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Republic of Iraq

Ministry of Higher Education

and Scientific Research

Thi-Qar University

College of Education for Pure Sciences

Minimizing Three Simultaneous Criteria in

Machine Scheduling Problem

A Thesis Submitted to

The Department of Mathematics, College of Education

for Pure Sciences, University of Thi-Qar in a Partial

Fulfillment of the Requirements for the Degree

of Master of Sciences in Mathematics

by Jafar Saleh Aneed

(B.Sc. 2010)

Supervised by

Assist Prof. Professor

Dr. Mohammed K.Z. Al-Zuwaini Dr. Kadhem M. H. Al-Mousawi

2013 A.D 1434 A.H

ہ ہ ہ ھ ھ ھ ھ ے ے ۓ چ

ۓ ڭ ڭ ڭ ڭ ۇ ۇ ۆ

چۆ

صدق اهلل العلي العظيم

( 91 ) اآليةسورة النمل من

الإهداء

ىل , عزمه يثين أن ادلهر يس تطع ومل, بصربه الصعاب وجابه, معره س نني وهبين من اإ

العزيز وادلي... الشموخ رمز

ىل من الساهرة العني فاكنت وحناهنا حهبا حبر يف وترعرعت دهما من سقتين من اإ

العزيزة وادليت... يل يدعو اذلي والقلب أجيل

ىل ال عزاء أ خوايت و أ خويت... حيايت رايحني و احلب مشوع اإ

ىل البحث هذا أ متام يف ساعدين من مجيع اإ

ىل أ صدقايئ مجيع اإ

الوفاء بعض فيه لعل هجدي مثرة... مجيعا هلم أهدي

جعفر صاحل

ACKNOWLEDGEMENTS

ACKNOWLEDGEMENTS

All the praise and thanks be to ALLAH, the most gracious and

most merciful, for his grace that enabled me to continue the

requirements of my study.

My sincerest gratitude is due to my supervisors, Dr. Mohammed

K. Al-Zuwaini and Dr. Kadhim M. Al-Mousawi for their patience,

help and prudent guidance throughout this work. Their support and

guidance has allowed me to complete one of my goals in my life.

Without their help, completion of this goal would have been more

difficult.

Words of thanks should go to the head and the staff of the

Department of Mathematics, and all my M.Sc. course teachers to

whom I owe much.

My thanks and priding to my colleagues in the higher studies.

Also, my thanks go to everyone who helped me throughout the

fulfillment of my research, especially Dr. Rabee Hadi Jari, Firas Sabar

alhussinawi, Sami Mezal Araibi, and Hussein Jameel.

I cannot find words to express my indebtedness to my father, my

mother, my brothers, and my sisters, for their love, support and

inspiration at every step of my success and failure, without which

none of this would have been possible.

Jafar

Supervision Certification

We certify that this thesis entitled (Minimizing Three Simultaneous

Criteria in Machine Scheduling Problem) was prepared under my

supervision at the Department of Mathematics/ College of Education for

Pure Sciences/ University of Thi-Qar, in a partial fulfillment of the

requirements for the degree of Master of Sciences (M.Sc.) in

Mathematics.

Signature:

Supervisor: Assist prof. Dr. Mohammed Kadhim Z. Al-Zuwaini

Date: / / 2013

Address: College of Computer Sciences and Mathematics, Thi-Qar University

Signature:

Supervisor: Professor Dr. Kadhem Mahdi Hashim Al-Mousawi

Date: / / 2013

Address: College of Education for Pure Sciences, Thi-Qar University

Head recommendation of the mathematics department

In view of the available recommendations; I forward this thesis for

debate by the examining committee.

Signature:

Name: Assist prof. Dr. Rabee Hadi Jari

Head of the Department of Mathematics,

College of Education for Pure Sciences, Thi-Qar University

Date: / / 2013

Committee Certificate

We are the examining committee, certify that after reading this thesis

entitled (Minimizing Three Simultaneous Criteria in Machine Scheduling

Problem) and examining the students (Jafar Saleh Aneed) in its contents. We

think in our opinion it has met the requirements for the degree of Master of

Sciences (M.Sc.) in Mathematics, with ( ) grade.

Chairman Member

Signature: Signature:

Name: Name:

Scientific Position: Scientific Position:

Date: / / 2013 Date: / / 2013

Member Member (Supervisor)

Signature: Signature:

Name: Name: Dr. Mohammed K. Al-Zuwaini

Scientific Position: Scientific Position: Assist Professor

Date: / / 2013 Date: / / 2013

Member (Supervisor)

Signature:

Name: Dr. Kadhim M. Al-Mousawi

Scientific Position: Professor

Date: / / 2013

Approved for the University committee on Graduate studies.

Signature:

Name: professor Dr. Mohammed Jassim Mohammed Al-Mousawi

Dean of the College of Education for Pure Sciences

Date: / / 2013

Abstract j I

Abstract

In this thesis, the problem of scheduling n jobs on a single machine

is considered to minimize Multiple Objective Function (MOF). There are

two aims for this study, the first one is to find the optimal solution for

the sum of completion times, maximum earliness and maximum tardiness

with unequal release dates, no preemption is allowed, this problem de-

noted by 1/ri/∑n

i=1Ci + Emax + Tmax, or 1/ri/∑n

i=1Ci + ETmax to the

best of our knowledge this problem is not studied before. The second aim

is to find the near optimal solution for the same problem by using neural

networks.

For the first aim, a Branch and Bound (BAB) algorithm is proposed

with two lower bounds (LB1, LB2) and four upper bounds (UB1, UB2, UB3,

UB4) that are introduced in this thesis, in order to find the exact (opti-

mal) solution. Nine special cases are derived and proved that yield optimal

solutions without using (BAB) algorithm. Three dominance rules are sug-

gested and proved which help in reducing the number of branches in the

search tree. Results of extensive computational tests show the proposed

(BAB) algorithm is effective in solving problems with up to (30) jobs at

a time less than or equal to (30) minutes. In general, this problem is

strongly NP-hard.

For the second aim, since our problem is strongly NP-hard, we apply

the neural network method to find near optimal solution. Computational

experience is found that this neural network method can solve the problem

with up to (8) jobs with reasonable time. We observed from computational

experience, the neural networks method is a very good process, it gives a

near optimal value for the objective function.

II

List of Symbols and Abbreviations

ANN Artificial Neural Network

BAB Branch and Bound

Completion time of job i

The maximum completion time (makespan)

DP Dynamic programming

Due date of job i

Deadline of job i

EDD Earliest Due date

The maximum earliness

Maximum function

Flow shop scheduling

H1 First Hidden layer

H2 Second Hidden layer

ILB Initial Lower Bound

I Input layer

JIT Just in time

Job shop scheduling

LB Lower bound

MEDD Modified Earliest Due Date

MOF Multiple Objective Function

MST Minimum Slack Time

MRST Minimum Remaining Slack Time

m Number of machines

NP Non-deterministic polynomial

NN Neural Network

n Number of jobs

O Output layer

Open shop scheduling

Processing time of job i

pmtn Preemption

prec Precedence constraint

P Polynomial time

Release date of job i

SPT Shortest Processing Time

Slack time of job i

SRD Shortest release date

SRPT Shortest Remaining Processing Time

The maximum tardiness

TSP Traveling Salesman Problem

UB Upper Bound

Weight of job i

The Total Flow Time

Contents

Abstract I

List of Symbols and Abbreviations II

Contents III

List of Figures VI

Introduction 1

1 Description of Machine Scheduling Problem 4

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Machine Scheduling Problem . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Single Machine Scheduling . . . . . . . . . . . . . . . . . . . . . . 5

1.2.2 Parallel Machine Scheduling . . . . . . . . . . . . . . . . . . . . . 6

1.2.3 Flow Shop Scheduling (Fm) . . . . . . . . . . . . . . . . . . . . . 6

1.2.4 Job Shop Scheduling (Jm) . . . . . . . . . . . . . . . . . . . . . . 6

1.2.5 Open Shop Scheduling (Om) . . . . . . . . . . . . . . . . . . . . . 7

1.3 Regular and Non-regular Performance Measure of Machine Scheduling

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Basic Scheduling Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5 The Classification Problem (α/β/γ) . . . . . . . . . . . . . . . . . . . . . 10

1.5.1 Machine Environment (α) . . . . . . . . . . . . . . . . . . . . . . 10

1.5.2 Job Characteristics (β) . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5.3 Optimality Criteria (γ) . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5.4 Examples of Scheduling Problems . . . . . . . . . . . . . . . . . . 12

1.6 Assumptions About Jobs . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

III

1.7 Assumptions About Machines . . . . . . . . . . . . . . . . . . . . . . . . 13

1.8 Sequence Rules for Machine Scheduling Problems . . . . . . . . . . . . . 14

1.9 Problem Classes and Computational Complexity . . . . . . . . . . . . . . 14

1.10 Solution Approaches for Machine Scheduling

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.10.1 Complete Enumeration . . . . . . . . . . . . . . . . . . . . . . . . 16

1.10.2 Branch and Bound Methods . . . . . . . . . . . . . . . . . . . . . 17

1.10.3 Dynamic Programming Method . . . . . . . . . . . . . . . . . . . 18

1.10.4 Heuristic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Branch and Bound Method to Minimize Three Criteria 21

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Decomposition of Problem (S) . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5 Dominance Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.6 Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.7 Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.7.1 The First Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . 38

2.7.2 The second Lower Bound . . . . . . . . . . . . . . . . . . . . . . . 39

2.8 Branch and Bound Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 40

2.9 Computational Experience . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.9.1 Test Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.9.2 Computational Experience with the Lower and Upper Bounds of

(BAB) Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3 Neural Networks to Solve Scheduling Problems 46

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.1.1 Artificial Neural Network (ANN) . . . . . . . . . . . . . . . . . . 46

3.1.2 Architecture of Neural Network . . . . . . . . . . . . . . . . . . . 49

3.1.3 Network Learning Methods . . . . . . . . . . . . . . . . . . . . . 53

3.1.4 The activation functions . . . . . . . . . . . . . . . . . . . . . . . 55

3.1.5 Applications of Neural Networks . . . . . . . . . . . . . . . . . . . 55

3.1.6 Beginning of Neural Nets . . . . . . . . . . . . . . . . . . . . . . . 56

IV

3.2 Related Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.3 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.4 Multilayer Perceptron NN for Single Machine . . . . . . . . . . . . . . . 59

3.5 Proposed Neural Network Design . . . . . . . . . . . . . . . . . . . . . . 64

3.6 Artificial Neural Network Training . . . . . . . . . . . . . . . . . . . . . 65

3.7 The Matrix of Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4 Conclusions and Future Work 71

4.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

References 73

V

List of Figures

2.1 BAB method without dominance rule . . . . . . . . . . . . . . . . . . . . 35

2.2 BAB method with dominance rule . . . . . . . . . . . . . . . . . . . . . . 36

2.3 (SRPT) rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.4 (MEDD) rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.1 A simple artificial neuron. . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2 Single Layer Net. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.3 A multilayer neural net. . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.4 Error-correction learning diagram. . . . . . . . . . . . . . . . . . . . . . . 53

3.5 Activation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.6 The proposed design of ANN . . . . . . . . . . . . . . . . . . . . . . . . 65

3.7 Input of 3 jobs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.8 Input of 4 jobs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

VI

Introduction j 1

Introduction

Operations Research (OR) is one of the modern practicing sciences

which have witnessed wide success in different fields of life. OR has arisen

during the second world war when the military administration in Britain

gave a team of scientists and researchers the task of studying the techno-

logical and strategic problems specially in terrestrial and voyage defense.

