MACHINE BREAKDOWNS IN DYNAMIC FLEXIBLE JOB SHOPS

WITH SEQUENCE-DEPENDENT SETUPS:

A BIO-INSPIRED APPROACH

by

Cory Jamar Weathers

A thesis submitted to the graduate faculty

in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

Department: Industrial and Systems Engineering

Major: Industrial and Systems Engineering

Major Professor: Dr. Bala Ram

North Carolina Agricultural and Technical State University

Greensboro, North Carolina

2006

ii

School of Graduate Studies

North Carolina Agricultural and Technical State University

This is to certify that the Master’s Thesis of

Cory Jamar Weathers

has met the thesis requirements of

North Carolina Agricultural and Technical State University

Greensboro, North Carolina

2006

_________________________________ _________________________________

Major Professor Committee Member

_________________________________ _________________________________

Committee Member Department Chairperson

_________________________________

Dean of Graduate Studies

iii

DEDICATION

To our God above, who enables, humbles, and strengthens me.

To my parents, whose love and support has always been and continues to be a source of

my motivation.

To Stevy, for more than she realizes.

To my grandmothers, whose caring, pride, and integrity guide me in all that I do.

iv

BIOGRAPHICAL SKETCH

Cory Jamar Weathers was born on February 24, 1982, in Baltimore, Maryland. He

received the Bachelor of Science degree in Integrated Science and Technology from

James Madison University in 2004. He has worked as an engineering/operations intern

for the U.S. Army Evaluation Center, Agwater Technologies, LLC., and General Mills,

Incorporated. He has been working as both a teaching and research assistant in the

Department of Industrial and Systems Engineering at North Carolina Agricultural and

Technical State University. His research interests include discrete-event simulation,

supply chain systems, and intelligent agent applications. He is a member of the Institute

of Industrial Engineers (IIE) and the National Society of Black Engineers (NSBE). He is

also a member of the Alpha Pi Mu and Phi Kappa Phi honor societies.

v

ACKNOWLEDGMENTS

I would like to express my sincere appreciation to my advisor, Dr. Bala Ram, for

his patience, guidance, and encouragement during my master’s program. He has been a

fine example of what an educator and advisor should be to his students. His commitment

to mentorship is something that I can only hope to try to emulate.

I would also like to thank Dr. Paul Stanfield and Dr. Xiaochun Jiang for serving

as my thesis committee members. They have shown their sincere interest in my success

as a student by challenging me and constantly encouraging me to look more closely at the

details.

I wish to thank the entire faculty and staff of the Industrial and Systems

Engineering Department at North Carolina Agricultural and Technical State University

for their overall support and encouragement.

I offer my gratitude to Dr. Anne Henricksen and Dr. Robert Koolvord for their

help with ProModel.

Lastly, I offer a very special thanks to Dr. Barbara Gabriel and Dr. Okechi

Egekwu for they have been dear friends, mentors, and further sources of inspiration since

my first days at JMU.

vi

TABLE OF CONTENTS

LIST OF FIGURES ......................................................................................................... viii

LIST OF TABLES............................................................................................................. ix

LIST OF SYMBOLS ......................................................................................................... xi

ABSTRACT..................................................................................................................... xiii

CHAPTER 1. INTRODUCTION .......................................................................................1

1.1. Job Shop Scheduling.................................................................................................1

1.2. Dynamic Job Shop Scheduling .................................................................................4

1.3. Problem Description .................................................................................................5

1.4. Statement of Research Objectives ............................................................................7

1.5. Organization of Thesis..............................................................................................8

CHAPTER 2. DYNAMIC JOB SHOP SCHEDULING WITH MACHINE

BREAKDOWNS – A REVIEW..................................................................9

2.1. Dynamic Job Shop Scheduling Subject to Machine Breakdowns ............................9

2.2. Multi-Agent System for Dynamic Job Shop Scheduling........................................13

2.3. The Response Threshold Model for Dynamic Scheduling – A Review.................18

2.4. Parameter Modeling for RTM-DS..........................................................................23

CHAPTER 3. INTELLIGENT MACHINE BREAKDOWN-HANDLING

STRATEGY...............................................................................................25

3.1. Description of Scheduling Environment.................................................................25

3.2. Dataset Generation..................................................................................................26

3.3. Machine Breakdown-Handling Strategy, RTM-DS-BD.........................................33

vii

CHAPTER 4. COMPUTATIONAL COMPARISON OF TWO SCHEDILING

MODELS ..................................................................................................40

4.1. Introduction to Auction-Based Scheduling Model ..................................................40

4.2. Comparison Methodology .......................................................................................42

4.3. Experimental Results ...............................................................................................47

4.3.1. Main Effect of MDL on NST...........................................................................47

4.3.2. Interaction Effects on NST ..............................................................................47

4.3.3. Results of Post Hoc Analysis for Remaining Metrics .....................................50

4.3.4. RTM-DS-BD Rerouting Policy Results...........................................................53

CHAPTER 5. SUMMARY, CONCLUSIONS, AND FUTURE WORK ........................54

5.1. Research Summary ..................................................................................................54

5.2. Research Contribution .............................................................................................60

5.3. Future Work .............................................................................................................61

REFERENCES ..................................................................................................................64

APPENDIX........................................................................................................................67

viii

LIST OF FIGURES

FIGURES PAGE

2.1. Multi-Agent Scheduling System Architecture...........................................................13

2.2. Machine Agent Architecture......................................................................................15

2.3. Multi-Agent Scheduling Paradigm ............................................................................17

3.1. Configuration of the FMS Model ..............................................................................26

3.2. (a) RTM-DS and RTM-DS-BD Normal Routing (without breakdowns), (b) RTM-

DS-BD Reactive Routing (with breakdowns)............................................................38

4.1. Negotiation Scheme of the Auction-Based Scheduling Model .................................41

4.2. Mean Number of Setups by Model............................................................................49

ix

LIST OF TABLES

TABLES PAGE

3.1. Machine Capability Matrix for 6 Machines and 4 Operation Types

(Mean number of alternative machines: 4) ................................................................27

3.2. Sequence of Operation Types in a Job Type .............................................................27

3.3. Part Type/Processing Time Data for Two Alternative Machines..............................28

3.4. Part Type/Processing Time Data for Three Alternative Machines ............................29

4.1. Summary of Values for Independent Variables.........................................................44

4.2. Simulation Results for Benchmark (Two alternative machines) ...............................45

4.3. Simulation Results for Benchmark (Three alternative machines) .............................46

4.4. Significant Interactions Yielded from 6-way ANOVA on NST................................48

4.5. MDL*NAM*SBL*MTTR Sliced by NAM*SBL*MTTR ........................................48

4.6. MDL*NAM*SUT*MTTR Sliced by NAM*SUT*MTTR........................................49

4.7. MDL*NAM*ULZ Sliced by NAM*ULZ .................................................................49

4.8. MDL*ULZ*MTTR Sliced by ULZ*MTTR..............................................................49

A1. ANOVA RESULTS: Number of Setups....................................................................68

A2. ANOVA RESULTS: Throughput..............................................................................69

A3. ANOVA RESULTS: Number of Late Jobs ...............................................................70

A4. ANOVA RESULTS: Average Lateness ....................................................................71

A5. ANOVA RESULTS: Average Tardiness...................................................................72

A6. ANOVA RESULTS: Average Due Date Deviation ..................................................73

x

A7. ANOVA RESULTS: Average Wait Time .................................................................74

A8. ANOVA RESULTS: Makespan ................................................................................75

xi

LIST OF SYMBOLS

iN Number of job types in the shop

jN Number of operation types in the shop

kN Number of machines in the shop

Ji Job type i

Oj Operation type j

Mk Machine k

MTBF Mean time between machine failures

MTTR Mean time to repair a broken machine

Iijk Response intention of machine agent k to operation type j in job i

Si(t) Intensity of the stimulus associated with job i

Wk Sum of the processing time for all operations in the input buffer of

machine k

θjk Response threshold of machine agent k to operation type j

Pijk Mean processing time of machine k on operation type j in job i

θjk Response threshold of machine agent k to operation type j

θmin Minimal response threshold of machine agents

θmax Maximal response threshold of machine agents

α Learning factor of machine agents

Qij Probability of the success of wasp i in confronting wasp j

Fk Force of machine agent k

xii

η A positive parameter used to determine ijQ

SBL Shop breakdown level

xiii

ABSTRACT

Weathers, Cory J. MACHINE BREAKDOWNS IN DYNAMIC FLEXIBLE JOB

SHOPS WITH SEQUENCE-DEPENDENT SETUPS: A BIO-INSPIRED APPROACH.

(Major Advisor: Bala Ram), North Carolina Agricultural and Technical State

University.

Flexible job shops are characterized by versatility and robustness in terms of how

dynamically arriving jobs requiring various operations in various sequences are

successfully routed from entry to completion on the shop floor. Efficient scheduling of

operations in these types of systems requires methods that are distributed and adaptable.

Research and practice suggest that dynamic job shop scheduling, when achieved through

the coordination of multiple software agents, offers a feasible solution. This research

study builds upon previous research conducted at North Carolina Agricultural and

Technical State University that examined a bio-inspired multi-agent scheduling

mechanism termed the Response Threshold Method for Dynamic Scheduling (RTM-DS).

An adapted version, termed the Response Threshold Model for Dynamic Scheduling

Subject to Breakdowns (RTM-DS-BD), is evaluated and compared to the performance of

a multi-agent scheduling system coordinated using a contract-net protocol. The results

show that RTM-DS-BD significantly outperforms the contract-net method, particularly

when applied to scheduling in job shops subject to high machine setup times and high

levels of machine breakdowns.

1

CHAPTER 1

INTRODUCTION

This chapter contains an introduction to job shop scheduling. The chapter also

introduces the concept of multi-agent systems as applied to scheduling. In particular, two

multi-agent coordination strategies are presented. Following these introductions, the

research problem addressed in this thesis is defined.

1.1 Job Shop Scheduling

In scheduling, resources are allocated to a preset number of competing tasks over

a specified period of time (Du & Pinedo, 1995). Of the estimated 40,000 metal part

manufacturing facilities in the United States, most operate as “job shop” environments.

The term “job shop” refers to manufacturing environments where the flow of production

materials can be different for each of a number of different products. Much work in the

field of operations research has been devoted to optimizing the performance of job shop

scheduling (JSS). This is largely due to the considerable cost reductions that may be

achieved in doing so and the complexity of the problem.

Three major issues of concern in JSS are as follows:

1. Combinatorial explosiveness: Many JSS problems are NP-hard or NP-complete.

In solving an NP-hard problem, the time required for calculation of an optimal

solution increases exponentially with the size of the problem (Van Dyke Parunak,

1991). For example, a problem containing n jobs and m machines may have (n!)m

2

solution possibilities. It is impractical to use optimal solution-generating

algorithms to solve realistic scheduling problems due to the processing time that it

would require to determine such solutions.

2. Multi-criteria performance measures: Many of the metrics used to compare

performance between scheduling methodologies (machine utilization, average

queue length, average tardiness, etc.) are co-related, and in some cases,

competing. In such cases, the difficulty of optimization is undoubtedly increased.

3. Scheduling exceptions: Realistic scheduling environments always contain some

level of unpredictability. For example, machines may break down, rush orders

may arrive, orders may be cancelled, and production priorities may change.

These exceptions may cause a predictive schedule generated for use in a static

production environment to become invalid. Effective scheduling methods must,

therefore, recover from dynamically occurring exceptions in a timely, efficient

manner.

In the past fifteen years, multi-agent system (MAS) approaches to manufacturing

scheduling and job floor control have gained much attention. Distributed architectures,

autonomous decision-making, and self-configuration all characterize MAS. These

characteristics enable MAS to perform real-time planning and scheduling with abundant

robustness and flexibility.

Coordination among intelligent agents allows MAS to achieve responsiveness,

robustness, and flexibility. Specification of the guidelines by which this coordination

occurs in MAS has been a major challenge in agent-based scheduling research. A good

3

coordination policy can lead to efficient global system performance and also effectively

adapt to changing shop conditions (Cicirello & Smith, 2003).

One coordination strategy is the auction-based method. The auction-based

method has been applied to a number of scheduling scenarios and has proven to be

particularly effective in solving dynamic resource allocation problems. However, two

major drawbacks can be seen when there are a large number of agents in such systems.

First, the bidding process in an auction-based model requires that messages be sent back

and forth between agents for one or more rounds. If the system is large, excessive

communication occurs and the decision-making process is slowed drastically. Second, in

the auction-based method agents compete against each other for the right to perform a

given task. This competition often causes scheduling conflicts and other chaotic activity

within such systems.

Another MAS coordination method, termed the response threshold model

(RTM) for dynamic scheduling, has gained increased attention (Campos et al., 2000). The

model is applied to a single-stage flow shop with machines that operate in parallel. The

RTM borrows principles from the division of labor that is seen in social insect colonies.

Agents in the RTM are coordinated using a method of self-organization which differs

greatly from that of the auction-based method. A similar approach based on RTM,

termed the RTM-DS, performs very well on several metrics for scheduling in dynamic

flexible job shops with sequence-dependent setups (Yu, 2005).

4

1.2 Dynamic Job Shop Scheduling

Most modern manufacturing environments are considered dynamic job shops,

which consist of several machines and jobs arriving continuously (Holthaus, 1999). Each

of these jobs usually consists of a set of operations that must be performed by the

machines in a specified order. Determination of the order in which jobs are completed is

typically achieved with respect to some given scheduling objective(s). Scheduling

objectives may be to decrease mean and maximum flow time, to decrease variance of

flow time, to minimize mean and maximum tardiness, to decrease percentage, and so

forth.

Additionally, job arrival times, routings, and processing times remain unknown

until jobs actually arrive on the shop floor. Therefore, only the jobs that have arrived in

the shop can be considered for scheduling. It is here where comparisons between

principles of real time control and dynamic job shop scheduling (DJSS) can be drawn.

DJSS does not create production schedules (Vieira, Herrmann, & Lin, 2003).

Instead, predictive scheduling creates schedules. DJSS describes a method whereby jobs

are dispatched when necessary and information available at the time of dispatching is

used for scheduling. It is, therefore, necessary that DJSS systems be both robust and able

to schedule in real-time. Unfortunately, most DJSS systems that have been developed

have lacked this robustness and responsiveness due to their centralized nature.

5

1.3 Problem Description

As production environments evolve toward mass customization with diverse

product lines and short production lead times, the need for flexibility and robustness to

exceptions in scheduling systems has increased dramatically. The historical job shop was

one where job routes were fixed and known prior to production scheduling. However,

with the use of multi-purpose, flexible machining centers in contemporary job shops,

solutions to scheduling problems must consider multiple job routing possibilities. Such a

job shop is commonly referred to as a flexible job shop. Such job shops are characterized

by the fact that they allow for an individual operation to be processed on a number of

alternative machines. A dynamic scheduling problem for a flexible job shop subject to

exceptions, in the form of machine breakdowns, is studied in this research.

Manufacturing literature usually describes scheduling in terms of two binary

categories: static or dynamic, and deterministic or stochastic. The accepted definition of

each of these categories often depends on the intent of a given study. For the purposes of

this research, the terms are defined as follows.

� “Static” refers to environments in which jobs to be processed arrive all at once,

prior to the start of operations.

� “Dynamic” refers to environments in which jobs arrive individually, over the

course of an operating period(s).

� “Deterministic” refers to environments in which there is a generally high degree

of certainty among system data. The exact values of job processing times, setup

times, and due dates are known prior to the start of operations.

6

� “Stochastic” refers to environments where the abovementioned system data

values are not known until after the events occur. However, descriptive

probability distributions and relevant parameters may be known beforehand.

This research addresses the scheduling problem as it appears in a dynamic environment

that can be either deterministic or stochastic. The scheduling problem is summarized as

follows:

� The job shop contains a set of n types of jobs {J1, J2, …, Jn} and m machines.

� Each type of job requires w operations {O1, O2, …, Ow}, each of which may be

done on a set of alternative machines.

� Each machine is capable of processing multiple operations.

� The order of required operations for each job is fixed and is known a priori.

� A machine can process, at most, one operation at a time.

� Jobs arrive at the shop floor (scheduling environment) in a dynamic fashion and

job details (operation types, operation orders, number of required operations, etc.)

remain unknown until arrival.

� The time between machine breakdowns is stochastic.

� Machine repair times for machines are stochastic.

� Flow-time and tardiness-based objectives are considered for measuring

scheduling system performance.

7

1.4 Statement of Research Objectives

The objective of this research is to develop and evaluate a multi-agent scheduling

system for dynamic flexible job shops with sequence-dependent setups subject to

machine breakdowns. The system is expected to be capable of providing a good balance

of solution quality, efficiency and robustness. This research is based on a similar effort by

Yu (2005) and is composed of the following efforts:

(a) Study promising approaches reported in the literature to deal with manufacturing

scheduling when breakdowns occur.

