Single CNC machine scheduling with controllable processing times and multiple due dates

a thesis

submitted to the department of industrial engineering

and the institute of engineering and science

of bilkent university

in partial fulfillment of the requirements

for the degree of

master of science

By

Mehmet O˘

guz Atan

July, 2004

Assoc. Prof. M. Selim Akt¨

urk(Advisor)

I certify that I have read this thesis and that in my opinion it is fully adequate,

in scope and in quality, as a thesis for the degree of Master of Science.

Asst. Prof. Mehmet R¨

u¸st¨

u Taner

I certify that I have read this thesis and that in my opinion it is fully adequate,

in scope and in quality, as a thesis for the degree of Master of Science.

Asst. Prof. Yavuz G¨

unalay

Approved for the Institute of Engineering and Science:

Prof. Mehmet B. Baray

Director of the Institute

CONTROLLABLE PROCESSING TIMES AND

MULTIPLE DUE DATES

Mehmet O˘

guz Atan

M.S. in Industrial Engineering

Supervisor: Assoc. Prof. M. Selim Akt¨

urk

July, 2004

In order to reflect the characteristics of a modern manufacturing environment,

elements of customer satisfaction and the competition between firms should be

considered simultaneously. Manufacturers should be careful on deciding which

orders to accept, and should pay attention on the weighted earliness and

tardi-ness penalties they will be due, while considering the priorities of the customers.

Customers that want to minimize the risk of deviation from a delivery date, offer

multiple due dates to the manufacturer, each coming with a distinct price for

the order that is decreasing as the date gets later. Manufacturers that use

flexi-ble manufacturing systems have the capability to control the processing times of

jobs, by changing the machining conditions at the expense of tooling costs. In

this study, we consider the problem of scheduling a set of jobs on a single CNC

machine, while maximizing the total profit that is composed of sum of prices of

scheduled jobs less the sum of total weighted earliness/tardiness cost, tooling cost

and machining cost. This problem is NP-hard since the total weighted tardiness

problem is NP-hard alone. Furthermore, because of the nature of the tooling cost,

the problem is nonlinear. We propose a number of ranking rules and scheduling

algorithms. Using these rules and algorithms, we construct a single-pass heuristic

algorithm that determines the processing times for each job and schedules them

simultaneously, to maximize the overall profit.

Keywords: Scheduling, Single Machine, Total Weighted Tardiness and Earliness,

Multiple Due Dates, Controllable Processing Times, Heuristics, Order Rejection.

KONTROL ED˙ILEB˙IL˙IR ¨

URET˙IM ZAMANLARI VE

B˙IRDEN C

¸ OK TESL˙IM TAR˙IH˙I VARLI ˘

GINDA CNC

TAKIM TEZGAHI C

¸ ˙IZELGELEMES˙I

Mehmet O˘

guz Atan

End¨

ustri M¨

uhendisli˘

gi, Y¨

uksek Lisans

Tez Y¨

oneticisi: Do¸c. Dr. M. Selim Akt¨

urk

Temmuz, 2004

Modern bir ¨

uretim ortamının ¨

ozelliklerini yansıtabilmek i¸cin m¨

u¸steri

mem-nuniyeti ve firmalar arası rekabet unsurlarının birarada g¨

oz¨

on¨

unde

bulundurul-ması gerekir. ¨

Ureticiler m¨

u¸steriyi memnun edebilmek i¸cin m¨

u¸steri ¨

onceliklerini

hesaba katıp hangi sipari¸sleri kabul edeceklerinin kararını verecekleri gibi ¨

odemek

durumunda kalacakları toplam a˘

gırlıklı gecikme ve erken bitirme maliyetlerini de

dikkate almalıdırlar. Teslim tarihinin sapması ihtimalini g¨

oz¨

on¨

unde bulunduran

m¨

u¸steriler, ¨

ureticiye herbiri di˘

gerinden farklı, zaman i¸cinde gittik¸ce azalan fiyatlar

sunan birden ¸cok teslim tarihi ¨

onerir. Esnek ¨

uretim sistemleri kullanan ¨

ureticiler

i¸sleme ko¸sullarını de˘

gi¸stirerek kesici u¸c maliyeti kar¸sılı˘

gında ¨

uretim zamanlarını

kontrol etme kabiliyetine sahiptir. Bu ¸calı¸smada, toplam kazancı en¸coklayarak bir

grup i¸sin tek bir CNC takım tezgahında ¸cizelgelenmesini g¨

oz¨

on¨

une aldık. Toplam

kazancı ise ¸cizelgelenen i¸slerin toplam fiyatından toplam a˘

gırlıklı gecikme ve erken

bitirme, kesici u¸c ve i¸sleme maliyetlerini ¸cıkararak hesapladık. Tek ba¸sına toplam

a˘

gırlıklı gecikme probleminin NP-zor olmasından dolayı, bizim inceledi˘

gimiz

prob-lem de NP-zor’dur. Ayrıca, kesici u¸c maliyetinin do˘

gasından dolayı da problem

do˘

grusal de˘

gildir. Bu ¸calı¸smamızda farklı ¨

oncelik kuralları ve ¸cizelgeleme

algo-ritmaları ¨

oneriyoruz. Bu kural ve y¨

ontemleri kullanarak i¸sleri ¸cizelgelerken aynı

anda ¨

uretim zamanlarını da belirleyen bir yordamlama algoritması olu¸sturuyoruz.

Anahtar s¨

ozc¨

ukler : C

¸ izelgeleme, Tek Makine, Toplam A˘

gırlıklı Gecikme ve Erken

Bitirme, Birden C

¸ ok Teslim Tarihi, Kontrol Edilebilir ¨

Uretim Zamanları,

Yor-damlama, Sipari¸s Reddi .

I would like to express my sincere gratitude to Assoc. Prof. Selim Akt¨

urk for his

supervision and encouragement in this thesis work. I am grateful for his invaluable

contribution to my graduate study. His endless patience and understanding let

this thesis come to an end.

I am indebted to Asst. Prof. Mehmet R¨

u¸st¨

u Taner and Asst. Prof. Yavuz

G¨

unalay for accepting to read and review this thesis and for their valuable

com-ments and suggestions.

I would like to express my sincere thanks to Mustafa Rasim Kılın¸c, Z¨

umb¨

ul

Bulut, Ay¸seg¨

ul Altın and Salih ¨

Oztop for being such good friends to me. We

shared a lot and it would be hard to bear with all this thesis time without their

friendship and morale support.

I would like to thank to Kamer Kaya, Sinan G¨

urel, G¨

une¸s Erdo˘

gan, Ayten

T¨

urkcan, Banu Y¨

uksel and Taylan ˙Ilhan for sharing their creative ideas and

technical knowledge with me throughout my graduate study.