The aim of this team has been to find the best use to resources in addition

to study the action of the new kind of pomps. The form of this group was

regarded the first birth to what is called (Operations Research) in 1941

[34].

This study focuses on one of the most important subjects in Opera-

tions Research which is called (Machine Scheduling Problem). To solve

machine scheduling problems one tends to use optimization algorithms

which always find an optimal solution. However, not for all optimization

problems, polynomial time optimization algorithms can be constructed.

This is because, some of these problems, which cannot be solved in poly-

nomial time algorithms are NP-hard. In practice this means that solving

large instances of such problems optimality requires impracticable running

times. The NP-hard of a problem suggests that it is impossible to find

an optimal solution without the use of essentially implicit enumeration al-

gorithms (branch and bound or dynamic programming algorithms). But

this enumeration algorithm may be unable to solve problems with more

than a handful of jobs, and the solution generated by simple heuristics

may be far from the optimum [12].

Introduction j 2

Because the one machine problem provides a useful laboratory for the

development of ideas for heuristics and interactive procedure that may

prove to be useful in more general models, we consider the one machine

case in this study, with multi-criteria.

There are two approaches for the multi-criteria problem; the hierar-

chical approach and the simultaneous approach. First, in the hierarchical

approach, one of two criteria is considered as the primary criterion and

the other one is considered as the secondary criterion. The problem is to

minimize the primary criterion while breaking ties in favor of the schedule

that has the minimum secondary criterion value. Second, in the simul-

taneous approach there are two types; the first one typically generates

all efficient schedules and selects the one that yields the best composite

objective function value of these criteria. The second is to find the sum

of these objectives [31].

Scheduling has received much attention in the literature since the pio-

neering work of Johnson in 1954. In the first 30 years, it has been usual to

consider only one objective function as performance criterion. However, in

many practical situations a decision-maker has to take into account simul-

taneously several objectives. Therefore, the investigation of multi-criteria

scheduling problems has begun about 30 years ago with a growing interest

nowadays [56].

The starting point of artificial neural networks was the McClloch-

Neuron Pitts in 1943, which demonstrated how a network of neuron could

exhibit learning behavior. Neural networks are categorized by their archi-

Introduction j 3

tecture (number of layers ), topology (connectivity pattern, feed forward

or recurrent, etc.), and learning regime. The main advantages of artificial

neural networks technology are ( it is fast, it possess learning ability, it

adapts to the data, it is robust, and it is appropriate for nonlinear mod-

elling) [8].

This thesis consists of four chapters:

Chapter one: Gives a full description of machine scheduling problems,

including historical background, a number of assumptions for machines,

jobs and other assumptions, classification and representation of schedul-

ing problems are also mentioned. The well known methods for finding the

exact solution such as BAB algorithm and DP are discussed.

Chapter two: Is devoted to the problem of a single machine scheduling

with unequal release dates to minimize the sum of total completion time,

maximum earliness, and maximum tardiness by using branch and bound

algorithm. Special cases, heuristic methods (upper bounds), lower bounds

was also included. Computational experiences for the BAB algorithm was

found.

Chapter three: Presents the neural networks method to find near op-

timal solution. Computational experience is found that this local search

method can solve the problem with up to (8) jobs with reasonable time.

Chapter Four: Gives conclusions and an outlook on future work.

Chapter 1

Description of Machine Scheduling

Problem

1.1 Introduction

There are many definitions for machine scheduling, but the simplest

one for understanding is that, the allocation of resources over time to

perform a collection of tasks (Baker [9]). Resources and tasks are called

machines and jobs respectively and both of them can take many forms.

For example: We can consider a computer (or computers) as a machine

(or machines) and the programs that are to be run on that computer

(or computers ) as the jobs. Another example: We can consider hospital

equipments as a machines and the patients in that hospital as the jobs.

Generally speaking, scheduling means to assign machines to jobs in order

to complete all jobs under the imposed constraints. The problem is to find

the optimal processing order of these jobs on each machine to minimize

the given objective function. There are two general constraints in classical

4

Chapter One: Description of Machine Scheduling Problem j5

scheduling theory [12]. Each job is to be processed by; at most, one

machine at a time and each machine is capable of processing at most

one job at a time. A schedule is feasible if it satisfies the two general

constraints, and also if it satisfies the various requirements relating to

the specific problem type. The problem type is specified by the machine

environment, the job characteristics and an optimality criterion.

1.2 Machine Scheduling Problem

Within manufacturing scheduling, there are many different types of

problem classes. These include single machine, parallel machine, flow

shop, job shop, and open shop. Each of these problem classes is unique,

and each has its own constraints and objectives [21]. A more detailed

description of each problem class is given in the following subsections.

1.2.1 Single Machine Scheduling

The single machine scheduling problem involves scheduling a set of

tasks to a single resource. This is accomplished by determining a sequence

that includes each task, and then assigning the tasks to the resources.

Each task can be given a priority, ready time, processing time and due

date. The value of the performance measures can be computed on the

base of this information and the sequence of tasks. This problem grows in

complexity at an exponential rate as the number of tasks to be scheduled

increases [3].


1.2.2 Parallel Machine Scheduling

Parallel machine scheduling involves scheduling a set of tasks on two

or more machines that work in parallel with each other. The machines

perform identical operations and may or may not operate at the same

pace [20].

1.2.3 Flow Shop Scheduling (Fm)

A flow shop scheduling consists of two or more machines and a set of

jobs that must be processed on each of these machines. This arrangement

is called a flow shop because the products flow along a specific unidirec-

tional path. Each product must be processed on each machine in the

same order e.g. 1st-machine 1, 2nd-machine 2,. . ., mth-machine m. The

processing times for each job can vary from machine to machine and the

processing times on each machine can vary from job to job [22].

1.2.4 Job Shop Scheduling (Jm)

A job shop consists of two or more machines that perform specific

operations, and a set of jobs that must be processed on some or all of these

machines. Unlike the flow shop, there is no fixed path that the products

must follow through the system. Therefore the order of operations is not

fixed. This type of layout is typically used when the product variety is

high and the product volume is low [22].


1.2.5 Open Shop Scheduling (Om)

In this case, the ordering for each job is not identical and the sequence

of n jobs on each machine is different, and each job has to be processed

on each machine, but there is no particular order to follow, the open shop

is not as well research as the flow shop and job shop [32].

1.3 Regular and Non-regular Performance Measure

of Machine Scheduling Problems

A measure of performance is said to be regular if it is a non-decreasing

function of job completion times and the scheduling objective is to min-

imize the performance measure. Examples of regular measures are job

flow time F, schedule makespan (Cmax) and tardiness based performance

measures. A non-regular performance measure is usually not a monotone

function of the job completion times. An example of such a measure is job

earliness [23]. Some times the objective function containing more than one

criteria; one is regular and the other is non-reguler as the multiple function

(∑Fi + Emax).

1.4 Basic Scheduling Concepts

We start with introducing some important notations, where we con-

centrate on the performance criteria without elaborating on the machine

environments. We assume that there are n jobs, which we denoted by

j1, . . . , jn these jobs are to be scheduled on a set of machines that are


continuously available from time zero onwards and that can handle only

one job at a time.

We state here the notation that is used for the single machine, jobs

ji(i = 1, ..., n) has:

Preemption (pmtn) [58]: Preemption (or job-splitting) is allowed dur-

ing the processing of a job, if the processing of the job can be interrupted

at any time (preempt) and resumed at a later time, even on a different

machine. The amount of processing already done on the preempted job is

not lost.

Processing time (pij)[48]: The processing time of job j on machine i.

The subscript i is omitted if job j is only to be processed on one given

machine or on m parallel machines.

Due date (dj)[44]: The date when the job should ideally be completed,

the completion of job after its due date is allowed, but a penalty is in-

curred. When the due date absolutely must be met, it’s referred to as

deadline (dj), and when due date is constant for all jobs, then called com-

mon due date.

Release date (rj)[44]: Also known as ready time, the point of time that

job arrives at the machine and thus processing starts.

Weight (Wj)[3]: Denoting the importance of a job j relative to another

job.

Now for a given sequence of jobs (1, 2, · · · , n) the following can be

computed for job j.

1) The completion time Cj, Cj =∑j

i=1 pi.


2) The flow time Fj = Cj − rj.

3) The lateness Lj = Cj − dj.

4) The tardiness Tj = max{Cj − dj, 0}.

5) The earliness Ej = max{dj − Cj, 0}.

6) The unit penalty

Uj =

1 if Cj > dj

0 otherwise

7) The idle time

Ij =

rj if j = 1

rj − Cj−1 if rj > Cj−1, j = 2, ..., n

0 otherwise

Let σ be a given sequence, the following performance criteria appear

frequently in the literature [29].

• Cmax(σ)=maxj∈σ{Cj}(maximum completion time or makespan).

• Emax(σ)=maxj∈σ{Ej}(maximum earliness).

• Lmax(σ)=maxj∈σ{Lj}(maximum lateness).

• Tmax(σ)=maxj∈σ{Tj}(maximum tardiness).

•∑

j∈σWjCj(σ)(total weighted completion times).

•∑

j∈σWjEj(σ)(total weighted earliness).

•∑

j∈σWjTj(σ)(total weighted tardiness).

•∑

j∈σWjUj(σ)(weighted number of tardy jobs).


1.5 The Classification Problem (α/β/γ)

In this thesis, we adopt the terminology of Graham et. al. [26] to

classify scheduling problems. Suppose that m machines Mi (i = 1, . . . ,m)

have to process n jobs Jj (j = 1, . . . , n). A schedule problem type can

be specified using a three-field classification (α/β/γ) composed of the

machine environment, the job characteristics ,and the optimality criterion

.

1.5.1 Machine Environment (α)

The first field α = α1α2 describes the machine environment parameters

α1 ∈ {φ,P,Q,R,O,F, J}, which characterized the type of machine used

[15]:

• α1 = φ: for single machine.

• α1 = P: for identical parallel machine problem.

• α1 = Q: for uniform parallel machine problem.

• α1 = R: for unrelated parallel machine problem.

• α1 = O: for open shop problem.

• α1 = F: for flow shop problem.

• α1 = J: for job shop problem.

α2 ∈ {φ,m}

• α2 = φ: the number of machines is assumed to be variable.


• α2 = m: the number of machines is equal to m (m≥ 2).

1.5.2 Job Characteristics (β)

The second field β ⊆ {pmtn, rj, βprec} indicates certain job character-

istics [2].

• pmtn is present, the preemption are allowed, the processing of any job

may be interrupted at no cost and resumed at a later time. Otherwise,

no preemptions are allowed, once a job is started on a machine Mi,

the job occupies the machine until it is completed.

• rj is present, then each job may have different release dates. Other-

wise, all jobs arrive at time 0.

• If a precedence constraint is present, then there is a precedence rela-

tion (≺) among the jobs, i.e. if Jj ≺ Jk, then Jj must be completed

before Jk can be started. If βprec = chain, then (≺) forms chains.

If βprec = tree, then (≺) forms a tree. If βprec = prec, then (≺) is

an arbitrary partial order. If βprec is not present, then jobs can be

processed in any order.