(b) Develop a promising approach for machine breakdown recovery as applied to

dynamic flexible job shops, in the context of the RTM-DS approach of Yu (2005).

(c) Enhance the problem set generating approach used by Yu (2005) to include

machine breakdowns.

(d) Integrate the approach from (b) into a discrete-event simulation model adapted

from Yu (2005) and Siwamogsatham and Saygin (2004).

(e) Integrate the problem sets generated in (c) into discrete-event simulation models

adapted from Yu (2005) and Siwamogsatham and Saygin (2004).

(f) Design a set of computational experiments to compare the performance of the

models in (e) on a set of metrics similar to those used by Yu (2005), on the

problems generated in (c).

(g) Report on the results of the computational experiments in (f).

8

1.5 Organization of Thesis

The remainder of this thesis is organized as follows. Chapter 2 consists of a

review of manufacturing scheduling in the presence of machine breakdowns. It also

contains an introduction to the application of multi-agent systems to DJSS and a review

of the RTM-DS model. A discussion of the method that was designed for the intelligent

handling of machine breakdowns within the RTM-DS context is also included. Chapter 3

details the job shop scheduling test bed that was designed for this research. Chapter 4

provides a computational comparison of the performance of the RTM-DS multi-agent

coordination strategy to the performance of an auction-based coordination strategy, when

both models are used for scheduling in the test bed discussed in Chapter 3. The design of

experiments and experimental results are also discussed in Chapter 4. Lastly, Chapter 5

provides the conclusions that can be drawn from this research and details the research

contributions made by this work. In addition, future relevant research directions are

suggested.

9

CHAPTER 2

DYNAMIC JOB SHOP SCHEDULING WITH MACHINE

BREAKDOWNS – A REVIEW

In manufacturing, exceptions can be thought of as differences between the actual

and the expected state of the production system (Bruccoleri, Amico & Perrone, 2003).

These exceptions may come in the forms of resource failures, changes in job priorities,

dynamic introduction of new jobs, order cancellations, increases in job arrival rates,

changes in part mixes, and/or reworks due to quality issues. Though often ignored in the

scheduling literature, recovery from exceptions must be accounted for in any work whose

results are to be applied to real-world production scheduling. Machine breakdowns are

among the most common exception types. Results from past research efforts using

simulation as a modeling and analysis tool are discussed below. Research efforts offering

interesting strategies for improving the performance of job shops subject to machine

breakdowns are also discussed. Also included are discussions concerning multi-agent

systems applied to DJSS, in general, as well as MAS coordination via the RTM-DS

model, in particular.

2.1 Dynamic Job Shop Scheduling Subject to Machine Breakdowns

Simulation is able to mimic system behavior to the user’s desired level of detail.

It has, therefore, been used extensively to study the scheduling of dynamic job shops.

Many studies on dynamic job shop scheduling use simulation models to evaluate their

10

approaches (Yu, 2005). Abdin (1986) studied a flexible manufacturing system (FMS)

with machine breakdowns. In this work, scheduling was accomplished through

simulation using SLAM II simulation language. System performance was measured in

terms of resource utilization, mean flow time, total production, work-in-progress, and

makespan. This paper is one of the earliest simulation-based efforts aimed at optimizing

system performance in environments with exceptions. Dutta (1990) investigated

uncertainties such as machine failures, dynamic introduction of new jobs, and dynamic

increase in job priority in simulated FMS job shop environments. A production control

mechanism that monitors the shop floor for exceptions and takes corrective action based

on a knowledge-based heuristic strategy was proposed. Experimental results show that

simple and generic design strategies for the knowledge base can provide the basis for

effective and robust control behavior.

Wu and Wysk (1988, 1989) studied an FMS scheduling problem with dynamic

job arrivals and no disturbances. A multi-pass scheduling algorithm combined with a

partial simulation window strategy was periodically triggered to minimize the mean flow

time and the mean tardiness. Current shop status information was used to simulate

alternative dispatching rules and the best rule was chosen for scheduling. Experimental

results indicated that the use of a multi-pass approach yielded better results than the use

of a single rule for the entire horizon.

Perry and Uzsoy (1993) modeled a semiconductor testing facility as a dynamic

job shop with sequence-dependent setup. They combined a decomposition approach for

a static problem with an event-driven rescheduling (EDR) technique to handle

11

dynamically occurring system events. Matsuura et al. (1993) studied a similar dynamic

job shop problem. Full new schedules were generated to minimize the makespan in

response to disturbances including machine breakdowns and rush jobs. Simulation was

used to test the rules of first-come-first-served (FCFS) and shortest process time (SPT),

and then to switch between these two rules. Bengu (1994) constructed a simulation-

based scheduler for a dynamic flow-line environment. Machine breakdowns were

investigated as the reason for invoking local rescheduling decisions. Apparent tardiness

cost (ATC) was used as the scheduling rule in order to minimize mean weighted

tardiness.

Kim and Kim (1994) studied an FMS with dynamic job arrivals. A real-time,

simulation-based scheduling methodology was used to address several objectives. Full

new schedules were generated in response to machine breakdowns and rush jobs. The

system evaluated various dispatching rules and selected the best one for a specified

criterion. Holthaus (1999) presented simulation-based analysis of dispatching rules for

scheduling in dynamic job shops with interruptions on the shop floor. Flowtime and due

date-based objectives were considered. The relative performance of several dispatching

rules was evaluated. The results of the study revealed that the relative performance of

scheduling rules was affected by changing the levels of breakdown parameters. With

respect to due date-based objectives, the performance of the analyzed scheduling rules

were shown to be sensitive to the percentage of time the machines had failures and the

mean time to repair (MTTR) broken machines.

12

Brennan and O (2000) also presented a simulation test bed to evaluate a multi-

agent manufacturing system. The holonic manufacturing concept was applied to the shop

floor scheduling and control system. Arena discrete event simulation software was used

to simulate the system. The COM/DCOM (Component Object Model/Distributed

Component Object Model) approach was selected to implement agents. The impact of

dynamic job routing and machine breakdowns was investigated to analyze the model’s

performance. Kutanoglu and Sabuncuoglu (2001) tested several schedule recovery

policies for dynamic job shops subject to machine breakdowns. Their results showed that

rerouting newly arriving jobs to alternative machines should be the preferred strategy

when machines are broken often and machine repair times are short. The authors found

that “no reaction” is not an appropriate strategy for reactive scheduling. Interestingly, the

authors suggested, as an example of a likely feasible reactive policy, a strategy where the

decision as to whether job rerouting would be beneficial was made selectively and

dynamically for each directly affected job. This suggestion fits well with the strategy for

reacting to machine breakdowns within the context of the RTM-DS proposed in Chapter

3 of this thesis.

Chen and Chen (2003) studied dynamic scheduling problems in random flexible

manufacturing systems that had jobs with varying part types, alternative routing, random

part arrival time, and machine breakdowns. The authors proposed an adaptive scheduling

approach to make decisions about part/machine scheduling and operation/tool

assignments on a rolling horizon basis, taking machine availability into account.

Experimental results indicated that the proposed scheduling mechanism significantly

13

outperformed dispatching heuristics under different shop load levels and machine

downtime levels. The mechanism’s relative performance improved further when there

were more frequent disruptions, such as machine breakdowns and random job arrivals.

However, the authors suggested that frequent rescheduling to react to disruptions can

reduce system predictability, and hence reduce system effectiveness.

2.2 Multi-Agent System for Dynamic Job Shop Scheduling

In this research a multi-agent system-based approach is designed to address the

scheduling of dynamic flexible job shops subject to machine breakdowns. The approach

comprises two basic components: the multi-agent system architecture and coordination

among agents. The scheduling system is structured using a decentralized heterarchical

multi-agent architecture as shown in Figure 2.1 (Yu, 2005).

Figure 2.1. Multi-Agent Scheduling System Architecture

14

The system consists of multiple machine agents (MA) and one facilitator agent

(FA). An MA represents each machine in the system. There is no resource related to the

FA.

Each MA has three functions:

• Monitoring information for its machine: queue length (in the form of processing

time or number of operations), machine status (busy, idle, or breakdown), and the

type of operation being processed.

• Foraging jobs for its associated machine: When new jobs are available for

processing, the MA decides whether to offer to process them or not.

• Using appropriate dispatching rules to sequence jobs in the queue of its associated

machine.

The FA has three functions:

• Broadcasting incoming jobs that need routing

• Routing the jobs to a machine queue according to a specific rule (e.g., the lowest

threshold value).

• Deciding the winner of contests when more than one MA is competing for the

same job.

All MAs in the system possess the same architecture. This architecture, which is

adapted from Yu (2005), consists of six generic features (Figure 2.2): a mental model,

learning ability, sensors, effectors, a knowledge base, and an inference engine. The

functions of each feature are presented below:

15

Figure 2.2. Machine Agent Architecture

• The sensor perceives changes to the status of its associated machine and receives

stimuli from jobs and the type of operation to be routed.

• The effector allows an MA to act on the environment.

• The knowledge base contains knowledge about the rules to be used in sequencing

jobs in the queue, and the algorithms used to compute the parameters of the

coordination model.

• Reinforcement learning allows an MA to update its mental model.

Reinforcement Learning

- learning coefficient

- forgetting coefficient

- threshold update, (etc.)

- new threshold

Knowledge Base

- threshold rules

- response algorithms

- force algorithms

- sequencing rules, (etc.)

Mental Model

- objectives

- beliefs

- capabilities

- behavior rules

Inference Engine

- response decision

- queue sequencing

Sensor Effector

Shop Floor Environment

Threshold Update, Algorithms,

Rules

New

Threshold

Machine Status, Queue Status, Stimuli, Operation Type,

Threshold

Response Algorithms,

Sequence Rules, (etc.)

Stimuli, Machine Status, Queue Status, Operation

16

• The inference engine uses both the knowledge base and the mental model to make

a decision by looking for beliefs that match specified rule conditions.

• The mental model is similar in structure to the one described in Acronymics, Inc.

(2004). It contains each MA’s beliefs, capabilities, behavior rules, and intentions.

• Beliefs are a fundamental part of the MA’s mental model. Beliefs represent the

current information of the machine’s queue size, status, current operation type,

time to finish current operation, theshold for an operation, and so forth. Beliefs

are updated as the machine’s status changes.

• Capabilities specify the types of operations that an MA can perform, and the

corresponding processing times.

• Behavioral rules determine the actions an agent takes throughout its execution.

Behavioral rules compare the set of possible responses with the current

environment as described by an MA’s current beliefs. If a rule’s conditions are

satisfied by the environment, then the rule is applicable and the actions it specifies

are performed.

• Intentions specify the objectives to be achieved such as minimizing total weighted

tardiness, maximizing throughput, or achieving both simultaneously.

The scheduling of dynamic flexible job shops with sequence-dependent setups is

addressed using two rules in this study: the machine selection rule (MSR) and the queue

sequencing rule (QSR), which are shown in Figure 2.3 (Yu, 2005).

MSR is accomplished through coordination among all of the MAs. When an MA

perceives that an incoming job (either a new job or an existing job) is waiting for routing,

17

it looks at the type of operation to be processed. If the MA can process the operation, it

decides whether or not to offer to perform it using a threshold-based model. In the

selection logic, if more than one MA wants to process the same job, the winner is decided

using the method that is introduced in Chapter 3. When no MA picks the job, the FA is

activated and it routes the job to a machine queue.

QSR is implemented by every MA. An MA monitors the status of its associated

machine. Whenever the machine completes an operation and its queue is not empty, the

MA selects the next operation from the queue using some queue sequencing rule.

Sequencing rules are stored in the knowledge base of each MA.

Figure 2.3. Multi-Agent Scheduling Paradigm

Incoming Operations

MA n MA 2 MA 1

Machine 1

A

••• •••

••• •••

A

C

Machine 2

D

B

B

Machine n

E

G

G

A B

QSR QSR QSR

MSR

18

The proposed scheduling system structure has all of the advantages of a multi-

agent system introduced in Chapter 1:

• All MAs concurrently process information, thereby increasing computational

speed.

• MAs do not communicate directly with each other, thereby reducing

communication bandwidth requirements.

• MAs are independent of each other. Therefore, a failure of one MA does not

necessarily affect the other MAs. As such, the system is not relatively insensitive

to machine breakdowns and is more robust.

• Each MA is autonomous and reactive. The integration of sensing, processing, and

effecting features into each MA improves the responsiveness of the system.

• The loose coupling of MAs makes it very easy to add a new MA to a system and

to remove an existing MA from a system; this means that the system is scalable.

2.3 The Response Threshold Model for Dynamic Scheduling – A Review

Division of labor has been found to be a governing mechanism in the daily actions

of social insects such as ants, bees, and wasps. It allows many different tasks to be

performed simultaneously by many different individuals within social insect societies.

Bonabeau et al. (1997) pointed out that in these societies, the proportions of workers

performing the different tasks can vary in response to internal perturbations or external

challenges, and that the robustness achieved through insect workers' behavioral flexibility

increases the efficiency of their society.

19

Theraulaz et al. (1991) presented a self-organization model for task differentiation

based on wasp colony behavior. In this study, the authors modeled a colony’s self-

organized allocation of tasks using response thresholds. A form of this model – adapted

by Yu (2005) and referred to as the Response Threshold Model for Dynamic Scheduling

(RTM-DS) – is introduced below. The model consists of (i) response intention, which

determines how likely a wasp is to forage for a particular nest zone, (ii) a reinforcement

learning mechanism, which tracks wasp foraging tendencies and results in task

specialization, and (iii) dominance, which is a model parameter used to determine the

outcome of ties among two or more wasps possessing the same response threshold

against foraging for a particular nest zone.

Response Intention:

Wasps feed larva in a nest according to zones. Each wasp maintains a “response

threshold” for each zone of the nest. This response threshold is a wasp’s resistance

against foraging for a particular zone. Broods located in zones emit pheromone-based

stimuli representing the level of feeding demand for each brood. The zones that wasps

forage for are determined by combinations of their thresholds for particular zones and the

levels of stimuli being emitted from those zones. Lower threshold values lead to higher

likelihood of wasps foraging particular zones, while higher stimuli values also lead to

higher likelihood of foraging for those zones. The converse of this is also true; high

threshold values and low stimulus values lead to lower likelihoods of a wasp foraging a

particular zone. The likelihood of a wasp foraging for a particular zone is given by

20

Equation (2.1) (Yu, 2005). In the scenario described below, each wasp is analogous to a

machine agent, while each operation of a job is analogous to a feeding zone:

ijkjkki

iijk

pWtS

tSI

222

2

)(

)(

+⋅+=

θ (2.1)

Iijk is the response intention of agent (wasp) k to operation type j in job i, and Si(t)2

denotes the intensity of stimulus associated with job i. Wk is the sum of the processing

time for all operations in the queue of machine k. θjk is the response threshold of

machine agent k to operation type j. Pijk is the mean processing time of machine k on

operation type j in job i. Therefore, individual wasps with a higher Iijk are more likely to

respond at a lower level of stimulus.

Response intention indicates how strong the “will” of a machine is to process an

incoming operation. Its value depends on a machine’s total set of conditions including its

capability (expressed by pijk in Equation (2.1)), its availability (expressed by Wk in

Equation (2.1)), and its specialization (expressed by θjk in Equation (2.1)) (Yu, 2005).

Machine-Centered Reinforcement Learning:

Every time a machine k finishes a job type j, it decreases its threshold for job type

j by a learning step α:

θkj = θkj - α (θmin < θkj < θmax) (2.2)

At the same time, machine i increases its thresholds for other job types it can

process by α:

θks = θks + α (θmin < θkj < θmax, s ≠ j) (2.3)

21

This learning strategy updates thresholds from a machine-centric viewpoint. A

machine updates its thresholds only after it completes a job, and all thresholds associated

with the machine are updated simultaneously. According to Equation (2.1), it can be

inferred that the shorter the processing time of operation type j on machine k, the higher

the probability that operation type j is routed to machine k, and that machine k decreases

its threshold to operation type j more frequently. This results in machine k becoming

specialized in the operation type(s) that take the shortest processing time on machine k.

Dominance Contests:

In addition to the response threshold, machines (wasps) maintain another

parameter termed “force”. Force represents the ability of an individual wasp to dominate

in a contest with any of its nest mates (other machines on a shop floor). When two

individual wasps engage in a contest, the winner is chosen stochastically. The probability

of the success of wasp i in confronting wasp j is determined by Equation (2.4) (Bonabeau

et al., 1999)

)Fη(F ji1

1−−

+=

eQij (2.4)

where Fi is wasp i’s force, Fj is wasp j’s force, and η is a positive parameter used to

determine the outcome.

