Dynamic lot-sizing and scheduling in a cellular manufacturing system

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DYNAiMIC LOT-SIZING AND SCHEDULING IN A

CELLULAR MANUFACTURING SYSTEM

A THESIS

SUBMITTED TO THE DEPARTMENT OF INDUSTRIAL ENGINEERING

AND THE INSTITUTE OF ENGINEERING AND SCIENCES OF BILKENT UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE

By

Oiiiaii Dagliogliigil

August, 1993

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>b2>L,

Ill

I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a thesis for the degree of M aster of Science.

Assist. Prof. M. Selim Aktürk (Principal Advisor)

I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a thesis for the degree of M aster of Science.

ssoc. Proh^^îIle'lTial Dinger

I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a thesis for the degree of M aster of Science.

< d : .

Assist. Prof. Ihsan Sabuncuoglu

Approved for the Institute of Engineering and Sciences:

Prof. M eh m o t^ aray

ABSTRACT

DYNAMIC LOT-SIZING AND SCHEDULING IN A

CELLULAR MANUFACTURING SYSTEM

Oiiiaii Dagiioglugil

LI.S. in Industrial Engineering

Supervisor: Assist. Prof. M. Selim Aktiirk

August, 1993

111 most of the production systems, lot-sizing and scheduling decisions are given in different levels of hierarchy, however there is a strong interaction between these decisions. Most of the existing models do not utilize the shop floor con­ ditions in lot-sizing and scheduling decision dynamically, even though such a. decision might improve the system performance. In tins study, a dynamic lot-sizing and scheduling algorithm is suggested for cellular manufacturing sys­ tems, wliich utilizes a h}d)rid simulation/analytic modelling approach. The performance of the proposed algorithm is compared with the push and pull systems under different shop floor conditions.

K e y words: Dynamic Lot-sizing, Scheduling, Hybrid Simulation/Analytic Model, C'ellular Manufacturing Systems.

ÖZET

HÜCRESEL ÜRETİM SİSTEM LERİNDE DİNAMİK PARTİ

BÜYÜKLÜĞÜ VE Ç İZELG ELEM E

(Jıiıaıı Dağiıoğhıgil

Endüstri Mühendisliği Bölümü Yüksek Lisans

Tez Yöneticisi: Yrd. Doç. Dr. M. Selim Aktürk

Ağustos, 1993

Parti büjâiklüğü ve rizelgeleıne kararlan arasında kuvvetli bir etkiledim olmasına rağmen, bu kararla.]· üretim sistemlerinin eoğunda farklı öncelge se- viyeku'inde verilir.

Ateh'c şartlan, mevcut üretim sistemlerinin çoğunda “ parti bÜ3n’iklüğü ve çizelgeleme kararlarmda” sistem performansını arttırıcı etkisi olmasına rağmen, dinamik olarak kullanılmazlar. Bu çalışmada, hücresel üretim sistemleri için karma benzetim/analitik modelleme yöntemi kullanılarak dinamik bir parti fn'iyüklüğü ve çizelgeleme algoritması önerilmiştir. Önerilen algoritmaıım |)cr- formansı, değişik atelye şartlarında itme ve çekme sistemleriyle karşılaştın 1ınıştır

A n a h ta r sözcükler: Dinamik Parti Büyüklüğü, Çizelgeleme, Karma Ben- zetim/Analitik Model, Hücresel Üretim Sistemleri.

VI

ACKNOWLEDGEMENT

I would like to ex])res.s rny gratitude to Assist. Prof. Selim Aktiirk due to his supervision, guidance, understanding and encouragement throughout the development of this thesis. I am also indebted to Assoc. Prof. Cemal Dinger and Assist. Prof. Ihsan Salnmcuoğlu for showing keen interest to the subject matter and accepting to read and review this thesis.

1 would like to extend my dee])est gratitude and thanks to my parents and Müge Tiii'ker foi· their morale support a.nd encouragment.

I greatly appreciate my research group friends Selçuk y\vci, Okan Balkös(‘, İhsan Durusoy, Elif Görgülü and Haluk Ydmaz, my classmate' Mehmet Özkan and Alper Erdoğan lor their valuable remarks and patience in any way dui iiig my iVI.S. studies.

Contents

1 I n t r o d u c tio n a n d L i t e r a t u r e R e v ie w 1 1.1 Introduction... 1 1.2 Literature Review ... .3 1.2.1 Introduction... 3 1.2.2 Pu.sli S y s te m s ... 1 1.2.3 Pull Systems o 1.2.4 Other Ap])roaclies... S 1.2.5 Dynamic I^ot-Sizing and Scheduling... II 1.2.6 Hybrid Simulation/Ana.lytic Models 12 1.2.7 Lead ddme Estimation 14 1.2.8 Crou|) 'Pechnology and Cellular Manul'actui'ing... Hi 1.3 Conclusion... 18

2 P r o b l e m S t a te m e n t 20 2.1 Introduction... 20

2.2 D}aianiic Lot-Sizing and Scheduling... 20

2..'I Why Simulation/Analytic Approach... 22

2.4 A.ssumptions... 21

2.5 Simulation/Analytic. Model 25 2.5.1 Performance Mea.sure,s 25 2.5.2 Dyna.mism... 26

2.5.3 Using Machine Information (Utilization,s)... 27

2.5.4 Using Queue Information... 29

2.6 A lg o rith m ... 30 2.6.1 Introduction... ■... 30 2.6.2 M e rg in g ... 31 2.6.3 S p littin g ... 35 2.6.4 Determination of Piuameters 36 2.6.5 Conclusion... 39 3 P u sh a n d P u ll S y s te m s 4 0 3.1 Introduction... 40

.3.2 Review of Push and Pull S y s te m s ... 40

3.3 The Manufacturing (Jell 42 3.4 Simulation Model of The Push System 42 3.5 Simulation Model of The Pull S y stem ... 44

CONTENTS

3.6 Conclusion 46

4 E x p e r i m e n t a l D esig n

Introduction

4.2 Expcriinonta! Design

4.2.1 Recognition of and .Statement of The l^roblem 4.2.2 Choice of luictors and Levels

4.2.3 Selection of Response' Variables

4 7 47 18 . . . '48 . . . 49 62 4.2.4 Choice of Experimental Design 63 4.2.5 Poirforming The E x p e rim en t... 63

,3 Data Analysis ·")·)

4.3.1 Analysis of Variance 0 0

4.3.2 Analysis of Push S y s te m ... 66 4.3.3 Analysis of Pull System 68 4.3.4 Analysis of Iljdrrid Simulation/Analytic Model (MSAM) 69 4.4 Computational .Analysis... 61 4.5 Conclusion and Recommendations... 66

5 C o n c lu s io n 6 7

List of Figures

2.1 HYBRID Siinulat,ion/Anal3 tic Model

■) 9 _{Rescheduling Set} 2..'3 linpleinentation of Rule 1 2.4 An exani])le of merging . 2.5 Implementation of Rule 2 31 32 33 33