1.5.3 Optimality Criteria (γ)

The third field (γ) specified the objective function or the optimality

criterion, the value we wish to optimize. The parameter γ ∈ {fmax,∑fj},

which are defined as follows:

fmax ∈ {Cmax, Fmax, Lmax, Emax, Tmax}


or∑fj ∈ {

∑Cj,

∑WjCj,

∑Ej,

∑WjEj,

∑Tj,

∑WjTj,

∑Ui,

∑WiUi, }

or fj is equal to sum or weighted sum for two or more of objective function.

1.5.4 Examples of Scheduling Problems

i. 1/ri/∑n

i=1Ci + Tmax: is the problem of scheduling jobs with release

dates on a single machine to minimize bi-criteria (the total completion

time and maximum tardiness).

ii. F2/rj/∑n

j=1Cj: is the problem of scheduling jobs with release dates

on a two machine flow shop to minimize the total completion times.

iii. 1/prec/∑n

i=1WiCi: is the problem of scheduling jobs with precedence

constraints, on a single machine to minimize the total (weighted)

completion time.

1.6 Assumptions About Jobs [49]

J1: The set of jobs is fixed and known.

J2: All jobs are available at the same time and are independent some

times, each job, j=1,· · ·,n is characterized by a release date rj and a

due date dj (all data being integer).

J3: Each job has the same degree of importance. This assumption is

sometimes, relaxed (i.e. Job j has a weight Wj represent the impor-

tant of job j ).


J4: Any operation started is not interrupted by other operations and

continues to its completion. This assumption is sometimes relaxed

(i.e. job splitting is allowed).

J5: Each job can be in one of three states:

i. Waiting for the next machine.

ii. Being operated on by a machine.

iii. Having passed its last machine.

J6: Each job can be processed by only one machine at the same time.

1.7 Assumptions About Machines [49]

M1: The number of machines is known and fixed.

M2: All machines are available at the same instance and are independent

of each other.

M3: Each machine can be in one of three statuses:

i. Waiting for the next job.

ii. Operating on a job.

iii. Having finished its last job.

M4: All machines are equally important.

M5: Each machine has to process all jobs assigned to it.

M6: Each machine can process not more than one job at a time.


1.8 Sequence Rules for Machine Scheduling Prob-

lems

1) The EDD rule, that is sequencing the jobs in non-decreasing order of

their due date dj, which solves the problem 1//Lmax. This rule also

minimizes Tmax for the 1//Tmax problem [35].

2) The SPT rule, that is, sequencing the jobs in non-decreasing order of

their processing times pj. This rule solves the problem 1//∑n

j=1Cj

[52].

3) The MST rule, which is sequencing the jobs in non-decreasing order

of their slack time sj = dj − pj. In single machine environment with

ready time set at zero, which solves 1//Emax problem [30].

4) The SRPT rule, that is sequencing the jobs in non-decreasing order

of shortest remaining processing times , (this rule is well known to

minimize 1/rj, pmtn/∑n

j=1Cj problem) [9].

5) The MEDD rule, that is sequencing the jobs in non-decreasing order

of smallest remaining due dates, (this rule is well known to minimize

1/rj, pmtn/Tmax problem) [10].

1.9 Problem Classes and Computational Complexity

Some problems are polynomially solvable, which means that there ex-

ists an algorithm requiring a computing effort that could be shown to grow

as a polynomial in the size of the problem. Other problems can only be


solved by an enumerative algorithm, which in essence has an exponential

time complexity [12]. The problems that have known polynomials algo-

rithms are said to be well solved. The complexity theory is concerned

with the decision problem instead of optimization problem. A decision

problem is a question to which the answer is either ”yes” or ”no”, the

question is: does there exist a solution with an objective function value

less than or equal to fixed value k? (The optimization problem is trans-

formed into a finite series of decision problems by varying the value k).

Decision problems can be classified into three classes:

1) The class P (Polynomial), which contains all decision problems that

are polynomially solvable.

2) The class NP (Non-deterministic polynomial), which contains deci-

sion problems which can be easily verified whether a given solution

is ”yes” or ”no” answer. It is clear that the class P is a subclass

from class NP . The scheduling problems that will be considered in

this thesis can be solved by non-deterministic algorithm and thus is

a member of NP . Cook [19] proved there are hardest problems in

NP , such problems are called NP-Complete.

3) The class Open, which contains all decision problems that did not

prove so far P or NP , for instance the problem 1/dj = d/∑n

j=1WjEj

+W′

jTj is open [15].

A problem P is NP-complete if the existence of polynomial algorithm

for P implies the existence of a polynomial algorithm for any problem in


NP (i.e. P=NP). The location of the border line separating the easy

problem (in P) and the hard one (in NP-Complete) has been under wide

investigation by many researches, and turns out that a minor change in the

value of an easy problem parameters often transforms this problem into

a hard one. An optimization problem is called NP-hard if the associated

decision problem is NP-Complete. Since the computation time needed to

solve a scheduling problem is very important, recent development in the

theory of computational complexity has applied to machine scheduling

problems has aroused the interest of many researchers.

1.10 Solution Approaches for Machine Scheduling

Problems

In this section a discussion is carried out for the most well known

methods that have been used to solve machine scheduling problems. Both

enumerative approaches BAB and DP and heuristic approaches are con-

sidered for solving the machine scheduling problems.

1.10.1 Complete Enumeration

Complete enumeration methods generate one by one, all feasible sched-

ules and then pick the best one. For a single machine problem of n jobs

there are n! different sequence. Hence for the corresponding m machines

problem, there are (n!)m different sequence. This method may take con-

siderable time as the time as the number (n!)m is very large even for


relatively small values of n and m [34].

1.10.2 Branch and Bound Methods

BAB methods can be used for solving many combinatorial optimiza-

tion problems. These methods are example of implicit enumeration ap-

proach, which find an optimal solution by examining subsets of a feasible

solution. These procedures can be conveniently represented as a search

tree. Each node of the search tree corresponds to subset of feasible solu-

tions to a problem. A branching rule specifies how the feasible solutions

at a node are partitioned into subsets, each corresponding to descendant

node of the search tree.

The scheduling problems that we consider require an objective func-

tion, to be minimized. A lower bound scheme associates the LB with each

node of the search tree. The idea is to eliminate any node for which the

lower bound is greater than or equal to the value of the best known fea-

sible solution.

The branching rule describes how feasible solution at a node is parti-

tioned into subsets. There are several types of branching rules for schedul-

ing problems, the most common of which are forward branching and back-

ward branching. In forward branching rule jobs are sequenced one by one

from the beginning, while in a backward branching rule the jobs are se-

quenced one by one from the end.

The bounding rule calculates the LB on the optimal solution for each

subproblem generated by the branching rule. The well known methods of


obtaining lower bounds for machine scheduling problems are:

1. Relaxation of constraints.

2. Relaxation of objectives.

3. Langrangian relaxation.

4. Dynamic programming state-space relaxation.

To minimize an objective function of particular scheduling problem,

first the UB of the minimum of this objective function is needed. This UB

is the value for trial solution. At the beginning the trial solution may be

found using a heuristic procedure.

Finally, the branch and bound method can be improved by applying

dominance rules that discard nodes before computing their lower bounds.

These dominance rules are computationally useful as they reduce storage

requirements on the computer as well as reducing computation time [16].

1.10.3 Dynamic Programming Method

Fundamentals of dynamic programming were elaborated by Bellman

in 1950s [11]. This method was used, for machine scheduling problems,

for the first time, by Held and Karp [27]. The name Dynamic Program-

ming is slightly missleading, but generally accepted. A better description

would be recursive or multistage optimization. Since it interprets opti-

mization problem as a multistage decision problem. It means that the

problem is divided into a number of stages, and at each stage a decision

is required which impacts on the decision to be made in later stage [24].


If dynamic programming is applied to a combinatorial problem, then in

order to calculate the optimal criterion value for any subset of size K,

first, the optimal value for each subset of size K − 1 have to be known.

Thus if our problem is characterized by a set of n elements, the number

of subset considered 2n. It means that dynamic programming algorithms

are of exponential computational complexity. More precisely, Dynamic

programming method starts with an initial subproblem, which is easy to

solve at each iteration, it determines the optimal solution for a subprob-

lem which is larger than all previously solved subproblems by utilizing all

the information about the solution of these subproblems. This continues

until the original problem is solved.

1.10.4 Heuristic Methods

It is clear that, to solve a scheduling problem one tends to use im-

plicit enumerative approaches to find an optimal solution. However, these

approaches have two disadvantages. Firstly, they are mathematically com-

plex and thus a lot of time to be invested. Secondly, when it concerns an

NP-hard problem, the computational requirements are enormous for large

sized problem, to avoid these drawbacks can appeal to heuristics. Reeves

[50] defined the heuristic method as follows:

A heuristic is a technique which seeks good (i.e. near optimal) solution

at a reasonable computational cost without being able to guarantee either

feasibility or optimality, or even in many cases to state how close to opti-

mality a particular feasible solution.


In recent years, the improvement in heuristic methods has become

under the name of (Local Search Methods).

Chapter 2

Branch and Bound Method to

Minimize Three Criteria

2.1 Introduction

In general, multi-criteria scheduling refers to the scheduling problem

in which the advantages of a particular schedule are evaluated using more

than one performance criterion. The managerial relevance of considering

multiple criteria for scheduling has been cited in the production and opera-

tions management literature since the 1950’s. Smith (1956)[52] shows that

the choice of a criterion will affect the characteristics of a ”best schedule”;

different optimizing criteria will result in very different schedules. Van

Wassenhove and Gelders (1980)[57] provide evidence that a schedule that

performs well using a certain criterion might yield a poor result using

other criteria. Hence, lack of consideration of various criteria may lead to

solutions that are very difficult to implement in practice. Although the

importance of multi-criteria scheduling has been recognized for many years

21

Chapter Two: Branch and Bound Method to Minimize Three Criteria j 22

(French, 1982[23]; Nelson et al., 1986[47], and George and Paul 2007[25]),

little attention has been given in the literature to this topic. From the

problem complexity perspective, the multiple-criteria problem becomes

much more complex than related single-criteria counterparts (Lenstra et

al., 1979[40] and Nagar et al. (1995)[46]) review the problem in its general

form, where as Lee and Vairaktarakis (1993)[41] review a special version

of the problem, where one criterion is set to its best possible value and the

other criterion is tried to be optimized under this restriction. Hoogeveen

(2005)[28] studies a number of bi-criteria scheduling problems. Also, there

are some papers about this object (Cheng et al. 2008[14], [25], and Azi-

zoglu et al. 2003 [5]).

In this chapter the problem of scheduling n independent jobs on a

single machine is considered to minimize MOF, the sum of total comple-

tion time, maximum earliness and maximum tardiness by using the BAB

method. This problem is denoted by 1/ri/∑n

i=1Ci + Emax + Tmax.