The model of the division of labor via response threshold provides an adaptive

approach for coordination in multi-agent systems. Campos et al. (2000) developed an

RTM-based algorithm for dynamic flow shop scheduling with parallel machines and

single-stage jobs. This problem, originally presented by Morley and Ekberg (1998),

22

consists of assigning trucks to painting booths in a truck-painting facility where the

objective is minimization of both the total makespan and the number of paint flushings.

In the approach of Campos et al. (2000), a global demand, given by the sum of the

priorities of the unassigned trucks in each particular color, is established for each color.

A booth maintains a threshold for each paint color. A derivation of Equation (2.1) is used

to determine painting assignments. Color demands and booth thresholds are updated

following each assignment. Ties are broken using heuristics. The model is benchmarked

against Morley and Ekberg’s (1998) market-based approach. The best combinations of

values for the parameters of both approaches are determined using a genetic algorithm.

Cicirello and Smith (2001) presented an effort on the application of RTM in a study

which developed wasp-like agents used to solve a scheduling problem with parallel

machines and multiple single-stage job types. In their approach, each machine in the

system has an associated agent termed a routing wasp. Each routing wasp is in charge of

assigning jobs to the queue of its associated machine. Each routing wasp has a set of

response thresholds for the job types that its machine can process. The study explored

different force representations and different dominance contest strategies. The wasp-like

model is shown to be self-adaptive to changes in product demand levels. The authors’

wasp-based strategy is applied to the truck-painting problem and benchmarked against

Morley’s market-based strategy.

A more recent effort by Kittithreerapronchai and Anderson (2003) focused on the

work presented by Campos and Bonabeau (2000). Discrete-event simulation was used to

investigate two different threshold reinforcement schemes: local update and global

23

update. Experimental results indicate that the global update strategy yields faster

specialization than the local update strategy.

The three efforts described above and the one by Yu (2005) are the only

contributions to the application of the RTM model to dynamic job shop scheduling found

in the literature. All of these efforts focus on the same parallel flow shop problem with

the exception of Yu (2005), where the author demonstrates how an RTM-inspired model

can be applied to a multi-machine job-shop.

2.4 Parameter Modeling for RTM-DS

Yu (2005) suggested that parameter setting is vital to the performance of the

RTM-DS model. The author found it useful and appropriate to choose model parameter

values (S0, α, and θmax) according to scheduling problem parameters, such as number of

machines, average processing time, average setup time, and so forth. Through

experimentation using simulation optimization, the author was able to determine good,

problem-dependent formulas to be used in setting the three parameters.

Yu (2005) performed the simulation optimization experiments in Promodel using

Simrunner. Simrunner is a decision support tool that utilizes evolutionary algorithms in

helping a user to locate optimal solutions to simulation model parameters. The objective

of this simulation effort was to determine good settings for the three parameters (α, S0, &

θmax) that would minimize the number of setups incurred during simulation for the

problem described in later sections. This research relies on the results of Yu’s effort for

24

setting of the values of the three parameters, as applied to scheduling a job shop subject

to breakdowns.

In summary, the above review of the theory of scheduling dynamic job shops

subject to breakdowns is a direct motivation for the research study presented in this

chapter. Multi-agent systems and the RTM-DS approach, which are the key ideas used in

this research, are also introduced. Though bio-inspired techniques are attractive because

of the robustness and flexibility that they may offer, few research efforts reported in the

literature address the performance of such techniques in dynamic job shops subject to

machine failures. Additionally, there have been no reports addressing the application of

any RTM-inspired coordination methods on dynamic job shops subject to machine

failures. Issues on the application of RTM to such job shops, and a strategy for agents to

deal with unexpected events, have not been addressed in the literature. This research

addressed the scheduling of dynamic job shops and the findings are presented in the

following chapters.

25

CHAPTER 3

INTELLIGENT MACHINE

BREAKDOWN-HANDLING STRATEGY

This chapter discusses the scheduling problem for which the RTM-DS

coordination strategy was designed. The problem is characterized by a dynamic job shop

with sequence-dependent setups where all machines are subject to stochastic usage-based

failures. The techniques used to generate representative problem sets and an intelligent

strategy for handling machine breakdowns in scheduling are presented. The intelligent

strategy is based on the RTM-DS approach of Yu (2005). The modified approach is

termed the Response Threshold Model for Dynamic Scheduling Subject to Breakdowns

(RTM-DS-BD).

3.1 Description of Scheduling Environment

The flexible manufacturing system, to which the RTM-DS model is applied

(shown in Figure 3.1), consists of six machines. Each machine has a dedicated input

buffer (Yu, 2005). There is a central output buffer which stores work-in-process parts.

The product mix is composed of six different part types, each of which has an equal

probability of arriving at the shop floor. Processing times are deterministic,

transportation times are assumed to be negligible and part inter-arrival rates are

exponentially distributed. A sequence-dependent setup is required between any two

differing consecutive operations on the same machine.

26

Figure 3.1. Configuration of the FMS Model

3.2 Dataset Generation

Datasets which represent a typical job shop are generated for the purpose of

evaluating the effectiveness of the RTM-DS model. The datasets for the six types of jobs

are created via the following three steps:

1. Machine Capability Matrix Generation

The machine capability matrix consists of “0”s and “1”s. A “1” means that the

machine is able to process the operation type, and the “0” means that the machine is

unable to process the operation type. As an example, Table 3.1 gives the machine

capability matrix for a “six machines and four operations” problem. The “1” in each row

is generated by a U[2,6] distribution, which means that there are four alternative

machines for every operation type.

Loading

Machine 1 Machine 2 Machine 3

Unloading

Machine 4 Machine 5 Machine 6

I I I

I I I

Central Buffer

I Machine input buffer

27

Table 3.1. Machine Capability Matrix for 6 Machines and 4 Operation Types. (Mean

number of alternative machines: 4) Operation Type M1 M2 M3 M4 M5 M6

1 1 0 1 0 1 1

2 0 1 1 1 0 0

3 0 1 0 1 0 1

4 1 1 1 0 1 1

2. Generation of Operation Sequences

The sequence of operations for each job type was generated randomly using a

method where each possible sequence had an equal probability occurring. The six job

types each required random generation of sequences of operation types. One example of

such a sequence is shown in Table 3.2.

Table 3.2. Sequence of Operation Types in a Job Type Step Number Operation Type

1 2

2 3

3 1

4 4

3. Assignment of Operation Processing Times

The machine capability matrix is used to generate a corresponding job processing

time matrix. Processing times are generated using a U[4,8] distribution. A macro

implemented via VBA in Microsoft Excel is used to generate the machine capability

matrix and job processing times, simultaneously. Tables 3.3 and 3.4, presented by Yu

(2005) are generated in this manner.

28

Table 3.3. Part Type/Processing Time Data for Two Alternative Machines

Part Type Operation # Operation Type 1 2 3 4 5 6

1 3 4 - 3 - - -

2 1 - - - - 4 3

3 2 2 - - 3 - -

4 4 - 2 - - 1 -

A

5 5 6 - - 5 - -

1 1 - - - - 6 4

2 3 10 - 11 - - -

3 2 2 - - 2 - - B

4 4 - 9 - - 8 -

1 5 4 - - 5 - -

2 1 - - - - 4 5

3 2 4 - - 6 - -

4 4 - 6 - - 6 -

C

5 3 5 - 6 - - -

1 3 5 - 8 - - -

2 4 - 4 - - 3 -

3 2 6 - - 7 - - D

4 1 - - - - 7 11

1 1 - - - - 13 12

2 4 - 8 - - 6 -

3 5 8 - - 10 - -

4 2 6 - - 5 - -

E

5 3 2 - 5 - - -

1 6 - 7 - - - 7

2 3 4 - 2 - - -

3 1 - - - - 7 8

4 5 9 - - 11 - -

5 2 3 - - 4 - -

F

6 4 - 5 - - 6 -

Note: Cell entries under the alternative machines column denote the processing time (min) and those

marked with ‘-’ imply the inability of the machine to process the specified operation.

29

Table 3.4. Part Type/Processing Time Data for Three Alternative Machines

Part Type Operation # Operation Type 1 2 3 4 5 6

1 3 4 - 3 - 5 -

2 1 - 6 - - 4 3

3 2 2 - 4 3 - -

4 4 - 2 - - 1 2

A

5 5 6 - - 5 - 6

1 1 - 4 - - 6 4

2 3 10 - 11 - 10 -

3 2 2 - 1 2 - - B

4 4 - 9 - - 8 10

1 5 4 - - 5 - 2

2 1 - 6 - - 4 5

3 2 4 - 4 6 - -

4 4 - 6 - - 6 6

C

5 3 5 - 6 - 6 -

1 3 5 - 8 - 7 -

2 4 - 4 - - 3 4

3 2 6 - 6 7 - - D

4 1 - 9 - - 7 11

1 1 - 12 - - 13 12

2 4 - 8 - - 6 3

3 5 8 - - 10 - 8

4 2 6 - 6 5 - -

E

5 3 2 - 5 - 4 -

1 6 7 - 6 - - 7

2 3 4 - 2 - 4 -

3 1 - 5 - - 7 8

4 5 9 - - 11 - 10

5 2 3 - 4 4 - -

F

6 4 - 5 - - 6 6

Note: Cell entries under the alternative machines column denote the processing time (min) and those

marked with ‘-’ imply the inability of the machine to process the specified operation.

30

4. Inter-Arrival Time Assignment

According to job shop scheduling literature, the inter-arrival time of jobs is best

represented by an exponential distribution (Ramasesh, 1990). With average machine

processing times established, the average job arrival rate must be selected so that the

machine utilization is less than 100%. Otherwise, the number of jobs in the input buffers

in front of the machines will grow without bound. Equation (3.1) is used to generate the

mean inter-arrival time of each job (Rangsaritratsamee, Ferrell & Kurz, 2004).

k

gp

UNv

µµ

λ==

1 (3.1)

where

v = Mean inter-arrival time,

U = Shop utilization,

λ = Mean job arrival rate,

pµ = Mean processing time per operation (including setup),

gµ = Mean number of operations per job,

kN = Number of machines in the shop.

Yu (2005) suggests that the value of pµ is difficult to determine a priori. Because

the problem involves a sequence-dependent setup, different machine selection rules and

input buffer sequencing rules may lead to a different mean number of setups, and

therefore different pµ . Further, even under the same machine selection rule, the

possibility of alternate machines for the various operations complicates the computation

31

of pµ , especially since the job route cannot be known in advance (Yu, 2005). For this

research, results from experiments by Yu (2005) are used to set values for pµ used in

computation of v.

5. Due-Date Assignment

Due date is assigned using the Total Work (TWK) rule (Kunnathur, Ahmed &

Charles, 1996); (Holthaus, 1999); (Rangsaritratsamee, Ferrell & Kurz, 2004). The TWK

rule is as follows:

∑+=jk

ijkii PKRd (3.2)

where

id = Due date of job i ,

iR = Arrival time of job i .

ijkP = Mean processing time of machine k on operation type j in job i.

K = Tightness factor that reflects the amount of expected delay a job will experience.

The TWK rule states that the due date of a job equals the sum of the job arrival

time and a multiple of the total job processing time. Based on the parameters, pµ and

gµ , Equation (3.2) can be restated as follows:

gpii KRd µµ+= (3.3)

32

The parameter K is generated at simulation run-time via a uniform distribution with a

range of 2 to 6. A K value of “2” represents a loose scheduling environment while a K

value of 6 represents a tight scheduling environment.

6. Machine Breakdown Parameters

In order to evaluate the performance of the RTM-DS model in a job shop subject

to machine breakdowns, it is necessary to enhance the problem sets used by Yu (2005) to

include parameters for machine breakdowns. The method used to generate these

parameters is presented below.

Mean Time Between Failures, Mean Time to Repair, and Shop Breakdown Level

In the job shop scheduling literature, machine breakdown parameters typically

include mean time between failures (MTBF) and mean time to repair (MTTR). MTBF

refers to the average time between successive machine failures, while MTTR is defined

as the average time required for corrective maintenance of a given set of machines upon

failure. It is important to note that in this research, MTBF is dependent upon actual

machine usage. For example, the mere availability of a machine does not necessarily

affect its frequency of failure. Instead, frequency of failure is dependent upon the time

that machines spend actually operating. It is believed that this view is appropriate for

production environments containing devices such as flexible machines, which are subject

to mechanical wear. Both MTTR and MTBF are often used to make reliability

predictions and to calculate the typical availability of a production system.

In this research, shop breakdown level (SBL) is used for the average machine

availability and reflects the probability that machines are inoperable during a given

33

scheduling period. Based on values used for MTTR and SBL in the literature, MTTR is

chosen as an integer multiple of p , the average processing time, and SBL is chosen to be

between 0 and 1. MTTR and MTBF occur according to exponential and lognormal

distributions, respectively. Equation (3.4) from Holthaus (1999) shows how MTBF can

be computed from MTTR and SBL.

MTTRSBL

MTTRMTBF −= (3.4)

3.3 Machine Breakdown-Handling Strategy, RTM-DS-BD

The following discussion explains the proposed method for handling machine

breakdowns via RTM-DS-BD. The method draws inspiration from Kutanoglu and

Sabuncuoglu (2001) where the authors suggested a reactive strategy that selectively and

dynamically decided if rerouting upon machine breakdown would be beneficial for each

directly affected job. In RTM-DS-BD, machine agents representing broken machines

continue to participate in the intention update and reinforcement learning processes. The

fact that MAs of broken and non-broken machines continue to participate in these

processes is particularly beneficial to the RTM-DS-BD scheduling system. Continued

participation helps to minimize the degradation of MAb intention and MAb specialization.

Such degradation would occur if MAb was not allowed to participate in the update

processes while non-broken MAs continued participation. Because one of the major

strengths of the RTM-DS coordination strategy is that it enables reduction of machine

setups via MA specialization, it was necessary to design RTM-DS-BD such that it reacts

34

to machine breakdowns in an intelligent manner by utilizing the specialization

information that is accumulated by MAs prior to breakdown. Additionally, since

operation completion times are of importance in scheduling, it is useful to consider the

effects of machine breakdowns on completion times for interrupted operations. The

completion times for interrupted operations are affected by a number of factors such as

the amount of work (in time units) currently in a machine’s input buffer, the remaining

time for any current operations and repair times.

The notation below is used to describe the breakdown handling strategy:

b = index of broken machine

*j = index of the operation interrupted by breakdown

*jA = set of alternative machines for operation *j

bt = clock time of breakdown for machine b

r

bt = repair time for the broken machine b

nowt = current time

s

kt = setup time on machine k before next operation

c

jkt = estimated completion time of operation j on machine k

ijkp = processing time for operation j of job i on machine k

m

ijkp = remaining processing time (following breakdown) for operation j of job i on

machine k

35

a

jkt = when breakdown occurs, the next closest clock time when machine k will be

ready to process operation j (For a broken machine, a

jbt = bt + r

bt ; for a good

machine idle at time t = bt , a

jkt = bt ; for a good machine busy at time t = bt ,

a

jkt = nowt + m

ijkp )

For example, assume that operation j* is pre-empted when machine b breaks

down and that since the operation type does not change, machine b does not require a

setup to process the interrupted operation (operation j*) after machine b is repaired.

When a breakdown occurs, MAb will notify the facilitator agent (FA) of the breakdown.

FA will then broadcast information regarding operation j* to all MAs. MAs will use this

information to compute their intention to complete operation j*, considering setup, time

required to complete their current operation (if busy), and the prorated time to complete

the remainder of operation j*. The computation of the earliest completion time for j* on

a good machine k is expressed as:

kij

bij

m

bijs

k

a

k

c

kjp

p

pttt *

*

*

* ⋅++= ( *jAk ∈ , bk ≠ ) (3.5)

The computation of the estimated earliest completion time for *j on the broken machine

b is expressed as:

m

bij

r

bb

c

bjpttt ** ++= (3.6)

Consequently, the earliest completion time for *j on either a good machine or a broken

machine is affected by current shop conditions which, in turn, help determine the

machine input buffer to which *j will be routed.

36

Figure 3.2 illustrates the differences between the RTM-DS (without breakdowns)

and the RTM-DS-BD. Figure 3.2 (a) shows the job routing strategy used by both RTM-

DS and RTM-DS-BD when a job operation goes through a machine with an interruption

due to a breakdown. Upon arrival at the shop floor, the job is sent immediately to the

central buffer where the FA uses the MSR to route the job to a selected machine input

buffer. After waiting in the buffer, the job is processed by the selected machine for its

first of a given number of required operations. This process is repeated until the job

receives all of its required processing. After job processing is complete, the job departs

from the shop floor via the output buffer.

Figure 3.2 (b) shows the job routing strategy used by RTM-DS-BD when a job

operation is interrupted by a machine breakdown. Upon arrival to the shop floor, the job

is sent to the central buffer where the FA uses MSR for a routing decision. Once routed

to a machine’s input buffer, the job will wait until either it reaches the front of the queue,

– after which the part would proceed to receive its first of a series of operations – or the

machine is interrupted by a breakdown. Upon breakdown, the job is sent from the

machine input buffer to the central buffer to be rerouted to either a non-broken machine

or a broken machine. Whether a job is routed to a broken or non-broken machine

depends on the intention values that each alternative machine’s respective machine agent

currently calculates for the current operation of the rerouted job.

