A common soluton methodology cannot be appled for all the three types of problem, snce the fundamental approach to solvng these problems s dfferent for each one. For example, n case of a mult-level mult-product model, lot-szng methodology should consder addtonal parameters such as the dynamc nteracton among dfferent products, part commonalty, and global optmzaton, apart from the parameters consdered n the SLLS algorthm. Therefore, the SLLS algorthm s lmted to snglelevel sngle-product models. The use of analytc queung theory for tacklng sngle-level mult-product models and mult-level mult-product models s beyond the scope of ths research. 62

63 Chapter 5: Modelng a manufacturng system n MPX Ths secton descrbes the bref dscusson on manufacturng system modelng n MPX. Complete detals are llustrated n MPX (2001). The followng subsectons dscuss the ndvdual features of the man menu bar, n MPX Input menu bar Input menu bar conssts of fve submenus. They are: 1. General data 2. Labor data 3. Equpment data 4. Product data 5. Bll of materal Snce a bll of materal s not requred for modelng the problem under study, the followng sub sectons descrbe the dscusson only on the remanng submenus General data The General data submenu s used to defne the followng data. Fgure 16 dsplays a General data submenu n MPX. Basc tme unt of operatons, flow tme and producton perod Total-workng hours per day Producton perod.e., tme perod over whch demand apples Utlzaton lmt for all workstatons Percentage of varablty of ndvdual parameters 63

64 Chapter 5: Modelng a manufacturng system n MPX Fgure 16: Dsplay of General data sub menu n MPX Labor data Labor submenu s used to defne the followng data. Fgure 17 dsplays a Labor submenu n MPX. Number of people n each group of labor Percentage of over-tme for each group Percentage of unavalable tme 64

65 Chapter 5: Modelng a manufacturng system n MPX Fgure 17: Dsplay of Labor submenu n MPX Equpment data The Equpment submenu s used to defne the followng data. Fgure 18 dsplays Equpment submenu n MPX. Number of workstatons n each equpment group Type of each equpment group,.e., standard, delay Percentage of overtme for each group of equpment Mean-tme to falure for each group of equpment Mean-tme to repar for each group of equpment Labor group assgned for each group of equpment 65

66 Chapter 5: Modelng a manufacturng system n MPX Fgure 18: Dsplay of Equpment submenu n MPX Product data The Product submenu s used to defne the followng data. Product type End demand of each product type Lot sze of each product type The Operaton routngs s a submenu n Product submenu. It s used to defne the followng data for each product type. Operaton routngs Name of equpment nvolved n each operaton Equpment setup tme of each operaton (accounted for a lot) Equpment runtme of each operaton (accounted for each part) Labor setup tme (accounted for a lot) 66

67 Chapter 5: Modelng a manufacturng system n MPX Labor runtme of each operaton (accounted for each part) Scrap rate at each operaton The followng example descrbes how the problem under study s modeled n MPX. Let lot 1 of sze 500 and lot 2 of sze 300 have to be defned n MPX model. In the Product submenu, enter lot 1 and lot 2 as two dfferent types under the product name column. End demand of lot 1 and lot 2 are defned as 500 and 300, respectvely. The lot szes of lot 1 and lot 2 are defned as 500 and 300, respectvely. Operaton routngs and processng parameters for both product types wll be defned smlarly. If tme phasng s requred between lot 1 and lot 2, then t wll be acheved by the technque descrbed n Secton Fgure 19 dsplays product submenu n MPX. Fgure 19: Dsplay of Product submenu n MPX 67

68 Chapter 5: Modelng a manufacturng system n MPX 5.2. Run menu bar The Run menu bar s used to execute the model. It has an opton of checkng the model for error before startng the model executon. It also has an opton of fndng the explanaton for errors f any errors are found. It also has some other advanced features whch are not requred for modelng the problem under study Output menu bar The Output menu bar s used to dsplay MPX results. It has an opton of dsplayng the results assocated wth labor, equpment, and product types. Estmated values of the WIP and lead-tme of a lot are dsplayed n Product submenu, n the Output menu bar. The processng tme of ndvdual operatons s also found n the Product submenu, n the Output menu bar. Fgure 20 dsplays the Output menu bar n MPX. Fgure 20: Dsplay of Output menu bar n MPX 68