2.2 Problem Formulation

Single machine scheduling models seem to be very simple but are very

important for understanding and modeling multiple machines models. A

set N = {1, 2, · · · , n} of n independent jobs has to be scheduled on a sin-

gle machine in order to optimize a given criterion. This study concerns

the one machine scheduling problem with multiple objectives function de-

noted by (1/ri/∑n

i=1Ci + Emax + Tmax). In this problem, preemption is

not allowed, no precedence relation among jobs is assumed, only one job


i can be processed at a time. Each job i has a release date ri at which

it cannot be processed before, needs pi time units to be processed on the

machine, and ideally should be completion at its due date di. For each

job i can be calculate the slack time si = di − pi. The objective is to find

a schedule to minimize the sum of total of completion times (∑n

i=1Ci),

maximum earliness (Emax) and maximum tardiness(Tmax). The problem

(1/ri/∑n

i=1Ci + Emax + Tmax) can be stated as follows:

A set of n independent jobs N = {1, 2, · · · , n} are available for process-

ing at time ri, job i(i = 1, 2, · · · , n) is to be processed with uninterruption

on a single machine that can handle only one job at a time, requires pro-

cessing time pi, and ideally should be completed at its due date di. For

a given sequence π of the jobs, completion time of job i, Cπ(i), earliness

Eπ(i), and the tardiness Tπ(i) are given by:

Cπ(1) = rπ(1) + pπ(1)

Cπ(i) = max{Cπ(i−1), rπ(i)}+ pπ(i) , i = 2, · · · , n

(2.1)

Eπ(i) = max{dπ(i) − Cπ(i), 0} , i = 1, · · · , n (2.2)

Tπ(i) = max{Cπ(i) − dπ(i), 0} , i = 1, · · · , n (2.3)

The problem is strongly NP-hard because the problems 1/ri/Tmax [48]

and 1/ri/∑n

i=1Ci, [38] are stronglyNP-hard and the problem 1//∑n

i=1Ci+

Emax is NP-hard, ([1, 3, 37]).

The aim is to find a sequence π that minimizes the total cost R =∑ni=1Cπ(i) +Emax(π) + Tmax(π). The mathematical form of this problem,

which denoted by (S) can be stated as follows:


minR = minπ∈δ{∑n

i=1Cπ(i) + Emax(π) + Tmax(π)}

s.t

Cπ(i) ≥ rπ(i) + pπ(i) i = 1, 2, ..., n

Cπ(i) ≥ Cπ(i−1) + pπ(i) i = 2, 3, ..., n

. . . (1)

Eπ(i) ≥ dπ(i) − Cπ(i) i = 1, 2, ..., n

Tπ(i) ≥ Cπ(i) − dπ(i) i = 1, 2, ..., n

Cπ(i) > 0, Eπ(i) ≥ 0, Tπ(i) ≥ 0, rπ(i) ≥ 0, pπ(i) > 0 i = 1, 2, ..., n

. . . (S)

where π(i) denotes the position of job i in the ordering π and δ denotes

the set of all enumerated schedules.

2.3 Decomposition of Problem (S)

In this section, the problem (S) decomposed into three subproblems

with a simple structure. Some results are stated which help in solving the

problem (S).

The problem S can be decomposed into three subproblems say (SA1),

(SA2) and (SA3) where:

N1 = minπ∈δ{∑n

i=1Cπ(i)}

s.t

Cπ(i) ≥ rπ(i) + pπ(i) i = 1, 2, ..., n

Cπ(i) ≥ Cπ(i−1) + pπ(i) i = 2, 3, ..., n

rπ(i) ≥ 0, pπ(i) > 0 i = 1, 2, ..., n

. . . (SA1)


N2 = minπ∈δ{Emax(π)}

s.t

Cπ(i) ≥ rπ(i) + pπ(i) i = 1, 2, ..., n

Cπ(i) ≥ Cπ(i−1) + pπ(i) i = 2, 3, ..., n

Eπ(i) ≥ dπ(i) − Cπ(i) i = 1, 2, ..., n

Eπ(i) ≥ 0, rπ(i) ≥ 0, pπ(i) > 0 i = 1, 2, ..., n

. . . (SA2)

N3 = minπ∈δ{Tmax(π)}

s.t

Cπ(i) ≥ rπ(i) + pπ(i) i = 1, 2, ..., n

Cπ(i) ≥ Cπ(i−1) + pπ(i) i = 2, 3, ..., n

Tπ(i) ≥ Cπ(i) − dπ(i) i = 1, 2, ..., n

Tπ(i) ≥ 0, rπ(i) ≥ 0, pπ(i) > 0 i = 1, 2, ..., n

. . . (SA3)

Theorem (2.1)[4]

N1 + N2 + N3 ≤ R where N1, N2, N3 and R are the minimum objective

function values of (SA1),(SA2),(SA3) and (S ) respectively.

2.4 Special Cases

A machine scheduling problem of type NP-hard is not easily solv-

able and it is more difficult when the objective function is multi objective.

Using some mathematical programming techniques to find the optimal so-

lution for this kind of problem as: Dynamic programming and branch and

bound method. Sometimes special cases for this problem can be solved.


A special case for scheduling problem means finding an optimal schedule

directly without using mathematical programming techniques. A special

case if it exists depends on satisfying some conditions in order to make

the problem easily solvable [33]. These conditions depend on the objective

function as well as the jobs. In this section some special cases of problem

(S) are given.

Case(1): The SRD schedule gives an optimal solution for problem (S) if

pi = p and di = ip for all i in SRD.

Proof:

Since di = ipg∀i in SRD, then Emax = 0 and Tmax =∑n

i=1 Ii. Then

the problem (S) reduced to 1/ri, pi = p/∑n

i=1Ci + Tmax.

Now, since pi = p for all i in SRD, then∑n

i=1Ci =∑n

i=1

∑ij=1 Ij +

(n2+n2 )p. But (n

2+n2 )p is constant, then

∑ni=1Ci ≡

∑ni=1

∑ij=1 Ij (i.e. a

schedule that is optimal solution with respect to∑n

i=1Ci is also optimal

with respect to∑n

i=1

∑ij=1 Ij). But

∑ni=1

∑ij=1 Ij ≡ Cmax [51].

Carliar, (1982)[13] show, that SRD schedule is optimal schedule for Cmax.

Hence SRD rule gives an optimal solution for problem (S). �

Case(2): If p1 ≤ p2 ≤ · · · ≤ pn, r1 ≤ r2 ≤ · · · ≤ rn and Ci = dig∀i

in a schedule SPT , then SPT is an optimal solution for problem (S).

Proof:

Since Ci = dig∀i in SPT , then Emax = Tmax = 0. The problem (S)

reduced to 1/ri/∑n

i=1Ci, but this problem solved in SPT rule [2]. �


Case(3): If Ci = dig∀i in a schedule π and the preemptive is allowed,

then π gives an optimal solution for the problem 1/ri, pmtn/∑n

i=1Ci +

Emax + Tmax.

Proof:

SinceEmax = Tmax = 0 in π, then the problem (S) reduced to 1/ri, pmtn/∑ni=1Ci, but this problem was solved by SRPT rule [9]. Then π gives an

optimal solution for the problem 1/ri, pmtn/∑n

i=1Ci + Emax + Tmax pro-

vided that Ci = dig∀i ∈ π. �

Case(4): If in SPT schedule ri = rg∀i and satisfy Just In Time(JIT ),

then SPT gives an optimal solution for the problem 1/ri = r/∑n

i=1Ci +

Emax + Tmax.

Proof:

From (JIT ) we get Emax = Tmax = 0, then the problem (S) reduced

to 1/ri = r/∑n

i=1Ci. But this problem was solved by (SPT ) rule. Then

SPT gives an optimal solution for the problem 1/ri = r/∑n

i=1Ci+Emax+

Tmax. �

Case(5): Any schedule gives an optimal solution for the problem (S),

if ri = r, pi = p and di = dgg∀i = 1, 2, · · · , n.

Proof:

Since∑n

i=1Ci = nr+ (n2+n2 )p, Emax = max{d− (r+ p), 0} and Tmax =

max{(r+np)−d, 0} in any schedule. Then any schedule is optimal for the

problem 1/ri = r, pi = p, di = d/∑n

i=1Ci+Emax+Tmax (because the three


quantities are constants). �

Case(6): For the problem S, if ri = r, pi = pg∀i and

i. If Emax(EDD) = Emax(MST ), then EDD schedule gives the optimal

solution.

ii. If Tmax(MST ) = Tmax(EDD), then MST schedule gives the optimal

solution.

Proof: (i)

Since any sequence gives an optimal solution for 1/ri = r, pi = p/∑n

i=1Ci

problem and EDD rule gives an optimal solution for 1/ri = r, pi = p/Tmax

problem and MST gives an optimal solution for 1/ri = r, pi = p/Emax

problem. But Emax(EDD) = Emax(MST ). So EDD gives an opti-

mal solution for 1/ri = r, pi = p/∑n

i=1Ci + Emax + Tmax provided that

Emax(EDD) = Emax(MST ). �

Proof: (ii)

Since any sequence gives an optimal solution for 1/ri = r, pi = p/∑n

i=1Ci

problem and MST rule gives an optimal solution for 1/ri = r, pi = p/Emax

problem and EDD gives an optimal solution for 1/ri = r, pi = p/Tmax

problem. But Tmax(MST ) = Tmax(EDD). So MST gives an opti-

mal solution for 1/ri = r, pi = p/∑n

i=1Ci + Emax + Tmax provided that

Tmax(MST ) = Tmax(EDD). �

Case(7): For the problem S, if ri = r, di = dg∀i = 1, 2, · · · , n, and


i. If∑n

i=1Ci(MST ) =∑n

i=1Ci(SPT ), then MST schedule is the opti-

mal solution.

ii. If Emax(SPT ) = Emax(MST ), then SPT schedule is the optimal

solution.

Proof: (i)

From conditions ri = r, di = dg∀i = 1, 2, · · · , n, any order gives optimal

solution for problem 1/ri = r, di = d/Tmax. Now, since∑n

i=1Ci(MST ) =∑ni=1Ci(SPT ), thenMST minimize of the problem 1/ri = r, di = d/

∑ni=1Ci+

Emax+Tmax. �

Proof: (ii)

From conditions ri = r, di = dg∀i = 1, 2, · · · , n, any order gives opti-

mal solution for problem 1/ri = r, di = d/Tmax. Now, since Emax(SPT ) =

Emax(MST ), then SPT minimize of the problem 1/ri = r, di = d/∑n

i=1Ci+

Emax+Tmax. �

Case(8): If ri = r and SPT schedule gives di + pj ≤ djg∀gi ≺ j and

Tmax(SPT ) = Tmax(EDD), then SPT is optimal solution for the problem

(S).

Proof:

Since di+pj ≤ djg∀gi ≺ j in SPT schedule then di−pi ≤ dj−pjg∀i ≺

jg(pi > 0). Thus SPT gives optimal solution for both criteria Emax and∑ni=1Ci, and from condition Tmax(SPT ) = Tmax(EDD). Then SPT is op-

timal solution for the problem (S). �


Case(9): For the problem S, if ri = r, di = dg∀i = 1, 2, · · · , n, then any

schedule σ = (1, 2, · · · , n) with C1 < d < Cn, where C1 is Cmin and Cn is

Cmax is optimal with value of∑n

i=1Ci+Emax+Tmax is∑n

i=1Ci(σ)+Cn−C1.

Proof:

It is clear that Emax + Tmax is minimum if C1 < d < Cn with value is

Cn − C1 by considering the three possible position for d.