For jobs that have experienced a fraction of an operation prior to a machine

breakdown, RTM-DS allows for immediate routing from the current (broken) machine to

the central buffer for rerouting to an alternative machine. Since the part has received a

37

fraction of its current operation at the time of breakdown, it requires only the remaining

operation processing time. A following section on rerouting jobs from the input buffers

of broken machines describes this process in greater detail.

When an operation is uninterrupted, both RTM-DS and RTM-DS-BD allow for

routing decisions to be made dynamically, based on current shop floor conditions, as seen

in Figure 3.2 (a). However, unlike RTM-DS, RTM-DS-BD allows for such decisions to

be made in response to machine breakdowns.

Rerouting of Jobs Interrupted by Machine Breakdowns

Within the context of the RTM-DS, when machine b breaks down, both the job

currently being processed (JI) and any jobs in the input buffer of that machine (JQ) are

affected. In order to minimize the negative effects on the system due to the breakdown,

these jobs need to be rerouted. Simultaneously, the agent for the broken machine (MAb)

needs to adjust its mental model (Figure 2.2).

Rerouting of Jobs from Input Buffers of Broken Machines

Upon machine breakdown, operations affected by breakdown (j* and JQs) are

sent to the central buffer and rerouted to machines selected in accordance with the RTM-

DS-BD strategy. Both jobs affected by breakdown and newly arriving jobs may be

routed to MAb, since MAb continues to participate in the threshold update process during

breakdown. MAb keeps all of its specialization information and updates its threshold

using the original learning rules. Reconsider Equation (3.1), which was originally used to

compute response intention for all MAs in Yu (2005) when the job shop was not subject

38

to machine breakdowns. Since the job shop in this research is subject to machine

breakdowns, MAb will use a modified version of this equation to calculate its intention.

Figure 3.2. (a) RTM-DS and RTM-DS-BD Normal Routing (without breakdowns),

(b) RTM-DS-BD Reactive Routing (with breakdowns)

Operation:

- Sequence-dependent Setup

- Processing Time

(see Tables 3.3 & 3.4; pijk)

- Reinforcement Learning

(see Equations (2.2) & (2.3))

Machine Input Buffer:

- QSR (see Figure 2.3)

Central Buffer:

- MSR (see Figure 2.3)

- Intentions Evaluated

(see Equation (2.1))

Job Arrival

All Operations

Complete?

Job Departure

N

Y

Operation:

- Sequence-dependent Setup

- Processing Time

(see Tables 3.3 & 3.4; pmijk)

- Reinforcement Learning

(see Equations (2.2) & (2.3))

Machine Input Buffer:

- QSR (see Figure 2.3)

Central Buffer:

- MSR (see Figure 2.3)

- Intentions Evaluated

(see Equation (3.12))

Job Arrival

Was Operation

Interrupted?

Job Departure

N

Y

All Operations

Complete?

Modify Process

Time:

-Update pijk-(see

-Equation (2.1)

-to pmijk- (see

-Equation (3.9))

Was Selected

Machine Interrupted?

Y

Y

N

(a) (b)

N

Operation:

- Sequence-dependent Setup

- Processing Time

(see Tables 3.3 & 3.4; pijk)

- Reinforcement Learning

(see Equations (2.2) & (2.3))

Machine Input Buffer:

- QSR (see Figure 2.3)

Central Buffer:

- MSR (see Figure 2.3)

- Intentions Evaluated

(see Equation (2.1))

Job Arrival

All Operations

Complete?

Job Departure

N

Y

Operation:

- Sequence-dependent Setup

- Processing Time

(see Tables 3.3 & 3.4; pijk)

- Reinforcement Learning

(see Equations (2.2) & (2.3))

Machine Input Buffer:

- QSR (see Figure 2.3)

Central Buffer:

- MSR (see Figure 2.3)

- Intentions Evaluated

(see Equation (2.1))

Job Arrival

All Operations

Complete?

Job Departure

N

Y

Operation:

- Sequence-dependent Setup

- Processing Time

(see Tables 3.3 & 3.4; pmijk)

- Reinforcement Learning

(see Equations (2.2) & (2.3))

Machine Input Buffer:

- QSR (see Figure 2.3)

Central Buffer:

- MSR (see Figure 2.3)

- Intentions Evaluated

(see Equation (3.12))

Job Arrival

Was Operation

Interrupted?

Job Departure

N

Y

All Operations

Complete?

Modify Process

Time:

-Update pijk-(see

-Equation (2.1)

-to pmijk- (see

-Equation (3.9))

Was Selected

Machine Interrupted?

Y

Y

N

Operation:

- Sequence-dependent Setup

- Processing Time

(see Tables 3.3 & 3.4; pmijk)

- Reinforcement Learning

(see Equations (2.2) & (2.3))

Machine Input Buffer:

- QSR (see Figure 2.3)

Central Buffer:

- MSR (see Figure 2.3)

- Intentions Evaluated

(see Equation (3.12))

Job Arrival

Was Operation

Interrupted?

Job Departure

N

Y

All Operations

Complete?

Modify Process

Time:

-Update pijk-(see

-Equation (2.1)

-to pmijk- (see

-Equation (3.9))

Was Selected

Machine Interrupted?

Y

Y

N

(a) (b)

N

39

In particular, MAb will use a revised processing time in calculating its response intention.

This calculation is explained with the following example:

Suppose that at time nowt , j* or any other operation j is ready to be routed and that

MAb is capable of performing the operation type. Upon receiving the information on the

operation via FA, MAb reads its mental model and updates the processing time from

ijbp to *

ijbp (in order to account for the time required for repair of MAb) in the following

manner:

)(* r

bbnowijbijb tttpp −−+= (3.7)

where

ijbp = the original processing time of operation j of job i on machine b.

MAb computes its new response intention using Equation (3.8):

2*22

2

)()(

)(

ijbjbbi

iijb

pWtS

tSI

+⋅+=

θ (3.8)

When machine b is fixed, MAb reverts back to using ijbp for updating its intention to

perform operations.

40

CHAPTER 4

COMPUTATIONAL COMPARISON OF

TWO SCHEDULING MODELS

This chapter benchmarks the RTM-DS-BD against a previously reported auction-

based approach to scheduling for dynamic flexible job shops. The computational

comparison aims to measure the efficiency of the RTM-DS-BD methodology presented

in Chapter 3, and to evaluate the performance of the RTM-DS-BD when compared to the

auction-based approach. Various metrics are used to evaluate both models when applied

to a scheduling problem in which machines are subject to breakdowns.

4.1 Introduction to Auction-Based Scheduling Model

MacChiaroli and Riemma (2002) reported a market-like model for dynamic

operations scheduling. Intelligent agents in the model developed a schedule based on an

iterative bidding process. Siwamogsatham and Saygin (2004) evaluated the same

auction-based model (ABM) by comparing the performance of the ABM with the

performance of various dispatching rules. The negotiation methodology used in the

ABM is shown in Figure 4.1 (Siwamogsatham & Saygin, 2004).

In ABM, parts are represented by part agents and machines are represented by

resource agents. These agents negotiate via the popular contract-net protocol. Upon

completion of a task, resource agents announce their availability to all part agents.

Interested part agents make proposals to purchase service from available and eligible

41

Figure 4.1. Negotiation Scheme of the Auction-Based Scheduling Model

resources, according to part process plans.

The resource agents receive proposals and construct offers taking into account all

of the proposals that they have received and the level of service that they can offer.

Service level is rated in terms of the expected completion time for a requested operation.

The primary goal of a resource agent is to increase its reward level, which comprises the

value of the proposals that it receives. As the number of proposals that a resource agent

receives increases, so does the value of its offers. Also, the value of a proposal increases

Wait for

Announcement

Renegotiation

Proposal

Construction

Task Offer

Evaluation

Task

Commitment

Availability

Announcement

Proposals

Evaluation and

Task Offer

Submission

Task Offer

Selection

Task Offer

(Acceptance or

Refusal)

Part Agent Resource Agent

42

as a job becomes more critical in terms of its completion time. More critical parts must

offer higher rewards in order to obtain service from the resource.

Part agents evaluate offers from resource agents and select the resource that can

offer the earliest completion time. If parts and resources do not reach an agreement in the

first step, an iterative re-negotiation process begins. In this process, proposals are

increased and offers are reduced up to a predetermined limit until a predetermined

number of iterations have occurred.

4.2 Comparison Methodology

Both models are implemented in a ProModel simulation model to solve the FMS

scheduling problem presented by Siwamogsatham and Saygin (2004). The FMS

configuration is shown in Figure 3.1 in the previous chapter. A similar scenario was used

by Caprihan and Wadhwa (1997) to investigate the impact of routing flexibility on the

performance of an FMS.

Model performance is measured in terms of eight performance metrics: makepan,

average tardiness, average lateness, average due date deviation, average throughput,

average wait time, total number of late jobs and total number of setups. These same eight

metrics were used in Yu (2005) for computational comparison. The definitions of these

metrics are listed below:

• Makespan – simulation clock time upon completion of the last part in the system

• Tardiness – max {0, completion time - due date}

• Lateness – completion time - due date

43

• Due date deviation – |completion time - due date|

• Throughput – production rate (i.e., number of parts produced per unit of time)

• Wait time – the time in system for a part minus its total processing time

• Late job – any job that is completed after its due date

• Setups – number of setup is incremented whenever a machine performs two

consecutive operations of a different type

A 26 factorial design is used for this experiment. Six two-level independent

variables and eight dependent variables are utilized. Eight performance metrics

(makespan, tardiness, lateness, due date deviation, throughput, wait time, number of late

jobs, and number of setups) have been selected as dependent variables. The six

independent variables include Model (MDL), Number of Alternative Machines (NAM),

Setup Time (SUT), Shop Utilization (ULZ), mean time to repair (MTTR), and shop

breakdown level (SBL). MDL has two levels: RTM-DS-BD and auction-based model

(ABM). NAM, SUT and ULZ each have two levels: low and high. The eight dependent

variables (performance metrics) are the same as those used in Yu (2005). The first four

independent variables and their values are the same as those used in Yu (2005). MTTR

and SBL are adapted from Holthaus (1999) and are also designed with two levels, low

and high. Two alternative machines are considered as low NAM and three alternative

machines are considered as high NAM. A setup time of 4.5 minutes is considered as low

SUT while 19 minutes is considered as high SUT. Shop utilization of 0.7 is considered a

low ULZ and 0.9 is considered a high ULZ. The values for MTTR and SBL were chosen

in accordance with the experiments performed by Holthaus (1999), where an MTTR of

44

5 p was considered low while an MTTR of 10 p was considered high. SBLs of 10% and

2.5% were considered high and low, respectively. Table 4.1 summarizes the values of

each level for all of the independent variables.

Table 4.1. Summary of Values for Independent Variables

Independent

Variable: NAM SUT ULZ MTTR SBL

low value 2 4.5 70% 5 p 2.5%

high value 3 19 90% 10 p 10%

There were a total of sixty-four treatment combinations. Each treatment

combination included one thousand job arrivals and was replicated ten times for a total of

six-hundred and forty trials. The results of the six-hundred and forty trials are

summarized in Tables 4.2 and 4.3. The last column in both tables shows the percentage

of jobs that were rerouted to alternative machines following machine breakdowns. Since

the main objective of this experiment was to compare the RTM-DS-BD model with the

ABM model, the data from the six-hundred and forty replications was analyzed using

ANOVA to determine if there was a significant difference between the two scheduling

models on any or all of the eight performance metrics. For each dependent variable, the

test hypotheses were:

H0: there is no significant difference between RTM-DS-BD and ABM.

H1: RTM-DS-BD significantly outperforms ABM.

45

Table 4.2. Simulation Results for Benchmark (Two alternative machines)

Model

MDL NAM SUT ULZ MTTR SBL TRP NLJ AVL AVT ADD AWT NST MKS RE-ROUTE

%

rtm-ds-bd low low low high high 6.61 317.41 56.36 113.37 170.38 190.73 447.47 9385.59 42.27%

abm low low low high high 6.02 176.24 -24.45 37.67 99.73 183.26 998.85 9382.04

rtm-ds-bd low low low high low 4.84 44 -91.68 4.19 100.06 65.16 477.68 11943.57 17.94%

abm low low low high low 4.87 25.87 -94.54 1.92 98.38 62.82 1068.36 11817.87

rtm-ds-bd low low low low high 6.64 465.7 140.65 183.62 226.58 289.13 250.74 9543.41 11.23%

abm low low low low high 5.48 353.01 229.61 268.46 307.32 917.81 1145.82 11817.74

rtm-ds-bd low low low low low 4.82 174.5 -55.89 15.41 86.71 94.92 329.82 11927.27 9.15%

abm low low low low low 4.85 63.44 -77.57 7.72 93.01 80.24 1043.47 11929.09

rtm-ds-bd low low high high high 6.65 221.6 28.21 94.38 160.56 160.02 620.75 9464.87 43.89%

abm low low high high high 6.02 168.41 -27.85 36.74 101.33 171.58 1014.92 9553.52

rtm-ds-bd low low high high low 5.41 41.3 -93.57 4.71 102.99 62.48 632.17 11768.31 30.61%

abm low low high high low 4.81 26.35 -95.86 1.83 99.53 64.39 1064.55 11894.26

rtm-ds-bd low low high low high 6.62 254.13 -24.5 36.12 96.73 119.25 290.24 9195.08 38.34%

abm low low high low high 5.46 364.13 284.49 322.78 361.06 1065.18 1160.35 12234.11

rtm-ds-bd low low high low low 4.81 102.43 -75.17 8.92 93.01 77.65 387.72 11977.52 28.05%

abm low low high low low 4.88 63.81 -79.03 7.55 94.12 89.45 1060.31 12102.43

rtm-ds-bd low high low high high 3.01 102.42 -87.57 5.05 97.66 68.01 405.74 22202.25 14.23%

abm low high low high high 2.42 276.98 145.03 184.84 224.65 338.94 1028.45 22692.58

rtm-ds-bd low high low high low 2.43 28.83 -104.58 1.71 106.78 54.05 388.16 27918.62 16.35%

abm low high low high low 1.81 105.55 -50.25 15.82 81.85 111.32 1060.05 28663.76

rtm-ds-bd low high low low high 3.02 161.14 -67.41 11.05 89.51 83.96 355.73 21905.23 9.66%

abm low high low low high 2.42 286.06 186.71 225.95 265.18 667.3 1046.47 24630.06

rtm-ds-bd low high low low low 2.44 73.62 -93.73 3.08 99.86 61.72 363.74 28123.76 10.56%

abm low high low low low 1.81 160.23 -14.45 42.07 98.59 140.92 1055.35 28523.21

rtm-ds-bd low high high high high 3.6 207.32 261.77 326.19 389.5 903.14 984.79 20549.21 28.31%

abm low high high high high 2.42 278.92 120.91 161.56 202.22 403.42 1042.69 23062.89

rtm-ds-bd low high high high low 3.62 210.04 268.61 332.48 395.22 919.49 994.45 20519.91 32.26%

abm low high high high low 1.86 120.39 -41.18 22.83 86.83 119.49 1064.01 28709.46

rtm-ds-bd low high high low high 3.68 213.61 276.03 339.11 401.03 936.77 1004.75 20374.81 22.17%

abm low high high low high 2.42 336.76 383.59 416.79 450.21 896.72 1050.76 24677.46

rtm-ds-bd low high high low low 2.49 61.72 -94.89 3.09 101.06 60.14 379.92 28341.58 22.95%

abm low high high low low 1.8 162.29 -9.76 45.87 101.51 149.43 1054.53 28784.89

Problem Parameters Performance Metrics

NAM - Number of Alternative Machines, SUT - Setup Time (minute), ULZ - Shop Utilization, MDL - Model Type, MTTR – Mean

Time to Repair Broken Machines, SBL – Shop Breakdown Level, TRP - Throughput Rate (parts produced per hour), NLJ - Number of

Late Jobs, AVL -Average lateness (minute) AVT - Average Tardiness (minute), ADD - Average due date deviation (minute), AWT - Average Wait Time (minute), NST - Number of Setups, MKS - makespan (minute), RE-ROUTE % - percentage of job rerouted to

alternative machines upon breakdown

46

Table 4.3. Simulation Results for Benchmark (Three alternative machines)