27

4.1 Tardine,s.s and Flow Time Plots G2

List of Tables

2.1 Notations and abbreviations used in aigorith n i... 30

d.l Factors lor Pusli and Pnll Systems oO 4.2 R o u tin g s... ol 4.3 I'bxed Parameters for Push and Pull Sy stem s... 52

4.4 Performance Measures for High Demand Rate, No Bottleneck 51 4.5 Performance Measures for High Demand Rate, 2 Bottlenecks . . 55

4.6 Summaiy of .ANOVA Tables for Push System 56 4.7 Summary of .ANOVA Tables for Pull S y s te m ... 58

4.8 Summary of .ANOVA Tables for The Proposed M o d e l... 60

4.9 Averages L· Ranges of Performance Measures... 61

4.10 Normalized Vahu's of Performance M easu res... 65

A.l Analysis of Variance for hdovv T i m e ... 71

A.2 Analysis of Variance for M ak e-sp an .... ... 71

A.3 Analysis of Variance for T ard iness... 72

LIST OF TABLES Xlll

C hapter 1

Introduction and L iterature

Review

1.1

In tro d u ctio n

In t.lie production [ilanning literature, tliere are several approaches to lot-sizing and scheduling problems. Push and pull systems are dill’ercnt in their opera­ tional chai'acterislics, but they face similar problems in lot-sizing and

schedul-iVIatiU'ials Requirements Planning (MRP) emerged in lat(' 1960’s and Inonghl some im])rovemeiits to the production .systems. However MRP users faced im- ])lementation ])rol)lems, because in MRP safety stocks, lot sizes and lead tinuis are taken as fixed. Therefore MRP cannot rodate the sj'stem ])aram('lers to changing shop floor conditions.

The .lust In Tinu! (.JIT) system aro.se in .Japan in 1960’s. In the operational level, .JIT is a. ])ull system, whereas MRP is a push system. .JIT ])hiloso])hy is "])roduce the required items, in the required quantities at the required time”. In JIT systems, production flow is controlled by cards, also called kanban. Kanbans pull the required parts form the system, whenever demand arrives.

CHAPTER 1.

INTRODUCTION AND LITERATURE REVIEW

Number of kanbans and lot sizes are generally determined by the Toyota for­ mula. The lot sizes depend on estimated lead times in Toyota formula. There­ fore lead time estimation is an important problem for pull systems, which is also ti'ue for |)ush s\'stems due to the reasons cited above. There are sev­ eral alternative systems to push and pull systems in the literature, such as CONWIP, which suggests combining some of the push and pull features, how­ ever it lacks detailed simulation studies to justify their results.

The systems discussed so far have some common drawbacks, such as lead times are not estimated dynamically, systems are not adapting to changing shop floor conditions, and the interaction between lot-sizing and scheduling decisions are not utilized. To overcome these problems, the lot-sizing and scheduling decisions can l)e giv(>n jointly and they should be updated dynamically.

Optimized Production Technology (OPT) is developed by G’oldratt, and it ((uestions the reasons behind having single lot size throughout the system. OPT gives alternative lot size definitions and emphasize that lead times should not be predetermined, but constructed dynamically.

In this study, we will propose a dynamic lot-sizing and scheduling algorithm for cellular manufacturing systems. This algorithm will be utilized in a hybrid simulation/analytic model. The production system will be represented by a simulation model and it will interact with the proposed algorithm. The algo­ rithm will estimate the lead times dynamically, and use the shop floor informa­ tion to gimerate the lot sizes and schedules. The multi-objective performance measure of the proposed algorithm will be compared with tlie traditional pu,sh and pull .systems results.

In .section 1.2, a brief literature review of push systems, pull systems and other ap])roaches will be i)resented. Then dynamic lot-sizing and scheduling literature, and hybrid simulation/analytic models will be discussed, khnally the lead time estimation rules and Group Technology is reviewed. In chapter 2, hybrid simulation/analytic modeling approach and proposed algorithm will be disemssed, the assumptions, rules and estimation of parameters are also discussed in the same chapter.

In chapter 3, after a In ief review of push and pull systems, operating con­ ditions and assumptions of the simulation models for l)oth systems are given. .An experimental design is developed in chapter 4, and ANOVA results are presented with cominitational analysis. Finally in cha];ter 5, the conclusion of the study is present(vj with the po.ssible extensions of this study.

1.2

L ite ra tu re Review

1.2.1 Introduction

CH A Prim l.

INTHODUCTION AND LITERATURE REViEW

3

There arc differeiU approaches both in theory and practice for multi-item jn'o- dnetion .systems. .Some of these approaches are totally different in their 0])cr- alional rules like push sj'stems, ])ull .systems or sj'stems that use soni(i aspects of both push and pull systems (mixed systems). Lot-sizing and scheduling are important ])roblems in all approaches and directly aflTct the systems perfor­ mances.

Push systems, pull systems and mixed systems have different api)roachcs for lot-sizing and sclieduling. Characteristics of these approaches and related studies are discussed in sections 1.2.2, 1.2.3 and 1.2.4, respectively. There are other api)roaches which try to .solve lot-sizing and scheduling problems simultaneously and come up with dynamic lot-sizing and scheduling policies. In si'ction 1.2.5, the structure and interaction of lot-sizing and scheduling is reviewed, and tlui im])ortant as])ects are emphasized for a. dynamic lot-sizing and scheduling api)roach.

In section 1.2.6, mathematical and simulation models are reviewed, and ".Simulation/Analytic Models" are discussed. Lead time estimation is an im- portant issue in production systems and it directly affects system imrformances. Different rules and current literature on the lead time estimation are discussed in .section 1.2.7. Group Technology and Cellular Manufacturing concepts, and basic assumptions related to our study are given in section 1.2.8.

1.2.2 Push Systems

Material K.e(|uirenients Planning (M RP) came up as a good rsolution to the manuractnring problems in the late 1960’s and it was superior to the finite sdualuling ])roc.(idures. MRP systems are conventional push systems, hricdly MRP systems take actual demands into account and trigger the parts of the end products by using tin* lead time oif-.setting procedure, then the parts are pushed through the system.

CHAPTim 1.

INTRODUCTION AND LITERATURE REVIEW

4

MRP faced many problems in the implementation. In MRP systems, safety stocks, lot siz(!s and lead times are taken as fixed. The interdependence between the shop floor conditions and lead times are ignored, and MRP becomes unable to relate the estimaXed lead times to capacity loadings. The interaction between lot sizes and lead times is also ignored which degrades the effectiveness of the MRP .systems.

In the literature there are several studies that aim to im]:>rove the perfor­ mance of MRP systems. A class of a])proaches try to find o])timal solutions to multi-stage lot-sizing problems in push systems. Crowston and Wagner [3] have develo])ed an algorithm for a gcuieral multi stage assembly, and Zang- vvill [32] studied a serial assemldy process. These studies use mixed integer ])rogramming so they are a])plicable to small sized problems.