I would like to take this opportunity to thank Damla Erdo˘

gan, Emre

Kara-mano˘

glu, Aykut ¨

Ozsoy, Esra B¨

uy¨

uktahtakın, Halil S

¸ekerci, G¨

okhan Metan,

K¨

ur¸sad Derinkuyu, Emrah Zarifo˘

glu, Filiz G¨

urtuna and Bala G¨

ur. It was their

support that motivated me in all my desperate times.

Finally, I would like to express my deepest gratitude to mom, dad and my

brother Levent, for their love and understanding. Without them, this thesis would

be impossible.

1

Introduction

1

2

Literature Review

5

2.1

Due Date, Pricing and Order Rejection Considerations . . . .

6

2.2

Controllable Processing Times . . . .

16

2.3

The Total Weighted Earliness and Weighted Tardiness Problem .

20

2.4

Summary

. . . .

21

3

Problem Statement and Modeling

23

3.1

Problem Definition . . . .

24

3.2

Calculation of Lower and Upper Bounds for Processing Times . .

26

3.2.1

Mathematical Model of SMOP . . . .

26

3.2.2

Cost Components in SMOP . . . .

29

3.3

Model Definition . . . .

31

3.4

Mathematical Formulation . . . .

34

3.4.1

High Competition in the Market . . . .

34

vi

3.4.2

Rejection Flexibility with Loss of Goodwill Cost . . . .

35

3.5

Characteristics of the Problem . . . .

36

3.6

Summary

. . . .

46

4

Solution Procedure

47

4.1

Obtaining an Initial Schedule

. . . .

47

4.1.1

Sequencing the Jobs

. . . .

48

4.1.2

Initial Scheduling Algorithms . . . .

50

4.2

Improvement Methods . . . .

61

4.2.1

Increasing the Number of Jobs in the Schedule . . . .

61

4.2.2

Final Scheduling Phase . . . .

63

4.3

Summary

. . . .

66

5

Numerical Examples

67

5.1

Numerical example with small problem size

. . . .

67

5.1.1

Schedule-ahead case

. . . .

68

5.1.2

Back-forth case . . . .

70

5.1.3

Crush-back-forth case . . . .

71

5.1.4

Ranked-insertion case . . . .

73

5.1.5

Rankless-insertion case . . . .

74

6

Experimental Design

89

6.1

Experimental Setting . . . .

90

6.2

Results for Initial Scheduling Stage . . . .

93

6.3

Results for Improvement Algorithms

. . . 100

6.4

Analysis of the Results . . . 104

6.5

Utilizing the MINOS Solver . . . 111

6.6

Summary

. . . 114

7

Conclusion

116

7.1

Contributions . . . 117

7.2

Future Research Directions . . . 119

A Notation

128

B Results for Initial Scheduling

131

C Average Deviations at Initial Schedule

150

D Results for Improvement Algorithms

157

E Average Deviations after Improvement

176

F Rankings for Algorithm Combinations

183

3.1

Feasible region

. . . .

28

3.2

Cost components in SMOP . . . .

30

3.3

Revenue function . . . .

33

3.4

Critical points and Same revenue points

. . . .

38

3.5

Critical processing time . . . .

41

5.1

Schedule constructed by Schedule-ahead

. . . .

70

5.2

Schedule constructed by Back-forth . . . .

71

5.3

Schedule constructed by Crush-back-forth

. . . .

72

5.4

Schedule improved by Rankless-insertion . . . .

75

6.1

Average values for total profit at the initial schedule . . . .

94

6.2

Average values for total profit after improvement

. . . 102

6.3

Average improvement percentage with respect to average

addi-tional CPU usage at different improvement steps . . . 115

4.1

Dispatching rules . . . .

49

5.1

Job data used in the small example . . . .

68

5.2

Job data used in the example . . . .

76

5.3

Orderings obtained from ranking rules

. . . .

78

5.4

Schedule obtained by back-forth algorithm . . . .

80

5.5

Schedule obtained by crush-back-forth algorithm . . . .

81

5.6

Schedule obtained by schedule-ahead algorithm

. . . .

82

5.7

Total profit at the initial schedule . . . .

82

5.8

Schedule obtained by back-forth and rankless-insertion algorithms

84

5.9

Schedule obtained by schedule-ahead and rankless-insertion

algo-rithms . . . .

85

5.10 Schedule obtained by schedule-ahead and ranked-insertion

algo-rithms . . . .

86

5.11 Improvement amounts at all stages . . . .

87

6.1

Experimental design factors . . . .

90

x

6.2

Technological exponents and coefficients of the available tools

. .

93

6.3

Summary of total profit values after initial scheduling . . . .

94

6.4

Summary of CPU values after initial scheduling . . . .

96

6.5

Deviation averages in percentages at initial schedule . . . .

97

6.6

Averages for the number of scheduled jobs . . . .

98

6.7

Averages for the maximum number of scheduled jobs for all factor

combinations

. . . .

99

6.8

Performance of ranking rules under schedule-ahead algorithm . . .