Model

MDL NAM SUT ULZ MTTR SBL TRP NLJ AVL AVT ADD AWT NST MKS RE-ROUTE

%

rtm-ds-bd high low low high high 6.21 284.1 30.69 95.27 159.85 183.74 333.32 10378.31 17.59%

abm high low low high high 4.24 481.62 871.47 890.27 909.06 2621.03 1642.84 15652.89

rtm-ds-bd high low low high low 5.41 310.32 42.32 100.63 158.95 190.78 312.43 11072.84 21.75%

abm high low low high low 4.8 189.78 -11.59 42.01 95.62 167.05 1384.29 12015.81

Problem Parameters Performance Metrics

rtm-ds-bd high low low low high 6.21 199.24 -6.32 62.57 131.44 135.65 499.2 10514.99 16.14%

abm high low low low high 4.26 501.23 1357.18 1372.78 1388.38 3963.73 1688.05 18408.75

rtm-ds-bd high low low low low 5.47 118.21 -44.02 40.15 124.32 99.68 625.42 10726.87 13.18%

abm high low low low low 4.82 370.47 177.91 208.28 238.66 655.21 1463.07 13781.98

rtm-ds-bd high low high high high 6.36 158.28 -54.47 23.11 100.69 95.45 432.43 10325.41 43.76%

abm high low high high high 4.82 459.95 780.37 802.89 825.41 2549.19 1638.35 15645.92

rtm-ds-bd high low high high low 5.41 168.16 -54.15 20.11 94.35 95.14 351.71 10781.03 39.74%

abm high low high high low 4.87 187.65 12.26 69.74 127.22 275.32 1385.42 12197.62

rtm-ds-bd high low high low high 3.62 102.41 -82.01 6.61 95.22 71.26 395.32 18160.43 32.09%

abm high low high low high 4.21 483.04 1307.83 1327.47 1347.12 3803.21 1672.27 18143.29

rtm-ds-bd high low high low low 2.43 58.53 -98.87 2.58 103.97 58.63 387.46 25681.51 29.43%

abm high low high low low 4.27 390.34 338.09 370.05 402.86 985.05 1473.65 14286.86

rtm-ds-bd high high low high high 2.45 99.24 -85.12 5.93 96.99 69.95 366.71 24297.51 21.92%

abm high high low high high 1.84 460.93 1711.51 1725.15 1738.79 4448.81 1573.26 33033.78

rtm-ds-bd high high low high low 2.48 116.32 -80.74 7.39 95.52 72.54 359.43 25548.71 23.66%

abm high high low high low 1.87 334.01 248.73 273.62 298.51 493.37 1382.38 29494.51

rtm-ds-bd high high low low high 2.42 77.77 -87.77 4.73 97.24 68.08 451.92 24503.84 17.83%

abm high high low low high 1.83 467.21 2335.11 2348.57 2362.03 6168.53 1584.34 36570.86

rtm-ds-bd high high low low low 2.41 42.97 -97.22 2.74 102.71 60.09 477.92 25522.82 15.32%

abm high high low low low 1.89 382.43 467.37 486.38 505.39 1026.51 1396.33 30954.96

rtm-ds-bd high high high high high 2.44 78.13 -88.69 5.49 99.67 66.88 421.94 24470.71 39.34%

abm high high high high high 1.8 461.74 1744.61 1758.92 1773.24 4625.58 1576.13 33342.07

rtm-ds-bd high high high high low 2.42 78.11 -90.78 4.71 100.21 64.55 355.91 25795.19 41.22%

abm high high high high low 1.81 345.56 221.4 245.37 269.35 456.03 1385.32 29351.31

rtm-ds-bd high high high low high 4.87 43.65 -94.97 2.92 100.81 60.27 452.62 16891.34 30.23%

abm high high high low high 1.85 469.97 2518.23 2531.85 2545.46 6708.51 1605.65 37894.36

rtm-ds-bd high high high low low 5.43 20.92 -103.25 1.92 107.1 54.58 483.34 10722.43 28.54%

abm high high high low low 1.83 392.71 584.39 603.39 622.49 1266.14 1421.85 32035.61 NAM - Number of Alternative Machines, SUT - Setup Time (minute), ULZ - Shop Utilization, MDL - Model Type, MTTR – Mean Time to Repair Broken Machines, SBL – Shop Breakdown Level, TRP - Throughput Rate (parts produced per hour), NLJ - Number of

Late Jobs, AVL -Average lateness (minute) AVT - Average Tardiness (minute), ADD - Average due date deviation (minute), AWT -

Average Wait Time (minute), NST - Number of Setups, MKS - makespan (minute), RE-ROUTE % - percentage of job rerouted to alternative machines upon breakdown

47

4.3 Experimental Results

A 6-way ANOVA was performed on each dependent variable (performance

metric). Since the procedure used to determine whether RTM-DS-BD outperforms ABM

for all eight metrics is the same, the analysis of only one dependent variable, NST, is

described here. This analysis serves as an example of the type of analysis that was

performed for the other seven performance metrics.

4.3.1 Main Effect of MDL on NST

Although ANOVA indicates that there is a significant main MDL effect on NST

(F(1, 32)= 95.28, p<0.0001), interaction between MDL and other factors was also found.

Hence, post hoc analysis was needed. The following section discusses the post hoc

analysis in detail.

4.3.2 Interaction Effects on NST

Table 4.4 shows the significant interactions found via ANOVA. There is neither

a 6-way nor a 5-way significant interaction between the independent variables for NST.

However, there are two 4-way and two 3-way significant interactions. They are among

MDL*NAM*SUT*MTTR (F(1, 32)= 3.94, p<0.05), MDL*NAM*MTTR*SBL (F(1,

32)= 7.98, p<0.05), MDL*NAM*ULZ (F(1, 32)=4.25, p<0.05), and MDL*ULZ*MTTR

(F(1, 32)=6.03, p<0.05), respectively. Since there are significant interactions, post hoc

analysis using slicing is used to determine whether there is a main MDL effect. Slicing

analyzes the impact of one factor (in this case, MDL) by fixing the values of each of the

other factors involved in the interaction.

48

Table 4.4. Significant Interactions Yielded from 6-way ANOVA on NST

Source DF Mean Square F Value Pr > F

MDL*NAM*MTTR*SBL 1 177606.3 7.98 0.0049

MDL*NAM*SUT*MTTR 1 87812.40 3.94 0.0475

MDL*NAM*ULZ 1 94567.80 4.25 0.0398

MDL*ULZ*MTTR 1 134354.80 6.03 0.0143

ERROR 576 22265.50

Slicing Operations for NST

Since MDL is the factor of concern, the following slicing is performed:

1. MDL*NAM*MTTR*SBL is sliced by NAM, SBL, and MTTR for NST

2. MDL*NAM*SUT*MTTR is sliced by NAM, SUT, and MTTR for NST

3. MDL*NAM*ULZ is sliced by NAM and ULZ for NST

4. MDL*ULZ*MTTR is sliced by ULZ and MTTR for NST

Tables 4.5 through 4.8 show, respectively, that NST for RTM-DS-BD (NST≈466) is

significantly lower than NST for ABM (NST≈1288), despite interaction. Figure 4.2

serves as an illustration of the difference, in terms of NST, between the two models.

Table 4.5. MDL*NAM*SBL*MTTR Sliced by NAM*SBL*MTTR NAM MTTR SBL DF F Value Pr > F

2 high high 1 916.02 <.0001

2 high low 1 448.55 <.0001

2 low high 1 1359.96 <.0001

2 low low 1 1002.48 <.0001

3 high high 1 252.01 <.0001

3 high low 1 291.79 <.0001

3 low high 1 553.75 <.0001

3 low low 1 425.26 <.0001

49

Table 4.6. MDL*NAM*SUT*MTTR Sliced by NAM*SUT*MTTR

NAM SUT MTTR DF F Value Pr > F

2 low high 1 600.04 <.0001

2 low low 1 1062.21 <.0001

2 high high 1 726.25 <.0001

2 high low 1 1292.26 <.0001

3 low high 1 331.4 <.0001

3 low low 1 422.01 <.0001

3 high high 1 217.63 <.0001

3 high low 1 557.47 <.0001

Table 4.7. MDL*NAM*ULZ Sliced by NAM*ULZ NAM ULZ DF F Value Pr > F

2 high 1 1510.38 <.0001

2 low 1 2113.99 <.0001

3 high 1 663.9 <.0001

3 low 1 827.08 <.0001

Table 4.8. MDL*ULZ*MTTR Sliced by ULZ*MTTR NAM ULZ DF F Value Pr > F

2 high 1 680.44 <.0001

2 low 1 1485.68 <.0001

3 high 1 1128.65 <.0001

3 low 1 1692.64 <.0001

2 Scheduling Models Comparison

(Number of Setups)

0

200

400

600

800

1000

1200

1400

ABM RTM-DS-BD

NST

Figure 4.2. Mean Number of Setups by Model

50

4.3.3 Results of Post Hoc Analysis for Remaining Metrics

For the other seven performance metrics, the same post hoc analysis with slicing

was performed. The following sections describe, considering the significant interactions

found for each metric, the conditions under which RTM-DS-BD does not significantly

outperform ABM. For each metric, RTM-DS-BD significantly outperformed ABM using

all combinations of the factors other than those listed as the exceptional conditions.

Post Hoc Results for MKS

RTM-DS-BD does not significantly outperform ABM (for the MKS metric) under

the following conditions:

1. When NAM and SUT are low and ULZ and MTTR are high

2. When NAM, SUT, and ULZ are low and MTTR is high

3. When NAM, SUT, and SBL are low

RTM-DS-BD significantly outperforms ABM under all other combinations of factors in

terms of MKS.

Post Hoc Results for TRP

RTM-DS-BD does not significantly outperform ABM (for the TRP metric) under

the following condition:

1. Where NAM, SUT, and SBL are low

RTM-DS-BD significantly outperforms ABM under all other combinations of factors in

terms of TRP.

51

Post Hoc Results for AWT

RTM-DS-BD does not significantly outperform ABM (for the AWT metric)

under the following conditions:

1. When NAM and SBL are low, and SUT and MTTR are high

2. When NAM, MTTR, and SBL are low and SUT is high

3. When NAM and SUT are low, and MTTR and SBL are high

4. When NAM, SUT, and SBL are low and MTTR is high

5. When NAM, SUT, SBL, and MTTR are all low

RTM-DS-BD significantly outperforms ABM under all other combinations of factors in

terms of AWT.

Post Hoc Results for ADD

RTM-DS-BD does not significantly outperform ABM (for the ADD metric) under

the following conditions:

1. When NAM and SBL are low and SUT is high

2. When NAM, SUT, and SBL are low

3. When MTTR and SBL are high and NAM is low

4. When NAM and SBL are low and MTTR is high

5. When NAM, SBL, and MTTR are low

RTM-DS-BD significantly outperforms ABM under all other combinations of factors in

terms of ADD.

52

Post Hoc Results for AVT

RTM-DS-BD does not significantly outperform ABM (for the AVT metric) under

the following conditions:

1. When NAM is low and MTTR is high

2. When NAM, SUT, and SBL are all low

RTM-DS-BD significantly outperforms ABM under all other combinations of factors in

terms of AVT.

Post Hoc Results for AVL

RTM-DS-BD does not significantly outperform ABM (for the AVL metric) under

the following conditions:

1. When NAM, SUT, and SBL are low.

RTM-DS-BD significantly outperforms ABM under all other combinations of factors in

terms of AVL.

Post Hoc Results for NLJ

RTM-DS-BD does not significantly outperform ABM (for the NLJ metric) under

the following conditions:

1. Where NAM and ULZ are low

2. Where NAM, SUT, and SBL are low and MTTR is high

3. Where NAM, SUT, AND MTTR are low and SBL is high

4. Where NAM, SUT, and SBL are low

RTM-DS-BD significantly outperforms ABM under all other combinations of factors in

terms of NLJ.

53

4.3.4 RTM-DS-BD Rerouting Policy Results

Experimental results show that an average of 25% of jobs that are affected by

breakdowns on their current machines are rerouted to alternative machines having higher

intentions to perform the operations required by the affected jobs. When the shop has a

higher NAM, rerouting to alternative machines with higher intentions for the required

operations occurs 27% of the time versus 24% of the time when the shop has a lower

number of alternative machines. Further, when both SBL and MTTR are high, rerouting

occurs roughly 40% of the time versus roughly 20% of the time when either or both SBL

and MTTR are low.

54

CHAPTER 5

SUMMARY, CONCLUSIONS, AND FUTURE WORK

Contemporary manufacturing systems are highly dynamic due to various internal

and external uncertainties and disturbances such as machine breakdowns. As

manufacturing environments evolve towards diversified product lines and the increased

necessity of short lead-times, effective and efficient production gains significant

importance. Therefore, research that offers good methods of reacting to disruptions is of

great importance for the successful implementation of scheduling systems in real-world

production environments.

The overall aim of this research was to investigate how dynamic job shop

scheduling subject to machine failures may be handled by multi-agent systems in order to

meet the needs outlined above. This chapter summarizes the major tasks in this research,

details the contributions and conclusions made, and suggests further research directions.

5.1 Research Summary

Chapter 1 presented the motivation for this research by introducing the difficulties

associated with job shop scheduling and the special needs involved in scheduling for a

dynamic job shop. Namely, a scheduling problem for dynamic flexible job shops with

sequence-dependent setups was defined. The chapter also outlined the seven specific

objectives of this research.

55

Chapter 2 presented a comprehensive literature review on all the published

research that is relevant to this work and addressed three issues: simulation-based

scheduling approaches for dynamic job shop scheduling in the presence of machine

breakdowns, multi-agent systems architecture, and agent coordination via the response

threshold model for dynamic scheduling. Past work on dynamic job shop scheduling

subject to exceptions was introduced, including suggested methods for recovery from

such disturbances. The review revealed that the multi-agent systems approach has

become a popular method for manufacturing scheduling. Because of their distributed,

autonomous nature and communication capabilities, multi-agent systems are well suited

for the dynamic scheduling of complex manufacturing systems and have proven effective

in a wide range of production scheduling applications. Although recent research efforts

have addressed flexibility and robustness in flexible manufacturing environments, very

few have offered approaches using multi-agent systems that are capable of recovering

from machine breakdowns in an efficient manner. Further, no recently reported research

efforts have fully demonstrated such capabilities in a multi-agent scheduling system in

which agents coordinate via an indirect communication method such as the response

threshold model.

Chapter 2 also introduces the RTM-DS model by Yu (2005). The model, adapted

from a similar model by Bonabeau et al. (1997) mimics the division of labor witnessed in

social insect societies for the purpose of achieving high levels of flexibility and

robustness in manufacturing scheduling. In comparing a job shop to a wasp nest,

machines are analogous to wasps, jobs are analogous to feeding zones, and individual

56

operations for jobs are analogous to individual larvae. The primary objective of each

machine is to perform job operations in an efficient manner. Response threshold,

response intention, reinforcement learning, and dominance are introduced as the primary

attributes which govern the interaction between the machines and the jobs which arrive at

the shop floor. Equations used to calculate each attribute are also introduced in Chapter

2.

The first half of Chapter 3 described the flexible manufacturing system to which

the RTM-DS coordination strategy is applied. The system consists of six flexible

machines, each of which has a dedicated input buffer. Jobs arrive to the shop floor

according to an exponentially distributed inter-arrival time and require an average of four

total different operation types. Each machine is capable of performing an average of

three different types of operations. Jobs arrive at the shop floor via an input station and

move immediately to a central work-in-process buffer from which they are routed to

alternative machine input buffers according to the RTM-DS coordination strategy. After

parts undergo their last operation, they leave the shop floor through an output station.

The methods for generating the scheduling dataset are also presented in this chapter.

Each dataset consists of a machine capability matrix, random operation sequences

(according to part type), operation processing times (according to part type and machine

type), inter-arrival times, job due-dates, and breakdown parameters. All of the dataset

parameters except the breakdown parameters, mean time between failure and mean time

to repair, are inspired by techniques used by Yu (2005). The method for generating the

breakdown parameters is taken from a similar method by Holthaus (1999).

57

The second half of Chapter 3 describes an intelligent strategy, RTM-DS-BD, for

reacting to machine breakdowns within the RTM-DS context. The method is based on a

strategy in which broken machine agents continue to participate in the intention update

process for having jobs routed to their input buffers. In this manner, when machines

experience breakdown, the system still considers the machines eligible to process jobs.

The intention parameter of the RTM-DS has been modified to account for the time that a

broken machine will require to finish an operation, considering its current broken state

and the time that it will require for repair. The jobs that are directly impacted by a

machine breakdown are any jobs that broken machines were processing at the time of

breakdown (interrupted jobs) and any jobs waiting in the input buffers of the broken

machine. This strategy requires that both interrupted jobs and any jobs in the input

buffers of broken machines be sent to the central buffer in order to potentially be rerouted

to machine input buffers which have higher intentions for completing the current required

operations of the interrupted jobs.