Other class of approaclies are heuristics and some of them are quite ])0]mlar such as ])art period balancing or Wagner-Whitin Algorithm, however they ig­ nore the interde])endencies between proces.ses and apply single stage lot-sizing sequentially. Karinarkar et al. [14] developed a heuristic lot-sizing policy for a multi-item multi-nmchine manufacturing cell which they call Q-LOd\S. Q- LOTS u.ses open qucuiing networks and a nonlinear programming model. The residts are com])ared with simulation results and the model performs well in ceidain cases. Results of the study reveal that Work In Process (W IP) levels are due to queuing dodays which also cause long lead »times, and the.se delays are directly related to lot sizes.

Sadowski, Waller and McNeely [25] ]:>roposed a framework to calculate an elFective production capacity l)y constrained machine model in an MRP en­ vironment. The aim of their model is to calculate the machine requirements at the end product level by ignoring non-key machines and estimating tlie ef- hictive capacity using the key machines. In this model, miml:)ei· of products, machines and operations are known beforehand. Deterministic machine times are considei’ed rar.her than flow of jjarts. The estimated cajjacity fractions are found and corniJared with several threshold utilization values to determine the key machines. This model performs well as a strategic planning tool especicdl}' when the system is overloaded. This study emphasizes the importance of bot­ tleneck machines in system evaluation, and it shows that bottleneck machines dictate the tlirough])ut.

Karmarkar, Kekre and Kekre [13] studied lot-sizing in multi-item multi­ machine job shcjps. The}^ represent the shop performance using a queuing net­ work model which is then embedded into an optimization model that searches for optimal lot sizes and minimum WIP levels. They found that waiting times and lead times are functions of lot sizes, and these affect the inventoiy costs, schedule, performance and part coordination for assembly.

There is a trade off, since both small and large lot sizes give high lead times. Large lot sizes tie up machines and cause queue build up at the machine. Re­ ducing lot size causes the total work load at machines to increase because of the increased number of set-u])s. The increased work load causes saturation effect dis])layed b}^ queuing system and waiting times increase rapidly. This behavior shows that lot-sizing ])olicies have a. major impact on system performances.

1.2.3

Pull Systems

CHAPTER 1.

INTRODUCTION AND LITERATURE REVIEW

5

The .Japanese .Just-In-Time (JIT ) system arose in Toyota in 1960’s. JIT is a manufacturing philosophy with a sim])le goal: “Pro'duce the required items, in the required quantities, at the lequired time and in high quality”. In JIT philosopliy inventory is the root evil. The ideal inventory at each stage and

ideal lot size are both equal to one, and set-up time is equal to zero.

In the operational level, JIT is a pull system. In a pull system, the suc­ ceeding stage demands from the j:)receding stage only according to the rate and time the succeeding stage consumes the items. Item request from the pieceding stage is done via cards caJled “Kanban”. Kanbans flow u|)streain while the items flow downstream through the end of production stages. JIT cannot be applied to all production systems successfully, since it should be implemented in a repetitive manufacturing environment with smooth demand as suggested by seveial authors. Tlie details of J I T s}'stem are given by Hall [9] and Schönberger [27].

The J I T system can I.)e cither a dual card or a single card system. In a dual card Kanban system production is controlled with two kanbans. One kanban, called “Production Kanban”, accompanies the containers as they are being produced. When the irroduction of a container is completed and demand from the succeeding steige occurs by a “Withdrawal Kanban”, the production kanban is replaced with the withdrawal kanban and withdrawal kanban accomj)anies the container to succeeding stage. In a single card Kanban .system there is only one type of kanban, which is the ])roduction kanban. When a work center requires parts, the ])roduction kanbans are used to accpiire the required parts from the preceding work centers. When there is a match between the kanban and the part in the out stock queue of work center, the part is pulled to the succeeding stage and a signal is send to the preceding work center with a. kanban card. Number of kanbans is fixed in the short run and determines the maximum level of inventory.

Number of kanbans and lot sizes are calculated by using the following “Toy­ ota” formula:

CHAPTER I.

IN m O D U CriO N AND LITERATURE REVIEW

6

Ya = DL{\ -h n)

where

a : Lot, size

D : jJ)cina.!id

L : Lead lime cv: Safely fact-or

“y""rt”is called Ihe maximum invenlory level. Once maximum invenlory level is delermiued, “V’”and “a" have lo he delerniined accordingly. Clearly bolli number of kanbans (V) and lol size (a) are closely dei)endent on Lead Times (L). Though L is a slochaslic measure, an eslimaled L can be used for delermining and “a”.

Karmarkar and Kekre [16] sludied dual and single card Kanban systems in a two sta.ge Kanban cell by using Markovian Processes. They investigated the effect of lot-sizing on inventory and backorder costs. The ma.jor results of this study are the following ;

• Lot size associated with each card has a significant effect on perfonnance of Kanban systems

• The effect of numiter of cards is also significant

• There is an interaction Itetween the number of cards and lot size

CHAPTFAl L

INTRODUCTION AND LITERATURE REVIEW

7

This stud}^ em])ha.sizes the effect of lot-sizing on lead times and inventory levels in pull .systems, and shows how the ])erformance of the system changes with different lot-sizing ])blicies. (Ihanging either lot size or number of cards affects the maximum inventory level, and that changes the sho]> floor condi­ tions. In a similar study Ity Karmarkar [17], lot-sizing policy, card counts and capacity are given as the factors that affect the lead time and inventory le\ els. Huang, Rees and Taylor [11] developed a simulation model to investigate the effects of variable processing times, stage bottlenecks, variable demand

rates and combination ot these factors in a multi-line multi-stage Kanban pro­ duction systems. Using different processing time distributions they found that vciriability in processing times increases the overtime required. The increas­ ing number of kanbans do not decrease the overtime required after some value which indicates that there is a tradeoff value for the bound on inventory level. The Vciriability in demand rate and the combined affects are also significant in increasing the overtime. This results show tha.t different number of kanbans and the shop floor conditions are significant in determining the performance of the Kanban systems.

1.2.4

Other Approaches

CHAPTER I.

INTRODUCTION AND LITERATURE REVIEW

8

In the literature there are other studies that aim to comlrine the useful features of ])ull and ])ush systems in a cellular manufacturing environment. Briefly push system permits effective use of demand information, and pull features ])rovide detailed information and incentives in. the cell level. Using features of both systems can improv'e the system performance if the shop floor conditions such as U-slia]:)e layout or a stable demand pattern are already accomplished. .Some of the existing studies are discussed in more detail in the following pages.

Karmarkar [15] developed a model to combine the MRP at the plant wide level with Kanban ap])roach at the cell level. This approach is applicable to repetitive manufacturing with assemidy stages. Briefly, the .system uses MRP to generate gross reciuirements and use these values to calculate numlrer of kanbans at the cell level. However the lead time offsetting problems still exist in the .system. The interesting ])oint in this approach is the fact that the number of kanbans in each cell can change dynamically depending on the sho]> floor conditions which is adjusted by the cell supervisor. That means the fixed inventory level can be adaptive to the shop lloor conditions. However there is not any analytical derivation or any practical study of how the number of kanbans can be adjusted dynamically.