1. If C1 < d < Cn ⇒ Emax + Tmax = d− C1 + Cn − d = Cn − C1.

2. If d < C1 < Cn ⇒ Emax + Tmax = 0 + Cn − d > Cn − C1.

3. If C1 < · · · < Cn < d ⇒ Emax + Tmax = d− C1 + 0 > Cn − C1.

Also it is clear that SPT schedule σ is optimal for∑

σ∈S Ci(σ) as required�.

2.5 Dominance Rule

Because of branching scheme, the size of the search tree is directly

linked to the length of the current sequence (which represents the number

of nodes). Hence, a preprocessing step is performed in order to remove

as many positions as possible. Reducing the current sequence is done by

using several dominance rules. Dominance rules usually specify whether

a node can be eliminated before its lower bound is calculated. Clearly,

dominance rules are particularly useful when a node can be eliminated

which has a lower bound that is less than the optimum solution. Some of

dominance rules are valid for minimization of the sum of total completion

time, maximum earliness, and maximum tardiness.

As in the preprocessing step, similar dominance rules are also used


within the branch and bound procedure to cut nodes that are dominated

by others. These improvements lead to very large decrease in the number

of nodes to obtain the optimal solution [33].

Below three of dominance rules are stated in order to decrease the

number of nodes in search tree as well as decreasing the time.

Dominance Rule(1): If δk be a partial sequence which it’s jobs are

scheduled, K ⊂ N . For i,j∈ K = N−K,and let τ be the completion time

of last job in δk. If pi ≤ pj, di ≤ dj, Si ≤ Sj, and τ > max{ri, rj}. Then

job i proceed job j in the optimal solution for the problem (S).

Proof:

Let (δk, j, i) be the schedule which is obtained by interchanging jobs i

and j in (δk, i, j). All jobs other than i and j have the same completion

time in (δk, i, j) as in (δk, j, i). So the difference in completion time be-

tween (δk, i, j) and (δk, j, i) depends only on the completion time of jobs i

and j.

The total completion time of jobs i and j in (δk, i, j) is:

Ci + Cj = 2τ + 2pi + pj (2.4)

The total completion time of jobs i and j in(δk, j, i) is:

C′

i + C′

j = 2τ + 2pj + pi (2.5)

From (2.4) and (2.5), we get∑

i∈(δk,i,j)Ci −∑

i∈(δk,j,i)C′

i = pi − pj ≤ 0

Then∑

i∈(δk,i,j)

Ci ≤∑

i∈(δk,j,i)

C′

i (2.6)


The maximum earliness of (δk, j, i) is:

E′

max = max{Ec, E′

j, E′

i}, where Ec = maxd∈δk{Ed}

E′

j = max{dj − C′

j, 0}

E′

i = max{di − C′

i, 0}

Since C′

j ≤ C′

i and di ≤ dj, then E′

i ≤ E′

j

So E′

max = max{Ec, E′

j}.

The maximum earliness in (δk, i, j) is:

Emax = max{Ec, Ei, Ej}, where Ei = max{di − Ci, 0}

Ej = max{dj − Cj, 0}

But Ei ≤ E′

j because Si ≤ Sj, and since C′

j ≤ Cj, then Ej ≤ E′

j.

Therefore Emax = max{Ec, Ei, Ej} ≤ max{Ec, E′

j} = E′

max.

Then Emax ≤ E′

max (2.7)

For schedule (δk, i, j): The maximum tardiness is:

Tmax = max{Tm, Tj, Ti}, where Tm = maxs∈δk{Ts}

Tj = max{Cj − dj, 0}, and

Ti = max{Ci − di, 0}

For schedule (δk, j, i): The maximum tardiness is: T′

max = max{Tm, T′

j, T′

i},

where

T′

j = max{C ′

j − dj, 0} and

T′

i = max{C ′

i − di, 0}

Since C′

j ≤ C′

i and di ≤ dj, then T′

i ≥ T′

j. So T′

max = max{Tm, T′

i}.

Since Ci < C′

i ⇒ Ti < T′

i . But T′

i > Tj because Cj − dj ≤ C′

i − di.

Therefore T′

max = max{Tm, T′

i} ≥ max{Tm, Tj, Ti} = Tmax

Then T′

max ≥ Tmax (2.8)


From (2.6), (2.7), and (2.8) we get∑i∈(δk,i,j)

Ci + Emax + Tmax ≤∑

i∈(δk,j,i)

C′

i + E′

max + T′

max. �


scheduled, K ⊂ N . For i,j∈ K = N − K, and let τ be the completion

time of last job in δk. If pi ≤ pj, ri ≤ rj ≤ τ and di ≤ dj ≤ τ . Then job i

proceed job j in the optimal solution for the problem (S).

Proof:

Let (δk, j, i) be the schedule which is obtained by interchanging jobs i

and j in (δk, i, j).

The earliness of jobs i and j in (δk, i, j) and (δk, j, i) is equal to zero, since

τ ≥ dj.

From (2.4) and (2.5) we get∑i∈(δk,i,j)

Ci ≤∑

i∈(δk,j,i)

C′

i (2.9)

For schedule (δk, i, j): The maximum tardiness is:

Tmax = max{Tm, Tj, Ti}, where Tm = maxs∈δk{Ts},

Tj = max{Cj − dj, 0}, and

Ti = max{Ci − di, 0}

For schedule (δk, j, i): The maximum tardiness is:

T′

max = max{Tm, T′

j, T′

i}, where

T′

j = max{C ′

j − dj, 0} and

T′

i = max{C ′

i − di, 0}

Since C′

j < C′

i and di ≤ dj, then T′

i ≥ T′

j. So T′

max = max{Tm, T′

i}.


Since Ci < C′

i ⇒ Ti < T′

i . But T′

i ≥ Tj since di ≤ dj.

Therefore T′

max = max{Tm, T′

i} ≥ max{Tm, Tj, Ti} = Tmax

Then T′

max ≥ Tmax (2.10)

From (2.11) and (2.12) we get∑i∈(δk,i,j)Ci + Emax + Tmax ≤

∑i∈(δk,j,i)C

′

i + E′

max + T′

max. �


scheduled, K ⊂ N . For i,j∈ K = N − K, if pi ≤ pj, Si ≤ Sj, j, i are

early in (δk, i, j) and (δk, j, i) respectively and τ > max{ri, rj}. Then job

i proceed job j in the optimal solution for the problem (S).

Proof:

The tardiness of jobs i and j in (δk, i, j) and (δk, j, i) are equal to zero,

because the jobs j, i are early. Also we have since pi ≤ pj.∑i∈(δk,i,j)

Ci ≤∑

i∈(δk,j,i)

C′

i. (2.11)

The maximum earliness in (δk, i, j) is:

Emax = max{E,Ei, Ej}, where E = maxs∈δk{Es},

Ei = max{Si − τ, 0}

Ej = max{Sj − Ci, 0}

The maximum earliness in (δk, j, i) is:

E′

max = max{E,E ′

j, E′

i}, where E′

j = max{Sj − τ, 0}

E′

i = max{Si − C′

j, 0}

Since Si ≤ Sj, then E′

j ≥ E′

i. So E′

max = max{E,E ′

j} and E′

j ≥ Ei. But

E′

j ≥ Ej (since Ci > τ).


Therefore Emax = max{E,Ei, Ej} ≤ max{E,E ′

j} = E′

max.

Then Emax ≤ E′

max (2.12)

From (2.13) and (2.14) we get∑i∈(δk,i,j)Ci + Emax + Tmax ≤

∑i∈(δk,j,i)C

′

i + E′

max + T′

max. �

Example(2.1): The dominance rule illustrate in five jobs scheduling

problems

i 1 2 3 4 5

ri 0 2 5 3 8

pi 3 5 6 2 1

di 6 9 11 5 4

Solution:

51

63

ILB

UB

50 67 89 63 92

1 2 3 4 5

57 72 51 75

2 3 4 5

56 59 63

2 3 5

56

56

67

67

3

5

5

3

Figure 2.1: BAB method without dominance rule


51

63

ILB

UB

50 67 89 63 92

1 2 3 4 5

57 72 51 75

2 3 4 5

56 59 63

2 3 5

56

56

5

3

Figure 2.2: BAB method with dominance rule

The optimal schedule is (1, 4, 2, 5, 3) with∑5

i=1Ci + Emax + Tmax = 56

2.6 Upper Bound

In this section, four heuristic methods are used for ordering the jobs

and evaluating the cost of problem (S).

Heuristic (1): Order the jobs according to SPT rule, and find UB1 =

(∑n

i=1Ci + Emax + Tmax)(SPT ).

Heuristic (2): Order the jobs according to MST rule, and find UB2 =

(∑n

i=1Ci + Emax + Tmax)(MST ).

Heuristic (3): Order the jobs according to EDD rule, and find UB3 =

(∑n

i=1Ci + Emax + Tmax)(EDD).


Heuristic (4): Order the jobs according to SRD rule, and find UB4 =

(∑n

i=1Ci + Emax + Tmax)(SRD).

The heuristic which gives a minimum cost of the problem (S) among these

heuristics is chosen to be an upper bound, (i.e. UB = min{UB1, UB2, UB3,

UB4}). This UB is then used in a root node of the search tree in a branch

and bound method.

Example(2.2): The upper bound was illustrated in four jobs scheduling

problems

i 1 2 3 4

ri 0 3 3 5

pi 4 2 6 5

di 8 12 11 10

Solution:

The SPT schedule is (2,1,4,3) then UB1 =∑4

i=1Ci + Emax + Tmax = 64.

The MST schedule is (1,3,4,2) then UB2 =∑4

i=1Ci +Emax + Tmax = 55.

The EDD schedule is (1,4,3,2) then UB3 =∑4

i=1Ci +Emax + Tmax = 58.

The SRD schedule is (1,2,3,4) then UB4 =∑4

i=1Ci + Emax + Tmax = 52.

Hence UB = min{UB1, UB2, UB3, UB4} = 52.

It should be noted that an optimal sequence is (1,2,4,3) for this ex-

ample, and the optimal value is 50 which is obtained by using complete

enumeration.


2.7 Lower Bound

Deriving a lower bound for a problem (S) that has a multiple objective

function is very difficult since it is not easy to find the minimum cost for

the three objectives. Since the problem (S) is strongly NP-hard may be

find a lower bound that gives minimum value for one of them but not all.

In this section two lower bounds LB1 and LB2 are derived for problem

(S).

2.7.1 The First Lower Bound

The first lower bound is based on decomposing (S) into three sub-

problems (SA1), (SA2) and (SA3) as shown in Section (2.3), then N1 was

calculated to be the lower bound for (SA1) by SRPT rule [9], N2 was

calculated to be the lower bound for (SA2) by MRST rule [10], N3 was

calculated to be the lower bound for (SA3) by MEDD rule [10] and then

applying Theorem(2.1) to get the first lower bound for problem (S).

Example(2.3): The first lower bound was illustrated in four jobs schedul-

ing problems

i 1 2 3 4

ri 0 2 12 8

pi 6 3 7 9

di 8 5 9 11

Solution:

For the relax problem 1/ri, pmtn/∑n

i=1Ci, the SRPT rule shown in

Figure 2.3


Figure 2.3: (SRPT) rule

C1 = 9 C2 = 5 C3 = 19 C4 = 25 So N1 =∑4

i=1Ci = 58.

For the relax problem 1/ri/Emax, we assume that r = max1≤≤4{ri}

r = ri, i = 1, · · · , 4, then the problem reduce to 1/ri = r/Emax which was

solved by MST rule.