In Chapter 4, RTM-DS-BD was benchmarked with the ABM of Siwamosatham

and Saygin (2004). The experimental design consisted of two levels each for the

independent variables: NAM, ULZ, MTTR, SUT, and SBL. Settings for high and low

levels of each independent variable were chosen in accordance with studies reported by

Yu (2005) and Holthaus (1999). A total of sixty-four experiments were performed using

the two scheduling models. The settings for the RTM-DS-BD parameters (S0, α, and

θmax) were chosen using the results of experiments reported in Yu (2005). A comparative

58

analysis based on eight scheduling metrics was conducted. The results indicate that, in

most cases, RTM-DS-BD outperforms ABM on each of the eight chosen metrics.

Although the main effects show that RTM-DS-BD outperforms ABM on each

performance metric, some significant interaction between the independent variables was

shown. Therefore, post hoc analysis was required to determine whether RTM-DS-BD

outperforms ABM regardless of the combinations of the independent variables.

The results of the post hoc analysis show that RTM-DS-BD clearly outperforms

ABM in terms of the number of setups, NST. This result is consistent with the results of

Yu’s (2005) comparison between RTM-DS and ABM. It reinforces the fact that RTM-

DS-BD, whose coordination strategy is based largely on the division of labor

mechanisms seen in social insect societies, is very effective in terms of reducing the

number of required machine setups and, therefore, facilitating machine specialization.

For each performance metric except NST, due to the existence of some interaction

effects, RTM-DS-BD cannot be claimed to unconditionally outperform ABM. In

general, across all eight performance metrics, the following job shop characteristics

should exist in order for RTM-DS-BD to outperform ABM: 1) higher levels of

flexibility, 2) higher setup times, 3) higher machine repair times, and 4) higher levels of

breakdowns. The first of these requirements illustrates that the availability of alternative

machines for operations affected by breakdowns enhances RTM-DS-BD performance.

The last three requirements all relate to the availability of machines in job shops subject

to breakdowns and the impact of availability on RTM-DS-BD performance. For

instance, if the time required to set up machines between differing successive operations

59

is high, then machines are less often available to perform required operations. The same

is true of repair times and shop breakdown levels; the higher the levels of these factors

are, the lower shop availability is. Therefore, RTM-DS-BD is shown to perform

significantly better than ABM in job shops that are flexible, yet subject to low machine

availability.

Statistical results also show that upon the occurrence of breakdowns at their

current machines, affected jobs are rerouted to alternative machines about 20% more

frequently (39% of rerouting occurrences) when both shop breakdown level and machine

repair times are high than when either shop breakdown level and/or machine repair times

are low (20% of rerouting occurrences). This is further support that the RTM-DS-BD

rerouting policy is especially effective in flexible job shops with low availability since

operations are always sent to machines which have the highest intention of performing

those operations. Further, although in this research shop breakdown level (SBL)

represents the average breakdown level across all machines in the shop, likely results for

shops where individual machines may experience differing levels of breakdowns are

implied. Based on the experimental results, it is believed that in such shops, machines

that are more frequently broken would rely more heavily on the RTM-DS-BD rerouting

and intention update strategies for maintaining specialization. This is an additional

benefit offered by RTM-DS-BD.

60

5.2 Research Contribution

Flexible and robust operations scheduling is essential for enabling modern

manufacturing organizations to compete globally. The major contribution of this

research is the development of an intelligent, reactive approach to handling machine

breakdowns within the context of a multi-agent system based approach. Upon

breakdown, directly affected jobs are potentially rerouted to non-broken, more capable

machines. This is considered an intelligent strategy because broken machine agents are

allowed to have jobs routed to their input buffers despite their broken states, and are,

therefore, able to retain any specialization that they have achieved during the course of

their participation in the scheduling process. This is very beneficial in terms of reduction

of machine setups and overall robustness of the scheduling system.

Another contribution of this work has been to enhance the problem set generating

approach used by Yu (2005) to include the machine breakdown parameters, mean time to

repair (MTTR) and mean time before failure (MTBF). This was accomplished by

combining the Yu (2005) generation approach with a strategy found in the literature

(Holthaus, 1999) that models mean time to repair as a function of the average processing

time for operations on the shop floor, and combines MTTR with an additional breakdown

parameter to yield MTBF. The additional breakdown parameter, SBL, denotes the

percentage of scheduling time that machines in the job shop are not likely to be available

due to breakdown(s).

An additional contribution of this effort is the application of a multi-agent systems

approach for scheduling of a dynamic flexible job shop with sequence-dependent setups

61

subject to machine breakdowns. The multi-agent systems approach offers advantages

such as autonomy, distributed intelligence, and responsiveness to manufacturing

scheduling. This research effort is one of few reported attempts to apply multi-agent

systems to the scheduling of dynamic flexible job shops subject to machine breakdowns.

This research demonstrates that multi-agent coordination via the RTM-DS-BD

coordination strategy offers significant improvements in the performance of agent-based

manufacturing scheduling techniques.

Lastly, both the problem set enhancements and the intelligent breakdown

handling strategy were implemented in a simulation test bed, and a set of computational

studies were designed and used to measure the performance of the strategy on a number

of performance metrics.

5.3 Future Work

While this research offers promising results from original and interesting work

using multi-agent systems for scheduling in flexible job shops subject to breakdowns,

there are a number of issues which remain to be addressed. Some of these issues are

discussed below.

RTM-DS-BD Applied to a Single Queue, Multi-Server Environment

The scheduling environment in this study featured a multi-queue, multi-server

scenario. Jobs were sent from a central buffer, via a machine selection rule, to individual

machines queues. In this scenario, jobs required both routing and sequencing.

Alternatively, if machines did not have individual queues, jobs would be routed directly

62

to machines as soon as machines were available. The process of routing jobs from

machine queues (at the time of machine breakdown) to the central buffer would also be

eliminated in the “single queue, multi-server” scenario. It would be useful to analyze the

performance of the RTM-DS-BD strategy with such a scenario.

Investigation of Operation-Centered Reinforcement Learning

In this research, each time an MA finishes an operation, its threshold for the

finished operation decreases while its threshold for other operations increases. This

process is referred to as machine-centered reinforcement learning. An alternative to

machine-centered reinforcement learning is operation-centered learning, where all MA

thresholds are updated as soon as operations are ready to be routed to machine input

buffers. It is believed that machine-centered learning creates stronger MA specialization

than operation-centered learning and, therefore, fewer machine setups (Yu, 2005).

Especially in cases where setup times are high, fewer setups lead to better overall system

performance. However, such strong specialization may lead to excessive wait times

when setup times are short and, therefore, do not compensate for wait times.

Investigation of a system that combines both operation-centered and machine-centered

learning may be worthwhile since such a system would exploit the advantages of both

learning methods.

Application of RTM-DS-BD to Other Scheduling Scenarios Subject to Exceptions

Job shop scheduling is only one type of resource allocation problem that considers

the assignment of competing tasks to resources under various constraints. While the

RTM-DS-BD model yields good results for the scheduling of dynamic flexible job shops,

63

it may also be successfully applied to other types of resource allocation problems such as

supply chain systems coordination, telecommunications routing, contact center

operations, and emergency response systems. The robustness and flexibility of RTM-

DS-BD may offer value to the coordination of such systems.

Exceptions occur in many real world resource allocation problems. For instance,

supply chain systems may experience disruptions in shipment processes due to weather

disasters. Ambulatory services may experience sudden changes in response priorities due

to large-scale emergencies. A host of examples of such scenarios are feasible for

application of similar multi-agent coordination strategies.