99

6.9

Summary of total profit values after improvement . . . 101

6.10 Summary of CPU times after improvement . . . 103

6.11 Deviation averages from the best objective function value and the

greatest CPU time after RDI/RLI applied . . . 103

6.12 Average number of jobs in the schedule after RDI/RLI is applied

104

6.13 Averages for the additional CPU used to obtain the additional

profit, in percentages . . . 105

6.14 Average percentages of cost items in total cost . . . 107

6.15 Descriptives for Single-pass results by N . . . 108

6.16 Paired Samples Statistics for results by COV-SA-RDI combination 109

6.17 Paired Samples Statistics for results of different algorithms . . . . 110

6.18 ANOVA for results of best algorithms for all factors . . . 110

6.19 Examples of MINOS results . . . 112

6.21 Deviations from the best objective function value at MINOS stage 114

B.1 Results for SA under COV, ATC-2, ATC-5 orderings, 1 of 6

. . . 132

B.2 Results for BF under COV, ATC-2, ATC-5 orderings, 1 of 6 . . . 138

B.3 Results for CBF under COV, ATC-2, ATC-5 orderings, 1 of 6

. . 144

C.1 Average of 5 replications for each factor combination, for COV and

ATC orderings

. . . 151

C.2 Average of 5 replications for each factor combination, for ATC-2

and LPT orderings . . . 152

C.3 Average of 5 replications for each factor combination, for SPT and

WPD orderings . . . 153

C.4 Average of 5 replications for each factor combination, for WSPT

and EDD orderings . . . 154

C.5 Average of 5 replications for each factor combination, for ATC-3

and ATC-4 orderings . . . 155

C.6 Average of 5 replications for each factor combination, for ATC-5

and ATC-6 orderings . . . 156

D.1 Results for BF-RLI under COV, ATC-2, ATC-5 orderings, 1 of 6 . 158

D.2 Results for SA-RLI under COV, ATC-2, ATC-5 orderings, 1 of 6 . 164

D.3 Results for SA-RDI under COV, ATC-2, ATC-5 orderings, 1 of 6 . 170

E.1 Average of 5 replications for each factor combination, for COV and

ATC orderings

. . . 177

E.2 Average of 5 replications for each factor combination, for ATC-2

and LPT orderings . . . 178

E.3 Average of 5 replications for each factor combination, for SPT and

WPD orderings . . . 179

E.4 Average of 5 replications for each factor combination, for WSPT

and EDD orderings . . . 180

E.5 Average of 5 replications for each factor combination, for ATC-3

and ATC-4 orderings . . . 181

E.6 Average of 5 replications for each factor combination, for ATC-5

and ATC-6 orderings . . . 182

F.1 Rankings of the results of improvement algorithms . . . 184

F.2 Rankings of the results of all possible algorithm combinations

. . 185

G.1 CPU used to obtain more profit for BF-RLI, in percentages, 1 of 6 187

G.2 CPU used to obtain more profit for SA-RLI, in percentages, 1 of 6 193

G.3 CPU used to obtain more profit for SA-RDI, in percentages, 1 of 6 199

Introduction

Noteworthy outgrowth in manufacturing technologies is driving the firms in vast

majority of industries to improve their services and products. Moreover, practice

of e-commerce and resulting global business environment give rise to a

compe-tition between enterprises. In previous decades, firms have competed generally

on introduction of new products, offering product and service proliferation, or

on prices. However, in this last decade, any firm that enjoys the advantages

of modern technology, is able to come up with various products while incurring

small amount of costs to produce them. Consequently, the competition between

companies has moved on to the customer satisfaction perspective, as companies

have become indifferent in terms of product variety and manufacturing costs.

For manufacturers to have satisfied customers that would like to continue

working with them in the future, they need to be careful about two crucial

fac-tors. First, the manufacturer should accept the jobs that a customer orders,

since customers desire stability and do not want to be rejected. Secondly, the

customers require reliable delivery times for meeting their schedules. However,

preventing tardiness to satisfy the customer may cause orders to be produced

early. In this case, the manufacturer incurs additional inventory, transportation

or insurance costs. Hence, producing the order just in time is the main policy for

a manufacturer. On the other hand, each customer has a different priority with

respect to the manufacturer. Consequently, all of these reasons channel the

man-ufacturers to consider the weighted earliness and weighted tardiness objectives

simultaneously. Moreover, a manufacturer should also decide on which orders to

reject since he has a limited capacity.

In make-to-order environments, customers generally prefer their orders to be

delivered as early as possible. However, one thing that is as important as the

shortness of the lead-times is the accuracy of them. Although customers want to

be quoted short lead-times, they also want their orders to be delivered on time,

since they plan their related business according to those lead-time quotations. On

the other hand, customers know that the manufacturer has a limited capacity and

there is a risk for occurence of unexpected events at the manufacturer side that

may influence the deliveries, such as congestion in the production. Therefore, in

order to minimize the risk of a deviation in the delivery time which may affect

the customer negatively, customers offer a number of alternative due dates to

the manufacturer. Delivering the job on an earlier due date provides a greater

price for the job, and the manufacturer is penalized according to the

agreed-upon due date that was specified at the order acception stage. On the other

hand, customers generally specify a deadline, beyond which they will not buy the

order.

If a manufacturer has a flexible manufacturing system, he will be more

capa-ble of meeting due date requirements. This capability is obtained by controlling

the processing times via a readily available feature of modern CNC machines. We

can increase or decrease the processing times of jobs by changing the machining

conditions such as feed rate and cutting speed. However, increasing the

process-ing speed, consequently decreasprocess-ing the processprocess-ing time, results in increased tool

wear and additional tooling costs. Therefore, a manufacturer should pay

atten-tion to tooling costs, processing times, order rejecatten-tion possibility, and due dates

collectively, while scheduling a job.

In this study, we consider the problem of scheduling a set of jobs on a single

CNC machine while maximizing the total profit, which is composed of sum of

the prices of scheduled jobs less sum of weighted earliness/tardiness cost, tooling

cost, and machining cost for each job. This problem is NP-hard since the total

weighted tardiness problem is NP-hard alone. Moreover, the problem is

non-linear because of the nature of the tooling cost. Processing times of the jobs are

controllable and can take any value between the lower and upper bounds that

are obtained by solving a machining conditions problem. Furthermore, we have

the flexibility to reject the orders, and each order comes with a distinct set of

multiple due dates and a deadline. Thus, finding an efficient algorithm that will

provide the exact solution for the problem in a reasonable computation time is

almost impossible.

In this study, we first propose ranking rules to be utilized by the scheduling

al-gorithms for determining the order in which jobs are going to be processed. Then,

we provide initial scheduling algorithms that determine the processing time and

start time of each job simultaneously to construct a schedule in reasonable

com-putational times. We also provide efficient improvement algorithms that improve

the initial schedule both in terms of the number of scheduled jobs and the total

profit. The improvement algorithms have the capability of changing the

process-ing time of a previously scheduled job as well as determinprocess-ing the processprocess-ing time

and start time of a new job simultaneously. Considerable amount of improvement

is realized in reasonable CPU times by use of these improvement algorithms. We

also utilize MINOS solver to compress the schedule by decreasing the processing

times. This helps to obtain higher revenues and improves the objective function

value using reasonable computational effort.

We analyze the results obtained by using different algorithms and construct

a single-pass heuristic algorithm to solve the problem. The heuristic is composed

of four stages including the ordering, initial scheduling, improvement, and

MI-NOS. Our experiments show that the heuristic we suggest is able to improve the

solution quality significantly at each stage without using considerable amount of

computational effort. Therefore, we can conclude that for this difficult problem,

the heuristic we suggest finds good solutions in reasonable time, even in instances

with large problem sizes.

rejection considerations, controllable processing times, and the total weighted

earliness and weighted tardiness problem is presented. In Chapter 3, we provide a

detailed definition of the problem with related mathematical models. We propose

our solution procedure in Chapter 4, along with a number of heuristic algorithms.

In Chapter 5, we provide numerical examples for clarification of the algorithms

we propose. Experimental design and computational results are given in Chapter

6. Finally, we present the conclusion of this study along with the contributions

and future research directions.

Literature Review

In literature, scheduling problems have been studied extensively. There are a large

number of research papers dealing separately with due date selection and pricing,

order rejection criterion, flexible manufacturing systems and controllability of

processing times, and total weighted earliness and weighted tardiness criteria.