C1 = 37 C2 = 15 C3 = 22 C4 = 31

E1 = 0 E2 = 0 E3 = 0 E4 = 0 So N2 = Emax = 0.

For the relax problem 1/ri, pmtn/Tmax, the MEDD rule shown in

Figure 2.4

Figure 2.4: (MEDD) rule

C1 = 9 C2 = 5 C3 = 19 C4 = 25

T1 = 1 T2 = 0 T3 = 10 T4 = 14 So N3 = Tmax = 14.

Then LB1 = N1 +N2 +N3 = 58 + 0 + 14 = 72.

2.7.2 The second Lower Bound

The second lower bound can be calculated for the problem (S) by using

the relaxation of constraints of objective function as follows:

For the problems (SA1) and (SA3), we assume that r∗ = min1≤i≤n{ri}


and ri = r∗, i = 1, 2 . . . , n, to get the problems 1/ri = r∗/∑n

i=1Ci and

1/ri = r∗/Tmax, which are solved by SPT and EDD rules respectively.

For the problem (SA2), we assume that r = max1≤i≤n{ri} and r = ri, i =

1, 2, · · · , n, to get the problem 1/ri = r/Emax, which was solved by MST

rule and then applying Theorem (2.1) to get the second lower bound for

problem (S).

Hence the lower bound is LB = max{LB1, LB2}.

Example(2.3):The second lower bound illustrates in four jobs scheduling

problems

i 1 2 3 4

ri 0 1 2 1

pi 4 2 6 5

di 8 12 11 10

Solution:

Let r∗ = min1≤i≤4{ri} = 0, then SPT gives the schedule (2, 1, 4, 3) with

N1 =∑4

i=1Ci = 36 and the EDD gives the schedule (1, 4, 3, 2) with

N3 = Tmax = 5.

Let r = max1≤i≤4{ri} = 2, then MST gives the schedule (1, 3, 4, 2) with

N2 = Emax = 2.

Then LB2 = N1 +N2 +N3 = 36 + 2 + 5 = 43.

2.8 Branch and Bound Algorithm

In this section,a description of BAB algorithm is given and its imple-

mentation. The heuristic method is applied at the top of search tree (root

node) to provide the UB on cost of an optimal schedule is obtained by


choosing the upper bound from Section (2.6). Also at the top of the search

tree an ILB on the cost of an optimal schedule is obtained by choosing

the better of two lower bounds from Section (2.7). The algorithm uses a

forward sequencing branching rule for which nodes at level k of the search

tree corresponds to initial sequences in which jobs are sequenced in the

first k positions. The branching procedure describes the method to par-

tition a subset of possible solution. These subsets can be treated as a set

of solutions of corresponding subproblems of the original problem. The

bounding procedure indicates how to calculate the LB on the optimal so-

lution value for each subproblem generated in the branching process. The

search strategy describes the method of choosing a node of the search tree

to branch from it; we usually branch from a node with smallest LB among

the recently created nodes [4].

2.9 Computational Experience

An intensive work of numerical experimentations has been performed.

Subsection (2.9.1) shows how instances (test problems) can be randomly

generated.

2.9.1 Test Problems

There exists in the literature a classical way to randomly generate test

problems of scheduling problems.

• The processing time pj is uniformly distributed in the interval [1, 10].


• The release date rj is uniformly distributed in the interval [0, αP ],

where [α=0.125, 0.25, 0.50, 0.75, 1.00] and P =∑n

i=1 pi.

• The due date dj is uniformly distributed in the interval [P(1-TF-

RDD/2), P(1-TF+RDD/2)]; where P =∑n

i=1 pi. Depending on the

relative range of due date (RDD) and on the average tardiness factor

(TF).

For both parameters, the values 0.2, 0.4, 0.6, 0.8 and 1.0 are considered.

For each selected value of n, where n is the number of jobs, five problems

were generated.

2.9.2 Computational Experience with the Lower and Upper

Bounds of (BAB) Algorithm

The BAB algorithm was tested by coding it in MATLAB 7.10.0 (R2010a)

and implemented on Intel(R) Core(TM)2 Duo CPU T6670 @ 2.20 GHZ,

with RAM 2.00 GB personal computer.

Table (2.1), shows the results for problem (S) obtained by BAB al-

gorithm. The first column ”n” refers to the number of jobs, the second

column ”EX” refers to the number of examples for each instance n, where

n ∈ {5, 10, 15, 20, 25, 30}, the third column ”Optimal” refers to the opti-

mal values obtained by BAB algorithm for problem (S), the fourth column

”UB” refers to the upper bound, the fifth column ”ILB” refers to the ini-

tial lower bound, the sixth column ”Nodes” refers to the number of nodes,

the seventh column ”Time” refers to the time cost ’by second’ to solve the

problem, the last column ”Status” refers to the problem solved ’0’ or not


’1’. The symbols ”*” refers to the UB gives an optimal solution and ”**”

refers to the ILB gives an optimal solution. The BAB algorithm was

stopped when the sum of ”status column≥ 3”. A condition for stopping

the BAB algorithm was determined and considering that the problem is

unsolved (state is 1), that the BAB algorithm is stopped after a fixed pe-

riod of time, here after 1800 second (i.e. after 30 minutes).

If the value of UB=ILB then the optimal is UB and there is no need

to branch the serch tree of BAB algorithm.

From Table (2.1), it is noticed that the heuristic of upper bound gives

good results. It gives the value for objective function equal to optimal or

near optimal values.

Table (2.1): The performance of initial lower bound, upper bound, num-

ber of nodes and computational time in second of (BAB) algorithm

for (S).

n EX Optimal UB ILB Nodes Time Status

1 127 127* 125 36 0.00673 0

2 57 58 57** 14 0.0035 0

5 3 87 93* 87** 22 0.0080 0

4 77 77* 76 14 0.0025 0

5 60 60* 60** 0 0.0008 0

1 352 401* 336 1396 0.1488 0

2 201 226 196 1547 0.1583 0

10 3 337 368 327 414 0.0449 0


Table (2.1) continued

n EX Optimal UB ILB Nodes Time Status

4 389 408 376 1265 0.1385 0

5 146 195 143 67 0.0128 0

1 660 744 644 12529 1.5424 0

2 557 672 549 16211 2.3152 0

15 3 720 803 720** 210 0.0337 0

4 353 378 331 273 0.0473 0

5 567 599 532 252493 35.0969 0

1 939 1030 935 5277 0.8577 0

2 1074 1281 1064 7024 1.1478 0

20 3 769 912 762 7535 1.6953 0

4 943 1144 913 2676012 479.7449 0

5 777 929 769 12812 2.2995 0

1 1502 1908 1463 4230692 434.4436 0

2 1649 2021 1570 3974179 398.7797 0

25 3 1877 2040 1746 17151377 1800.0001 1

4 1363 1547 1359 487324 51.7407 0

5 1847 1963 1757 17791999 1800.0011 1

1 2883 3278 2767 16378467 1800.0003 1

2 2106 2578 2076 5231216 568.0532 0

30 3 1973 2420 1878 16315296 1800.0001 1

4 1873 2418 1796 4174669 470.0744 0

5 2086 2599 2056 665806 73.6937 0


Table (2.2) summarizes Table (2.1)

Table (2.2): Summary of Table (2.1) of (BAB) algorithm

n Av.Nodes Av.Time Unsolved problem

5 17.2 0.164 0

10 937.8000 0.1007 0

15 5.6343 7.8071 0

20 1031581 70.1028 0

25 8.7271 294.9880 2

30 8.5531 370.6000 2

Table (2.2) is the summary of Table (2.1), and shows the average of

nodes and computational times for the solved problems. It also shows the

unsolved problems among the 5 problems of each n, where n ∈ {5, 10, 15, 20,

25, 30}.

Chapter 3

Neural Networks to Solve

Scheduling Problems

3.1 Introduction

3.1.1 Artificial Neural Network (ANN)

Artificial neural network is an information processing system that has

certain performance characteristics in common with biological neural net-

works [54]. Below are some characteristics that are similar in both artificial

neural network and biological neural networks [39]:

1. The processing element (neuron) receives many signals.

2. Signals may be modified by a weight at the receiving element.

3. The weighted inputs are the processing element sums.

4. Under appropriate circumstances (sufficient input), the neuron trans-

mits a signal output.

46

Chapter Three: Neural Networks to Solve Scheduling Problems j 47

5. The output from a particular neuron may go to many other neurons

(the axon branches).

6. Information processing is local.

7. Memory is distributed:

• Long-term memory resides in the neurons synapses or weights.

• Short-term memory corresponds to the signals sent by the neu-

rons.

8. A synapse strength may be modified by experience.

9. Neurotransmitters for synapse may be excitatory or inhibitory.

Also, ANN can be thought of as ” black box” devices that accept

inputs and produces outputs [42]. ANN was inspired by the manner in

which the heavily interconnected, parallel structure of the human brain

processes information. There are collections of mathematical processing

units , known as neurons, which has a propensity for storing, making easily

available, experiential knowledge, emulate some of the observed proper-

ties of biological nervous systems and draw on the analogies of adaptive

biological learning [43]. A neural network element (neuron) is the smallest

processing unit of the whole network essentially forming a weighted sum

and transforming it by activation function to obtain the output in order

to gain sufficient computing power, several neurons are interconnected

together [54]. ANN resembles the brain in two respects [43]:


1. Knowledge is acquired by the network from its environment through

learning processes, where, knowledge in a NN is represented in the

values of the weights and biases, which form part of large and dis-

tributed network.

2. Inter-neuron connection strengths, known as synaptic weights, are

used to store the acquired knowledge.

Artificial neural networks have been developed as generalizations of

mathematical models of human brain or neural biology, based on the as-

sumptions that [39]:

1. Information processing occurs at many simple elements called neu-

rons.

2. Signals are passed between neurons over connection links.

3. Each connection link has an associated weight.

4. Each neuron applies an activation function (usually nonlinear).

A neural net consists of a large number of simple processing elements

called (neurons, units, cells, or nodes). Each neuron is connected to other

neurons by means of directed communication links, each with an asso-

ciated weight. The weights represented information being used by the

net to solve a problem. A neural network can be applied to a wide vari-

ety of problems, such as storing and recalling data or patterns, classify-

ing patterns, performing general mappings from input patterns to output

patterns, grouping similar patterns, or finding solutions to constrained


optimization problems. Each neuron has an internal state, called its acti-

vation or activity level, which is a function of the inputs it has received.

Typically, a neuron sends its activation as a signal to several other neu-

rons. It is important to note that a neuron can send only one signal at a

time, although that signal is broadcast to several other neurons.

For example, consider a neuron Y , illustrated in Figure (3.1), that

receives inputs from neurons X1, X2, and X3. The activations (output

signals) of these neurons are x1, x2, and x3, respectively. The weights on

the connections from X1, X2, and X3 to Y are w1, w2, and w3, respectively.

The net input, y−in = (net), to neuron Y is the sum of input signals from

neurons X1, X2,and X3 multiplied by the weights on the connections be-

tween them, i.e.,

y−in = (net) =∑3

i=1wixi.

Figure 3.1: A simple artificial neuron.