64

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67

APPENDIX A

SAS OUTPUT: ANOVA FOR EIGHT

PERFORMANCE METRICS

68

Table A1. ANOVA Results: Number of Setups

Source DF Type I SS Mean Square F Value Pr > F

MDL 1 108116525.8 108116525.8 4855.78 <.0001

NAM 1 12396406.3 12396406.3 556.75 <.0001

MDL*NAM 1 5116061.7 5116061.7 229.78 <.0001

SUT 1 28774.9 28774.9 1.29 0.2561

MDL*SUT 1 81933.7 81933.7 3.68 0.0556

NAM*SUT 1 13062.4 13062.4 0.59 0.4440

MDL*NAM*SUT 1 106777.1 106777.1 4.80 0.0289

ULZ 1 706022.0 706022.0 31.71 <.0001

MDL*ULZ 1 568652.0 568652.0 25.54 <.0001

NAM*ULZ 1 84693.4 84693.4 3.80 0.0516

MDL*NAM*ULZ 1 94567.8 94567.8 4.25 0.0398

SUT*ULZ 1 26047.5 26047.5 1.17 0.2799

MDL*SUT*ULZ 1 37134.7 37134.7 1.67 0.1971

NAM*SUT*ULZ 1 904.2 904.2 0.04 0.8404

MDL*NAM*SUT*ULZ 1 1250.8 1250.8 0.06 0.8127

MTTR 1 1017498.4 1017498.4 45.70 <.0001

MDL*MTTR 1 2227769.2 2227769.2 100.05 <.0001

NAM*MTTR 1 68538.9 68538.9 3.08 0.0799

MDL*NAM*MTTR 1 96799.5 96799.5 4.35 0.0375

SUT*MTTR 1 4987.0 4987.0 0.22 0.6362

MDL*SUT*MTTR 1 153372.1 153372.1 6.89 0.0089

NAM*SUT*MTTR 1 48604.3 48604.3 2.18 0.1401

MDL*NAM*SUT*MTTR 1 87812.4 87812.4 3.94 0.0475

ULZ*MTTR 1 110488.5 110488.5 4.96 0.0263

MDL*ULZ*MTTR 1 134354.8 134354.8 6.03 0.0143

NAM*ULZ*MTTR 1 1138.9 1138.9 0.05 0.8212

MDL*NAM*ULZ*MTTR 1 3784.3 3784.3 0.17 0.6803

SUT*ULZ*MTTR 1 73.9 73.9 0.00 0.9541

MDL*SUT*ULZ*MTTR 1 255.7 255.7 0.01 0.9147

NAM*SUT*ULZ*MTTR 1 1985.5 1985.5 0.09 0.7653

MDL*NAM*SUT*ULZ*MTTR 1 11011.9 11011.9 0.49 0.4822

SBL 1 254715.2 254715.2 11.44 0.0008

MDL*SBL 1 713012.8 713012.8 32.02 <.0001

NAM*SBL 1 430432.8 430432.8 19.33 <.0001

MDL*NAM*SBL 1 441814.1 441814.1 19.84 <.0001

SUT*SBL 1 444.8 444.8 0.02 0.8877

MDL*SUT*SBL 1 57750.9 57750.9 2.59 0.1078

NAM*SUT*SBL 1 6840.2 6840.2 0.31 0.5796

MDL*NAM*SUT*SBL 1 3247.7 3247.7 0.15 0.7027

ULZ*SBL 1 5071.7 5071.7 0.23 0.6334

MDL*ULZ*SBL 1 4732.6 4732.6 0.21 0.6449

NAM*ULZ*SBL 1 8019.8 8019.8 0.36 0.5486

MDL*NAM*ULZ*SBL 1 1200.3 1200.3 0.05 0.8165

SUT*ULZ*SBL 1 70.1 70.1 0.00 0.9553

MDL*SUT*ULZ*SBL 1 134.8 134.8 0.01 0.9380

NAM*SUT*ULZ*SBL 1 354.8 354.8 0.02 0.8996

MDL*NAM*SUT*ULZ*SBL 1 25.0 25.0 0.00 0.9733

MTTR*SBL 1 38417.4 38417.4 1.73 0.1895

MDL*MTTR*SBL 1 168.7 168.7 0.01 0.9307

NAM*MTTR*SBL 1 3351.1 3351.1 0.15 0.6982

MDL*NAM*MTTR*SBL 1 177606.3 177606.3 7.98 0.0049

SUT*MTTR*SBL 1 8277.6 8277.6 0.37 0.5423

MDL*SUT*MTTR*SBL 1 63448.0 63448.0 2.85 0.0919

NAM*SUT*MTTR*SBL 1 61628.0 61628.0 2.77 0.0967

MDL*NAM*SUT*MTTR*SBL 1 1294.7 1294.7 0.06 0.8095

ULZ*MTTR*SBL 1 8141.7 8141.7 0.37 0.5456

MDL*ULZ*MTTR*SBL 1 2415.3 2415.3 0.11 0.7420

NAM*ULZ*MTTR*SBL 1 1479.9 1479.9 0.07 0.7966

MDL*NAM*ULZ*MTTR*SBL 1 1615.4 1615.4 0.07 0.7878

SUT*ULZ*MTTR*SBL 1 892.3 892.3 0.04 0.8414

MDL*SUT*ULZ*MTTR*SBL 1 3160.8 3160.8 0.14 0.7065

NAM*SUT*ULZ*MTTR*SBL 1 5119.8 5119.8 0.23 0.6317

MD*NA*SU*ULZ*MTT*SBL 1 5106.7 5106.7 0.23 0.6322

69

Table A2. ANOVA Results: Throughput

Source DF Type I SS Mean Square F Value Pr > F

MDL 1 0.02081641 0.02081641 1083.13 <.0001

NAM 1 0.00446266 0.00446266 232.20 <.0001

MDL*NAM 1 0.00669516 0.00669516 348.37 <.0001

SUT 1 0.41769141 0.41769141 21733.5 <.0001

MDL*SUT 1 0.00165766 0.00165766 86.25 <.0001

NAM*SUT 1 0.00078766 0.00078766 40.98 <.0001

MDL*NAM*SUT 1 0.00123766 0.00123766 64.40 <.0001

ULZ 1 0.00004516 0.00004516 2.35 0.1259

MDL*ULZ 1 0.00005641 0.00005641 2.93 0.0872

NAM*ULZ 1 0.00002641 0.00002641 1.37 0.2416

MDL*NAM*ULZ 1 0.00001891 0.00001891 0.98 0.3217

SUT*ULZ 1 0.00000766 0.00000766 0.40 0.5282

MDL*SUT*ULZ 1 0.00006891 0.00006891 3.59 0.0588

NAM*SUT*ULZ 1 0.00000766 0.00000766 0.40 0.5282

MDL*NAM*SUT*ULZ 1 0.00004516 0.00004516 2.35 0.1259

MTTR 1 0.00062016 0.00062016 32.27 <.0001

MDL*MTTR 1 0.00043891 0.00043891 22.84 <.0001

NAM*MTTR 1 0.00000391 0.00000391 0.20 0.6523

MDL*NAM*MTTR 1 0.00000391 0.00000391 0.20 0.6523

SUT*MTTR 1 0.00026266 0.00026266 13.67 0.0002

MDL*SUT*MTTR 1 0.00054391 0.00054391 28.30 <.0001

NAM*SUT*MTTR 1 0.00003516 0.00003516 1.83 0.1767

MDL*NAM*SUT*MTTR 1 0.00000766 0.00000766 0.40 0.5282

ULZ*MTTR 1 0.00000766 0.00000766 0.40 0.5282

MDL*ULZ*MTTR 1 0.00000391 0.00000391 0.20 0.6523

NAM*ULZ*MTTR 1 0.00000766 0.00000766 0.40 0.5282

MDL*NAM*ULZ*MTTR 1 0.00001266 0.00001266 0.66 0.4174

SUT*ULZ*MTTR 1 0.00001891 0.00001891 0.98 0.3217

MDL*SUT*ULZ*MTTR 1 0.00000016 0.00000016 0.01 0.9282

NAM*SUT*ULZ*MTTR 1 0.00004516 0.00004516 2.35 0.1259

MDL*NAM*SUT*ULZ*MTTR 1 0.00004516 0.00004516 2.35 0.1259

SBL 1 0.01691266 0.01691266 880.01 <.0001

MDL*SBL 1 0.00606391 0.00606391 315.52 <.0001

NAM*SBL 1 0.00199516 0.00199516 103.81 <.0001

MDL*NAM*SBL 1 0.00252016 0.00252016 131.13 <.0001

SUT*SBL 1 0.00252016 0.00252016 131.13 <.0001

MDL*SUT*SBL 1 0.00206641 0.00206641 107.52 <.0001

NAM*SUT*SBL 1 0.00070141 0.00070141 36.50 <.0001

MDL*NAM*SUT*SBL 1 0.00058141 0.00058141 30.25 <.0001

ULZ*SBL 1 0.00005641 0.00005641 2.93 0.0872

MDL*ULZ*SBL 1 0.00000766 0.00000766 0.40 0.5282

NAM*ULZ*SBL 1 0.00001266 0.00001266 0.66 0.4174

MDL*NAM*ULZ*SBL 1 0.00003516 0.00003516 1.83 0.1767

SUT*ULZ*SBL 1 0.00001266 0.00001266 0.66 0.4174

MDL*SUT*ULZ*SBL 1 0.00000141 0.00000141 0.07 0.7869

NAM*SUT*ULZ*SBL 1 0.00003516 0.00003516 1.83 0.1767

MDL*NAM*SUT*ULZ*SBL 1 0.00004516 0.00004516 2.35 0.1259

MTTR*SBL 1 0.00008266 0.00008266 4.30 0.0385

MDL*MTTR*SBL 1 0.00021391 0.00021391 11.13 0.0009

NAM*MTTR*SBL 1 0.00000391 0.00000391 0.20 0.6523

MDL*NAM*MTTR*SBL 1 0.00013141 0.00013141 6.84 0.0092

SUT*MTTR*SBL 1 0.00013141 0.00013141 6.84 0.0092

MDL*SUT*MTTR*SBL 1 0.00008266 0.00008266 4.30 0.0385

NAM*SUT*MTTR*SBL 1 0.00000391 0.00000391 0.20 0.6523

MDL*NAM*SUT*MTTR*SBL 1 0.00005641 0.00005641 2.93 0.0872

ULZ*MTTR*SBL 1 0.00001891 0.00001891 0.98 0.3217

MDL*ULZ*MTTR*SBL 1 0.00003516 0.00003516 1.83 0.1767

NAM*ULZ*MTTR*SBL 1 0.00000391 0.00000391 0.20 0.6523

MDL*NAM*ULZ*MTTR*SBL 1 0.00001891 0.00001891 0.98 0.3217

SUT*ULZ*MTTR*SBL 1 0.00002641 0.00002641 1.37 0.2416

MDL*SUT*ULZ*MTTR*SBL 1 0.00000016 0.00000016 0.01 0.9282

NAM*SUT*ULZ*MTTR*SBL 1 0.00004516 0.00004516 2.35 0.1259

MD*NA*SU*ULZ*MTT*SBL 1 0.00001266 0.00001266 0.66 0.4174

70

Table A3. ANOVA Results: Number of Late Jobs

Source DF Type I SS Mean Square F Value Pr > F

MDL 1 5489859.709 5489859.709 895.61 <.0001

NAM 1 839416.621 839416.621 136.94 <.0001

MDL*NAM 1 3167459.505 3167459.505 516.74 <.0001

SUT 1 1865.444 1865.444 0.30 0.5814

MDL*SUT 1 391117.368 391117.368 63.81 <.0001

NAM*SUT 1 123956.771 123956.771 20.22 <.0001

MDL*NAM*SUT 1 289643.383 289643.383 47.25 <.0001

ULZ 1 17905.698 17905.698 2.92 0.0880

MDL*ULZ 1 37896.182 37896.182 6.18 0.0132

NAM*ULZ 1 34865.334 34865.334 5.69 0.0174

MDL*NAM*ULZ 1 58718.695 58718.695 9.58 0.0021

SUT*ULZ 1 15776.288 15776.288 2.57 0.1092

MDL*SUT*ULZ 1 1323.794 1323.794 0.22 0.6423

NAM*SUT*ULZ 1 23045.400 23045.400 3.76 0.0530

MDL*NAM*SUT*ULZ 1 18939.360 18939.360 3.09 0.0793

MTTR 1 735344.570 735344.570 119.96 <.0001

MDL*MTTR 1 2432.781 2432.781 0.40 0.5290

NAM*MTTR 1 3350.776 3350.776 0.55 0.4600

MDL*NAM*MTTR 1 1.850 1.850 0.00 0.9861

SUT*MTTR 1 163225.218 163225.218 26.63 <.0001

MDL*SUT*MTTR 1 4943.785 4943.785 0.81 0.3695

NAM*SUT*MTTR 1 537.014 537.014 0.09 0.7673

MDL*NAM*SUT*MTTR 1 4.266 4.266 0.00 0.9790

ULZ*MTTR 1 2807.342 2807.342 0.46 0.4988

MDL*ULZ*MTTR 1 13704.711 13704.711 2.24 0.1354

NAM*ULZ*MTTR 1 5742.913 5742.913 0.94 0.3335

MDL*NAM*ULZ*MTTR 1 9678.710 9678.710 1.58 0.2094

SUT*ULZ*MTTR 1 0.105 0.105 0.00 0.9967

MDL*SUT*ULZ*MTTR 1 33.402 33.402 0.01 0.9412

NAM*SUT*ULZ*MTTR 1 18158.784 18158.784 2.96 0.0858

MDL*NAM*SUT*ULZ*MTTR 1 10311.725 10311.725 1.68 0.1951

SBL 1 2908834.557 2908834.557 474.54 <.0001

MDL*SBL 1 187720.484 187720.484 30.62 <.0001

NAM*SBL 1 153837.820 153837.820 25.10 <.0001

MDL*NAM*SBL 1 19442.478 19442.478 3.17 0.0754

SUT*SBL 1 250303.645 250303.645 40.83 <.0001

MDL*SUT*SBL 1 1.891 1.891 0.00 0.9860

NAM*SUT*SBL 1 46007.224 46007.224 7.51 0.0063

MDL*NAM*SUT*SBL 1 100894.743 100894.743 16.46 <.0001

ULZ*SBL 1 30006.156 30006.156 4.90 0.0273

MDL*ULZ*SBL 1 20563.617 20563.617 3.35 0.0675

NAM*ULZ*SBL 1 1768.268 1768.268 0.29 0.5914

MDL*NAM*ULZ*SBL 1 17495.293 17495.293 2.85 0.0917

SUT*ULZ*SBL 1 11820.274 11820.274 1.93 0.1655

MDL*SUT*ULZ*SBL 1 2622.578 2622.578 0.43 0.5133

NAM*SUT*ULZ*SBL 1 1606.018 1606.018 0.26 0.6089

MDL*NAM*SUT*ULZ*SBL 1 2017.004 2017.004 0.33 0.5664

MTTR*SBL 1 1150.927 1150.927 0.19 0.6649

MDL*MTTR*SBL 1 24203.325 24203.325 3.95 0.0474

NAM*MTTR*SBL 1 35396.402 35396.402 5.77 0.0166

MDL*NAM*MTTR*SBL 1 127396.087 127396.087 20.78 <.0001

SUT*MTTR*SBL 1 64.408 64.408 0.01 0.9184

MDL*SUT*MTTR*SBL 1 2062.922 2062.922 0.34 0.5621

NAM*SUT*MTTR*SBL 1 49733.701 49733.701 8.11 0.0046

MDL*NAM*SUT*MTTR*SBL 1 57983.846 57983.846 9.46 0.0022

ULZ*MTTR*SBL 1 1146.051 1146.051 0.19 0.6656

MDL*ULZ*MTTR*SBL 1 7300.601 7300.601 1.19 0.2756

NAM*ULZ*MTTR*SBL 1 3341.721 3341.721 0.55 0.4606

MDL*NAM*ULZ*MTTR*SBL 1 325.513 325.513 0.05 0.8178

SUT*ULZ*MTTR*SBL 1 2865.614 2865.614 0.47 0.4944

MDL*SUT*ULZ*MTTR*SBL 1 7.172 7.172 0.00 0.9727

NAM*SUT*ULZ*MTTR*SBL 1 190.871 190.871 0.03 0.8600

MD*NA*SU*ULZ*MTT*SBL 1 6.302 6.302 0.00 0.9744

71

Table A4. ANOVA Results: Average Lateness

Source DF Type I SS Mean Square F Value Pr > F

MDL 1 49824411.67 49824411.67 2355.46 <.0001

NAM 1 26976140.81 26976140.81 1275.31 <.0001

MDL*NAM 1 32938048.41 32938048.41 1557.16 <.0001

SUT 1 3927861.06 3927861.06 185.69 <.0001

MDL*SUT 1 5993039.65 5993039.65 283.32 <.0001

NAM*SUT 1 3917264.00 3917264.00 185.19 <.0001

MDL*NAM*SUT 1 2228970.58 2228970.58 105.38 <.0001

ULZ 1 17430.42 17430.42 0.82 0.3644

MDL*ULZ 1 108960.19 108960.19 5.15 0.0236

NAM*ULZ 1 15550.21 15550.21 0.74 0.3916

MDL*NAM*ULZ 1 1183.42 1183.42 0.06 0.8131

SUT*ULZ 1 64299.95 64299.95 3.04 0.0818

MDL*SUT*ULZ 1 3935.07 3935.07 0.19 0.6664

NAM*SUT*ULZ 1 253.34 253.34 0.01 0.9129

MDL*NAM*SUT*ULZ 1 13048.71 13048.71 0.62 0.4325

MTTR 1 3418457.33 3418457.33 161.61 <.0001

MDL*MTTR 1 2863090.01 2863090.01 135.35 <.0001

NAM*MTTR 1 955707.86 955707.86 45.18 <.0001

MDL*NAM*MTTR 1 1054498.98 1054498.98 49.85 <.0001

SUT*MTTR 1 2729.68 2729.68 0.13 0.7196

MDL*SUT*MTTR 1 15133.66 15133.66 0.72 0.3980

NAM*SUT*MTTR 1 70497.43 70497.43 3.33 0.0684

MDL*NAM*SUT*MTTR 1 73534.91 73534.91 3.48 0.0628

ULZ*MTTR 1 42361.22 42361.22 2.00 0.1576

MDL*ULZ*MTTR 1 148141.15 148141.15 7.00 0.0084

NAM*ULZ*MTTR 1 15486.98 15486.98 0.73 0.3925

MDL*NAM*ULZ*MTTR 1 1008.77 1008.77 0.05 0.8272

SUT*ULZ*MTTR 1 27567.23 27567.23 1.30 0.2541

MDL*SUT*ULZ*MTTR 1 2610.73 2610.73 0.12 0.7255

NAM*SUT*ULZ*MTTR 1 4548.84 4548.84 0.22 0.6430

MDL*NAM*SUT*ULZ*MTTR 1 1160.52 1160.52 0.05 0.8149

SBL 1 26580704.69 26580704.69 1256.61 <.0001

MDL*SBL 1 21222159.81 21222159.81 1003.28 <.0001

NAM*SBL 1 10871555.16 10871555.16 513.96 <.0001

MDL*NAM*SBL 1 13551491.82 13551491.82 640.65 <.0001

SUT*SBL 1 1144479.05 1144479.05 54.11 <.0001

MDL*SUT*SBL 1 1974200.40 1974200.40 93.33 <.0001

NAM*SUT*SBL 1 1710215.35 1710215.35 80.85 <.0001

MDL*NAM*SUT*SBL 1 885409.12 885409.12 41.86 <.0001

ULZ*SBL 1 9549.18 9549.18 0.45 0.5019

MDL*ULZ*SBL 1 12065.90 12065.90 0.57 0.4504

NAM*ULZ*SBL 1 14264.03 14264.03 0.67 0.4119

MDL*NAM*ULZ*SBL 1 42253.25 42253.25 2.00 0.1581

SUT*ULZ*SBL 1 93897.55 93897.55 4.44 0.0356

MDL*SUT*ULZ*SBL 1 17706.42 17706.42 0.84 0.3606

NAM*SUT*ULZ*SBL 1 9652.36 9652.36 0.46 0.4996

MDL*NAM*SUT*ULZ*SBL 1 30814.71 30814.71 1.46 0.2279

MTTR*SBL 1 672853.77 672853.77 31.81 <.0001

MDL*MTTR*SBL 1 683354.50 683354.50 32.31 <.0001

NAM*MTTR*SBL 1 49879.97 49879.97 2.36 0.1252

MDL*NAM*MTTR*SBL 1 42680.74 42680.74 2.02 0.1560

SUT*MTTR*SBL 1 2200.55 2200.55 0.10 0.7472

MDL*SUT*MTTR*SBL 1 849.53 849.53 0.04 0.8412

NAM*SUT*MTTR*SBL 1 65657.39 65657.39 3.10 0.0786

MDL*NAM*SUT*MTTR*SBL 1 52234.59 52234.59 2.47 0.1166

ULZ*MTTR*SBL 1 43.58 43.58 0.00 0.9638

MDL*ULZ*MTTR*SBL 1 25849.85 25849.85 1.22 0.2694

NAM*ULZ*MTTR*SBL 1 16403.72 16403.72 0.78 0.3789

MDL*NAM*ULZ*MTTR*SBL 1 27737.87 27737.87 1.31 0.2526

SUT*ULZ*MTTR*SBL 1 28468.09 28468.09 1.35 0.2465