Since we consider the interaction between all of these issues in our study, we

want to present the literature about the theoretical background of these topics in

an organized manner.

We will start with the issues of due date selection, pricing, and order rejection

considerations in the following section. The reason we are presenting these

top-ics in the same section is that in studies they are generally mentioned together

as complementary parts. Secondly, we will discuss the literature on scheduling

with controllable processing times. Next, the literature dealing with the total

weighted earliness and weighted tardiness criteria will be presented. Finally, we

will mention the shortcomings of the literature that gave us the inspiration to

conduct this study.

2.1

Due Date, Pricing and Order Rejection

Considerations

In make-to-order environments, customers generally prefer their orders to be

de-livered as early as possible. However, one thing that is as important as the

shortness of the lead-times is the accuracy of them. Although customers want to

be quoted early due dates, they also want their orders to be delivered on time,

since they plan their related business according to those due date quotations.

Therefore, in order to prevent the manufacturers to quote unrealistic due dates

that they can never meet on time, the manufacturers are subject to penalty costs

in case of failure in terms of meeting an agreed-upon due date. Those penalties

are generally specified by a priori commitment between manufacturer and

cus-tomer, which is made for each order that is quoted a due date. In addition to

the tardiness penalties that may become as high as one million dollars per day

in aerospace industry [51], the companies that cannot meet specific due dates

are also subject to some intangible costs such as loss of good will and decreased

customer satisfaction, which in turn may be realized later as a decrease in the

market share. Furthermore, a company that does not quote accurate due dates

is subject to a large possibility of obtaining very small profits or none, at all [56].

On the other hand, the sensitivities of customers to quoted due dates may differ,

i.e., some customers might be ready to wait longer for their orders if they are

offered lower prices. Therefore, assigning an appropriate due date quote for each

customer becomes a crucial consideration for manufacturers, since the customer

will be lost otherwise. There are cases that customers dictate a deadline for

de-livery of their orders, while being ready to pay more for deliveries at preferred

due dates before the deadline. In fact, the due dates are quite often negotiable

in real life [29].

Nevertheless, an order may not be profitable for a manufacturer to produce,

if it has stringent due date requirements. In such a case, putting this urgent

order into the schedule in order to produce it before its deadline may cause

some other orders to be delayed. Unless the profit obtained by producing a rush

order is larger than the loss realized by delayed orders, it is meaningless for a

manufacturer to accept this urgent order. However, there are also some examples

of manufacturers that are able to control the processing times of jobs by using

different machining parameters [8, 32, 37]. Such a flexibility allow manufacturers

to prevent jobs to be late, and provides possibility to accept larger number of jobs

that include urgent ones, which in fact probably will result in increased revenues.

Nonetheless, controlling processing times are subject to additional manufacturing

costs generally due to quicker depreciation of tooling. On that account, accepting

or rejecting an order also seems to be an important decision. Actually, an order

rejection decision is imposed to a customer by quoting a long lead time or a high

price that is much more undesirable than the quotations of the other firms in the

market.

The concept of availability interval, which is the time period between arrival

of an order and the latest acceptable start time of it, is introduced in study of

Keskinocak et al. [33]. One offline model that assumes the manufacturer to know

complete information about orders a priori, and three online models that differ

from each other by the maximum allowable period for deciding the lead-time quote

are examined. These periods end at the beginning, somewhere interior, and at

the end of the availability interval in these three models. Moreover, the revenues

are also sensitive to lead-times. They offer different algorithms for these models,

and use competitive analysis to compare the performance of the algorithms for

online models with the optimal algorithm for the offline model. The study also

considers an enhanced model that includes an urgent customer class with higher

revenue contribution. It is shown that leaving space is necessary to obtain higher

revenues, but just for urgent jobs if there are any. On the other hand, they figure

out that it is not desirable to promise capacity beyond some certain period. One

other intuitive conclusion is that the quoting decision is made better when you

have more time to observe the order arrival characteristics.

In a recent study, the time window for each order is specified by the arrival

time, preferred due date, and latest acceptable due date [11]. A single-machine

production system with multiple products and negligible setups is considered.

Preemption is allowed, and different products provide different revenues. A mixed

integer program is used to maximize the total profit under capacity constraints.

It was observed that higher lead-time flexibility and higher partial fulfillment

flexibility resulted in higher profits. The reason of such an observation is the

increase in number of accepted orders.

Similarly, in the study of ElHafsi [21], a new order should be processed and

delivered in a specific time window. There exists a schedule when the job arrives,

and the manufacturing system consisting of several processing centers is subject

to setup times between production batches and to breakdowns that are followed

by constant repair times. The option of partial deliveries is also investigated. The

aim is to minimize the operating costs in this study. In case of urgent orders,

they are split among most available processing centers to achieve the smallest

possible lead-times.

Duenyas [18] presented the problem of determining lead-times to orders in a

dynamic, stochastic environment, with multiple customer classes differing from

each other by lead-time and price preferences. He formulates the problem as a

semi-Markov decision process. Quoting decision considers the system load, and

the first-come, first-served (FCFS) rule is applied in sequencing. The results

showed that customers that are ready to pay more for quick service are more

sensitive to changes in lead time and less likely to place an order when they are

quoted the same lead-time with an ordinary customer. In the second part of the

study, author introduces a heuristic that finds the sequence of orders. The major

assumption in this case is that the important fraction of the congestion cost in the

system is supposed to be realized by the existence of a new order in the system.

The lacking part of the study of Duenyas [18] was considered in the study

of Duenyas and Hopp [19], i.e., whenever a new order arrives to the system,

they used the information on how much time left until due date of each order

in the production queue. They provided the proof of the optimal policy when

the company has finite capacity, which sequences the orders according to the

earliest due date (EDD) rule. This policy is more realistic and applicable than

the previous heuristic solution presented in the study of Duenyas [18].

In some companies, the practice of scheduling newly arriving jobs is not

dy-namic, but rather realized periodically. In fact, decision of quoting lead-times and

scheduling the orders arriving over time is more accurate in periodical case, since

uncertainty about future arrivals is less in this case. However, if the rescheduling

period becomes very large, it will no longer be beneficial to schedule periodically,

as the jobs should wait a long time before being scheduled. In such a case, a

feasi-ble period to schedule the job may pass before the scheduling decision is realized.

Li and Cheng [36] worked on determining due dates and sequence of a number of

new jobs, while there are some old jobs already scheduled in the system. Their

objective is to minimize the maximum weighted tardiness and due-date

assign-ment cost that represent the lost of goodwill of customer as a result of the quoted

lead-time. Authors show that the problem is NP-hard, but it is possible to solve

it using a dynamic program if all old jobs have equal weight, and all new jobs

also have equal weight.