3.1.2 Architecture of Neural Network

The architecture of the neural network refers to its framework as well

as its interconnection scheme. The number of layers and the number of

nodes per layer often specify the framework [54]. Often, it is convenient


to visualize neurons as arranged in layers. Typically, neurons in the same

layer behave in the same manner. The behavior of neurons depends on

activation function and the pattern of weighted connections over which it

sends and receives signals.

The arrangement of neurons into layers and the connection patterns

between layers is called the net architecture [39]. These layers are:

Input Layer: A layer of neurons which are called input units or sensory

layer, that receives information from external sources, and passes this

information to the network for processing. These information may be

either sensory inputs or signals from other systems outside the one

being modeled [8]. The input layer perform not computation but

distributed information to other units in the successive layer [54].

Hidden Layer: Layer of the nodes which are called hidden units or pro-

cess layer, that receives information from the input layer and pro-

cesses them in a hidden way. It has no directed connections to the

outside world (inputs or outputs). All connections from the hidden

layer to other layers within the system [42]. This layer provide into

the networks the capability to map or classify nonlinear problems

[54]. The neural network may have one or more hidden layers and

some it do not have.

Output Layer: Layer of the nodes which are called output units or re-

sponse layer, that receives processing information and sends outputs


signals out of the system [42]. This layer encodes possible concepts

(or values) to be assigned to the instance under consideration. For

example each output unit represents a class of objective [54].

Hence, the neural networks can be classified with respect to the layers as

follows [39]:

i. Single Layer Net: A single layer net has one layer of connection weights.

Often, the units can be distinguished as input units, which receive

signals from the outside world, and output units, from which the re-

sponse of the net can be read. In the typical single layer net shown

in Figure (3.2), the input units are fully connected to output units

but the input units and the output units are not connected to other

units in the same layer.

Figure 3.2: Single Layer Net.


ii. Multilayer net: Is a net with one or more hidden layers (or levels) of

nodes (so-called hidden units) between the input units and the output

units. Typically, there is a layer of weights between two adjacent

levels of units (input, hidden, or output). Multilayer nets can solve

more complicated problems than can single layer nets, but training

may be more difficult. However, in some cases, training may be more

successful, because it is possible to solve a problem that a single layer

net cannot be trained to perform correctly at all [54]. The multilayer

nets illustrated in Figure (3.3).

Figure 3.3: A multilayer neural net.


3.1.3 Network Learning Methods [45]

Among the many interesting properties of a neural network is the

ability of the network to learn from its environment and to improve its

performance through learning. A neural network learns about its environ-

ment through an iterative process of adjustments applied to its weights

and thresholds. The types of learning are determined by the manner in

which the weights changes take place.

The learning process implies the following sequence of events:

1. The neural network is stimulated by an environment.

2. The neural network undergoes changes as a result of this stimulation.

3. The neural network responds in a new way to the environment, be-

cause of the changes that have occurred in its internal structure.

3.1.3.1 Error-correction learning

Figure 3.4: Error-correction learning diagram.


3.1.3.2 Supervised learning

An essential ingredient of supervised is the availability of an external

teacher, which is able to provide the neural network with a desired or tar-

get response. The network parameters are adjusted under the combined

influence of the training vector and the error signal. This adjustment is

carried out iteratively in a step-by-step fashion with the aim of eventu-

ally making the neural network emulate the teacher. This form of super-

vised learning is in fact an error-correction learning, which was already

described.

3.1.3.3 Unsupervised learning

In unsupervised or self-organized learning there is no external teacher

to oversee the learning process. In other words, there are no specific

samples of the function to be learned by the network. Rather, provision is

made for a task-independent measure of the quality of representation that

the network is required to learn and the free parameters of the network are

optimized with respect to that measure. Once the network has become

tuned to the statistical regularities of the input data, it develops the ability

to form internal representations for encoding features of the input and

thereby creates new classes automatically.


3.1.4 The activation functions [39, 45]

hhbh

Figure 3.5: Activation Functions

3.1.5 Applications of Neural Networks

The study of neural networks is an extremely interdisciplinary field,

both in its development and in its application. A brief sampling of some

of the areas in which neural networks are currently being applied suggests

the breadth of their applicability. The examples range from commercial


successes to areas of active research that show promise for the future. The

able of neural networks on solve the complex problems implies to use in

many applications, such as [39]:

Signal processing, Control, Pattern Recognition, Medicine, Speech pro-

duction, Speech Recognition, and Business.

3.1.6 Beginning of Neural Nets

The McCulloch-Pitts neuron is perhaps the earliest artificial neuron

in 1943. It displays several important features found in many neural net-

works. The requirements for McCulloch-Pitts neurons as follows [39]:

1. The activation of McCulloch-Pitts neuron is binary, and the neuron

either fires (has an activation of 1) or does not fire (has an activation

of 0).

2. McCulloch-Pitts neurons are connected by directed, weighted paths,

where the neurons are connected by directed paths (has weights).

3. A connected path is excitatory if the weight on the path is positive;

otherwise it is inhibitory. All excitatory connections into particular

neuron (one neuron) have the same weights.

4. Each neuron has a fixed threshold such that if the sum of net inputs

to the neuron is greater than the threshold, then the neuron is fire.

5. The threshold is set so that the inhibition neuron is absolute. That

is, any nonzero inhibition input will prevent the neuron from firing.


6. The signal takes one step to pass over one connection link from neuron

to other.

3.2 Related Works

Various applications, such as communications, routing, industrial con-

trol, operations research, and production planning employ scheduling con-

cepts. Most problems in these applications are confirmed to be NP-

complete or combinatorial problems. This fact implies that an optimal

solution for a large scheduling problem is rather time consuming. The

traveling salesman problem (TSP ) is a typical NP-complete problem,

comprising a Hamiltonian cycle, which seeks a tour that has a minimum

cost; obtaining the optimal solution is very time consuming. Various

schemes have been developed for solving the scheduling problem. Linear

programming is a widely used scheme for determining the cost function

based on the specific scheduling problem. Willems and Rooda translated

the job-shop scheduling problem into a linear programming format, and

then mapped it into an appropriate neural network structure to obtain

a solution [53]. Furthermore, Foo and Takefuji employed integer linear

programming neural networks to solve the scheduling problem by mini-

mizing the total starting times of all jobs with a precedence constraint

[55]. Meanwhile, Zhang et al. proposed a neural network method derived

from linear programming, in which preemptive jobs are scheduled based

on their priorities and deadline [18]. Additionally, Cardeira and Mammeri

investigated the multiprocessor real time scheduling by applying the k-


out-of N rule to a neural network [17]. Above investigations concentrated

on the preemptive jobs (processes) executed on multiple machines (mul-

tiprocessor) with job transfer permitted by applying a neural network.

Meanwhile, Hanada and Ohnishi [6] developed a parallel algorithm based

on a neural network for preemptive task scheduling problems by allowing

for a task transfer among machines. Park et al. [36] embedded a classical

local search heuristic algorithm into the TSP optimization neural network.

The next section introduces the problem statement. The neural network

method and the scheduling of some small systems are determined in the

section after.

3.3 Problem Statement

The machine scheduling problem studied in this section requires n in-

dependent jobs Ji(i = 1, 2, · · · , n) to be processed on a single machine

with the following assumptions:

i. Each job i has a release time ri.

ii. The single machine can process one job at most at a time.

iii. No preemption is allowed.

Let

π Schedule for the n jobs.

pi processing time of job i on the machine.


di due date of job i.

Ci completion time of job i, for the schedule π, Ci =∑i

j∈π,j=1 pj

Ei =max{0, di − ci}

Ti =max{0, ci − di}

The objective function can be written as:

f(π) =n∑i=1

Cπ(i) + Emax(π) + Tmax(π) (3.1)

In this chapter, we use multilayer perceptron neural networks as a

heuristic method for solving the problem (S).

3.4 Multilayer Perceptron NN for Single Machine

Multilayer perceptron neuron network consists of a set of sensory units

(source nodes) that constitute the input layer, one or more hidden asso-

ciative layers of computation nodes, and response units of computation

nodes. The input signal propagates through the network in a forward

direction, on a layer-by-layer basis. Multilayer perceptrons have been

applied successfully to solve a number of diverse, difficult problems by

training them in a supervised manner with a highly popular algorithm

known as the back-propagation algorithm. This algorithm is based on

error-correction learning rule. Error back-propagation learning consists of

two passes through the different layers of the network: a forward passes

and a backward pass.


In the forward pass, an activity pattern (input vector)is applied to

the sensory nodes of the network, and its effect propagates through the

network layer by layer. Next, a set of outputs is produced as the actual

response of the network. During the forward pass the synaptic weights of

the networks are all fixed.

During the backward pass, on other hand, the synaptic weights are

all adjusted in accordance with an error-correction rule. Specifically, the

actual response of the network is subtracted from a desired response to

produce an error signal. This error signal is then propagated backward

through the network against the direction of synaptic connections hence

the name (error back-propagation ). The synaptic weights are adjusted

to make the actual response of the network move closer to the desired

response in a statistical sense [7].

The structure of a Perceptron algorithm is presented in the following

steps [39]:

Step 0: Initialize weights and bias. (For simplicity, set weights and bias

to zero). Set learning rate α(0 < α ≤ 1). (set threshold value θ).

Step 1: While stopping condition is false, do Steps 2-6.

Step 2: For each training pair s:t, do Steps 3-5.

Step 3: Set activations of input units:

xi = si.

Step 4: Compute actual response of output unit:

y−in = b+∑n

i=1 xiwi


where n is the number of nodes in the input layer

y =

1 if y−in > θ

0 if −θ ≤ y−in ≤ θ

−1 if y−in < −θ

Step 5: Update weights and bias if an error occurred for this pattern.

If y 6= t,

wi(new) = wi(old) + αtxi,

b(new) = b(old) + αt.

else

wi(new) = wi(old),

b(new) = b(old).

Step 6: Test stopping condition:

If no weights changed in Step 5, stop; else, continue do steps (2-6).

The structure of Backpropagation algorithm is presented in the follow-

ing steps [39]:

Inputs : Training pairs of input and the desired output. Weights range,

sigmoid alpha value, learning rate, error limit.

Outputs : A learned neural network with updated weights .

Begin :

Step 0: Initialize weights. (Set to small random values).

Step 1: While stopping condition is false, do Steps 2-9.


Step 2: For each training pair, do Steps 3-8.

Feedforward:

Step 3: Each input unit (Xi, i = 1, · · · , n) receives input signal xi and

broadcasts this signal to all units in the layer above (the hidden

units).

Step 4: Each hidden unit (Zj, j = 1, 2, · · · , p) sums its weighted input

signals,

z−inj = v0j +∑n

i=1 xivij,

applies its activation function to compute its output signal for hidden

units,

zj = f(z−inj),

and sends this signal to all units in the layer above (output units).

Step 5: Each output unit (Yk, k = 1, . . . ,m) sums its weighted input

signals,

y−ink = w0j +∑p

j=1 zjwjk

and applies its activation function to compute its output signal,

yk = f(y−ink).

Error Backpropagation:

Step 6: Each output unit (Yk, k = 1, · · · ,m) receives a target pattern

corresponding to the input training pattern, computes its error infor-

mation term,

δk = (tk − yk)f′(y−ink),


calculates its weight correction term (used update wjk later),

∆wjk = αδkzj,

calculates its bias correction term (used update w0k later),

∆w0k = αδk,

and sends δk to units in the layer below.