MDL*SUT*ULZ*MTTR*SBL 1 1814.48 1814.48 0.09 0.7697

NAM*SUT*ULZ*MTTR*SBL 1 2442.03 2442.03 0.12 0.7341

MD*NA*SU*ULZ*MTT*SBL 1 9.98 9.98 0.00 0.9827

72

Table A5. ANOVA Results: Average Tardiness

Source DF Type I SS Mean Square F Value Pr > F

MDL 1 41368137.87 41368137.87 2036.39 <.0001

NAM 1 25780850.28 25780850.28 1269.09 <.0001

MDL*NAM 1 29188018.30 29188018.30 1436.81 <.0001

SUT 1 3733333.73 3733333.73 183.78 <.0001

MDL*SUT 1 5172488.20 5172488.20 254.62 <.0001

NAM*SUT 1 3708552.70 3708552.70 182.56 <.0001

MDL*NAM*SUT 1 2557338.81 2557338.81 125.89 <.0001

ULZ 1 29362.72 29362.72 1.45 0.2298

MDL*ULZ 1 89019.93 89019.93 4.38 0.0368

NAM*ULZ 1 16085.81 16085.81 0.79 0.3739

MDL*NAM*ULZ 1 537.78 537.78 0.03 0.8708

SUT*ULZ 1 42214.42 42214.42 2.08 0.1500

MDL*SUT*ULZ 1 8175.38 8175.38 0.40 0.5261

NAM*SUT*ULZ 1 768.71 768.71 0.04 0.8458

MDL*NAM*SUT*ULZ 1 15759.21 15759.21 0.78 0.3788

MTTR 1 3031313.17 3031313.17 149.22 <.0001

MDL*MTTR 1 2789177.32 2789177.32 137.30 <.0001

NAM*MTTR 1 1037847.30 1037847.30 51.09 <.0001

MDL*NAM*MTTR 1 1021076.49 1021076.49 50.26 <.0001

SUT*MTTR 1 12148.97 12148.97 0.60 0.4396

MDL*SUT*MTTR 1 18346.98 18346.98 0.90 0.3423

NAM*SUT*MTTR 1 71097.15 71097.15 3.50 0.0619

MDL*NAM*SUT*MTTR 1 71770.76 71770.76 3.53 0.0607

ULZ*MTTR 1 58071.45 58071.45 2.86 0.0914

MDL*ULZ*MTTR 1 116704.00 116704.00 5.74 0.0169

NAM*ULZ*MTTR 1 19798.38 19798.38 0.97 0.3240

MDL*NAM*ULZ*MTTR 1 449.82 449.82 0.02 0.8818

SUT*ULZ*MTTR 1 22488.82 22488.82 1.11 0.2932

MDL*SUT*ULZ*MTTR 1 4486.08 4486.08 0.22 0.6386

NAM*SUT*ULZ*MTTR 1 5985.79 5985.79 0.29 0.5875

MDL*NAM*SUT*ULZ*MTTR 1 2309.07 2309.07 0.11 0.7361

SBL 1 24137137.42 24137137.42 1188.18 <.0001

MDL*SBL 1 20691488.31 20691488.31 1018.56 <.0001

NAM*SBL 1 11504608.38 11504608.38 566.33 <.0001

MDL*NAM*SBL 1 13569183.31 13569183.31 667.96 <.0001

SUT*SBL 1 1310699.75 1310699.75 64.52 <.0001

MDL*SUT*SBL 1 2070635.63 2070635.63 101.93 <.0001

NAM*SUT*SBL 1 1611694.29 1611694.29 79.34 <.0001

MDL*NAM*SUT*SBL 1 946455.23 946455.23 46.59 <.0001

ULZ*SBL 1 3658.76 3658.76 0.18 0.6714

MDL*ULZ*SBL 1 5406.85 5406.85 0.27 0.6061

NAM*ULZ*SBL 1 9478.78 9478.78 0.47 0.4948

MDL*NAM*ULZ*SBL 1 49397.88 49397.88 2.43 0.1195

SUT*ULZ*SBL 1 78081.74 78081.74 3.84 0.0504

MDL*SUT*ULZ*SBL 1 25022.38 25022.38 1.23 0.2675

NAM*SUT*ULZ*SBL 1 5413.59 5413.59 0.27 0.6059

MDL*NAM*SUT*ULZ*SBL 1 40421.78 40421.78 1.99 0.1589

MTTR*SBL 1 718697.68 718697.68 35.38 <.0001

MDL*MTTR*SBL 1 683921.22 683921.22 33.67 <.0001

NAM*MTTR*SBL 1 63474.08 63474.08 3.12 0.0776

MDL*NAM*MTTR*SBL 1 54415.15 54415.15 2.68 0.1022

SUT*MTTR*SBL 1 216.66 216.66 0.01 0.9178

MDL*SUT*MTTR*SBL 1 41.46 41.46 0.00 0.9640

NAM*SUT*MTTR*SBL 1 45120.61 45120.61 2.22 0.1367

MDL*NAM*SUT*MTTR*SBL 1 47145.90 47145.90 2.32 0.1282

ULZ*MTTR*SBL 1 443.01 443.01 0.02 0.8827

MDL*ULZ*MTTR*SBL 1 15670.02 15670.02 0.77 0.3802

NAM*ULZ*MTTR*SBL 1 9190.52 9190.52 0.45 0.5015

MDL*NAM*ULZ*MTTR*SBL 1 35735.14 35735.14 1.76 0.1853

SUT*ULZ*MTTR*SBL 1 21490.76 21490.76 1.06 0.3041

MDL*SUT*ULZ*MTTR*SBL 1 3311.54 3311.54 0.16 0.6865

NAM*SUT*ULZ*MTTR*SBL 1 5614.83 5614.83 0.28 0.5993

MD*NA*SU*ULZ*MTT*SBL 1 525.97 525.97 0.03 0.8722

73

Table A6. ANOVA Results: Average Due Date Deviation

Source DF Type I SS Mean Square F Value Pr > F

MDL 1 33963292.47 33963292.47 1720.93 <.0001

NAM 1 24386617.09 24386617.09 1235.68 <.0001

MDL*NAM 1 25896268.25 25896268.25 1312.17 <.0001

SUT 1 3630204.09 3630204.09 183.94 <.0001

MDL*SUT 1 4316930.20 4316930.20 218.74 <.0001

NAM*SUT 1 3591571.94 3591571.94 181.99 <.0001

MDL*NAM*SUT 1 2830963.57 2830963.57 143.45 <.0001

ULZ 1 35291.03 35291.03 1.79 0.1817

MDL*ULZ 1 83781.99 83781.99 4.25 0.0398

NAM*ULZ 1 11265.92 11265.92 0.57 0.4502

MDL*NAM*ULZ 1 117.65 117.65 0.01 0.9385

SUT*ULZ 1 32461.51 32461.51 1.64 0.2002

MDL*SUT*ULZ 1 9080.43 9080.43 0.46 0.4978

NAM*SUT*ULZ 1 279.05 279.05 0.01 0.9054

MDL*NAM*SUT*ULZ 1 12999.09 12999.09 0.66 0.4174

MTTR 1 2593368.36 2593368.36 131.41 <.0001

MDL*MTTR 1 2792002.19 2792002.19 141.47 <.0001

NAM*MTTR 1 1075520.22 1075520.22 54.50 <.0001

MDL*NAM*MTTR 1 1034091.94 1034091.94 52.40 <.0001

SUT*MTTR 1 36491.87 36491.87 1.85 0.1744

MDL*SUT*MTTR 1 15640.25 15640.25 0.79 0.3737

NAM*SUT*MTTR 1 84444.15 84444.15 4.28 0.0390

MDL*NAM*SUT*MTTR 1 58465.52 58465.52 2.96 0.0858

ULZ*MTTR 1 64167.31 64167.31 3.25 0.0719

MDL*ULZ*MTTR 1 103157.03 103157.03 5.23 0.0226

NAM*ULZ*MTTR 1 17994.14 17994.14 0.91 0.3400

MDL*NAM*ULZ*MTTR 1 1120.90 1120.90 0.06 0.8117

SUT*ULZ*MTTR 1 24563.67 24563.67 1.24 0.2650

MDL*SUT*ULZ*MTTR 1 3604.30 3604.30 0.18 0.6693

NAM*SUT*ULZ*MTTR 1 4155.58 4155.58 0.21 0.6465

MDL*NAM*SUT*ULZ*MTTR 1 1537.85 1537.85 0.08 0.7802

SBL 1 21598681.83 21598681.83 1094.41 <.0001

MDL*SBL 1 20372916.25 20372916.25 1032.30 <.0001

NAM*SBL 1 11997039.96 11997039.96 607.89 <.0001

MDL*NAM*SBL 1 13755612.89 13755612.89 697.00 <.0001

SUT*SBL 1 1544440.88 1544440.88 78.26 <.0001

MDL*SUT*SBL 1 2102658.10 2102658.10 106.54 <.0001

NAM*SUT*SBL 1 1572834.30 1572834.30 79.70 <.0001

MDL*NAM*SUT*SBL 1 964186.34 964186.34 48.86 <.0001

ULZ*SBL 1 2122.63 2122.63 0.11 0.7431

MDL*ULZ*SBL 1 3604.68 3604.68 0.18 0.6693

NAM*ULZ*SBL 1 9626.28 9626.28 0.49 0.4852

MDL*NAM*ULZ*SBL 1 46715.86 46715.86 2.37 0.1245

SUT*ULZ*SBL 1 75767.45 75767.45 3.84 0.0506

MDL*SUT*ULZ*SBL 1 25759.94 25759.94 1.31 0.2537

NAM*SUT*ULZ*SBL 1 5144.50 5144.50 0.26 0.6099

MDL*NAM*SUT*ULZ*SBL 1 41509.99 41509.99 2.10 0.1475

MTTR*SBL 1 726621.89 726621.89 36.82 <.0001

MDL*MTTR*SBL 1 722776.34 722776.34 36.62 <.0001

NAM*MTTR*SBL 1 66423.72 66423.72 3.37 0.0671

MDL*NAM*MTTR*SBL 1 79953.10 79953.10 4.05 0.0446

SUT*MTTR*SBL 1 28.75 28.75 0.00 0.9696

MDL*SUT*MTTR*SBL 1 368.66 368.66 0.02 0.8913

NAM*SUT*MTTR*SBL 1 36641.23 36641.23 1.86 0.1735

MDL*NAM*SUT*MTTR*SBL 1 33446.85 33446.85 1.69 0.1935

ULZ*MTTR*SBL 1 668.96 668.96 0.03 0.8540

MDL*ULZ*MTTR*SBL 1 12631.45 12631.45 0.64 0.4240

NAM*ULZ*MTTR*SBL 1 7482.69 7482.69 0.38 0.5383

MDL*NAM*ULZ*MTTR*SBL 1 35609.56 35609.56 1.80 0.1797

SUT*ULZ*MTTR*SBL 1 21699.06 21699.06 1.10 0.2948

MDL*SUT*ULZ*MTTR*SBL 1 2468.51 2468.51 0.13 0.7237

NAM*SUT*ULZ*MTTR*SBL 1 6026.66 6026.66 0.31 0.5807

MD*NA*SU*ULZ*MTT*SBL 1 686.91 686.91 0.03 0.8521

74

Table A7. Results: Average Wait Time

Source DF Type I SS Mean Square F Value Pr > F

MDL 1 289711192.1 289711192.1 2599.61 <.0001

NAM 1 182691008.6 182691008.6 1639.31 <.0001

MDL*NAM 1 194782264.5 194782264.5 1747.80 <.0001

SUT 1 15014504.6 15014504.6 134.73 <.0001

MDL*SUT 1 18669926.0 18669926.0 167.53 <.0001

NAM*SUT 1 17168418.7 17168418.7 154.05 <.0001

MDL*NAM*SUT 1 14043094.3 14043094.3 126.01 <.0001

ULZ 1 298222.2 298222.2 2.68 0.1024

MDL*ULZ 1 496292.0 496292.0 4.45 0.0353

NAM*ULZ 1 133975.7 133975.7 1.20 0.2733

MDL*NAM*ULZ 1 26204.0 26204.0 0.24 0.6279

SUT*ULZ 1 180000.8 180000.8 1.62 0.2043

MDL*SUT*ULZ 1 72579.1 72579.1 0.65 0.4200

NAM*SUT*ULZ 1 16178.6 16178.6 0.15 0.7033

MDL*NAM*SUT*ULZ 1 94236.8 94236.8 0.85 0.3582

MTTR 1 21490640.1 21490640.1 192.84 <.0001

MDL*MTTR 1 19796690.5 19796690.5 177.64 <.0001

NAM*MTTR 1 6306827.6 6306827.6 56.59 <.0001

MDL*NAM*MTTR 1 6446658.7 6446658.7 57.85 <.0001

SUT*MTTR 1 24621.3 24621.3 0.22 0.6385

MDL*SUT*MTTR 1 82707.0 82707.0 0.74 0.3893

NAM*SUT*MTTR 1 699248.8 699248.8 6.27 0.0125

MDL*NAM*SUT*MTTR 1 733643.9 733643.9 6.58 0.0105

ULZ*MTTR 1 137383.3 137383.3 1.23 0.2673

MDL*ULZ*MTTR 1 252974.2 252974.2 2.27 0.1325

NAM*ULZ*MTTR 1 58162.5 58162.5 0.52 0.4703

MDL*NAM*ULZ*MTTR 1 11560.3 11560.3 0.10 0.7475

SUT*ULZ*MTTR 1 56243.1 56243.1 0.50 0.4777

MDL*SUT*ULZ*MTTR 1 26497.1 26497.1 0.24 0.6260

NAM*SUT*ULZ*MTTR 1 16402.2 16402.2 0.15 0.7014

MDL*NAM*SUT*ULZ*MTTR 1 75116.2 75116.2 0.67 0.4120

SBL 1 181192660.9 181192660.9 1625.86 <.0001

MDL*SBL 1 167349892.9 167349892.9 1501.65 <.0001

NAM*SBL 1 100454695.6 100454695.6 901.39 <.0001

MDL*NAM*SBL 1 106600609.3 106600609.3 956.54 <.0001

SUT*SBL 1 8038294.7 8038294.7 72.13 <.0001

MDL*SUT*SBL 1 10064531.3 10064531.3 90.31 <.0001

NAM*SUT*SBL 1 11221517.1 11221517.1 100.69 <.0001

MDL*NAM*SUT*SBL 1 9376564.5 9376564.5 84.14 <.0001

ULZ*SBL 1 237.9 237.9 0.00 0.9632

MDL*ULZ*SBL 1 31970.7 31970.7 0.29 0.5924

NAM*ULZ*SBL 1 23326.1 23326.1 0.21 0.6475

MDL*NAM*ULZ*SBL 1 83431.1 83431.1 0.75 0.3873

SUT*ULZ*SBL 1 358230.9 358230.9 3.21 0.0735

MDL*SUT*ULZ*SBL 1 210657.3 210657.3 1.89 0.1697

NAM*SUT*ULZ*SBL 1 117053.2 117053.2 1.05 0.3059

MDL*NAM*SUT*ULZ*SBL 1 225220.9 225220.9 2.02 0.1557

MTTR*SBL 1 6169578.2 6169578.2 55.36 <.0001

MDL*MTTR*SBL 1 5868442.8 5868442.8 52.66 <.0001

NAM*MTTR*SBL 1 373499.6 373499.6 3.35 0.0677

MDL*NAM*MTTR*SBL 1 339854.3 339854.3 3.05 0.0813

SUT*MTTR*SBL 1 9032.1 9032.1 0.08 0.7760

MDL*SUT*MTTR*SBL 1 8533.6 8533.6 0.08 0.7821

NAM*SUT*MTTR*SBL 1 561768.8 561768.8 5.04 0.0251

MDL*NAM*SUT*MTTR*SBL 1 547642.5 547642.5 4.91 0.0270

ULZ*MTTR*SBL 1 594.5 594.5 0.01 0.9418

MDL*ULZ*MTTR*SBL 1 9358.2 9358.2 0.08 0.7721

NAM*ULZ*MTTR*SBL 1 28633.9 28633.9 0.26 0.6124

MDL*NAM*ULZ*MTTR*SBL 1 66585.8 66585.8 0.60 0.4399

SUT*ULZ*MTTR*SBL 1 42630.3 42630.3 0.38 0.5365

MDL*SUT*ULZ*MTTR*SBL 1 13855.8 13855.8 0.12 0.7245

NAM*SUT*ULZ*MTTR*SBL 1 10834.7 10834.7 0.10 0.7553

MD*NA*SU*ULZ*MTT*SBL 1 39707.2 39707.2 0.36 0.5508

75

Table A8. ANOVA Results: Makespan

Source DF Type I SS Mean Square F Value Pr > F

MDL 1 1983078358 1983078358 3123.41 <.0001

NAM 1 1037868219 1037868219 1634.67 <.0001

MDL*NAM 1 1080818997 1080818997 1702.32 <.0001

SUT 1 38086482828 38086482828 59987.4 <.0001

MDL*SUT 1 135067367 135067367 212.74 <.0001

NAM*SUT 1 98023156 98023156 154.39 <.0001

MDL*NAM*SUT 1 75729990 75729990 119.28 <.0001

ULZ 1 3450910 3450910 5.44 0.0201

MDL*ULZ 1 3014979 3014979 4.75 0.0297

NAM*ULZ 1 569875 569875 0.90 0.3438

MDL*NAM*ULZ 1 135665 135665 0.21 0.6441

SUT*ULZ 1 1553130 1553130 2.45 0.1184

MDL*SUT*ULZ 1 135858 135858 0.21 0.6438

NAM*SUT*ULZ 1 713449 713449 1.12 0.2896

MDL*NAM*SUT*ULZ 1 898447 898447 1.42 0.2347

MTTR 1 155168692 155168692 244.40 <.0001

MDL*MTTR 1 131566705 131566705 207.22 <.0001

NAM*MTTR 1 23294047 23294047 36.69 <.0001

MDL*NAM*MTTR 1 24925921 24925921 39.26 <.0001

SUT*MTTR 1 407380 407380 0.64 0.4234

MDL*SUT*MTTR 1 85309 85309 0.13 0.7141

NAM*SUT*MTTR 1 2994579 2994579 4.72 0.0303

MDL*NAM*SUT*MTTR 1 5202704 5202704 8.19 0.0044

ULZ*MTTR 1 713149 713149 1.12 0.2897

MDL*ULZ*MTTR 1 1351042 1351042 2.13 0.1452

NAM*ULZ*MTTR 1 25950 25950 0.04 0.8399

MDL*NAM*ULZ*MTTR 1 2174024 2174024 3.42 0.0648

SUT*ULZ*MTTR 1 558208 558208 0.88 0.3488

MDL*SUT*ULZ*MTTR 1 380810 380810 0.60 0.4390

NAM*SUT*ULZ*MTTR 1 81024 81024 0.13 0.7210

MDL*NAM*SUT*ULZ*MTTR 1 3201291 3201291 5.04 0.0251

SBL 1 528510607 528510607 832.42 <.0001

MDL*SBL 1 964240090 964240090 1518.71 <.0001

NAM*SBL 1 529738838 529738838 834.36 <.0001

MDL*NAM*SBL 1 556312845 556312845 876.21 <.0001

SUT*SBL 1 240751627 240751627 379.19 <.0001

MDL*SUT*SBL 1 42031888 42031888 66.20 <.0001

NAM*SUT*SBL 1 50605281 50605281 79.70 <.0001

MDL*NAM*SUT*SBL 1 54054739 54054739 85.14 <.0001

ULZ*SBL 1 26082 26082 0.04 0.8395

MDL*ULZ*SBL 1 94066 94066 0.15 0.7004

NAM*ULZ*SBL 1 216097 216097 0.34 0.5599

MDL*NAM*ULZ*SBL 1 9784 9784 0.02 0.9013

SUT*ULZ*SBL 1 319305 319305 0.50 0.4785

MDL*SUT*ULZ*SBL 1 315818 315818 0.50 0.4809

NAM*SUT*ULZ*SBL 1 323275 323275 0.51 0.4758

MDL*NAM*SUT*ULZ*SBL 1 844043 844043 1.33 0.2494

MTTR*SBL 1 25397187 25397187 40.00 <.0001

MDL*MTTR*SBL 1 34057586 34057586 53.64 <.0001

NAM*MTTR*SBL 1 438546 438546 0.69 0.4063

MDL*NAM*MTTR*SBL 1 3119063 3119063 4.91 0.0271

SUT*MTTR*SBL 1 92972 92972 0.15 0.7021

MDL*SUT*MTTR*SBL 1 601851 601851 0.95 0.3307

NAM*SUT*MTTR*SBL 1 2909300 2909300 4.58 0.0327

MDL*NAM*SUT*MTTR*SBL 1 1538914 1538914 2.42 0.1201

ULZ*MTTR*SBL 1 954345 954345 1.50 0.2207

MDL*ULZ*MTTR*SBL 1 1863 1863 0.00 0.9568

NAM*ULZ*MTTR*SBL 1 55895 55895 0.09 0.7668

MDL*NAM*ULZ*MTTR*SBL 1 6288 6288 0.01 0.9208

SUT*ULZ*MTTR*SBL 1 496 496 0.00 0.9777

MDL*SUT*ULZ*MTTR*SBL 1 51009 51009 0.08 0.7769

NAM*SUT*ULZ*MTTR*SBL 1 6479 6479 0.01 0.9196

MD*NA*SU*ULZ*MTT*SBL 1 843239 843239 1.33 0.2496