A manufacturing environment that setups are realized when changing the

production to a different part type is examined in the study of Unal et al. [58].

Orders arrive to a single machine over time, and there is a set of jobs that has

already been scheduled for production. The objective in this study is to

min-imize the total weighted completion time or the maximum completion time of

the new jobs. Nevertheless, while trying to fulfill this objective, schedulers seek

to prevent additional setups and tardiness in the current schedule.

Neverthe-less, such a schedule may not exist, which in turn results in a revision that will

cause additional setups. The rescheduling event is realized periodically, instead of

scheduling each job one by one. In this way, a better solution that simultaneously

considers more than one job is achieved. However, the length of the scheduling

period is subject to the order arrival rate. As a result of the study, the authors

provide necessary and sufficient conditions for the existence of an undisrupted

schedule. On the other hand, polynomial time algorithms are introduced for the

total weighted completion time problem with unit weights and for the maximum

completion time problem. Authors also offer two heuristics to solve the total

weighted completion time problem, which is actually NP-hard.

Chand and Chhajed [9] examined the problem of determining optimal due

dates and the sequence of a number of jobs on a single machine, where the

processing times are known and deterministic. While assigning the jobs to the

predefined due dates, they try to minimize the penalties of earliness, tardiness,

and due date assignment. Predefining the number of due dates might have been

used to reflect the cases in real life that a company is available for production

for only some specific periods. Setups, preventive maintenance, or allocation of

machines to different productions in certain time periods may be the reasons of

using such specific due dates. When the length of the available period is long,

the predefined due date may be assigned to a greater number of jobs, which is

one of the cases considered in the paper. An optimal algorithm with the time

complexity of O(n log n) is introduced for this case. For the case that the number

of jobs to be assigned to specific due dates is not given, an optimal algorithm

was developed, which seems to be computationally inefficient for large number of

jobs and due dates.

A 100% reliable due date quotation problem is introduced in the study that

was realized by Kaminsky and Lee [29]. In this research, authors investigated

a system that consists of a single server with orders arriving over time. The

model tries to minimize the average quoted lead-times, which is a meaningful

objective if the company is in competition in terms of time-based figures such as

shorter lead-times. On the other hand, freedom of tardy jobs is guaranteed. An

on-line algorithm is provided, which works every time a job arrival or completion

occurs. Whenever a new order arrives, the sequencing procedure tries to increase

the sequence of the new job, by inserting it directly to the sequence, or moving

stepwise into front by realizing pairwise interchanges with the jobs that have

longer processing times. Nevertheless, the sequencing procedure should not cause

any job that is already scheduled to become tardy, i.e., if it is not possible to

insert the new job into the schedule without making any job tardy, the new job

should be sequenced at last. After the sequencing decision is made, the algorithm

assigns a lead-time quote to each new job, by considering the known processing

time and adding a slack to it. The reason of assigning slacks keeps the idea of

providing flexibility to the schedule in behind. Reserving capacity, i.e., leaving

space for future jobs will allow scheduler to achieve better objective values, when

he inserts profitable jobs in future. Four different slack assignment procedures

were described in the study. Assigning a constant slack to every job, which is

not a reasonable approach, is the first, while a procedure that assigns longer

slacks to longer jobs, a more reasonable one, is the second method. Authors

compared the algorithm with the first-come, first-served (FCFS) rule, which is

an upper bound and showed the effective performance of it. However, this model

is most appropriate for industries that have huge tardiness penalties, such as the

aerospace industry. Businesses with small tardiness penalties do not require 100%

reliable lead-times. It may be better to define different objectives in these cases.

A dynamic environment that jobs with randomly distributed processing times

are arriving continuously, constitutes the model for the study of Slotnick and

Sobel [51], which explores the value of shop information to the quoter of the

lead-times. Authors consider two cases, one with full information and the other with

just statistical information available to the decision-maker. The expected present

value of net profit and the long-run average profit per unit time is maximized

both in single and multiple customer classes cases. It is clear by the research

that, using more information on characteristics of the system, firms quote better

lead-times that result in higher revenues. Intuitively, it is more beneficial to quote

better lead-times when there are more customers in the system. Actually, this

is the case when profit margin is relatively low. Therefore, companies that have

low profit margin may consider using shop information more efficiently.

Sometimes, in order to prevent firms from quoting short lead-times that are

impossible to meet, firms are imposed to maintain a certain minimum service

level [44]. In this study congestion related costs are also considered while quoting

the lead-times. Another research that uses the serviceability level as a constraint

belongs to Spearman and Zhang [55]. However, the objective is to minimize the

number of tardy jobs, which may result in unrealistic, unethical policies. Since

the length of the tardiness is not important in this case, it is possible to quote

lead-times as optimal, which are well known to be unrealistic. Although it is

harder to deal with total tardiness than dealing with number of tardy jobs, it will

lead to more realistic and trustworthy solutions.

A method that can be used in many systems and requires data that firms

already gather is provided to meet service level constraints in the study of Hopp

and Sturgis [27]. However, it is assumed that the service level is indifferent to

the number of jobs in the system. Therefore, the reliability of the lead-time that

is quoted in an empty system is the same as the reliability of a quotation in

a congested system. Such a situation is not acceptable, if applicability of the

method in real life is requested.

Hegedus and Hopp [25] proposed a model for quoting due dates, where the

customers request due dates.

Service level, fill rate, and inventory costs are

considered in the study, and a due date setting policy, which is suitable for MRP

environments, is obtained. An interesting result of the study says that until the

importance of service level performance reaches a certain level, the customers are

always quoted the due dates they require.

Duenyas [18] assumes that there are multiple customer classes in the

manu-facturing environment, which differ from each other according to lead-time and

price preferences. He considers that same revenues are obtained from orders of

the same customer class. However, he suggests using the heuristic he introduced

for setting due dates as an intermediary for negotiating the price with customers.

In this case, if a customer states that he would order if he is quoted a due date

earlier than the company offered, the heuristic could be used to find the sequence

of the order for the customer-preferred due date, and the cost of processing

ear-lier - which is possibly due to the jobs that become tardy after the insertion of

our job to an earlier place in the sequence - can be calculated. Accordingly, the

customer is asked to pay the additional cost. Duenyas also suggests the usage of

the heuristic for quoting dynamic prices.

In a similar study, firms compete in a make-to-order environment, which

con-sists of different types that have distinct service requirements and sensitivities to

the delay time from order to delivery [35]. Firms compete in terms of prices and

production rates assigned for each customer type. However, in order to capture

the sensitivity of customer to the price quotation, the model uses a demand

func-tion that is strictly decreasing in the quoted price and delay time. With the same

logic, in order to capture the price sensitivity of the customers, Palaka et al. [44]

consider the demand as downward sloping in both price and quoted lead-time.