Step 7: Each hidden unit (Zj, j = 1, · · · , p) sums its delta inputs (from

units in the layer above),

δ−inj =∑m

k=1 δkwjk,

multiplies by the derivative of its activation function to calculate its

error information term,

δj = δ−injf′(z−inj),

calculates its weight correction term(used update vij later),

∆vij = αδjxi,

and calculates its bias correction term(used update v0j later),

∆v0j = αδj.

Update weights and biases:

Step 8: Each output unit (Yk, k = 1, · · · ,m) updates its bais and weights

(j = 0, · · · , p):

wjk(new) = wjk(old) + ∆wjk.

Each hidden unit (Zj, j = 1, · · · , p) updates its bias and weights (i =

0, · · · , n):

vij(new) = wij(old) + ∆vij

Step 9: Test stopping condition.


3.5 Proposed Neural Network Design

The proposed design of ANN is to use the permutations as an input to

it. The designed ANN is a supervised learning neural net. The extracted

values are better values of each input the desired output for each learned

object. The system of ANN design consists of four main layers (Input

layer I, two hidden layers H1, H2, and Output layer O), it can be shown

as I | H1 | H2 | O completely connected net and illustrated in Figure

(3.6), the design of each layer becomes:

• Input Layer (I):The input vector X is feeding into layer I. It has l

nodes, xi(i = 1, · · · , l).

• First Hidden Layer (H1):It has m hidden nodes, hj(j = 1, · · · ,m).

This number is practically chosen by made many training trials on

system even to reach to highest system accuracy).

• Second Hidden Layer (H2):It has t hidden nodes, zk(k = 1, · · · , t).

This number is practically chosen by made many training trials on

the system even to reach to highest system accuracy.

• Output Layer (O):It has s nodes, yr(r = 1, · · · , s).

• Interconnecting weights between the nodes of layer I and the nodes

of layer H1 are denoted as wlm.

• Interconnecting weights between the nodes of layer H1 and the nodes

of layer H2 are denoted as vmt.


• Interconnecting weights between the node of layer H2 and the nodes

of layer O are denoted as uts.

• Internal threshold value θ for every nodes in layers I,H1, H2, and O

in the interval [0.1,0.9].

• Internal bias value (b) for every layers.

Figure 3.6: The proposed design of ANN

3.6 Artificial Neural Network Training

At this phase the system will be trained to solve the new problems

that coming later from extracted data. At this time the initial weights


of ANN will be created randomly from the interval [-0.3,0.3], with error

limit = 10−6. These values have been chosen because better practical

results was achieved by using them.

The structure of system ANN training algorithm is presented in the

following steps:

Inputs: Data base buffer, contains training data set with Number Trial

recommended of n values and optimal value as desired output where

(n=3,4,5,6,7,8).

Outputs: Trained ANN with updated weights, saved in database buffer.

Begin

Step0: Open Data base buffer as BuffDB.

Step1: Create ANN with I|H1|H2|O design.

Step2: Create Sample Learning Pair Array SampleArr contains Input

Vector and Desired Output.

Step3: Initialize Weights vector randomly from [-0.3,0.3].

Step4: Set error limit e←− 10−6,itration←− 10000.

Step5: Initialize array Index of SampleArr array to the beginning of

it.

Step6: For each selected training objects record CurrRecord in BuffDB

do SampleArr [Index].

Step7: Read Input Vector from CurrRecord. SampleArr [Index].


Step8: DesieredOutput Read optimal from CurrRecord.

Step9: Index←− Index+1.

Step10: End For

Step11: Call Backpropagation Algorithm Save Current Network with

new values in BuffDB.

Step12: Update BuffDB Database file.

Step13: End.

Table (3.1): Results of neural networks learning.

n No. nodes H1 H2 No. Trial Error Iteration Threshold Accuracy

in I permutation No. Limit %

30 40 25 997 ∗ 10−6 614 0.8 97

3 9 30 40 6 50 995 ∗ 10−6 730 0.8 98

30 40 75 994 ∗ 10−6 800 0.8 98

30 40 100 992 ∗ 10−6 863 0.8 100

40 30 25 998 ∗ 10−6 1000 0.8 93

4 12 40 30 24 50 999 ∗ 10−6 1350 0.8 93

40 30 75 975 ∗ 10−6 1560 0.8 95

40 30 100 970 ∗ 10−6 2100 0.8 96

60 40 25 999 ∗ 10−6 613 0.7 92

5 15 60 40 120 50 983 ∗ 10−6 730 0.7 93

60 40 75 990 ∗ 10−6 819 0.7 95

60 40 100 950 ∗ 10−6 819 0.7 96


Table (3.1) continued

n No. nodes H1 H2 No. Trial Error Iteration Threshold Accuracy

in I permutation No. Limit %

40 20 25 1795 ∗ 10−6 1830 0.9 95

6 18 40 20 720 50 1773 ∗ 10−6 1900 0.9 96

40 20 75 1705 ∗ 10−6 1960 0.9 97

40 20 100 1683 ∗ 10−6 2000 0.9 98

40 20 25 975 ∗ 10−6 1350 0.8 95

7 21 40 20 5040 50 969 ∗ 10−6 1225 0.8 97

40 20 75 963 ∗ 10−6 1900 0.8 98

40 20 100 950 ∗ 10−6 2025 0.8 100

40 20 25 889 ∗ 10−6 5300 0.9 93

8 24 40 20 40320 50 713 ∗ 10−6 6300 0.9 95

40 20 75 725 ∗ 10−6 7020 0.9 97

40 20 100 727 ∗ 10−6 10000 0.9 98

3.7 The Matrix of Weights

In this section we describe the matrices weights for three and four jobs.

where the matrices weights are

W =

w11 w12 . . . w130

w21 w22 . . . w230

...... . . . ...

w91 w92 . . . w930

=

0.1079 −0.6495 . . . −0.5301

−0.8642 −1.1514 . . . −1.1809

...... . . . ...

−0.0182 1.2561 . . . 0.7903


Figure 3.7: Input of 3 jobs

V=

v11 v12 . . . v140

v21 v22 . . . v240...

... . . . ...

v301 v302 . . . v3040

=

−0.4903 −0.3486 . . . −0.6382

0.6706 0.6023 . . . −0.4330

...... . . . ...

0.0034 0.3282 . . . −0.2753

U=

u11 u12 u13

u21 u22 u23...

......

u401 u402 u340

=

2.5975 2.2437 −0.9240

1.7057 0.7443 −1.5584

......

...

1.7064 1.9374 1.6869

where the matrices weights are

W =

w11 w12 . . . w140

w21 w22 . . . w240

...... . . . ...

w121 w122 . . . w1240

=

2.1518 2.2450 . . . −0.4642

0.4515 4.3378 . . . −0.4591

...... . . . ...

0.6582 1.9460 . . . −0.2511


Figure 3.8: Input of 4 jobs

V=

v11 v12 . . . v130

v21 v22 . . . v230...

... . . . ...

v401 v402 . . . v4030

=

0.3416 0.9118 . . . −0.1471

−1.9146 −1.8133 . . . −0.2785

...... . . . ...

0.5719 0.1553 . . . 0.2418

U=

u11 u12 u13

u21 u22 u23...

......

u301 u302 u303

=

3.0227 −3.1546 −1.6656

3.2392 −3.5525 −2.1573

......

...

0.5255 0.2680 −0.8063

Chapter 4

Conclusions and Future Work

4.1 Conclusions

In this thesis, the problem of scheduling jobs on one machine for a

variety of three criteria was considered.

A Branch and Bound algorithm was proposed to find the optimal so-

lution for the problem 1/ri/∑n

i=1Ci+Emax+Tmax with two lower bounds

(LB1, LB2), four upper bounds (UB1, UB2, UB3, UB4) and three domi-

nances rule. Nine special cases for the last problem were derived and

proved.

For the multi-criteria scheduling problem 1/ri/∑n

i=1Ci +Emax +Tmax

the neural network method was proposed to find the near optimal for small

size problem.

4.2 Future Works

An interesting future research topic would involve experimentation

with the following problems:

71

Chapter Four: Conclusions and Future Work j 72

1) 1/ri/F (∑n

i=1Ci, Emax,∑n

i=1 Ui)

2) 1/ri/F (∑n

i=1Ci,∑n

i=1 Ui, Tmax)

3) 1/ri/F (Cmax, Tmax, Emax)

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المستخلص

واحدة ماكنة على Jobs)) النتاجات من n جدولة مسألة دراسة رسالةال ههذ في مت

لهذه .(MOF) (Multiple Objective Function)لتصغير دالة متعددة االهداف

مجموع مسألة لOptimal solution) )يجاد الحل االمثل إالهدف االول , الدراسة هدفان

أزمنة للنتاجات كونت عندماخير أكبر زمن تأو تبكيرزمن كبر أو لنتاجاتاأتمام زمن

هذه المسألة تمثل بالصيغة .بتجزئة عمل النتاجات السماح عدم متساوية و غير تحضير

(1/ / والهدف , (حسب علمنا)وهي مسألة لم تدرس من قبل ( + +

.بأستخدام الشبكات العصبية لمسألةنفس اول تقريبية لحل يجادإالثاني

من القيود الدنيا ينخوارزمية التفرع والتقيـد مع اثن اقترحنا ,األولبالنسبة للهدف

ه الرسالة المقترحة في هذ ( , , , )من القيود العليا وأربعة ( , )

والتي أنتجت حلول خاصة حاالت تسعةوبرهان تمكنا من اشتقاق كذلك. اليجاد الحل االمثل

قي تقليص عدد تساعد هيمنةللاعد وق ةثالث مع .بدون استخدام خوارزمية التفرع والتقيد مثلى

قيد ن خوارزمية التفرع والتبأ أثبتتنتائج االختبارات الحسابية .في شجرة البحث اتعتفرال

هذه .دقيقة (30)قل او يساوي أنتاج في وقت (30) لغاية ألةالمقترحة فعالة في حل المس

(.NP-hard Strongly) عام من النوع بشكل المسألة

(NP-hard Strongly) المعقد النوع من هي مسألتنا أن بما ,ما بالنسبة للهدف الثانيأ

عمليا .تقريبية حلول إليجاد Neural Networks))ة الشبكات العصبية قيطر باستخدام قمنا

حل تستطيعة الشبكات العصبية يقطر بأن وجد الحسابية التجارب خالل ومن

جيدةتكون ة الشبكات العصبيةيقطر أنكما الحظنا , معقول وبوقت تانتاج(8) لغاية المسألة

.قريبة من الحل االمثل لدالة الهدف لحلوإذ أنها تعطي , في بعض المسائل

مجهوريةالعراق

العلميوزارةالتعلميالعايلوالبحث

الرصفةلكيةالرتبيةللعلوم/ رجامعةذيقا

تصغريثالثةأ هدافمتساويةال مهيةيفمسأ ةلجدوةلاملاكنة

مقدمةرساةل

ىل جامعةذيقار/للعلومالرصفةلكيةالرتبية/تقسمالرايضيااإ

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منقبل

جعفرصاحلعنيد

أ رشاف

ادلكتورال س تاذادلكتورال س تاذاملساعد

املوسوياكظمهمديهامشمحمداكظمزغريالزويين

م3102 ھ0121

minimizing three simultaneous criteria in machine ...utq.edu.iq/final2/111.pdf · minimizing three...

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