CHAPTER

INTRODUCTION AND LITERATURE REVIEW

Spearnicui and Zazaiiis [30] compare-: Kanban and A4RP systems, and oiFer theoretical motivations for the superiority of ])iill systems. Tliey offer a system called Constant-VVork-ln-Process ((JONVVIP). In this system the kanban flow is from the last station to the first one, then the material is pushed between stations. Total numbei' of VVIP in the system cannot go l)eyond the determined level. The practical l.)enefits of this system are;

• CONW11^ can run in a moi*e relaxed ])ull environment • No stable i)roduct ]iiix is necessary

• Routing is imi)ortant not the specific part number

• Only a constant WIP is delined for entii'e line, numl^er of cards will not be determined for each station.

An open queueing network is used to model' push system and closed queue­ ing network is used for the pull system. The results cire derived for exponential processing times and Poisson ¿irrivals. Using steady state solutions the follow­ ing three conjectures are submitted:

• Congestion is less in pull systems • Pull systems are easier to control

• Benefits of ])ull environment owe more to l)ounded VVIP than pulling everyw’here.

The third conjectui*e is the most interesting one. It slates that o|)erating conditions are important in performance of production systems as well as the operating rules. Tlu' third conjecture is studied by an ex])onential (¡ONWIP and Kanban systems with infinite su])]>ly and demand, and using queuing net­ works Spearman and Zazanis [30] analytically showed that:

^‘Throughput of a Kanban system will not exceed that of an equivalent CONV\TP system almost surely.”

CHAPTER 1.

INTRODUCTION AND LITERATURE REVIEW

10

Since there are many assumptions in these analytic derivcitions, detailed simulation studious are necessary for their validation. Determiijation of tlie routings and lot sizes are not studied yet and open to further stud\’.

O ptim ized P rod u ction T echnology (O P T )

0])timized Production Technology (O P T ), also called Theory of Constraints (TOC), is developed by Coldratt [7j. The TOC perceives constraint as “any­ thing that limits a system from achieving higher performance versus its goal”. OPT a|)proach foc.us(*s on capacity constraints and bottleneck resourcccs. It uses some principles of common sense like:

• Bottlenecks are the key items in the production line • .An hour lost on bottleneck is an hour lost in the system • Bottlenecks govern both inventory and throughiiut

OPT questions the reasoning behind having a single lot size throughout the system, and distinguishes the need for two different lot sizes which are the transfer lot and the proce.ss lot. O PT gives the following definitions;

T ran sfer L o t: A lot from the part point of view, indicating how many parts to j)roduce before transferring to the next operation.

P ro cess L o t: A lot from a resource point of view, indicating how many parts to produce before |)roducing another part.

OPT offers that the |)rocess lots and transfer lots can be different, actually process lots should be multiples of transfer lots. This approach brings an alternative to MRP and .JIT fixed lot size concepts. In OPT, it is discussed that the ¡trocess lots should not be predetermined but constructed dynamically in scheduling and the lead times are results of these schedules.

OPT uses a Drum Buffer Rope (D BR ) approach [6]. DBR first recognizes the bottleneck re.source that dictates the rate of production in the entire line,

which is call(3cl the D ru m beat. Since inventoiy should be minimized, the only B u ffe r will be in front of the bottleneck resource to make sure that it docs not run out of work due to fluctuations, breakdowns or variation in process times. The R o p e is betwoien the buffer in front of the bottleneck and the raw matcn ial in front of the first opeiiition. The production rate of the first operation is dictated by the rate at which material in the buffer is consumed by bottleiu'ck. Furthermore the excess inventory is prevented in the entire line. OPT works by first locating l)ufFer in front of the bottleneck and use forward scheduling, then the rest of the liiu' is scheduled by using finite backward scheduling.

The concepts of OPT such as the distinction of transfer lot, process lot and variable lot sizes, ¡rutting bound on VVIP and focusing on bottleneck resources are important jroints. The conce])ts related to variable lot sizois are introduced by the OPT approa.ch. but Irounding VVIP is also common to both .JIT a.nd CONVVIP ap])roaches. However there are criticisms about OPd" a.|rproach since it is a black box, and there is no further academic study about it.

1.2.5

Dynamic Lot-Sizing and Scheduling

Lot-sizing and scheduling are two important decision prolrlems in all ¡rroduc- tions systems, however it is difficult to come up with simultaneous solutions. There is a vast amount of literature about lot-sizing on the one hand and scluxluling on th(' other hand, however there is few study that contains ele­ ments of I.)oth lot-sizing and scheduling. The scheduling literature nearly al­ ways assumes that lot-sizing decisions are already being made prior to schedul­ ing decisions.

CHAPTER 1.

[N'mODUCTION AND LITERATURE REVIEW

11

ggs [2] studied the intcnactions between lot-sizing and scheduling. A multi-item, multi-stage V4RP system was simulated. .Six scheduling rules and five lot-sizing rules were u.sed. He found a strong interaction between lot-sizing rules and scheduling rules. Specifically, he found that no single lot-sizing rule, no single scheduling rule, and no combination of rules were superior for different performance measures like number of late orders, units of stock out, avera.ge

CHAPTER. 1.

INTRODUCTION AND LITERATURE REVIEW

12

inventory, etc.

Con.sidering tlie lot-.sizing piol)leni.s, where principles of group technology are applied or machines are grouped into cells, might require a different ap­ proach than the traditional ones. Because no setups cir(' needed betw(;en two consecutively scheduled jobs from the same family, although a setiq:) is required between jobs of dilFerent families. Lot-sizing of similar jobs is niaiidy done to avoid setup times. However this is not tlie best strat(;gy, since creating the smaller lots or splitting the lots can inciea.se the service hwel and decrease the flow time of the parts .

In a cellular manufacturing system, Potts and Wassenhove [21] discussed that scheduling all jobs from the same family is not a good strategy, parti­ tioning each family of jobs into several batches and scheduling these liatches can be a better strategy, even though solving these type of problems requires a simultaneous approach to both,lot-sizing and scheduling.

In all multi-item production systems, lot-sizing and scheduling are two im- porta.nt decision inoblems. Since these decisions are interrelated, simultaneous solutions improve the system performance. An analytical iipproach to this problem is difficult but a simulation/analytic approach can be an alternative solution method. Our model will focus on this approach, the details oi this approach are discussed in the next section.

1.2.6

Hybrid Simulation/Analytic Models

There are two ty]:>es of mathematical models, which are analytic, models and simulation models. Analytic model is a set of equations which charactei'izes a system, and the .solution procedure uses either analytical equations or nu­ merical algorithms. A simulation model is an operating model of the system that mimics the system behavior. A simulation model is known as a flow chart model, and the solution procedure consists of running a computer model, col­ lecting data and analyzing it to get the .system behavior.