They subtract the price quoted to the customer multiplied with a price

elastic-ity factor, from the mean demand corresponding to zero price and zero quoted

time. Their objective in this study is to find the optimal price, quoted

lead-time, and capacity utilization values. They also consider the capacity expansion

case up to a certain limit. In addition, authors introduce sensitivity analyses for

different parameter values.

Ray and Jewkes [46] model an operating system that consists of a firm and

a number of customers, where the demand is sensitive to guaranteed delivery

time and price, which is actually determined by the length of the delivery time.

Moreover, in order to prevent quotation of unrealistic delivery times, they use

an internal target delivery time reliability level. The company is also able to

increase the capacity, which in turn adds an additional investment cost. Authors

find the optimal capacity increase rate, and the optimal price and delivery times

accordingly. They also show that if the price is found as an independent decision

variable, companies may realize two mistakes, which are quotation of shorter

delivery times to price sensitive customers that want to wait more to pay less; and

quotation of longer lead-times to lead-time sensitive customers that are ready to

pay more for quicker service. Economies of scale is also considered in the study, by

choosing the unit operation cost decreasing convex with respect to mean demand

rate.

Another study is about the make-to-order (MTO) firms that enter in a bidding

competition with the objective of determining the lead-time and the contract price

that maximize the expected contribution from the current bidding opportunity,

while ignoring any future bid requests [20]. It is assumed that the probability

of acceptance of a bid that a company makes is known. The authors present

an enumerative solution procedure that identifies the optimal contract price and

lead-time pairs for the bid.

So and Song [54] consider price, delivery time guarantee, and capacity

ex-pansion level as the decision variables, in order to maximize the expected net

profit per unit time subject to a reliability requirement. The company is also

able to increase the capacity, however, a linear expansion cost is incurred in this

situation. They analyze two cases; first with capacity expansion is not allowed,

and second with the price is fixed. In both cases, it is shown that the optimal

capacity expansion rate is either the upper bound for expansion or not expanding

at all.

An interesting approach for capturing the sensitivity of customers to the price

quotations is handled in the study of So [53]. Author defines the attraction of

firms, which plays role in the market share of the firms. Attraction of a firm

consists of price quotation, delivery time, and a fixed reputation figure. In order

to prevent unrealistic quotations, a service level constraint is used. Market share

of the firm is used as the demand it realizes, and the objective of the study

is to maximize the operating profit. A firm’s delivery time and price decisions

are found by considering that other firms’ decisions are known. Another game

theoretical method that finds the decisions of all firms at the same time is also

presented.

Wester et al. [62] consider a make-to-order manufacturing system

consist-ing of a bottleneck machine and various product types that require setups when

changing the production type. In this system, the orders with customer

deter-mined due dates are rejected if and only if they are likely to cause late deliveries.

They consider three approaches to order acceptance decision. In a monolithic

ap-proach, each time an order arrives, a detailed schedule containing previous orders

and the new one is prepared, and the order is accepted if no late deliveries are

observed. In a hierarchical approach, although a detailed schedule is available,

the acceptance criterion is work content, e.g., if sum of processing times for all

orders exceed a level, order is rejected. However, in a myopic approach, there are

no detailed schedules, and each time an order is completed the order that causes

minimum lateness on previous orders is selected. They also claim that monolithic

approaches are difficult to handle, since a rescheduling should be realized every

time an order arrives. However, it is better to use a monolithic approach when

the setups are large and due dates are tight, since it is more selective than the

other approaches.

compares it to the present schedule without the new order [52]. If the reward

of the new job is not worth changing the schedule, the order is rejected. In this

study, the acceptance policy is learned through training a neural network using

reinforcement learning.

Keskinocak et al. [33] consider the possibility of rejecting an order, if it is

not possible to start processing it without causing it to be completed late. In

these situations, a time that is longer than the maximum acceptable

lead-time for the customer is quoted. When the schedule is busy or the capacity is

reserved for future orders we expect a refusal for the order. One of the algorithms

introduced in the paper, algorithm Q-FRAC accepts an order if it guarantees a

certain portion of the maximum possible revenue. Otherwise, the order is rejected,

and capacity is reserved for future orders with larger revenue fraction. However,

the decision making of accepting or rejecting an order is considered in three cases.

In the first case, the decision should be made immediately when the order arrives.

In the second case, it is possible to make the decision in a pre defined time period

before the latest start time, and in the last case, it is decided at anytime until

the latest start time of an order. We should remark that, as the decision period

increases, the uncertainty about future arrivals decrease since it is possible to

realize more arrivals in a longer period. Therefore, the scheduling decision will

result in a better solution in cases of longer decision periods for acceptance. A

similar approach that rejects orders by assigning a lead-time that is longer than

the market standard can be found in the study of Duenyas and Hopp [19].

The study of Weng [61] figures out an optimal policy for jointly optimizing

the order acceptance rate and the manufacturing lead-times. In this way he tries

to control the congestion at the shop floor level, but assigning an acceptance rate

may not always be suitable, since in practice jobs differ from each other in terms

of workload, revenue, etc.

Roundy et al. [48] studied the order acceptance problem where they treat

time as a discrete variable measured in hours. They modeled a job insertion

problem, whereas they also developed a simulated annealing, a genetic algorithm

and an LP-based heuristic that look very promising.

It is possible to find a survey of scheduling research up to 1989, which consider

different due date determination decisions in the study of Cheng and Gupta [15],

while two of the earliest studies that deal with due date considerations were

realized by Seidmann and Smith [49], and Baker [7].

2.2

Controllable Processing Times

Although vast majority of the studies in scheduling literature assume that

pro-cessing times are fixed, in fact, utilization of flexible manufacturing systems makes

it possible to control the processing times of the jobs. By allocating additional

resources or by changing machining parameters such as cutting speed and feed

rate of a CNC turning machine, we can control the processing times. This

flexi-bility provides us to have alternative production schedules that are close to each

other especially in terms of the length of the time horizon used to schedule the

jobs. For instance, when a disturbance such as a machine breakdown or arrival

of an urgent order occurs, it is possible to obtain a schedule that is close to the

initial one by decreasing the processing times of the jobs.

Controllable processing times were considered first by Vickson [59, 60]. He

considers the single machine problem with the objective of minimizing the job

processing costs plus the total weighted flow cost. In his first study [59], the

job compression costs are assumed to be linear functions of the processing times.

He developed a heuristic algorithm and provided a performance bound. The

heuristic is found to give optimal solutions in most cases. In the second study

[60], he considers two single machine sequencing problems with the objective of

minimizing the total processing cost together with either maximum tardiness or

average flow costs.