CHAPTER I.

INmODUCmON AND LITERATURE REVIEW

13

The use of analytic models is usually preferred if desired results can be obtained by reasonable costs. However analjddc models are not api)licable to complex and dynamic |)roblems. In such problems, computer sinudation techniques evaluate a model numerically over a time period and'generate data, describing system’s operating characteristics. Analytic and simulation models can be regarded as two different a.])proaches that can be used in modeling. So it can be desirable to combine analytic, and simulation models into a hyl)rid model and use them if they are cost efficient.

Shanthikumar and Sargent [29] give the following description: A hyl)rid simulation/arndytic model is a. mathematical model which combines identifi­ able simulation and analytic models. The purpose of using a hybrid simula- tion/aiicdytic model in ])rol)lem solving are;

• Reduce the cost of developing a model solution

• Reduce the amount of computational effort instead of using a simulation model, or the numerical solution procedure of an analytic model when its com])utational effort is excessive

Shanthikumar and Sargent give' four classes of hybrid simulation/analytic models:

• A model whos(' beha.vior over time is obtained by altering betwecui using independent simulation and analytic models. •

• A model in which a. simulation model and an analytic model opcualc' in parallel over time with interactions through their solution procedures.

»

• A model in which a. simulation model is used in a subordina.te way for an analytic model of the total system.

y

• A model in which a. simulation model is used as an overall model of the total system, and it requires values from the solution ])rocedure ol an analytic model representing a portion of its input parameters.

CHAPTER 1.

INTRODUCTION AND LITERATURE REVIEW

14

A fourth type model could be more suitcd^le when a multi-item produc­ tion system is concerned. Rc])resenting such a system using complex (|ueuing networks or stochastic models is almost impossible. Howevei' if lot-sizing and scheduling problems can be solved using an efficient .solution algorithm, using it in conjunction with a simulation model to develop a hybrid simulation/analytic model can be a good solution methodology. This ap])roach will be utilized in our dynamic, lot-sizing and scheduling algorithm.

In the literature there arc' few hybrid models dea.ling with the queuing ])roblems. .An example of first class hybrid model is a single machine job shop subject to machine lailures and o|)portune maintenance by Shanthikumar [28]. There are simulation-optimization studies which combine simula.tor generator and optimization subroutine. In such a study Bengu and Haddock [1] studied a single item continuous review inventory system where demand quantity and the lead time to receive an order quantity follow a probability distribution, and shortages are allowed. The model finds the optimal reorder point and reorder quantity. The simulation generator translates description of the system into a a simulation program using SIlMAN simulation language. Then a set of search procedures like sequential search, pattern search, and Nelder Sz Mead search are combined with the simulation program to assist in the optimization ])rocess. This study is a good example to show how a hybrid simulation/analytic model solves a complex problem by using SIMAN simulation language.

1.2.7 Lead Time Estimation

MRP .systems use estimated lead times to offset the demand figures. In MRP systems lead times an' fixed, and they are independent from the changing shop floor conditions. In JIT systems, again fixed lead times are used to calculate the lot sizes and the number of kanbans. However lead times are functions of lot-sizing policies, and traditional a.i)proaches ignore this dependency. Gol- dratt [6] discusses that “lead times are results of a schedule and cannot be predetermined”. Karmarkar [16] showed tluit “lead time increases with lot size

CHAPTER 1.

INTRODUCTION AND LITERATURE REVIEW

15

and this relation is asymptoticall}' linear on rays”. ReUrted studies of Kar- rnarkar [13, 14] indicate the relationship between the lead times and .system performances, which were discussed in section 1.2.2. In all i>roduction systems, long lead times inpjose costs due to high VVIP, poor due date performance and uncertainty. To overcome these problems, the lead times must be determined dynamically considering the shoj) floor conditions. In this section some of the lead time (estimation rukvs and their performances will l)e discussed.

In the literature, there are two major approaches. 4'he first once is the static approach which considers only the job characteristics such as processing times or number of ma.c.hin(es. Tlu' second one is the dynamic approach which considers the shop status ijiformation like queue lengths in addition t() job characteristics. Embedding dynamism in the estimation rules imiu'ove tlu' efficiency. Ragatz and Malrert [22] studied some common lead time estimation rules and gave a sinudation study for their jjerformance.

The followings are some of the well known static lead time (istimation rules:

• Total Work (flT'VK)

• Number of Operations (NOP)

• Total Work and Number of Operations (TW K +N O P)

Studies in the literature indicates that TW K ])erfoiins betto'.r than otluu· static rules. The results of a study by Kanet and Christy [12] confirms that when the criteria is tardimvss or .system inventory TW K dominates the' others. TW K rule is given by the following relationshi]):

L, = kP,

The lead time of job “i” is a function of the total processing time 'UÇ'

and ])arameter The constant “A:” is a policy parameter that I'eflects the e.xpected delay a job will face, and “A·” is determined for entire product mi.x. TW K will be used in the experimental design of this study to estimate the lead tiiiKis for pull and push systems.

CHAFTER 1.

INTliODUCTION AND LITERATURE REVIEW

16

Ragatz and Mabert [22] studied several dj'namiclead time estimation rules. In addition to job characteristics such as processing time data or number of machines in the sj'stem, these rules consider the number of parts in the cjueucs, the total processing times of the parts in the queue, etc. .Since thes(' figures change djmamically with changing shop floor conditions, the rules are reactive to the .system changes. 'Fhcse rules are the followings;

• .Jobs in Queue (.JIQ) • Work in Queue (WIQ)

• Weeks' .Jobs in System (W EEKS) • .Jobs in System (JIS )

• Response Maj)i)ing Rule (R,MR)

They showed that when the criteria is standard deviation of lateness, mean absolute missed due dates or mean tardiness, for different sequencing rules like SPT, FCJ’\S or MINSLK, .JIQ works successfully.

The .JI() is given by;

Li = kj Pi + k'l^JIQi)

When job is released to the shop, all the queues on jol) ¿’s routing are |)ooled for the number of jobs in queue {JlQ i). This shop status information is then combined with the process time of job i (Pi)· Constants and "k-i'^ are policy parameters and reflect the expected delay of jobs, d'his rule will be later utilized in our dynamic algorithm to estimate the lead times.

1.2.8

Group Technology and Cellular Manufacturing

Croup technology (GT) is a manufacturing philosophy whose main idea is to capitalize on similar activities [31]. Cellular manufacturing (CM), which is a

CH APrEli 1.

INTRODUCTION AND LITERATURE REVIEW

17

derivative of GT, is the physical division of the celluhir manufacturing facilities niadiinery into production cells.

CM is cin interesting topic for both practitioners and accidemics in the last years. .Japanese coni])anies use CM extensively in .JIT manufacturing. .Also new technologies sup])ort the CM approach. Support is in the form of evolving information based technologies, such as the use of robotics and automalcd material handling systems. Eincient use of these philosophies and technologies requires a cell-structure ap])roach to manufacturing. A cell has a modified flow shop structure. In a modified flow shop, parts can enter the shop at one of several machines, can progress through the shop by a limited number of jraths, can exit the' shop at one of several machines, and backtracking is not allowed. The inclusion of assembly/transfer lines makes cell production widely a])])licable.

In CM, each cell is designed to jiroduce a jrart family or a set of j)art families. A part family is defined by Greene [8] as a set of ])arts that require similar machinery, tooling, machine operations, jigs and fixtunis. The parts in the same family normally manufactured completely in a single cell. Some of the basic CM assumptions include:

• Ecvch cell is a modified flow shop

• The operations of a job should not split between cells

• For any jol) there is at least one feasible cell wlnu'e all operations can l.)e completed

• .Jobs can have more' than one feasible cell

• Classic assumi^tions pertaining to job scheduling such as precedence re­ lationship and non-interference constixunt.

The advantages of the CM when com])ared to job shops can be listed as follows:

CHAPTER L

INTRODUCTION AND LITERATURE REVIEW

I . ) lleduc(xl control

2) Reduced material handling

3) Reduced set-up time 4) Reduced WIP

5) Reduced tooling

The disadvantages are as follows: 1) Reduced slioj) flexibility

2) Possible reduced machine utilization

Suital)ility of MRP systems, or using Kanbcin systems to rcigulate the ma­ terial flows inside and between cells, and potential use of finite scheduling ¡procedures such as OPfl' for scheduling celluhir systems need to be investi­ gated. It is not ajpparent that how will the aspects of production planning like lot-sizing, scheduling, capacity planning, etc., can be adapted to CM [31].

1.3 Conclusion

In this chapter characteristics of push, ])ull and several alternative systi-'ins are discussed. Coniinon problems in lot-sizing, lead time estimation and schetluling are reviewed. EIFects of the lot-sizing ¡policies on different production system l)erformances are em])hasized l)y different approaches. Some characteristics of cellular manufacturing and dynamic lead time estimation rules are also re­ viewed.

The interaction between shoj) floor conditions and performance measures are given by several studies in the literature. Dynamic lot-sizing and schedul­ ing methods improve the system ]rerformances since the S3'stem adapts itself to changing shop floor conditions. Therefore, it is important to develop a dynamic

CHAPTER 1.

INTRODUCTION AND LITERATURE REVIEW

lot-sizing and sch(-;iduling algoritlini in cellular manufacturing environment us­ ing a hybrid simulation/analytic ai.)]-)roach, whicli will be discussed in the next chapter.

C hapter 2

Problem Statem ent

2.1

In trod u ction

111 this chapter, the problem statement and the Hybrid Aiialytical/Simulation model are discussed in detail. In section 2.2, the interaction lietween lot-sizing and scheduling is discussed, also the importance of dynamism and the tradeolTs which are closely related to shop floor conditions are emphasized. In section 2.3, the problem characteristics are discussed, and the advantages of a Hybrid Simulation/Analytical approach are cited. Then in section 2.4. the assumptions for the hybrid model are given. Section 2.5 describes the details of the hybrid model and discusses the performance measures, dynamic nature of the model and how to utilize the shop floor information. In the last section, the pro]:iosed dynamic lot-sizing and scheduling algorithm is discussed in detail.

2 .2

D ynam ic L ot-Sizin g and Scheduling

Joint determination of lot sizes and schedules in a multi-product multi-period production system is a substantial problem. Production scheduling is con­ cerned with the optimal allocation of scarce resources to different jobs over

CHAPTER 2. PROBLEAd STATEMENT

21

time. In production scheduling, time performance is taken to be province of sequencing models, which take job ])rocessing times as given. However, in practice processing times can be controlled by choice of lot sizes, which has a major im])a.ct on inake-s])aii, flow times and other measui'es. Since lot­ sizing and sclieduling strongly affect the shop floor conditions, ])erformance of a production system can be im]>roved by updating lot sizes and schedule:·: dynamically.

As discussed in section 1.2.5, there is a strong relationship between lot­ sizing rules and scheduling rules. Among tho.se rules, no pair is universally superior. When shop floor conditions change over time, dilferent lot-sizing and scheduling combinations perform better than others. Using fixed lot sizes and schedules “as in the MRP systems” makes the system una.ble to react to changing conditions and ignores the interactions that might be exploited to increase the overall .system effectiveness.

In multi-item production systems, scheduling of facilities often centers around some constrained “bottleneck” machine, required to ])rocess dilTerent jobs. Plenert [20] discussed that “no generalized solution exists for the bottlenecks that allow multiple ])roducts to be processed”. The lot sizes for different jobs going thi'ough the bottleneck are determined generally by rules which han­ dle each job independently without completely exploring the interrelationships between jobs.

For a highly utilized (bottleneck) machine in a multi-item production sys­ tem, merging lots or increasing lot sizes decreases the number of setups on the bottleneck, so setup utilization decreases. When merging is concerned, the question of “which/how many jobs will be merged” has to be answered. This merging process clc:arly changes the original schedule since some jobs shift forward or backward by skipping some others. Continuously applying this merging ])rocess is a. sim])le dynamic lot-sizing policy, and naturally it vqKlatt'.s scheduling.

In a model which allows lot splitting, for a low utilized machine, s])litting the lots increases number of setups and setup utilization for the succeeding

CHAPTER 2. PROBLEM STATEMENT

22

machines. Sub-lots can be transferred to the next machine and processed while other items from the same job but of a different sul)-lot are jn'ocessed on the ciirrejit machine. By tliis way, waiting times are shortened and flow time of parts decrease. Though lot s])littiiig does not always lead rescheduling immediately, since the lot variety and the number of lots in tlie system increase, possibility^ of improving the schedule also increases.

In a multiple criteria dynamic lot-sizing and scheduling model, there can l)e conflicting ])erformance measures. Suppose two lots are nnu'ged, either one lot moves several steps forward or other moves several steps backward. So number of setups reduces, make-s])an and ilow times of lots ])receding the merged lots also reduce. However earliness and tardiness might increase for some lots due to new schedule. One cannot sim])ly say whether the system is better off or not without com])aring the new schedule with the previous one. Therefore, there are several tradeoffs which can be explored by considering the current sho]) floor conditions.

2.3

W h y S im u latio n /A n aly tic A p p roach

Shop iloor data like utilizations or queue lengths are results of a])plied lot-sizing and scheduling rules. Dynamicall}' adjusting lot sizes and schedules, however, use shop floor data as feedback from the system. It is the cyclical natme of this situation that makes the ])rohlem so difficult to solve.

In the literature there are two major approaches to solve this problem: either simultaneous!}' solve for both lot sizes and schedules, or a sequential approach which divides the problem into two parts. Dilts ami Ramsing [4j showed that using Mixed Integer Programming (MIP) approach to solve the dynamic lot-sizing and scheduling problem has limited applicability, because the number of binary varialdes in MIP is sufficiently large, and its com])lexity is

0(n^T) for n job-T period problem. They discussed that the number of binary

CHAPTER 2. PROBLEM STATEMENT

₂₃

even when the ca])acity is sufficient, the computational time is prohibitive. In the sequential approach, the lot-sizing problem is considered a planning decision, and it is assumed to be solved at at higher level than the scheduling problem. The scheduling suliproblem is viewed as a low level decision jn'oblem that should be solved after the lot-sizing problem. However, there is a signif­ icant interaction between lot-sizing and scheduling as discussed b}' Dilts and Ramsing [4]. The secpiential approach is a simplification of the problem, so it does not come up with satisfactoiy solutions.

.-\s we discussed in section 1.2.7, when analytical models cannot be applied to complex problems, hybrid simulation/analytic models can be a good solution methodology. The features of the system can be modeled b}· the. simulation model without oversimplification.

Hybrid simulation/analytic model can be successfully used for dynamic lot­ sizing and scheduling in a multi-item multi-period production system. Consider the cyclic nature of the problem, lot sizes and schedules are input to the sys­ tem, and shop floor data is used for dynamically adjusting the lot sizes and schedules. If the system is modeled as a simulation model, and a dynamic lot­ sizing and scheduling algorithm is developed, these two models can be utilized simultaneously in a simulation/analytic model. The lot sizes and schedules can be used as an input to the simulation model and shop floor information can be used as an input to the lot-sizing and scheduling algorithm.

The lot-sizing and scheduling algorithm works at the end of T length in­ tervals, and the system updates it.self dynamically. .Selection of T value is an important decision making problem. There is a tradeoff in choosing T, such as the rescheduling interval T must be small enough to insert dynamism in the system, but small T increases computational effort. Large T reducois the dynamism and might decrease efficiency of the overall system, since the model cannot react to sudden changes.

This dynamic lot-sizing and scheduling algorithm with the simulation model can be classified as a class 4 hybrid simulation/analytic model discussed in

CHA PTER 2. PROBLEM STATEMENT

₂₄

section 2.7 by Shanthikimiar [29]. The.se type of models can be used if some “portion” of the system can be .solved using an analytical model. Dynamic lot­ sizing and scheduling algorithm corresponds to this portion.The system itself however is too comj^lex to use any analytical method such as ciueiiing networks, so simulation is the aippropriate way of representing the S3’stem.

2 .4

A ssum ptions

In this study, a dynamic lot-sizing and scheduling algorithm is develo])ed foj· a (Jellular Manufacturing (CM) environment. Some of the CM characteristics which can be utilized to simplify the problem are as follows:

• Modified flow shop

• Only major setu])s incur between families. Minor setups between parts of a given famih· can be ignored and they are included in the proccvssing times.

In a modified flow shop, parts can enter the shop at one of several machines, can progre.ss through the shop by a. limited number of paths, and can e.xit the shop at one of several machines. There are no backtracking cind no cycles, this naturally decreases the system complexity, and scheduling alternatives, How­ ever in a jol) shop, backti'acking and cycles are allowed, and there are more alternative routings compared with CM. So in a job shop system, scheduling problem is inucli more difficult. Fox [5] showed that “in a job shop, 85 or­ ders moving through 8 o])erations without alternatives, with a single machine substitution for each and no machine idle time have possible' schedules”.

Behind the difficulties of scheduling in a job shoj), when a dynamic al­ gorithm is concerned, the schedule has to be revised in every T length time epochs. This is simply .solving the lot-sizing and scheduling problem sequen­ tially in every T length periods during the rnake-span. This needs a significant computational effort and time.

CHAPrER 2. PROBLEM STATEMENT

25

In a CM system with n jiarts and k families, where; v is much larger than A:, setups occur between k families, but in a job sho]) with n parts, setups occur between n parts. .So alternatives for lot merging and scheduling increase drastically, which makes the problem more complex. Solving the most general iorm of the dynamic lot-si^iing and scheduling problem in a job shop is very difficult. However, the above characteristics of the CM .systems can be utilized to simplify this prol)lem.

Another reason for using CM .system is that, it can be used in both ])ush and pull systems. However, the Kanban .system, which is the shop floor con­ trol module of the pull systems, can only be implemented in CM environment, and is not applicable to job shops. In job shops the material flow is coni])lex, there are bactrackings. cycles and the inventory control is difficult which is not suitable for .HT philosophy. .So the performance of the dynamic lot-sizing and scheduling algorithm for a CM system can be compared with the results of both push and pull systems using a simulation analysis. Finally as discussed in section 1.2.8, the classical scheduling assumptions such as precedence re­ lationships and noninterference constraints should also be considered in CM systems.

2.5

S im u latio n /A n aly tic M odel

2.5.1 Performance Measures

In scheduling literature, objective functions generallj' include either job com­ pletion ba..sed objective.' such as minimizing make-span or a.verage flow time, or due date related oltjectives such as minimizing tardiness, earliness or maxi­ mum lateness. Performance measures of the proposed dynamic, lot-sizing and scheduling algorithm are as tollows:

• Make-S])an

CHAPTER 2. PROBLEM STATEML^NT

26

• Earliness • Tardiness

Tl)e objective is to minimize a weighted sum of these four measures, so both classes of objectives are comlnned in this model. Flow time is directly ])ropor- tional to the Work-in Process inventory (WIP) in the system, and it is related with the inventory holding cost. Make-span is related with the processing time, waiting time and number of setups in the system. Saving a setup reduces the make-span by the same amount, since the processing times are assumed as con­ stant. High earliness increases the finished goods inventory holding cost, while high tardiness decreases the service level, and indicates the positive deviation from the due dates.

2.5.2 Dynamism

.As discussed in section 2.2, a dynamic algorithm must update the lot sizes and schedules simultaneously. Proposed dynamic lot-sizing and scheduling algorithm interacts with the simulation model every T length periods. At any time l.\. if the lot sizes and schedule are given to the simulation model, which mimics the real multi-item ])roduction system, the model contiimes to execute, and at time H -b T the new shop floor information is send to lot- sizing and scheduling algorithm (analytic model). Then the proj)osed analytical procedure calculates the new lot-sizes and schedules immediately, and updates the parameters in the simulation model by changing the queue structures and item lot sizes. Meanwhile the simulation time does not alter. Then simulation model executes between /.j A T and i \ + 27\ and shop floor information is send to algorithm again at i\ -b 2T. then u])dated lot sizes and schedules are send

back to simulation model and so on, as shown in the'figure 2.1.

In the hybrid simulation/analytic model, shop floor information includes the following items;

CHAPTER 2. PR0BLEA4 STATEMENT

₂₇

Shop Поог information SIMULATION MODLL Interacting ANALY'riC MODHL Updated lot <fc schciliilcs ТГ ТГ Time Horizon

Figure 2.1: HYBRID Simulation/Anal}Muc Model • Utilization of all machines

• 'Related delta of parts in queue such as lot size, due date, its position in the queue, etc.

These two items are explained in the next two sections in more detail.

2.5.3

Using Machine Information (Utilizations)

Utilization is the rate of time the machine is busy. It has two parts, run time utilization for ])rocessing parts and setup utilization for setups, vvhicli can be stated as follows:

Utilization = Run Time Utilization + Set-up Utilization

In any manufacturing system, including CM systems, utilizations of ma­ chines can vary a great deal due to loading schemes, and the simulation model which mimics the CM system can have both low and high machine utiliza­ tions. As discussed in literature review chapter, bottleneck machines govern both inventory and throughput. OPT approach discusses that “an hour lost

CHAPTER 2. PROBLEM STATEMENT

₂₈

on the bottleneck machine is an hour lost on the S3'stem”. So determining the bottlenecks is the first step. A threshold value of 90 % is used, therefore machines utilized above 90 % are considered as Bottleneck Machines (BM). In a single bottleneck case, saving a setup on the bottleneck is savirig a setu]) for overall system, which consequently decreases the make-span. In a multi l>ot- tleneck case, similarly saving a setup on a bottleneck reduces the make-span and (low time of parts, however trade olFs must be considered when the objec­ tives related with the due date performance such as tardiness and earliness are concerned.

In section 2.2, the importance of splitting lots was introduced, in OPT a])proach, Goldratt discussed that utilizing the idle time of the loose machines, speed u]) the flow and reduce inventories, for this purpose process lots should be in multiples of transfer lots. Consider the following scenario. If the lot size is 71, once a setuj7 is incurred, all the parts in the lot have to wait for the part to be processed. If the lot is split to two, the first ?;./2 j^arts do not need to wait for the rest of the lot, so flow time decreases for these parts. However, since the lot size is halved the machine incurs two setuj)s that increases the flow times of succeeding parts and also setup utilization. Therefore, the tradeoff is between the increase in the setup time and the idle time of the machine.

A machine utilized below a threshold value of 50 % is called Loose Machine (LM), and lot splitting is ])erformed on a LM to utilize the idle time. A machine other than a LM or BM is called a Non-Bottleneck Machine (NBM). NBM’s are not utilized in the proposed model, because neither lot merging and scheduling, nor lot s])litting can significantly improve the system performance with this set of machines, but will definitely increase the computational time.

It is important to indicate that machines can shift from LM set to NBM set, or from BM set to NBM set in any time due to changes in sho]) floor conditions and the demand pattern, and the reverse is also true. Lot splitting increases the number of setups on a loose machine, therefore increases the setu]) utilization. If the utilization goes beyond 50 %, then it is considered as a NBM and no iteration takes place, so after a time period the utilization might begin

CHAPTER 2. PROBLPJM STATEMENT

₂₉

to decrease and it Ix^coines a loose machine again when utiliziition goes beyond 50 %. A similar argument holds for a bottleneck machine. These three sets, which are bottleneck machines, non-bottleneck machines and loose machines dyna,mically change overtime.

2.5.4 Using Queue Information

For each j)art in a (|ueue, the following data is recorded;

• Family number • Part number • Lot size • Ready time • Due date

Its position in (|ueue Processing and set)i|) tinu;s

Lead time estimation is very important in multi-item production systems. becau.se sclieduling decisions are dependent on lead time estimations as well as performance measures such as tardiness and earliness. .Jobs In Queue (-IIQ) is a. dynamic, lead time estimation rule which ])erforms bettei- than many others as di.scu.ssed in section 1.2.7. .JIQ rule uses process times of succeeding ]mrts including setups as jol) information, and total number of items in cpieue, that is found from the routing of the jrart, as shop floor information. So when the sequence of a ])art changes, the new lead time can be estimated dynamically.

Due date and the current time are known when the shop floor information is ])assed to the analytical procedure. Slack time of a part, which is defined as “the necessary time to finish all processes in the system” is necessary to

CHAPTER 2. PROBLEM STATEMENT

30

T Interaction Period Length BM Bottleneck Mcichine NBM Non-Bottleneck Machine LM Ijoose Machine

R.S Resclieduling Set s v Scheduling Value

Average Flow Time

TS Total Setup Time EAR Earl i ness

d'AR d ardiiKNSS

dhNOVV Chirrent Simulation Time N(A) Normalized value of A

Table 2.1 Notations a.ud abbreviations used in al<>;orithmO

estimate the possible earliness and tardiness. Since lot sizes and schedules ci.re continuously updated, the lead time of the parts also change. Using .JIQ rule, the algorithm calculates the estimated slack time for all the items in queue, so ecudiiiess and tardiness values can be calculated dynamically at each iteration (T length interval).

The details of the dynamic lot-sizing and scheduling algorithm are described in the. ne.xt section, for simplicity the notation and abbreviations that will be used in this chapter are summarized in table 2.1.

2 .6

A lgorith m

2.6.1 Introduction

In section 2.5, performance measures, dynamism, using machine information and queue information were presented. In this section, the details of the algo­ rithm will be described and estimation of the related parameters that are used in algorithm will be discussed.

CHAPTER 2. PROBLEM STATEMENT

31

г г т

_{□ Z I 3}

Machine Rcschciluling Sci

cove IN _QUHUE

I'^igure 2.2: Resell eel uling Set

2.6.2 Merging

On boUleiK'ck machines, increasing lot sizes via merging lots decreases tlu' number of setiijis and setu|) utilizations. Decreasing number of setups might imjiiove the flow time and tardiness performances for some ])arts. However during merging ])rocess, some parts move forward or backward in the queue, and this might a.fh'ct due date performances, so there is a tradeolf. As discussed in OPT, bottlenecks govern the through])ut in the system. .So the merging rule should im.])rove the ]K:'.rformances of the bottlenecks without decreasing certain ]>erformance measures like earliness. Our approach is similar to using process lots in OPT, in OPT process lots are multiples of transfer lots to save from .setups and increase the throughj)ut.

When the shop floor information is passed to the algorithm, first the ma­ chine utilizations are analyzed, and the machines that are utilized above 90 % are identified as BMs. Merging is ap])lied only to BMs as discussed in section 2.5.3.

Rescheduling all the parts in the queue might be unnecessary, because al­ gorithm interacts with tin' simulation model every T length intervals and all the parts in the cpieiie might not be processed during T. .So it is reasonable to schedule some ])a.i t of tiu' (pieue, which is defined ii.s the rescheduling set (HS) as shown in figun' 2.2.

The delinition of RS can be given as follows: The setup times and procc'ssing times arc* summed beginning from the first part in the queue. Whenever the sum is equal or greater than 1.5T, the counted parts form the RS. A multiple gieater than 1 has to l.)e used for T, because some setups are eliminated in rescheduling and the final schedule has to cover the period T. If the current