In the study of Daniels and Sarin [17], job processing times are treated as

decision variables that may be controlled through the assignment of an additional

resource. They consider the problem of joint sequencing and resource allocation

when the scheduling objective is to minimize the number of tardy jobs. They

suggest a constructive procedure for developing the tradeoff curve between the

number of tardy jobs and the total amount of allocated resource.

Nowicki and Zdrzalka [40] deal with a two machine flow shop scheduling

prob-lem such that processing times and sequence of the jobs are both decision

vari-ables. They assume that cost of processing a job is a linear function of its

process-ing time and the objective is to minimize the total processprocess-ing cost plus maximum

completion time cost. In this study, two heuristic methods are presented together

with their worst case analysis. Nowicki [39] considers the same problem with

re-lease times and delivery times. He presents a (ρ + 1/3)-approximation algorithm

for the problem, where ρ is the worst case performance bound of a procedure

for solving the pure sequencing problem. Zdrzalka [65] also studied the single

machine problem with release dates, delivery times and controllable processing

times. He suggested an approximation algorithm to minimize the schedule cost,

which has the performance guarantee of (ρ + 1/2). A survey of the studies that

deal with controllable processing times up to 1990 was provided by Nowicki and

Zdrzalka [41].

Panwalkar and Rajagopalan [45] treat the problem of finding the optimal

processing times, an optimal due date and an optimal sequence that will minimize

total processing, earliness and tardiness costs in a static single machine sequencing

problem. In this study, all jobs have a common due date and processing times are

controllable with linear costs. They suggest an algorithm with the complexity of

the order O(n log n) + O(n

3

_).

Liman et al. [37] discussed the problem of scheduling a set of jobs in a static,

deterministic environment, on a single machine, with a common due window and

controllable processing times. The objective is to minimize the weighted total

costs of earliness, tardiness, earliest due date, window size, and job compression.

Optimal job sequence and other values of schedule are obtained using the

Succes-sive Shortest Path Algorithm, after being formulated as an assignment problem.

Biskup and Jahnke [8] conducted an analogous research. However, they scheduled

the jobs that have a common due date, instead of a window, while the processing

times were reducible if all of the jobs were going to be included in this

reduc-tion. Polynomially solvable algorithms were provided for both the objective of

minimizing the total earliness and tardiness penalties, and for the objective of

minimizing the number of tardy jobs. This study is applicable in manufacturing

systems that the products have common setups.

A similar approach, which tries to capture the characteristics of the

prob-lems with controllable processing times, considers the processing times as

non-increasing linear functions of the equal amounts of resource allocated in the study

of Ng et al. [38]. Aim of the study is to schedule the jobs with a common due

date, on a single machine, while minimizing the scheduling, due-date assignment,

and resource consumption costs. Authors provide efficient algorithms with O(n

2

log n), which can solve the problems up to 5000 jobs in less than a minute.

A single machine scheduling model with controllable processing times having

linear costs was considered by Cheng et al. [14]. Objective is to minimize the

processing time compression costs and costs associated with the number of late

jobs. They present a number of computationally efficient heuristics that are able

to find near-optimal solutions.

Chen et al. [12] introduce a discrete model such that processing times can

be controlled among a finite number of specified settings. Their objective is

to minimize the total processing costs plus the cost associated with one of the

most common criteria such as makespan, maximum tardiness, total completion

time, weighted number of tardy jobs, and total earliness-tardiness penalty. In

this study, a pseudo-polynomial dynamic programming algorithm is proposed for

hard problems.

Ilhan [28] deals with minimizing the total weighted tardiness where the cost

function is convex. He proposes a DP-based heuristic to solve single CNC

ma-chine scheduling problem with a given sequence, and suggests a local search

al-gorithm that uses it as a base heuristic. On the other hand, Kayan and Akturk

[32] provide upper and lower bounds for the processing time of each job, where

the objectives are minimizing the manufacturing costs and minimizing a

regu-lar scheduling measure. The manufacturing cost is assumed to be comprised of

machining and tooling costs.

The studies in literature that deal with controllable processing times are not

restricted to only single machine problems. There are also papers considering flow

lines and parallel machines. Karabati and Kouvelis [31] handled the simultaneous

scheduling and optimal-processing-times selection problem in a multi-product

de-terministic flow line, which is operated under a cyclic scheduling approach. They

tried to achieve the desired production rate while minimizing the operating cost.

First, they solved the subproblem with a fixed sequence, and suggested a

row-generation solution procedure that can handle realistic-sized problems effectively.

They also presented an iterative solution procedure for the simultaneous

schedul-ing and optimal control-of-processschedul-ing-times problem. This procedure was shown

to find good local optimal solutions in reasonable computation times. Moreover,

they suggested a genetic algorithm for large problems.

Cheng and Shakhlevich [16] assume that all operations of a job can be

com-pressed by the same amount with an additional cost. They consider the

propor-tionate flow-shop problem with the objective of minimizing the makespan and

compression costs. The compression cost function is non-decreasing with respect

to the compression amount. For the bicriterion problem, they present an O(n log

n) algorithm, while for the single criterion problem that minimizes the makespan

plus compression costs, they provide an O(n

2

_{) algorithm.}

The non-identical parallel machine scheduling problem with the objective of

minimizing the total processing cost plus total flow time or total weighted

ear-liness and weighted tardiness was solved by Alidaee and Ahmadian [4]. They

solved each of these two problems by a polynomial time algorithm after

reduc-ing them to a transportation problem. Cheng et al. [13] solved the unrelated

parallel machine scheduling problem with a convex compression cost that is a

function of the compression amount. They assumed that jobs have a common

due date, and formulated the problem as an assignment problem. The problem

is solved in O(n

3

_{m + n}

2

_{m log(nm)) time. Zhang et al. [66] deal with the parallel}

machine total weighted completion time problem. They utilize convex quadratic

programming relaxation and derive a 3/2-approximation algorithm.

2.3

The Total Weighted Earliness and Weighted

Tardiness Problem

There are a large number of studies on the issue of earliness and tardiness

penal-ties. Among them, specifically the total weighted earliness and weighted tardiness

problem is our point of interest. However, in this section we also present studies

that come up with important results, which consider the total earliness and

tar-diness problem without weights and the studies that consider only the earliness

or only the tardiness costs.

Ow and Morton [43] analyzed the single machine total weighted earliness and

weighted tardiness problem with distinct due dates. They proposed two dispatch

priority rules. They also proposed a filtered beam search method to provide

near-optimal solutions with relatively small search trees. Fry et al. [22] suggest a

heuristic solution that requires an enumeration procedure to guarantee a global

optimum. They suggest that utilization of the earliest due date (EDD) rule to

sequence the jobs and then applying adjacent pairwise interchange provides the

best results.

Yano and Kim [64] provided a DP algorithm to obtain the optimal schedule,

given a fixed sequence of jobs. Azizoglu et al. [5] provided good lower and upper

bounds for the problem where the earliness and tardiness penalties are the same

given the assumption that no inserted idle time exists in the schedule. For the

same problem, Hoogeveen and van de Velde[26] suggested a branch and bound

algorithm, while also providing lower and upper bounds for the problem.

Oguz [42] analyzed the single machine total earliness and tardiness scheduling

problems with distinct due dates. She proposed efficient heuristic procedures for

an approximate solution. She also provided a dynamic programming formulation

for the exact solution to the problem. Garey et al. [23] considered the single

ma-chine scheduling problem with symmetric earliness and tardiness penalties. They

provide efficient algorithms to minimize the absolute discrepancies from the due

dates. There are similar studies that actually consider the total weighted

earli-ness and weighted tardiearli-ness problem by dealing with the amount of discrepancy

from a due date such as the studies of Achuthan et al. [1], Bagchi and Sullivan

[6], Kanet [30], Lakshminarayan et al. [34], and Sidney [50].

Cheng et al. [14], Alidaee and Ahmadian [4], Panwalkar and Rajagopalan

[45], Liman et al. [37], and Chen et al. [12] consider total earliness and

tardi-ness penalties in their studies. On the other hand, there are also studies that

consider only tardiness costs or only earliness costs. Rinnooy Kan et al. [47]

provided bounds for the single machine scheduling problem with the objective

of minimizing the weighted tardiness cost. Yang et al. [63] solved the problem

of minimizing the sum of weighted job tardiness and resource overtime costs on

a single resource. They provided a pseudopolynomial algorithm to be used to

solve the general scheduling problem using a priority sequencing rule. Then, a

local search algorithm is utilized to improve the initial solution. Problems with

high tardiness costs can be solved to obtain near-optimal solutions in reasonable

computation times. On the other hand, Chand and Schneeberger [10] consider

the single machine scheduling problem with the objective of minimizing the total

weighted earliness. However, they assume that no tardy jobs exist in the

sched-ule. In this study, both exact and approximate methods are introduced to solve

the problem.

2.4

Summary

In the literature, single machine scheduling problems have been studied

exten-sively. However, among the literature that deals with controllable processing

times, to the best of our knowledge, there are no studies that consider the total

weighted tardiness and weighted earliness penalties. Furthermore, vast majority

of the research that assumes the controllability of processing times restricts this

flexibility to a range, while associating a linear compression cost. On the other

hand, we relate the processing times and associated costs with tool and operation

parameters.

It is also possible to find a large number of studies dealing with different due

date and order rejection considerations. Some of these papers dictate a common

due date for all jobs to be scheduled, while most of them consider a single due date

or a deadline for each job. There are also some studies that provide due windows

for scheduling the jobs. However, there is no study that considers a combination

of a number of due dates and a deadline that will provide alternative decision

opportunities to the manufacturer. Instead of dictating a single due date to a

manufacturer, offering different due dates associated with different prices for the

orders will be advantageous both for the manufacturer and for the one who assigns

the order. The reason for that is the manufacturer will have different alternatives

to schedule the job and consequently he will have less difficulty in scheduling

orders from other customers. Such a production schedule with fewer restrictions

will provide larger profits to the manufacturer. On the other hand, the one who

orders the job will be more comfortable about the delivery of his order, since the

flexibility of choosing the appropriate due date will decrease the possibilities of

delay in order deliveries. Therefore, the losses due to unexpected delivery delays

will be minimized.

As a result, the topic of single machine scheduling with controllable

process-ing times under the total weighted earliness and weighted tardiness criteria in

existence of multiple due dates have not been considered in the literature. In

this thesis, we aim to show how the flexibility of controlling processing times

can be utilized to solve the single machine scheduling problem, while the

inter-actions between the realistic features such as earliness and tardiness penalties,

due dates and deadlines, and order acceptance or rejection decisions are carefully

investigated.

In this chapter, we presented a review of the existing literature related to our

study. We tried to point out the similarities and diversities of our study with the

researches that were performed until now. We also mentioned the reasons for our

motivation to carry out this study. In the next chapter, we define our problem in

detail, provide a mathematical formulation for the original problem and define a

number of different cases with specific features.

Problem Statement and

Modeling

In this study, we aim to schedule parts that are ordered by a number of customers,

on a single CNC machine. The processing times can be controlled by changing

the machining conditions such as cutting speed and feed rate. The customers

are offering a number of different due dates to the manufacturer for each part

they order. Consequently, the price of a part differs according to the specific

due date the manufacturer satisfies. Actually, the price of a part decreases as

the due date gets later. The objective of the manufacturer is to decide on which

due date among the ones offered by the customer to choose for a specific part,

and to determine the processing time of the part (i.e., under which machining

conditions to produce the part) while maximizing the total profit in existence of

manufacturing costs and weighted earliness/tardiness costs.

We will start this chapter by defining our problem in detail, mentioning the

assumptions used throughout the study, and clarifying the notation. Next, we

will provide the necessary machining condition calculations that we will utilize

while determining the processing time of the jobs. After that, the

mathemati-cal formulation for the original problem and for different cases of the problem

will be presented. We will complete this chapter by investigating the special

characteristics of the problem.

3.1

Problem Definition

In this problem, we have N jobs to schedule on a single CNC machine, where each

job corresponds to a single cutting operation. The tool that is going to be used

in the operation and the cutting parameters such as surface roughness, depth of

cut, length and diameter of the generated surface are specified. Furthermore, the

customer that orders a job also offers the manufacturer a number of due dates

and a deadline. Consequently, the price of a part differs according to the specific

due date the manufacturer satisfies. Actually, the price of a part decreases as

the due date gets later. The objective of the manufacturer is to decide on which

due date among the ones offered by the customer to choose for a specific part,

and to determine the processing time of the part (i.e., under which machining

conditions to produce the part) while maximizing the total profit in existence of

manufacturing costs and weighted earliness/tardiness costs. The manufacturing

cost consists of two parts; tooling cost and machining cost. In addition, while

determining the machining conditions, there are some constraints on machining

power, tool life and surface finish to be considered carefully. The processing

times of jobs can be chosen between the upper and lower bounds, which are

calculated using the tooling and operation parameters. The detailed information

about the calculations of upper and lower bounds on processing times and the

manufacturing costs are presented in Section 3.2. In Section 3.3 the model is

defined clearly along with the related notation. Mathematical formulations of

the problem corresponding to the different cases are provided in Section 3.4, and

a detailed investigation of the problem characteristics is realized in Section 3.5.

Assumptions

The assumptions about the system characteristics considered in this study are

provided below: