paul a. stephenson, m.b.a., m.sc.,collectionscanada.gc.ca/obj/s4/f2/dsk1/tape9/pqdd... · as either...

NEW DOMINANCE ORDERS

FOR SCHEDULING PROBLEMS

BY

PAUL A. STEPHENSON, M.B.A., M.Sc., B.Sc.

A Thesis Submitted to the School of Graduate Studies

in Partial Fulfillment of the Requirements for the Degree

Doctor of Philosophy

McMaster University @ Copyright by Paul A. Stephenson, February 1999.

National Library l*l of Canada Bibiiothèque nationale du Canada

Acquisitions and Acquisitions et Bi bliographic Services sewices bibliographiques

395 Wellington Street 395. rue Wdlïngtori ûttawaON K l A W OctawaON K 1 A W Canada CaMda

The author has granted a non- exclusive Licence aliowing the National Library of Canada to reproduce, loan, distribute or s e l copies of this thesis in microfom, paper or electronic fonnats.

The author retaùis ownership of the copyright in this thesis. Neither the thesis nor substantial extracts f?om it may be printed or otherwise reproduced without the author's permission.

L'auteur a accordé une Licence non exclusive permettant a la Bibliothèque nationale du Canada de reproduire, prêter, distribuer ou vendre des copies de cette thèse sous la fome de microfiche/film, de reproduction sur papier ou sur format électronique.

L'auteur conserve la propriété du droit d'auteur qui protège cette thèse. Ni la thèse ni des extraits substantiels de celle-ci ne doivent être imprimés ou autrement reproduits sans son autorisation.

Nl3W DOMINANCE OFLDERS FOR SCHEDULING PROBLEMS

DOCTOR OF PHILOSOPHY (1999) McMaster University . - (Business Admuwtrat ion) H d t o n , Ontario

SUPERVISOR.

NUMBER OF PAGES:

New Dominance Orders for Scheduling Problems

Paul A. Stephenson M.B.A., M.Sc., B.Sc. (McMaster)

Dr. George Steiner

ix, 95

Abstract

A sequencing problem involves 6nding a sequence in which to process a set of jobs

that minimizes a given cost function. The primary ciBicul@ with such problems is

that as the number of jobs increases the nwnber of sequenoes gmws astronomicaily

large, and the problems become intract able.

Pairwise job interchange is one of the most commonly used solution techniques

for sequencing problems. It compares the cost of sequences that differ only in the

interchange of two jobs. In this way the cost function indicates a preference for the

ordering of certain pairs of jobs such that if a pair of jobs is not in preference order,

then the jobs can be interchangeci with no increase in cost. The traditional method of

pairwise job interchange assumes that either there are no intermediate jobs (adjacent

painuise interchange) or that an interchange can be performed no matter what the

intermediate job6 are (nonadjacent painvise intemhange). in this thesis we introduce

a generalization that permits the pairwise interchange of a pair of jobs provided that

the intermediate j o b belong to a restricted subset of jobs (abset-mtricted painuise

intemhange) . We use subset-restricted pairwise interchange to derive a dominance or der on

the jobs. This is a partial ordering of the jobs that consists of pairs whûse relative

order can be fixed in an optimal sequence. The search space can then be reduced to

consist of only those sequences tbat satisfy the relative job orderings in the dominance

order. We apply this technique to certain on* and twemachine sequencing problems,

and show that the use of our dominance orders significantly reduces the wmputation

time necessary to solve these problems.

Acknowledgement s

1 would like to gratehilly acknowledge the tremendous assistance 1 have received hom

my supenisor Dr. George Steiner. This assistance has ranged hem, correcting my

grammar on numerous occasions, to 'stretching the truth' on even more numerous

progress reports. 1 thank you for introducing me to the subject of Scheduling, your

extensive know ledge, and your (almost ) infinite patience.

1 would iike to acknowledge the many hiends who have made my experience

as a gaduate student at McMaster mernorable. 1 want to thank John Harding for

his early programming assistance and for the many trips to Brandon University to

work on this material. Thanks are also due to Sam and Mary for ailowing us to

get in some much needed table-hockey on these trips. 1 would also like to thank my

very good fiiends from the Mathematics Department: Spiro Daoussis, Bob StamiEar,

Mike Pfau, Tarit Saha, and Eric Derbez. You guys belong in the circus.. .right next

to the dog-faced-boy (Fkanco Guerriero). n o m the Business School 1 want to thank:

Yavuz Günalay, Kalyan Gupta, Brian Denton, Halit Oster, Laurent Lapierre, Kaywan

A-fkharnie, Mustafa Karakul, and Erdem Gilridliz. The worst softball team deserves

the worst captain, thanks guys. Thanks also to some of the &ends who helped to

make the Phoenix my second home: Russ Riddel, Tammy Ivanco, Jonathan Wilson,

Marcy Lumsden, and Chris O'Donovan.

1 would especidy like to express my gratitude to Susan Larson. Her patience

and support during the dlncult times will never be forgotten.

1 also want to acknowledge Karmen BIeile for her continued companionship.

Finally, 1 am very grateful to my family for their many years of support and

encouragement. I'm glad that we're ali around to see this day.

Contents

1 Introduction 1

....................................... 1.1 Methods of sequenchg and scheduling 1

......................................................... 1.2 Preview of the thesis 6

1.3 Preliminary definitions and notation ......................................... 8

....................................................... 1.3.1 Sequencing notation 8

............................................................. 1.3.2 Partial orders 10

2 Minimizing maximum cost for single-machine sequencing problems wit h release times 13

................................................................. 2.1 Introduction 13

............................................................. 2.2 Previous results 14

................................................. 2.4 A general dominance order -22

...................................................... 2.4.1 Linearly ordered 2, 26

............................................. 2.4.2 Weighted maximum lateness - 3 2

3 Minimizing makespan in the two-machine flow-shop with release times 38

................................................................ 3.1 Introduction -38

.......................................................... 3.2 Dominance results -39

3 .2.1 Subset-restricted interchange ............................................ - 4 0

3.2.2 New dominance order for F2 Irj/ C,, .................................. -43

3.3 Branch and bound .......................................................... - 5 0

3.3.1 Branching rule ........................................................... - 5 0

3.3.2 Bounds ................................................................... 51

3.3.3 Decomposition and dominance .......................................... - 5 2

3.4 Computational experiment ................................................. - 5 3

3 .4.1 Teçt problems ........................................ ... .................. 53

3.4.2 Fksults .................................................................. -53

.................................................................... 3.5 Si in imary 55

4 Minimizing maximum lateness in the two-machine flow-shop with release t imes 60

................................................................. 4.1 Introduction 60

.......................................................... 4.2 Dominance results - 6 2

............................................. 4.2.1 Subset-restricted interchange 62

.............................. 4.2.2 ~ e w dominance order for ~ l / r , , ~ e m / ~ ~ , 70

4.2.3 Fuzzy approximation of dominance ...................................... -74

........................................................... 4.3 Branch and bound 76

........................................................... 4.3.1 Branching d e -76

.................................................................. 4.3.2 Bounds -77

.......................................... 4 .3.3 Decomposition and dominance -78

4.4 Computational experiment ................................................. -79

4.4.1 Test problems ............................................................. 79

................................................................... 4.4.2 Results 79

5 Thesis surnmary

A Algorithm for F2/rj/Cmu

B Algorithm for R2/rj, L:,

Bibliography

List of figures

Directed graph G(s) for problem 1 /r j / f.. ................................. 17

..................................... Directed graphs G ( s 2 ) and G(si) for PI 18

...................................... Staircase stmcture for the general case 24

....................................................... Ideals and filtersi for + 27

......................................... Planar representation for Example 1 29


...................................................... ide& and filters for eC 31

............................. Staircase stmcture for problem 1 / r j /maxwjLj 34

............................... Directed graph G(s) for problem F2 /ri/ Cm, 41


Representation of the diagonal jobs A ...................................... -47

........................ Directed graph G(s) for problem FZ / r j . perm/ L, 65

..................................... Directed graphs G(s2) and G(sl) for SI 70

List of tables

Dominating pairs for Theorem 2.1. .......................................... 19

Dominating pairs for Theorem 2.2. ......................................... -20


......... Results of computationd experiment for problem l/îj/ max wjCj -37



.......................................... Dominating pairs for Theorem 3.3. 44

............. Results of computational expriment for problem F2 /rj/ Cm, -57

Dominating paths for Theorem 4.1. ........................................ -66

Dominating paths for Theorem 4.2. ........................................ -68

Results for (0.2,0.2) and (0.4,0.2). ......................................... -83

Results for (0.4,0.4) and R = 0.6. .......................................... -84

Results for R = 0.8 and R = 1.0. ............................................ 85

Chapter 1 Introduction

1.1 Methods of sequencing and scheduling

Sequencing and scheduling problems are of great practical importance in many ar-

eas , including production planning and cornputer systems. Broadly defineci, t hese

pro blems are concerneci wit h the optimal allocation of scarce resources to act ivit ies

over time [26]. A sequencing problem involves hding a sequence in which to process

a set of tasks, that minimizes a given cost function. A scheduling problem involves

detennining a detailed assignment of jobs to machines over a period of t h e , that

minimizes a given cost function. The assignment of jobs to machines is subject to

difierent technological constraints imposeci by the jobs t hemselves and the machine

environment. Given the variety of technologicai constraints and cost functions, the

number of different types of scheduling problems is practically uniimitecl. The job

shop scheduling model describes a very important class of scheduling problems. In

this model, there is a set of jobs that must be procesecl by a set of machines, in or-

der to minimize a single cost function that is usually a function of the job completion

times. Each job is broken down into separate operations that need to be perforrned

on Werent machines, in a prespeciûed order depending on the job. Each machine

can process only one job at a tirne, and each job can be processeci on only one rna-

chine at a tirne. Problems in which the machine order is the same over ail jobs are

called flow-shop problems. We will only be concemed with scheduling problems that

are sequencing problems; problems for which a sciiedule is completely specified by

the sequence in which the jobs are prccessed. We will focus on certain single-machine

scheduling problems and ~ e m a c h i n e flow-shop problems.

The theory of computational compl&w provides a mathematical framework in

which t O s t udy and compare different computat iond problems. (For an introduction

to the concepts involved the interested reader is referred to Lawler et al. [26] and the

book by Garey and Johnson [13].) According to this theory, problems are classifieci

as either 'easy' or 'hard' depending on the theoretical worst-case behavior of their

solution algorithms. The easy, or well-solved problems, are those for which there

exist solution algorithms whose running times are bounded by a polynomial in the

size of the problem input. The hard problems are those for which it is provably

doubtful that such polynomial time algorïthms exïst. For our sequencing problems

this is the class of "NF'-hard" problems. (Following the scheduling literature, we

use "NP-hard" to refer to the optimization version of an "NP-complete7' decision

problem.) For the class "NP-hard" it has been proven that, a polynomial algorithm

exists for a given problem if and only if polynomial algorithms exist for all problems

in the class. Since the class "NP-hard" contains many known hard problems, this

provides strong evidence that a polynomial algorithm probably does not exist for any

of these problems. This complexity result suggests that we must resort to solution

methods that are essentially enumerative in nature to solve such problems. This is

the case for the one- and tw+rnachine problems that we study in this thesis. For t hese

problems we employ a well-known enumerative method called branch and bound.

Branch and bound attempts to solve a sequencing problem by building a search

tree that consists of fixed partial sequences. The branching scheme determines how

to branch between k e d partial sequences, in order to traverse the tree in search of

an optimal sequence. The bounding scheme specifies upper and lower bounds on

the optimal completion of a given âued partial sequence. A partial sequence and

aU its descendants can be eliminated from further consideration if its lower bound

is greater than the best solution identified in the search tree up to this point. A

stopping d e , which is usuaily expressed in terms of the total computation time or

the total number of partial sequences examine&, curtails the search. If the aigorithm

terminates by examining or eliminating all partial sequences, then the best solution

identifieci is necessarily optimal. In this thesis, we examine new ways to further reduce

the search space for such algonthms, in order to reduce both storage requirements

and computation tirne. We derive a dominance order, which is a partial ordering of

the jobs that is obeyed by at least one optimal sequence. Partial sequences that do

not obey the dominance order can be eLiminated fkom consideration, reducing the

size of the search tree. One technique we use to derive our dominance order is called

pairwise job interchange.

Pairwise job interchange is one of the most commonîy used techniques for

sequencing and schediilinp problems. Pairwise job interchange compares the cost of

sequences that Mer only in the interchange of a pair of jobs. It tries to improve a

sequence by perfonning interchanges that result in no increase in m. The merent

m e s of pairwise interchanges can be considered as interchange operators that perform

the specific transfomat ion.

The interchange operators are clasified following Monma [30], both by the

type of interchange they perform and the relative position of the jobs. The jobs (be-

fore interchange) are classified as either adjacent or nonadjacent, and the type of

interchange is either a direct swap of a pair of jobs or the insertion of a job either

fornani or baclcvard in the sequence. The most commody used interchange opera-

tors in the scheduling literature are: Adjacent Paif'unge Intemhange (API), Painuése

Interchange (PI), BackWad Insertion (BI), and Fonuani Insertion (FI). To illustrate

these operators consider the sequence si = (XmYkZ) where m and k are individ-

ual jobs and X, Y, and 2 are (possibly empty) sequencg of jobs. If Y = 0, then

sl = (XmkZ), i.e., m and k are adjocent and API applied to sl gives sz = ( X k m Z ) .

I f Y # 0, then m and k are nonadjacent and PI, BI, and FI applied to sl give

respectively s3 = ( X k Y m Z ) , s, = ( X Y k m Z ) , and s5 = ( X k r n Y Z ) . Note that BI

involves removing job rn and inserting it backward in the sequence just after job k,

while FI is the dual operation of removing job k and inserting it forward just before

job m. Also note that if Y = 0, then PI, BI, and FI ail reduce to AI.

Pairwise job interchange has been used in various areas of scheduling theory.

Among t hese:

Proving the optimality of sequencing rules;

Proving precedence relations and dominance criteria;

a Computing lower bounds for enurnerative algorithms like branch and bound.

What all of these have in cornmon is the property that there exists a preference order

for the pairwise interchange of jobs for a given interchange operator, such that, if a

pair of jobs is not in preference order then applying the interchange operator to the

pair to make theh order consistent with the preference order results in no increase

in cost. In this way the sequencing function indicates a preference for the ordering

of certain pairs of jobs. This ordering can be defineci a priori, or it can be sequence-

or tirne-dependent (i-e., dependent on the start time of the pair of jobs). In general,

these preference orders can be very poorly behaved, they need not even be transitive.

The sequencing rules of Johnson [21], Smith [37], and Jackson [20], are exam-

ples of preference orders for the adjacent interchange of jobs that are complete and

transitive orders. This order is complete because every pair of jobs k and rn is ordered

by the sequence, and it is transitive because of the transitivity of the sequence. Recail

that in this case if k is preferred to m, then k and m can be interchanged if k and m

are adjacent and n precedes k. If there are n jobs, then any optimal sequence can be

transformeci into preference order by applying at m a t 1(2 adjacent interchanges,

proving the optimality of the sequence. Johnson's rule establishes such a sequence

for the twemachine maximum completion time flow-shop problem F2 / /Cm, . (We

use the standard notation to describe scheduling problems, and refer the reader to

[26] and [32] for any terminology not defineci here.) The d e s of Smith and Jackson

state the optimality of the weighted shortest processing time and earliest due date

sequences for the total weighted completion tirne and maximum lateness problems,

1 //C wjCj and 1 / /Lw , respectively.

Dominance criteria and p r d e n c e relations are partial orderings of the jobs

that are satisfied by at least one optimal sequence. These orderings can be used to

limit the search space, or provide branching rules for enurnerative techniques such as

branch and bound. They are most often deriveci using interchange operators where

the conditions that define the preference ordering for interchange are sequence or

time-dependent. Next we consider some examples to illustrate the different properties

of t hese preference orders. The precedence relation for total tardines 1 / /C Tj due

to Emmons [7], is an example where the preference order for nonadjacent pairaise

interchange (i-e., for the PI operator) is an incomplete and transitive order, or partial

order. This order does not depend on tirne, and is dehed a priori by the jobs'

parameters. By applying nonadjacent job interchanges any optimal sequence can be

transformeci into an optimal sequence which extends or satisfies the preference order.

Thus the preference order for the PI operator is a precedence relation. Rinnooy Kan

et al. [36] extend this to general nondecreasing costs fj(Cj), to minimize total cost

for the singlemachine scheduling problem 1 //x f, (Ci) . They derive dominance

criteria for an enurneration scheme that fUs a schedule from back to fiont. They

use a number of interchange operators includùig (PI) and (BI) to demonstrate their

dominance criteria. Here the conditions for the preference orders for interchange are

sequence-dependent, depending on the predecessor and successor sets as well as the

job parameters. For multiple machines Della Croce [5] derives dominance criteria

for partial sequences using (PI) and (FI). He does so for the tw-machine total

completion time problem, F2 / /C Cj , and the maximum completion time problem

with release timg, F 2 /ri/ Cm-, in flow-shops. For the A P I operator two different

approaches have been used to derive precedence relations. The first is a sequence

dependent approach, and the second is a tirnedependent approach. In the first case we

have the pyramid precedence orders of Ersclder et al. [8] and [9], for maximum lateness

wit h release times and maximum complet ion t ime wit h release times and deadlines,

i I r j / Lm= and 1 Ir,, &/ CM, respectively. The pyramid order, is a precedence

relation that is a subset of the preference order for adjacent interchange. This subset

defines precedence relations that are tme for nonadjacent jobs, not just adjacent jobs.

They £ind conditions on intermediate jobs t hat determine when adjacent preférences

extend to nonadjacent preferences. Given a sequence (XmYkZ) with k prefmed

LO m for adjacent interchange, they find conditions on the intermediate sequence Y

that permit k and m to be interchangeci using PI to obtain the dominant sequence

(X kY m Z ) . Thus, this approach c m be considered as sequence-dependent , depending

on the intermediate sequence. The parametric precedence relations of Szwarc et al.

[39], [38], and 1401 are examples where the preference order for the adjacent interchange

of jobs is an incomplete and intransitive order that depends on the start time of the

pair. Pairs that are ordered independently of start time are called globdy ordered

while the remaining pairs are said to be locaily ordered. Assuming that the global

order is decomposable, they split the problem into smaller subproblems and look at

the m k of global and local orderings on each part. It is the mixture of these two

types of orderings that is not transitive, and they try to h d a subset of this mixture

that is a precedence relation.

Lower bounds can be cornputeci for certain single machine scheduling problems

using the technique of Della Croce [5]. This technique is to repeatedly apply the time-

dependent precedence relations of Szwarc to compute a lower bound for a partial

sequencing of the jobs. He derives lower bounds for the total quadratic lateness

problem 1 //x L: , the total tardines problem 1 //x Tj , and certain multicriteria

problems.

In this thesis, we consider a generaiization of pairwise interchange, that is

a new sequencedependent approach which inwrporates subset restrictions on the

intermediate sequence. We cal1 this technique subset-rgtricted pairwise interchange.

We use this technique to derive new dominance orders for certain oneand twemachine

sequencing problems. These dominance orders are used to reduce branchhg in an

efficient branch and bound algorithm capable of solving large problem instanca.

1.2 Preview of the thesis

For the remainder of this Chapta, we present the preliminary definitions and notation

that will be useci throughout the thais. In the next two sections we give the basic

definitions and notation for sequencing and scheduling problems and partial order

theory, respectively. This is foUowed by a discussion of the fundamental concepts

and definitions, of subset-restricted pairwise interchange, which are formulated for a

general sequencing problem.

In Chapter 2, we derive new dominance results for the single machine schedul-

ing problem 1 I r j / fmm, using subset-restricted pairwise interchange. We derive a

dominance order that is a suborder of the adjacent interchange order for these prob

lems. This extends the pyramid p r d e n c e orders of Erschler et ai. (181 and [9] )

for 1 /ri / Lm, and 1 / r j , &/ Cm=, to the general problem 1 /rj/ fma, which includes

1 I r j / rnax wjCj and 1 /ri / rnax w, Lj as special cases. We do so by applying Mer-

ent interchange operators together with a new representation of the dominance order.

This new diagonal representation is not Iimited by the dimension of the adjacent inter-

change order like the pyramid representation, which was restricted to tslrrdimensiona.1

orders. We tested the effectiveness of the dominance order in a branch and bound

algorithm for the problem 1 / r j / rnax wjCj. The algorithm performed very weli solv-

ing 2381 of the 2 4 0 test problems, while reducing the total computation time by 58

percent.

In Chapter 3, we apply subset-restricted interchange to derive a new domi-

nance order for the lx-machine flow-shop problem F 2 /r i / Cm,, and incorporate it

into a very fast branch and bound aigorithm. We teçted the algorithm in a largescale

computational experiment. The algorithm solved, within a few seconds, over 95 per-

cent of the test problems, some with up to 500 jobs. Even for the unsolved problem,

we were within 3 percent of the optimal solution in the worst case. This means that

the algorithm has the potential of being used as a subroutine for F2 /r i / Cm, type

su bpro blems generated during the solution of more complicated problems. We also

found that the speed of the algorithm is largely due to the use of the dominance or-

der, whîch reduced the total computation time by around 80 percent. The experiment

also helped in classifying 'easy' and 'hard' instances of the problem, it indicated that

the hardest problems were those with range of release times approximately 1/2 the

expected total processing time of the jobs.

In Chapter 4, we derive new dominance results for the weii-knom two-machine

permutation flow-shop problem F2 / r j , p e m / Lm-. This tirne, we apply subset-

restricted interchange with a new interchange operator, which we cal1 shufRe inter-

change. With this operator we derive a new dominance order, which we use in a

branch and bound algorithm for the problem- The algorithm also features a new d e

composition procedure which fixes jobs at the beginning of the schedule, and was

quite effective at reducing the size of the problem. The algorithm performed weil,

very quickly solving 4,384 of the 4500 test problems which ranged in size fkom 20

to 200 jobs. Because of this, the algorithm also has great potential as a subroutine

for the solution of more complicated scheduling problems. We also found that the

dominance order reduced the total computation tirne by 15 to 20 percent.

Chapter 5 gives a brief sllmmary of the main results of the thesis, as well as

some directions for future research.

1.3 Preliminary definit ions and notation

1.3.1 Sequencing notation

Recall, we cail a scheduling problem a sequenczng prvblern and its objective a se-

quencing firnction if any schedule can be completely specified by the sequence in

which the jobs are performed. For the problems 1 I r j / fmar and F2 / r j / Cm=, it is

weli known that such optimal permutation schedules exist. For problem F2 / r j / Lm,,

we wiU restrict our attention to such permutation scheduies and denote the problem

by F 2 /rj, perrn/ Lm=. For each of these problems we let J = {1,2, . . . , n} be the set

of jobs to be sequenced. A sequence s on J is a fundion hom {1,2, . . . , n} to J repre-

sented by the n-tuple (s (1) , s (2) , . . . , s (n)), where s (i) is the ith job in sequence S.

For these problems the objective is to find a sequence s that rninimizes the maximum

cost taken over ail the jobs, where the c a t of a job is a nmction of its completion

t ime.

The singlemachine scheduling problem 1 /r i / f,, has the following defining

properties. Each job j requires p, units of processing time on the machine, it can

start processing on the machine at any t h e after its release time rj, however, once

its processing has started it cannot be intempted. The cost of job j is given by a

nondecreasing real dueci function fj(t) that gives the cost of completing job j at

time t . If are let C j be the completion time of job j, then its cost îs given by f j (Ci) .

An important example is the lateness objective with L j = Cj - di, where dj is the due

date for job j. Here the objective is t o find a sequence s that minimks mgc L,(i). l<t<n

For the two machine flow-shop problems F2 Irj/ C,, and F2 /rj , p e m / L,,

the jobs have release times rj and they must be p r o c d in the same sequence

without interruption h t on machine 1 and then on machine 2. For a given job j

its processing on machine 1 must be finished before its processing on machine 2 can

start. For job j, its processing t i m s on machines 1 and 2, are given by aj and bj ,

respectively. The completion time of job j on machines 1 and 2 wiil be denoted by

Cj and Cf, respectively. The completion time of job j is C j where now Cj = q. For the problem F2 I r j / Cm,, the cost of job j is C, and the objective is to fud

a sequence s that r n h k k s the completion tirne of the last job in the sequence,

Cs(,. For F2/rj,pwm/ Lm-, the CO& of job j is given by Lj = Cj - dj = - dj,

and the objective is the same as above for 1 Irj/ Lm,. We will choose to consider

these problems in th& equivalent delivery-tirne fom, denoted by 1 /ri/ L, and

F2 Irj, perm/ L:,, where qj = T-di is t.he delivery tirne for job j and T is a constant

with T 2 max{dj ( j E J ) . If we define L; = Cj+qj , then L; = C, +T - d j = Li +T,

and we see that the two objectives L,, and Lm, are equivalent.

For our problems, the dominance order will be a partial order dehed by the

parameters of the jobs. Thus we introduce certain definitions for partidy orderd

sets (posets) .

1.3.2 Partiai orders

By a partially ordered set we mean a pair P = ( X , S p ) consisting of a set X together

with a binary relation Lp on X x X, which is rdexive, antiçymmetric, and transitive.

For u, u E X , u sp v is interpreted as u is l e s than or equal to v in P. Similarly,

u < p v meam that u $ p v and u # v. The usual symbols $ and < will be reserved

for relations between r d numbers. A partial order P = (X, sp) is a linear order (or

complete order) if for every pair ( u , v ) E X x X either u S p v or v sp u. Given a

pair of partial orders P = (X, S p ) and Q = (X, Io) on the same set X, we c d Q

an extension of P ( P a suborder of Q ) ifu Sp v implies u Sq v for a.li u ,v E X .

A partial order Q = (X, SQ) is a h e u r &-on of a partial order P = (X, SP) ,

if Q is a linear order that extends P. Given two partial orders Pl = (X, SPI) and

9 = (X, <&), we can define the partial order Pl n PZ = (X, spi-), the intersection

of Pi and 4, where u <pIOf i v if and only if u SPI v and u v for all u, u E X.

The dimension of a partial order P = (X, S p ) , denoted by dim(P), is the smdest

1 such that there exists a set {QI, Q2, . . . , Q I } of linear extensions of P such that

P = nf,,Q,. A subset I Ç X is an ideal of P if for every v E I and u E X such

that u Sp v we have u E 1. Simiiarly, F C X is a jüter of P if for every u E F

and v f X such that u S p v we have v E F. For every v E X the principal ideal

I ( v ) is dehed by I ( v ) = {u E X lu sp v ) and the principnlfilter F (v ) is defined by

F (u) = {u E X IV sp u}. A partial order P on the job set of a sequencing problem

is called a dominance order if there is an optimal sequence that is a linear extension

of P.

1.3.3 Subset-restricted pairwise interchange

We follow Monma [30] in defining our interchange operators. Consider a sequence with

job r n preceding job k, of the form (XmY kZ) where X, Y, and 2 are subsequencg

of J. We define three types of interchanges of jobs k and m that leave k preceding rn

in the resulting sequence.

1 . Backward Insedion(B1) (XrnYkZ) -r ( X Y kmZ) 2. Fornani Imertion(FI) (XmYkZ) -r ( X k m Y Z ) 3. Painuise Interchonge(P1) (XmYkZ) + ( X k Y m Z )

If we let Y be the set o f jobs iri sequence Y (we do not distinguish between these),

then each of these interchanges reduces to adjacent pairwise interchange in the case

when Y = 0. This leads to the definition of the adjacent interchange order.

Definition 1.1 A partial order < is an adjacent interchange order for a sequenc-

zng functzon f if for al1 jobs k, rn and sequences X, Z k e n implies f ( X k m Z ) 5

f (XrnkZ) .

Note that all of the above interchanges involve interchanging k, m or both k and

m around sequence Y. Intuitively, whether or not an interchange of jobs k and

m with k e r n leads to a reduction in CO& (for a given sequencing function f and

adjacent interchange order t ), should depend on the composition of Y. This involves

placing restrictions on certain of the parameters of the jobs in Y . In the case when

interchangeability does not depend on the composition of Y, then 4 is a dominance

order for the sequencing problern, Le., there exists an optimal sequence that is a linear

extension of +. Such an example is the dominance order 4 defined by

k e n pk 5 Pm and dk 5 d,

for the total tardiness problem on a single-machine, 1 //x T, [7]. Note that here 4 is

the intersection of the 5, and Id orders. In general, an adjacent interchange order 4

is not necessarily a dominance order, as we s h d see later. We consider interchanges

that are restricted by conditions on Y and define the subset-restricted interchange

conditions as foilows.

Definition 1.2 An adjacent interchange orùer r together m-th the wllection of sub-

sets RP' = {R:& Ikern) sattPfes the Restricted Painvise Intemhange Condition

for a sequencing W t w n f if for al1 jobs k , m and sequences X, Y, Z k f m and

Y C R,P:, imply f ( X k Y m Z ) 5 f (XmYkZ) .

Definition 1.3 An adjacent intemhange order t together with

sets R ~ ' = {R:& l k t r n ) satisjks the Restricted Backward

for a sequencing hnction f if for dl jobs k , rn and sequences

Y E R , B : ~ imply f ( X Y k m Z ) 5 f (XrnYkZ) .

the collection of sub-

Insertion Condition

X, Y, Z k ~ m and

the collection of sub- Definition 1.4 An adjacent interchange order e together with

sets RF' = {R[<_ [ k e r n ) sathfies the Restricted Fonuard Insertion Condition for

a sequencing finctàon j if for dl jobs k, rn and sequenees X, Y, Z kern and

Y R , F : ~ imply f (XkmYZ) 5 f (XrnYkZ) .

The interchonge w~ for the pair kern are the sets of jobs that k and m

can be interchanged around without increasing cost. For PI , B 1, and FI these are PI given by Rkem, R&, and R:& respectively. In the following Chapters, we wili

use subset-restricted interchange to derive a dominance order + on the jobs, that is

a suborder of the adjacent interchange order 4.

Chapter 2 Minimizing maximum cost for singlemachine sequencing problems with release times

2.1

In this

change

Introduction

Chapter, we introduce a new technique, a generalization of pairwise inter-

that takes into consideration the composition of the intermediate squence.

This yields a preference order which permits the interchange of a pair of jobs pmvided

that the intermediate jobs belong to a restricted subset. The traditional methods of

pairwise job interchange can be viewed as special cases of this subset-restricted inter-

change, where the subsets are either uniformly the empty set (adjacent interchange)

or the entire job set (nonadjacent interchange). Ln general, an adjacent interchange

order is not a complete order and therefore it is not a dominance order, as we s h d

see for the 1 / t j !L, problem, using an example due to Lageweg et al. (231. We

prove, however, that ushg subseerestricted interchange for the class of regular, sin-

gle machine scheduiing problems 1 Irj/ f,,, we can derive a dominance order that

is a suborder of the adjacent interchange order. Recail, for these problerns, each job

j has an associated nondecreasing, real valued cost function fj (t), the cost of corn-

pleting j at time t, and the objective is to minimi.le the maximum c a t f,,. The

dominance orders we derive have the property that they are definecl in&pen&nUy of

the processing times. This makes them especially useful in applications with stochas-

tic or ill-dehed proogsing times. We use only certain extreme values of the other job

parameters that display a 'staircase-like' structure. In addition, the dominance orders

derived belong to a special class of partial orders, the interval orders [ll]. This also

leah to the complexity implication that the above problems are strongly NP-hard

for interval-ordered tasks. Our results can be vie& as a unifid tmtrnent of job

interchange, that generalztes well-known d e s for deriving dominance relations and

&ends the pyramid dominance orders of Erschier et al. ([8] and [9]) for 1 /rj /Lm,

to the general problem 1 /rj/ f-. We generalize the pairwise interchange and in-

sertion operatioas of Monma [30], and introduce 'pyramid-like' structures of higher

dimension than two, extending the 2-dimensionaï staircases of [8] and [9]. In our

unised theory, the problem 1 /rj /Lw represents the m& special case.

The Chapter is organized as follows. In the next section, we review some im-

portant previous results. In section 3, we define the adjacent interchange order 4 and

the interchange regions for 1 /rj / f,, . In section 4, we derive a dominance order 4

for the problem I /rj If,, , which includes the problems 1 / r j / L,, , 1 Ir,, q/ Cm,,

1 / r j /max wjCj , and 1 /tj /max wj L, as special cases. Ln section 5, we discuss the

results of a computational experiment for the problem 1 /+j /max wjCj . Finally, in

section 6 we siimmarize the results of the Chapter.

2.2 Previous results

Although the maximum cost objective is quite general, there are still some very im-

portant polynomially solvable general cases. The single machine problem with prece-

dence constraints 1 /prec / f,, , is solved in 0(n2) time by Lawler's algorithm [25].

Lawler's algorithm sequences the jobs fiom back to front, by repeatedly sequencing

last a maximal job in the precedence order which has the srnailest cost if put in the

last position. An interchange argument using BI shows that it is only necessary to

consider such schedules. Baker et al. [2] generaiize Lawler's algorithm to cover the

preemptive case with release timg 1 /ri, pmtn, precl f,, , which can be irnplemented

to run in 0(n2) time. That the nonpreemptive case 1 /tj / fmu is stmngly NP-hard

follows fiom the result for 1 /rj /Lm, due to Garey and Johnson [13]. This corn-

plexity result has led researchers to consider various enurneration and approximation

algorit hms.

Many branch and bound algorithms have been proposed for the maximum late

ness problem 1 /rj /L, ; exhibiting va.rious branching and lower bounding schemes.

Among these are the algorithms of Baker and Su [Il, McMahon and Florian [28], and

Lageweg et al. [23]. The relatively most effective is an algorithm due to Carlier [3]

, for the problem 1 /tj IL;, . The effectiveness of Carlier's algorithm is due largely

to the efficient binary branching d e he employs. The branching rule is based on the

dominance properties of a critical path in the ready-Jackson sequence; this is the se-

queace obtained by greedily sequencing the job j with the largest qj (Le., earliest due

date) fkom arnong the ready jobs. Zdralka and Grabowski [43] consider an extension

of this branching rule for the problem 1 lprec, rj / f,, , which they test for the prob

lems 1 / r , / L;, and 1 / r j /max wj L, . Erschler et al. [8] and [9], derive a dominance

order that is a suborder of the adjacent interchange order, which they use to reduce

branching in the algorithm of Baker and Su [IO].

Potts [33] develops an approximation algorithm A for 1 Irj / L;, , which iter-

atively tries to improve the 'ready-Jackson' sequence. Algorithm A has a worst case

performance guarantee of 3/2, i.e., L , ( A ) / L ~ , 5 3/2. Given the complexity of the

problem 1 /rj / L, , the best possible approximate result that can be obtained is a

polynomial approximation scheme. Hall and Shmoys present such a scheme for the

problem, in [19].

2.3 Interchange regions

In this section, we derive the interchange regions (subsets) for the general problem

1 /ri / f,, . Recall, in this problem, each job j has an associated nondixmasang, real-

valued CO& function fj, where f, (t) is the cost of completing job j at time t , and

the objective is to minimize f,, = max fj(Cj) over all sequences. We order the lIj<n

jobs according to >,, where f* 21 f, o fi ( t ) 2 fj ( t ) for al1 t >_ O. Note that, in

the general case, If does not order every pair i and j, it might be only a partial

order. Two speciai, linearly ordered cases of If occur for the lateness objective in

its delivery form , where fj (t) = t + q,, and the weighted completion time objective,

where f, ( t ) = wjt . Hall [17] considered the 2, order and noticed that the hear

ordering property makes it possible to extend Potts' [33] approximation algorithm for

the 1 /ri IL:, problem to the 1 / r j / f, , problem when 2 is a linear order.

The adjacent interchange order and the restricted subsets for i / r j / f,, are

defined below. We note that these definitions use no processing tzme infornation.

This means that all of the subsequent results are true irrespective of job processing

times.

Definition 2.1 Adjacent interrhange order: k<m o r k 5 r , and fk 2 f,.

Definition2.2 Given kem, define the following subsets of jobs: (i) R $ m = { j I r j s r m ~ f k 2 f f j } (ii) R ~ & = { j I r j ~ r m )

( 2 2 2 ) R,F:, = { j Ifk Lf f j 1- Ln general, an adjacent interchange order r is not necessarily a dominance

order, as it can be demonstrated by the instance of the maximum lateness problem

1 Ir j / L:,, shown in its equivaient delivery time form in Example 1 [23]. As we shall

see in the next section, the adjacent interdiange order for 1 / r j / L:',, is d&ed by

k+m @ r k 5 rm and qk > qm. For the 5 job example specified below we have 442,

however, the unique optimal sequence is (1,2,3,4,5) with L:, = 11. Thus 4 is not

a dominance order, since the unique optimal sequence is not a linear extension of 4.

Example 1 A 5 job pmblem to illustmte that r is not necessar-iiy a dominance

Next we prove that the adjacent interchange order f together with the above

collections of subsets satisfy the subset-restricted interchange conditions.

Theorern 2.1 Partial order + together with the collection of subsets IlPx = { R F & ~ 1 k r r n ) satisfies the Restricted Psi-e I n t d a n g e Condition for 1 / r j / f,, .

Proof. Given a sequence s, recall that fmu (s) = max fsCi) (c.~)), where Csu-) is the l g l n

completion time of job s(j) We construct a d i r d graph G (s) to eraluate f,, ( s )

(see Figure 2.1). Each job s(j) is represented by two nodes: the first one has weight

rS(j), followeà by the second node with weight p.^) G (s) has the property that the

complet ion time of job s( j) in sequence s is the length of the longest 'node-weighted'

path fiom O to s ( j ) . We represent the paths £rom O to s( j ) by pairs (s(i), s(j))

1 5 i 5 j 5 n, where s(i) is the first node on the path following the start node O and

s ( j ) is the last node of the path. Then by definition

-

and we can evaluate f,. (s) as the maximum over all such pairs ( s ( i ) , s ( j ) ) in G (s) -

Figure 2.1: Directeci graph G ( s ) for problem 1 /rj If,, .

Let si = ( X m Y k Z ) be a sequence with the property that k e m and Y C

R z , We apply pairwise interchange to sl and obtain sequene s2 = (XkYrnZ) .

We demonstrate that sa is not worse than sl by showing that for every pair of jobs

in s2 there exists a dorninatàng pair in sl with a not smaller f value. For example,

consider pair (k, m) in 8 2 , then it has the dominating pair (m, k) in sl (see Figure

2.2): That is,

which holds since k<m implies r k 5 r, and fk ( t ) 3 fm ( t ) for al1 t, fkom which it

~ O ~ O W S that f, rk+pm+ x &+pt , and findy o E Y

fk is nonddng.

Figure 2.2: Directed graphs G(s2) and G(sl) for PI.

Table 2.1 gives a dominating pair in si for each pair in s2. ( We use lower case

letters x , y, or z to refer to arbitrary generic elements of the subsequencg X,Y or 2,

respectively). The last three coliimns contain the arguments why they are dominating

pairs. I

Table 2.1: Dominating pairs for Theorem 2.1.

Theorem 2.2 Partial order 4 together with the collection of subsets RBI = { R : ; ~ 1 k e r n ) satisfies the Restricted Backward Insertion Condition for 1 / r j / f,, .

Proof. The proof is totally analogous to that for pairwise interchange. Table 2.2

gives the corresponding dominating pairs.

Theorem 2.3 Partial o h t together with the collection of subsets R ~ ' = { R : : ~ 1 k e r n )

satisjies the Restricted Forward Insertion Condition for 1 / r j / f,, .

Proof. The proof is totally analogous to that for pairwise interchange. Table 2.3

gives the corresponding dominating pairs. . Remark 1 We observe that for any krm, we have RZ, = ~f&, n R Z ~ . Thus

if Y E REm, then this implies not only f (XkYmZ) 9 f (XmYkZ), but also f ( X Y k m Z ) 5 f (XmYkZ) and f (XkmY 2) < f (XmY kZ) .

The preceding theorems could directly be used in branch and bound algorithms

for restricting the search space on sequences. This, however, would require branching

on sequences and storing for all pairs k e r n the subsets of Definition 2.2 and the

testing of membership in these, which would be time consuming and inefficient. In

the following sections, we show that there is a much more dective way to restrict the

search space, by proving that there is a dominance order on the jobs.

We also note that by simply modifying release times and processing times,

problems in which the jobs have setup times can be handled as well, Two forms of

job setups can be considered: either a setup pi is attached to job il Le., it cannot be

performed without the job being present, that is before r,; or it is detached, Le., it c m

be performed prior to ri, if the machine is idle and waiting to process job i. Attached

setups can be dealt with by using modified processing times ~ = p, + pi, while

detached setups can be handled using modified release times ri = max { O , ri - pi)

and processing tirne pi = pi + pi. Thus, our theory of dominance orders applies to

the case witb setups too.

2.4 A general dominance order

In this section, first we use subset-restricted interchange to derive a dominance order

4 for the general case of 1 /ri If,, , foilowed by various special cases containing

weil-known scheduüng problems. We can represent the adjacent interchange order + using a Sdimensional stmcture. The axes of the 3-dimensional space correspond to

the r , t, and f (t) values (see Figure 2.3). Jobs are represented by their fj(t) C U . in

this space resting on planes whose height equals their release t k . Recail that k r t n

r k 5 rm and fk 2 fm, that is, if k is on a lower r-plane than m and fk is above

fm, i.e., fk(t) 2 f,@) for al1 t. The precedence order 4 is defined in terms of certain

'extreme jobs' in e, or more precisely, certain extreme curves of the 3-dimensional

structure. (Some of these extreme c w e s might not correspond to real jobs.) To

identify these jobs, we introduce the least upper bound in II for a finite set of jobs

1 Ç J , as the nondecreasing function pmax f, defined as the pointwise mmïrnurn j € I

of the functions fi, i.e., with values fj(t))(t) = max fj(t) for t . A simple j € I je1

greedy procedure to define the set of kundary jobs M = {Mi li = 1,2,. . . , H + 1)

and the set of diagonal jobs A = {Di, Dz, . . . , DH), is presented below. Assume that

there are K distinct r values denoted by r' for i = 1,2, . . . , K, and r1 > r2 > - - - O - > rK.

Define the fundion = pmax { fj Irj = ri ), this is the letut upper bound in 5, of the

jobs on level ri.

Algorithm A for i / r j / f,, let ( r ~ , , f ~ ~ ) = ( r l , f l ) , l = 1 , i = 2

while i 5 K do

begin

begin

end;

end;

The procedure looks at the r-levels r' ( i = 1,2, . . . , K ) and compares the large& f

on this level ( f i ) with the largest f obtained so far ( f M 1 ) . I f f' represents a strict

increase compared to fMl , then (ri , f') becornes the new boundary pair (rM, +, , f Ml +, )

and ( rDl , f is the projection of ( r4 , f M l ) ont0 level ri. Note that these extreme

jobs are not necessarily real jobs, since they are defind using pointwise maximum.

Rather they are 'artificial jobs' used only for ordering purposes. The sets A and M

define a 'staircase-like' stnicture that contains a l l of the curves f , for j E J either

inside or on its surface (Figure 2.3 shows an example with 1A1 = 4). We augment the

partial order P = (J, f) with the diagonal set A and cal1 it PA = ( JA , eA), where

JA = J U A and +a is the order 6 extended to include the diagonal jobs, i.e., e~ is derived by applying Definition 2.1 to the extended set JA. Jobs k t m (k,m E J ) are

sepamted by A (are A-sepamted) if there exists a Di i E {1,2, . . . , H ) such that k is

in its principal ideal and m is in its principal filter, i.e., k E I(DJ and m E F(DJ

in PA. A induces a partition of P into separable and nonseparable pairs that can be

used to define 4.

Definition 2.3 Dominance order 4: For k e r n ( k , m E J ) , we d e f i e k 4 m if and

only if k and rn are sepamted by A.

It is well known that the main source of difficulty in all l/rj/ f,, problems is

the fact that at any time the machine becornes available, it may be better to wait for

a yet unreleased job rather than to schedule one of the jobs available. The partition of

P by A means that the only jobs for which it may be worth waiting are the ones which

are not separated by A kom the currently available jobs, that is 4 is a dominance

order.

Figure 2.3: Staircase structure for the general case.

Theorem 2.4 Partial onim + is a dominance order for 1 Irj /f,, .

Proof. We use subset-restricted interchange in the proof. The following observations

irnmediately follow fiom the construction of A:

Fùrthermore, for all y E J \ F (Di) and m E F (D,) we have r, 5 ro, 5 rm for any

i. This is the cruciui property used throughout our proof. Let s be any optimal

sequence. If every job in I ( D l ) is before every job in F ( D l ) , then all jobs separated

by Dl are already in 4 order, and consider I (4) and F ( 0 2 ) - Otherwise, let kl E

I ( D l ) be the last job in s that is after some job fi-om F ( D l ) , and let ml be the 1 s t

such job £rom F ( D l ) before ki. That is s = (X lmiY lk lZ i ) , where Zl n I ( D l ) = 0

and YI c J \ F (Dl) . By the above property, we have r,, 5 ro, 5 r,, for all

y1 E Yi , which implies that Yi C RFeml. Thus, by subset-restricted interchange,

we can insert ml bachard just after k1 to obtain the alternative optimal sequence

(XIYlk lmlZ1) . Following in this way, inserting the last job in F (Dl) bachard after

ktl until there are no such jobs, we obtain sequence sl which is an optimal sequence

with the property that I ( D l ) is before F (Di) . Continuing similarly, i f I (D2) is

before F (4) in sl , then all jobs separated by D2 are already in + order, and consider

1 (D3) and F (03). Otherwise, let k2 E 1 (a) be the last job in sl after some job fiom

F (4) \ F ( D l ) ( since I (D2) C I (Dl) and I ( D l ) is before F (Dl ) in si, k2 cannot

be after any job from F(D1) ), and let m2 be the last such job from F (D2) \ F (Dl ) .

Similady, we have sl = (X2m2Y2kzZ2), where Z2 n l (D2) = 0 and l$ C J\ F ( 0 2 ) . AS

above, we have that r, 5 th 5 r,, for ali y2 E f i , which implies that f i C REem2.

Thus we can insert m2 bachard just after k2 to obtain the sequence (X2Y2 k2m2 2 2 ) .

When a.U such jobs in F (D2) \ F ( D i ) have been inserted after hl we obtain sequence

s2, an optimal sequence with the property that 1 ( 0 2 ) is before F ( 0 2 ) and I ( D l ) is

before F (Dl ) . Continuing similarly, we obtain si for i = 3,4,. . . , H . Then s~ is an

optimal sequence with the property that I (Di) is before F (Di) for i = 1,2,. . . , H .

Thus s~ is a linear extension of 4, and we have that 4 is a dominance order indeed. I

Theorem 2.4 also has interesthg complexïty implications. Let us further aug-

ment PA by adding new least and greatest elements O and 1, and cal1 it PoVl =

(J0.1, f 0.1)~ where J0.i = J A U ( 0 ~ 1 ) and 40,l = e~ U ({O) x JA) U (JA x (1))- Then

4 admits an interval representation using intervals I j = [ x ( j ) , y ( j ) ] (j E J ) on the set

S = A U {O, 1) linearly ordered by GO,, , where for j E J, xi ' ) = max ( 1 E S 11 eOv1 j )

and y ( j ) = min ( 1 E S lj+l 1 ) . Partial orders (P, S p ) that possgs such an interval

representation on a linearly ordered set (where k S p rn H Ik lies to the left of I,)

are cded interval orders [1 11. This leads to the foilowing coroilary for 1 / r j / f ,, .

Corollary 2.1 Pmblem 1 /rj,prec/ f,, remazns NP-hard in the strong sense even

with .internai-orùer p d e n c e wnstmints.

2.4.1 Linearly ordered > f

In this section, we consider separately the special case where 3 is a linear order.

This means that for each pair of jobs i and j either fi ( t ) 2 fj ( t ) for all t, or

f, (t) 2 fi ( t ) for dl t. That is, we can completely arrange the jobs in nonincreasing f

order according to 21. We examine an quivalent pyramid representation for 4 ([8]

, [9], [29] and [121), and show how this representation does not extend to the nonlinearly

ordered general case.

For the linearly ordered case, we are able to represent the adjacent interchange

order in the plane using the r and f orders as the x and y axes respectively. Here jobs

are represented by points with preferences toward the origin, i.e., krm if k is closer

to the ongin than rn in both the r and f orders. The principal ideals and filters in 4

are represented by quadrants through these points. That is, let job i be represented

by the point (ri, f i ) . If we divide the plane into quadrants using the iines r = r, and

f = fi, then the SW and NE quadrants correspond to I (2) and F (i), the principal

ideal and 6lter in 4 for job i (see Figure 2.4).

This planar representation was used by Merce [29] to derive a dominance or-

der for the problem of minimixing the makespan in the presence of release tirnes and

deadlines (1 /ri, 5 /Cm ) . Fontan 1121 noted that if we consider due dates instead of

Figure 2.4: Ideals and filters for +.

deadlines then the same order is a precedence order for the lateness mode1 1 / r j /Lm, .

They did not consider the adjacent interchange order explicitly ([8] and [9]), rather

they defineci their order using certain extreme points in the plane, called siimmits.

These summits, represented by Si (i = 1 to N ), are the jobs that form a staîr-

case boundary in the plane and satisijr the property that their SE quadrant minus

the boundary is empty. The siimmits are completely ordered SI es2 4 . . . esN. For

each summit Si, Merce and Fontan define a pymmzd P(Si) , which is its N W quadrant

without its boundary iines but inc1uding Si. Jobs are classifieci using pairs that repre

sent their membership in pyramids. For j E J, they define u( j ) = min {i lj E P(S, ) ) , and v ( j ) = max {i 1 j E P(S,) } . With these, they define their partial order 4' by

k +' rn o v (k) c u (m). The planar representation for the %job problem of Exam-

ple 1 is shown in Figure 2.5. Jobs 3 and 5 are the sununits SI and Sz, respectively.

P(S1) = {l, 2,3,4) and P(S2) = { 5 } . We have v(i) = 1 for i = 1,2,3,4 and u(5) = 2,

which implis by definition of 4' that jobs 1,2,3 and 4 must precede job 5. Notice

that the unique optimal sequence (1,2,3,4,5) is a linear extension of +' (as we would

expect), but not a hear extension of e (as we saw earlier). Example 2 is another in-

stance of 1 /r,/ L,, this tirne with N = 9 summits. For this instance, the optimal

L;, = 114 and the sequence (1,S~,k,S2,S~,s4,s5,3,s6,s~,~,2,S8,Sa,4) is an optimal

sequence, which is also a linear extension of 4'. To further illustrate how 4' is defineci,

consider jobs k and m in Figure 2.6. Here k 4' m, since v ( k ) = 5 < 6 = u (m).

Example 2 A 15 job instance of 1 / r j / L;, with N = 9 summits.

Let us now apply our general dominance order 4 to this specid iinearly ordered

case: we consider a diagonal representation for 4, using the set of corner point

boundary jobs M = {Mi li = 1,2, . . . , H + 1 }, and the set of artificial jobs on their

inscribed diagonal A = {Dl, D2,. . . , DH) (see Figure 2.6). The set M C S is the

subset of boundary jobs with empty SE quadrants, and again we c d A the set of

diagonal jobs. We augment the partial order P = (J, e) by these diagonal jobs and

c d it PA = (JA, +), where JA = J U A and -CA is the planar order with these

diagonal jobs included. Jobs kem (k,m E J ) are separated by A (are A-separated)

if there exists a Di (i = 1,2, .. . , H ) such that k E I ( D i ) and m E F(Di) . Notice

that v ( k ) < u(m) when k and rn are A-separated. It can be easily v d e d for k e n

(k ,m E J ) that k +' m if and only if k and m are separated by A, i.e. k m. As

an example of this equivalence, consider again Example 1 in Figure 2.5: We see that

diagonal job Dl separates job 5 and jobs 1,2,3 and 4, and this is the only separation

present. Thus, by the diagonal representation of 4, jobs 1,2,3 and 4 must precede

job 5, and + and 4' define the same precedence orders indeed.

The pyramid representation used for 4' can be reinterpreted in partial order

terminology to explain why it does not extend to the nonlinearly ordered higher

dimensional case. The summits S. (i = 1,2, . . . , N) are marimal elements of a related

partial order fC, which is defineci by kecm o r k < r, and fk <I f,, the conjugate of

4 . Two partial orders on the same set are wnjugate if every pair of distinct elements

Figure 2.5: Planar representation for Example 1.

is comparable in exactly one of these partial orders. This is clearly the case if we

compare the principal ide& and filters for r and 4, using the planar representation.

For 4, these are the SW and NE quadrants, respectively, with the boundary lines

included. For eC, these are the NW and SE quadrants minus the boundary lines

(compare Figures 2.4 and 2.7). By a weii-known theorem of Dushnik and Miller

[6] kom partial order theory, eC exists if and only if dim(e) 5 2. In this context ,

the pyramids are just the principal id& of 4. The u(j) and u(j) defined in [9]

implicitly use the fact that 4 has an interval wntainment order representation, i.e.,

there exist intervals { I j 1 j E J ) such that itC j in Ii c I, for i, j E J. In the pyramid

representation of l/tj/Lmur, these iatefvals are just I j = [rj, d,] . On the other hand,

by the same theorem of Dushnik and Miller (61, a poset has an interval containment

representation iff its dimension is 2. Thus, the u(j) and u(j) can be defineci for any

problern for which dim(r)< 2, but it can be defined ody for such problems. Of

course, dim(6) 5 2 is equivalent to 2, king a linear order, so the u ( j ) and v (j) can

be dehed only in this case. Thus we see that the diagonal representation for 4 bas

Figure 2.6: PIanar representation for Example 2.

the advantage that it does not require that r possess a conjugate eC, and thus is not

msthcted by the dimension of 2 j.

It is interesting to note that the original proof due to Erschler et al. 191, for

1 /rj /Lm, , used painvise intmhange. This proof can also be modifieci to carry over

to other cost functions f when 2, is a linear order. We chose to present a proof

using backward insertion, however, because this extends to the (nonlinearly ordered)

general case. A proof using forward insertion can also be obtained by proceeding

in the opposite direction. This is due to the duality of the operations and regions

for the linearly ordered case. However, the proof based on forward insertion is not

extendable to the general case either, as the dualiS of regions no longer holds.

Alt hou& this section de& only with linearly ordereà 2 j, it covers a number

of weil-studied scheduling problems. In addition to the on- studied in ([SI and [9]),

we mention one further example.

Figure 2.7: Ideals and filters for ec.

Corollary 2.2 Partial order 4 is a dominance onier for 1 /rj/ max wjCj and, in

th& case, is the onier whzch orders the jobs in nonzncreaîing W j order.

T heorem 2 -4 and Corollary 2.1 also have interest ing complexity implications,

they yield the following tightening of previously known complexity results ([26] and

(171, respectively).

Corollary 2.3 Pmblerns 1 / r j , precl L, and 1 / r i , prec/ max wjCj remin NP-hard

in the stmng sense even with interval-order priecedence constmints.

Corollary 2.3 is interesting, as interval orders have a very special restricted

structure [Il], which d o g not seem to help in reducing the complexity of the schedul-

ing problems mentioned. This is in contrast with the result of [31] whicb shows

that P m / p j = 1, prec/ Cm, is polynomidy solvable for intenal-ordered precedence

constr aints.

Recall that when the adjacent interchange order + itself is a linear order, it

defines an optimal sequence. I t can easily be seen, that 4 is equivalent to 4 in this

case, and so it also defines an optimal sequence. This means that 4 also solves some

well-known special cases solved by Jackson's d e [20] or Lawler's method [25]: For

1 / r j / L:,, f i ( t ) = t + qi and the jobs are linearly ordered by fi f, o q, 2 q,. So

both r and 4 are linear orders and define an optimal sequence, for example, when

the j order is the same as the r order (the agreeably ordered case of 1 / + j / L;, in

which ri 5 rj w q, 2 qj); or one of the orders is trivial (Le., the j order is linear and

al1 the jobs have the same release tirne l / /L , , ) ; or al1 the jobs have the same cost

function (1 Irj, dj = d / L,,). Sunilar comments apply to the correspondhg speciai

CZ~S of 1 / r j / max wjCj and l / r j / f-.

2.4.2 Weighted maximum lateness

For this problem the Zr order is no longer linear, thus we proceed as in the generai

case. For the weighted maximum lateness problem, f j ( t ) = w,( t -d j ) for all j E J. Let

g, (t) = fj ( t )+L, where theconstant L ischosenso that L 2 max{wjdj [ j E J ) . That

i ~ , gj (t) = wjt + qj, where qj = L - wjdj 2 O . We see that fk 2, f, o (wk 1 wm)

and (qk 2 qm)- Thus, we cm represent the adjacent interchange order t and the

interchange regions using this 2-damensional reprsentation of zI. This meam that

our general definitions from before reduce to the following for 1 / r j /max w j L j -

We derive the A boundary by modifying the greedy p rodu re presented earlier.

Once again, we assume that there are K distinct r values r1 > r2 > - > rK and

let wi = maw {wj Ir, = r i } and qi = max {qj Irj = r ' ) for i = 1,2,. . . ,K. We use the

same notation as before for the boundary jobs Mi and the diagonal jobs Di-

begin

begin

end;

end;

If we represent f in %dimensions with q, w , and r as the x, y, and z axes, respec-

tively, then (qi, w') is the least upper bound accorcling to If of the jobs on the

plane r = r'. ( Thw is well defined by the finitenes of J . ) Symbolically, (pi, w') =

rnw {(qj, wj) Irj = ri), where we define rn- ( ( q j , w j ) lj E I ) = (max qj l max w j ) . I C I 1 E I

Taking least upper bounds, we greedily constmct the sets of possibly artificial jobs

A and M. These jobs are on the boundary of a step pyrcimid, which contains aU of

the original jobs inside or on its surface (see Figure 2.8 for an example with H = 6).

The definition of 4 is analogous to the general case: k 4 m if k and rn are A-

separated. n o m Theorem 2.4 we then have the following corollary for the problem

1 /rj / m a wjLj .

Corollary 2.4 Partial order 4 is o dominance order for 1 /rj /maxujLj .

2.5 Computat ional experiment

We consider a branch and bound algorithm for l / r j /maxwjCj that foll~ws Potts'

implementation [34]. Potts uses an 'adaptivey branching technique to ft: jobs at bot h

ends of the schedule. More precisely, each node of the search tree is represented by a

pair (al, oz), where al and 02 are the initial and finul partial sequences, respectively.

The immediate sucoessor of (ai, u2) is either of the form: (uli, 02) for a type 1 branch-

ing, or (ai, tuZ) for a type 2 branching, where i is an u n . job i E J \ (a1 ü oz).

Figure 2.8: Staircase structure for problem 1 / r j /max w, Lj .

The types of the branchings of the tree, are ail the same within a level, but in gen-

eral, they will ciiffer between levels. The type for a given level k is fixeci on the very

first visit to level k according to the following rule: branch in the direction of the

fewest number of ties at the minimiim lower bound- Let nl and n2 be the number

of ties at the minimum Iowa bound for type 1 and type 2 branchings, at level k. If

nl < n2 the next branching is of type 1, while if n2 < nl then the branching is of

type 2. If nl = na then the branching is the same as the previous level. Finaiiy, the

search strategy is to branch to the newest active node with the smallest lower bound.

The branch and bound algorithm was coded in Pascal and run on a Sun Sparc5

workstation. The experiment consisted of 24 groups with 100 problems in each group.

For each problem, with n = 100 or 500 jobs, 3n integer data (ri,pi, wi), were gener-

ated. Processing times pi and weights wi were uniformiy distributed between [l ,1001

and [l, W] for W = 10 and 100 respectively, while release times were uniformly dis-

tributed in [O, n * 50.5 - R] for R € {0.2,0.4,0.6,0.8,1 .O, 2.01, following the data gener-

ation technique used by Hariri and Potts [16]. Several versions of the algorithm were

tested both with and without the dominance order 4, for each group we preent only

the best results. An upper limit of 1,000,000 was used for the maximum number of

nodes in the branch and bound tree- The data in Table 2.4 shows that this limit was

exceeded only for a few problems, i-e., very few problems remaineci unsolved. Most

of these seem to be concentrateci in the groups d&ed by n = 100, W = 100 and

R = 0.4 or 0.8. Inspection of the results reveals that the number of problems solved

is roughly the same with or without using the dominance order 4. Nonetheless, the

dominance order seems to signiflrnntly impmve the running tirne of the algorithm.

The t img shown in Table 2.4 are the total times used for solving the 100 problems

in the group, using the format (min:sec) or (hrs:min:sec). The average CPU time for

the version with the dominance order is 42 percent of that for its counterpart without

it, resulting in a savings of 58 percent. This saving from using 4 appears to be due

to the reduction in the number of branchings and lower bounds that need to be com-

puted at each node. Note that if the precedence order 4 is used, then lower bounds

for type 1 and type 2 branchings, need not be calcuiated for ail the iinfixed jobs. For

type 1 branchings lower bounds o d y need to be calculated for those jobs with no un-

fixed predecessors in +, ie., the minimal jobs. Similady, for type 2 branchings lower

bounds only need to be calculated for thûse jobs with no iinfixed successors in 4, i.e.,

the maximal jobs. In siimmary, the branch and bound algorithm with the dominance

order 4 appears to be a very fast and effective solution method for large problems.

In this Chapter, we introduced a new method of pairwise job interchange, which

incorporates subset-restrictions on the intermediate sequence. We applied this tech-

nique to derive new dominance results for the problem 1 /rj / f,, , which includes

the problems 1 /ri IL, , 1 /ri, Cm,, 1 /rj /maxwjCj , and 1 /rj /max wjLj as

special cases. In particular, we derive a new dominance order 4 that is a suborder of

the adjacent interchange order. We extend the pyramid dominance order of Erschler

et al. ([8] and (91) for 1 / t j /Lw to the general problem l /rj/ f-. The order 4 is

shown to belong to a special class of partial orders, the interval orders. This has the

comp1exiW implication that these problems are stmngly NP-hanf for interval-ordered

jobs. We testeci the effectiveness of the dominance order in a braach and bound al-

gorithm for the problem 1 /rj /max wjCj . The results indicated that the dominance

order was quite effective for this problem, reducing the total computation time by 58

percent.

t

0:00:22 0:00:32 100 1 LOO 0:OO: 11 0:00: 14 100 1 100

1

CPU time Solved with 4 I without 4 with 4 1 without 4

20:33 35: 19 100 LOO 12:51 26:59 100 100 13:ûû 28:09 100 100

I

CPU time Solved with 4 - without 4 with 4 without 4

0:00:27 O:OO:35 100 100

0:39:461 0:57:51 1 100 1 LOO 1

Table 2.4: Results of computationd -riment for problem 1 /rj / max wjCj .

Chapter 3 Minimizing makespan in the two- machine flow-shop wit h release times

3.1 Introduction

Minimizing the makespan in a flow-shop environment is a classical scheduling prob

lem. The simplest case of this problem, the two -machine flow-shop problem F2/ /Cm,

on n jobs, is solved in O(n log n) time by Johnson's rule [21] . When arbitrary re-

lease times are added, the problem F2 / r j / Cm, becomes strongly NP-hard [27] .

This result has led researchers to consider different approximation and branch and

bound algorithms. Potts [35] developed several approximation algorithms and ana-

lyzed their worst case performance. The best of these algorithms has a worst case

performance ratio of 513 with t h e complexity 0(n3 Logn). Later, polynomial a p

proximation schemes were developed for the problem by Hali [18] and Kovalyov and

Werner [22]. Grabowski [14] presented a branch and bound algorithm for the prob

lem F2 I r j / Lm,, whi& can also be usêd for the problem F2 / r j / C-. Grabowski's

algorithm used a new type of branching scheme that exploitai certain dominance

properties of a critical path. Tadei et al. (411 tested several branch and bound alge

rithms for the problem F 2 Irj/ Cm,. They tested the effectiveness of va.rious lower

bounds and branchhg schemes. They have classifieci instances as "easy" or "hard ac-

cording to the distribution of the release times. Rom the easy class they have solved

probiems with up to 200 jobs. While fmm the hard category they were able to solve

instances with up to 80 jobs within 300 seconds. They have also found Grabowski's

dichotomic branching scheme l e s effective for the problem F2 Ir,/ Cm, than their

n-ary branching scheme.

In this Chapter, we consider a new branch and bound algorithm for solving the

problem F 2 I r j / Cm, with the foLlowing main features. We use an adaptive n-ary

branching rule, that fixes j ob at both ends of the schedule, similar to the one used

by Potts [34] for the problem Fm//C-. We present a new dominance order on the

job set, which is derived by using subset-restricted interchange. Although the proof

of validity for the dominance order is quite elaborate, its application requircs only a

very fast and simple procedure at the root of the branch and bound tree. We also

incorporate a simple decomposition procedure which proved to be especially effective

for problems with large job release times. We use four very quickly cornputable lower

bounds at each node of the tree. The algorithm represents an effective and very fast

tool for solving large instances of the strongly NP-hard problem F2 /ri/ Cm,. In a

large scale computational experiment, the algorithm has solved in a few seconds 1,714

of the 1,800 randomly generated test problems with up to 500 jobs. We have also

gainecl the insight that the relatively few problems which were left unsolved had their

parameter which determines the range of release t h concentrateci in a very narrow

interval. Even for the unsolved problems, the best solution found by the algorithm

was, on average, within l e s than 0.5% of the minimum schedule length.

The Chapter is organized as follows. In the next section we derive several new

dominance results for the problem F2 /ri/ Cm=. In section 3, we present the details

of our branch and bound algorithm, a pseudocode for it is found in Appendix A. In

section 4, we discuss the results of a large sale computationai experiment. In the

last section, we stimmarize our resdts.

3.2 Dominance results

In this section, we present a new dominance order 4 for problem F2 Ir,/ Cm,. We

derive 4 using the new technique of subset-mtncted interchange.

3.2.1 Subset-restricted interchange

RecaU that the problem FZ//C,, is solved by the Johnson d e r : first order the

jobs with a, 5 bj in nondecmming a order foilowed by the jobs with aj > b, in

nonincreasing b order (211. This ordering of the jobs is an adjacent interchange order

for the problem F2//Cma, and since it &O completely orders every pair of jobs, it

is an optimal ordering (Le. an optimal sequence). An adjacent interchange order

for the problem F 2 / r j / C,, is the intersection of the nondecreasing r order and

the Johnson order. Note that this order is no longer a complete order, rather it is

only a partial order. To emphasize the partitionhg of the jobs in the Johnson order

we define JaCb - = { j laj 5 b j ) and JaBs = { j laj > b j ) - N d , we formdy define the

adjacent interchange order 4 for the problem F2 /ri/ Cm=.

Definition 3.1 Adjacent Interchange Orderskrrn if rk 5 r, and

(2) k, rn E Jasb and a* 5 or, (fi) k E J& and n E J,>b or, (222) k , rn E Ja>b and bk 1 b,.

In general, if an adjacent interchange order is only a partial order (and not a

complete order), it need not be a dominance order. This is the case for t defineci

above, if we consider the instance of F2 I r j / C,, in Example 3. Here we have 3 4

(9 = ri = 10, 3 , l E Jo>b and 4 = 25 > 15 = bi), however, the unique optimal

sequence is (1,2,3,4) with C,, = 125. Thus we see that r is not a dominance

order since there is no optimal sequence that is a linear extension of r. We use

subset-restricted interchange to find a suborder of 4 that is a dominance order.

Example 3 A 4 job pmblem to illustmte that 4 is not necessarily a dominance

Next we distinguish between different type of pairs in < that have different

inter change properties.

Theorem 3.1 If k E Jocb - and at 5 %, then k e m tqether &th the set RZ, = J

satisJies the Restn'cted Forwuni Iruertion Condition.

Proof. Given a sequence s we constmct a directed graph G ( s ) to evaluate C,,(s).

Each job s( j ) is represented by three nodes with weights r . ~ ) , a . ~ ) , and bSLi) respec-

tively. G(s) has the property that Cmu(s) is the length of the longest 'nodeweighted'

pat h kom O to s(n) . Each of these paths can be identifieci by the pair (s( i) , s ( j ) ) , for

some i, j E [l, n], which are the endpoints of the horizontal segments of the path (see

Figure 3.1). Then by definition

G n a x (4 =

and we can evaluate Cm, (s) as the maximum over all such pairs (s (i ) , s ( j ) ) in G (s) .

Figure 3.1: Directed graph G ( s ) for problem F2 I r j / C-.


Let sl = (XmYkZ) be a sequence for pair k ~ r n with k E JaCb - and al. 5

. We apply forward insertion to sl and obtain sequence s2 = ( X k m Y 2). We

demonstrate that s2 is not worse than sl by showing that for every pair of jobs in sz

there exists a dominating pair and corresponding path in sl with a not smder C,,

value. Moreover, we show that the choice of the dominating pairs is independent

of the intermediate sequence Y, thus we can take the interchange region to be the

entire job set, Le., R L ' , = J . For example? consider pair ( k , k ) in S Z , then the

corresponding dominating pair in sl is (ml m) . That is,

since rk 5 rm and ak 5 a,,,.

Table 3.1 gives a dominating pair in si for each pair in s2. ( We use lower case

letters x, y or z to refer to arbitrary generic elements of the subsequences X,Y or 2,

respectively). The last three columns contain the arguments why they are dominating

pairs. I

Note that R:' , = J in Theorem 3.1 means that in fact there are no restric-

tions on the intermediate set Y, i.e., k can be inserted forward before m around any


subsequence Y & J. The next two theorems show cases when the intermediate set Y

must satisfy certain real restrictions for the interchange operators to be applicable.

Theorem 3.2 If nt E Ja,6 and bk 2 b,, then k e r n together uith the set R z , =

{ j Irj 5 r, ) satisfes the Restrïcted Backward Insertion Condition.

Proof. Similady, Table 3.2 gives a dominating pair in sl for each pair in ss. . Theorern 3.3 If k E Jas* and rn E then k e n together with the set R::, =

satisfies the Restricted Forruard Insertion Condation.

Proof. Table 3.3 gives the dominating pair in si for each pair in sz.

3.2.2 New dominance order for F2 I r j / C,,

Tadei et al. [41] have establishd a dominance order for problem F2 /rj/ Cm-, which

can be interpreted as a corollary of Theorem 3.1.

Corollary 3.1 [41] If r k 5 r,, k E - and ak 5 a,,, then job k priecedes rn in an

optimal solution.

Corollary 3.1 is indeed a conseQuence of Theorem 3.1, since its conditions imply

k e n , and so k and m satisfy all the conditions of Theorem 3.1. Therefore, FI c m

be applied to any sequence of the form ( X m Y k Z ) to insert k before rn around any

intermediate sequence Y. Since kem, it is clear that the dominance order of CoroUary

3.1 is a suborder of the adjacent interchange order 4. Note that this suborder does

not contain any pairs with both Cc and rn E Ja>b. Our dominance order wiil enrich

this suborder by extending it to pairs with k, rn E Jo,b too.

Before we derive our dominance order, we introduce a planar representation of

the adjacent interchange order r. We represent e in the plane with the x and y axes

replaced by the r and Johnson orders. A Job j is represented by the point (rj , aj) if

j E Jas* or ( r j , b j ) if if i JJob. Then k r m in this representation exactly if k is not

to the right or above m. This is demonstrateci for the &job problem of Example 3 in

Figure 3.2. The planar representation here impiics 2r4, 3 4 1 and 344.

Figure 3.2: Planar representation for Example 3.

The principal ide& and filters in 4 are obtained by dividing the plane into

quadrants using the line r = rj and the line a = a, if j E Jab-

Then the SW and NE quadrants correspond to I ( j ) and F b ) , the principal ideal

and filter in r for job j (see Figure 3.3). We add new comparabilities for the jobs in

J, ,b using subset-restricted interchange. These new comparabilities are defineci using

certain 'extreme jobs' in the planar representation of 4. These are the corner-point

boundary jobs M = {Mi li = 1,2,. . . , H + 1 }, i.e., the jobs with empty SE quadrants

that form a descending staircase, and the set of jobs on their inscribeci diagonal, the

diagonal jobs A = {Di, 4, . . . , DH). Note that the diagonal jobs are not necessarily

real jobs because of the way they are constructed, rather they may be 'artificial jobs'

and are used only for ordering purposes. We augment the partial order P = ( J , +)

by these diagonal jobs and c d it PA = (JA, eA), where Jp = J U A and is the

planar order with these diagonal jobs included. Jobs k e n (k,m E J ) are sepunzted

by A (are A-separateci) if there exists a Di (i = 1,2,. . . , H ) such that k is in its

principal ideal and m is in its principal filter, i.e., k E I(DJ and rn E F(Di) in PA.

A induces a partition of P into separable and nonseparable pairs. Notice that the

separable pairs can be of the three types in Definition 3.1, and are not restricted to

pairs k r m with k J,-,<b- - The new pairs we add are precisely the separable pairs in

P. We now forrnaily define our doniinance order +.

Figure 3.3: Representation of the diagonal jobs A.

Definition 3.2 Dominance order 4: For krm, we dejïne k m af

(2) k E Jas* and a k $4, or, 2 2 them ezists a Di in Pa such that k E I(D,) and rn E F(Di) .

To illustrate the definition of 4 consider the planar representation for Example

3 in Figure 3.2. Applying condition ( a ) to pairs k e r n with k E JaCb - we see that 2 e 4 is

the only such pair and since a2 = 20 5 25 = a4 we have 2 4 4. For condition ( i i ) here

there are 2 corner-point boundary jobs Ml = 4, M2 = 2 , and a single diagonal job

Dl in Ja,b with coordinates (20,20). We see that Dl separates jobs 2 and 3 from job

4, which gives us (again) 2 + 4 together with the new pair 3 4 4. F i n d y combining

these types of pairs we get that, jobs 2 and 3 precede job 4 in 4.

Theorem 3.4 Partial onfer 4 is a dominance order for F2 /r,/ C-.

Proof. Let s be an optimal sequence, we can assume by Theorem 3.1, that s is a

Linear extension of the suborder for condition ( 2 ) of the theorem. Thus we need to

prove the theorem only for pairs satisfying condition (iz).

The following observations immediately follow from the construction of A:

Furthermore, we have the foilowing crucial property: for aii y E J \ F (Di ) and

m E F (D;) we have r, 5 r ~ , < r, for any i. Consider a pair k and rn s a t i s h g

condition ( i i ) . If both k, rn E Ja<b, - then they also sath& condition (i), and k is

already before m in S.

Next we deal with pairs with k and rn satisfying condition ( i i ) and for which

both k , m E Jo,b. In the following, the notation SIJa,, is used to refer to S n Ja,b

for any S C J. If every job in I ( D l ) 1 is before every job in F ( D l ) , t hen all

jobs separated by Dl in JoTb are already in 4 order, and we consider I ( D 2 ) 1 Ja,b

and F (D*). Otherwise, let kl E I ( D l ) 1 ,-,, be the last job in s that is after some

job from F (Di), and let ml be the last such job from F ( D 1 ) before kl . That is

s = (X lmiY lk lZ l ) , where Zl n I (Dl)lJa,b = 0 and & c J \ F(D1) . By the

above property, we have r, 5 r ~ , 5 r,, for all y1 E YI, which implies that Yl Ç

BI Rklem1. Thus, by Theorem 3.2, we can insert ml bacinuard just after kl to obtain

the alternative optimal sequence (XiYi klml Z l ) . Note that this interchange cannot

violate condition (i) , because ml E J,,s and condition ( 2 ) could never require it to

precede any job. Following in this way, inserting the last job in F ( D l ) bucinuad

after ki until there are no such jobs, we obtain sequence sl which is an optimal

sequence with the property that 1 ( D l ) 1 is before F ( D l ) . Continuing with a

s i d a r argument, we obtain si for i = 2,3,. . . , L, where DL is the last diagonal job

in J&, and S L is an optimal sequence satisfying condition ( i ) with the additional

property that I (Dj) [ ja>, is before F (Di) for i = 1,2, . . . , L. This takes care of the

pairs with k, rn E &b.

Now we prove 4 is a dominance order for the remaining pairs with k E J,<* - and m E J&,. Let r n ~ be the £irst job in S L fiom F(DL) . If there are no jobs

from Jas6 &er m ~ , then SL also satisfics the property that I ( D i ) is before F ( D i )

for i = 1,2, . . . , L. Otherwise, let kL E Jas& be the first such job after r n ~ . By the

definition of DL we have kLemL- Then sr, = (XLmLYLkLZL) and by the choice FI of kL we have that YL C Jo>b = RkLfmL. Thus, by Theorem 3.3, we can insert

kL fornard just before r n ~ to obtain the sequence (XLkLrnLYLZL). Note that this

interchange does not violate condition (i), because YL c J,,& and condition ( i ) would

never require that kL follow any job from YL. Repeating, until there is no such

job kL, we obtain the sequence s', satisfying condition ( i ) with the property that

I (Di ) is before F ( D i ) for i = 1,2,. . . , L. We want to demonstrate that I ( D i ) is

before F ( D i ) for i = L + 1, . . . , H. Let k ~ + ~ be the last job in I(D-1) that is

after some job in F ( D L + ~ ) ~ ~ ~ > ~ , and let m ~ + l be the la& such job before kL+1. Then

s; = ( X L + ~ ~ L + ~ Y L + ~ ~ L + ~ ~ L + ~ ) where ZL+lWDL+l) = 0 and YL+1 C J\F(DL+l).

Note that YL+1 may contain jobs from both parts of the Johnson partition. k t Yi

be the first job in Jasb n YL+l after r n ~ + l . Then we have by our crucial property

that rVi 5 r ~ , + , < rmL+, , and since yi E Jas& and m ~ + l E Ja>b that B L + ~ .

F'urthermore, d the jobs between m ~ + l and Yi are in Ja>* = Ry):mL+i, and thus by

Theorem 3.3, we can insert yi fonuard before rn~+ l . By repeatedly doing this, we

end up with a sequence where there are only jobs from J&,b between m ~ + l and

Finally, again using Theorem 3.3, we can insert khi forwuni before m w l to obtain

a sequence with kGCl before r n - 1 . As for the previous case, these interchanges do

not violate condition (i) . We continue until there is no such khi and m h l and then

we obtain the sequence s ' ~ ~ that satisfies the property that I (Di) is More F (Di )

for i = 1,2, . . . , L + 1. We proceed in exactly the same way for the remaining cases

for L + 1 < j 5 H, and we end up with a sequence sH s a t i s w both conditions (i)

and (22). . 3.3 Branch and bound

In this section, we outline the basic components of the branch and bound algorithm

used to solve the problem F2 I r j / Cm-. A pseudocode for it is given in Appendix A.

3.3.1 Branching rule

We consider a variant of Potts' adaptive branching rule that fixes jobs at both ends

of the schedule [Ml. More precisely, each node of the search tree is represented by a

p" (ol, 4, where ol and 0 2 are the initial and final partial sequences, rapectively.

Let Si denote the set of jobs in ai for i = 1, 2 and let S be the set of iinfixed jobs

i.e., S = J\(Sl u Sz) We use 41, to refer to the restriction of 4 to the set S. An

immediate successor of (ol, u2) in the tree is either of the form (012, oz) for a type 1

branching; or (ci, io2) for a type 2 branching, where i is a minimal or maximal job in

4 I,, respectively. The types of the branchïnp are ail the same within a level of the

tree. The type for a given level k is fixed on the very first visit to level k according

to the following d e : branch in the direction of the fewest number of ties at the

minimum lower bound. Let nl and n2 be the number of ties at the minimum lower

bound for potential Srpe 1 and type 2 branchings, at level k. If ni < nz the next

branching is of type 1, while if na < nl then the branching is of type 2. If nl = n?

then the branching is the same type as at the previous level.

The search strategy is to branch to the newest active node with the smaliest

lower bound. We consider two rules to break ties between nodes with the same lower

bound. We assume that the jobs are index4 in increasing Johnson order. The rule

T2 breaks ties for type 1 or type 2 branchhgs by choosing the job with the smallest

or largest index, respectively. The other rule Tl brealcs ties for type 1 bran- by

choosing the job with the largest principal filter in i l , , the smaiiest index is used

as a further tie-breaker. Sirnilarly, for type 2 branchbgs the rule is to break ties by

choosing the job with the largest principal ideal in +., with the largest index as a

furt her t ie- breaker.

3.3.2 Bounds

Upper bounds are calculated for the mot node and for the first n nodes, afterward

the upper bound is evduated ody at leaf nodes. At the root, the upper bound is the

result of the improved ready- Johnson heuristic due to Potts 1351. This heuristic has

time complexity 0(n3 logn) and a worst case performance ratio of 5/3. For the first

n nodes, the upper bound is the 1ength of the sequence obtained by concatenating

the ready-Johnson sequence between ol and 02, which requins O(n log n) tirne. For

leaf nodes, there are no unfixeci jobs and the upper bound is just the length of the

sequence obtained by concatenating al and an.

Since the branch and bound tree may require the computation of lower bounds

for a huge number of nodes, it is very important that we use lower bounds whose

calculation requires only O(n) time per bound. (We have also experimented with some

potentially better lower bounds, requiring O(n log n) tirne, but they have noticeably

slowed down the algorithm without any substantial increase in the number of problems

solved.) We consider four lower bounds for each node (oi, a2). We calculate lower

bounds on the lengths of different paths for the iinfixed jobs S, and we combine these

in various ways with the actual lengths of the fixed sequences al and 02. For al

we look at the forward problem (with release times included) and we let C1(ol) and

C2 (ol ) be the completion t h e on machines 1 and 2, respectively. For 0 2 we consider

the reverse problem, in which each job j must be procgsed first by machine 2 for b,

time, followed by processing on machine 1 for a, and a 'delivery tirne' of rj. We define

the completion tirne C1(u2) and d ( u 2 ) analogously, and we define Cmu (oz) to be

the 'delivery completion tirne' for a 2 . For S we consider the forward problem, and

assume that these jobs cannot start on machines 1 and 2 before C1(ul) and C(ai),

respectively. Ignoring release t h e s for the jobs in S, let L1(S) be the completion

time on machine 2 of the Johnson sequence on S. Let L2(S) be the completion time

on machine 1 of the jobs in S sequenced in nondecreasing r order, that is the earliest

time at which the jobs in S can be completed on machine 1. Similarly, if we relax the

capacity constra.int on machine 1 and sequence the jobs in nondecreasing r + a order,

then the resulting completion time of this sequence on machine 2, L3(S) is a lower

bound on when the jobs in S can complete on machine 2. Our lower bounds are the

following:

LBl = Ll(S) + C2(u2)

LB2 = L2(S) + C' (4 LBj = L2 (S)+ m h iES bi + c2 (4

LB.4 = &(S) + C'(OZ).

Finaiiy, the best lower bound is the maximum of the above four lower bounds and

Cmax ( 0 2 ) - 3.3.3 Decomposition and dorninance

We find a starting sequence al by applying the following simple decomposition pro-

cedure, which was also used in [41]. Given a sequence s, if there exists a 2 5 k < n

sudi that min ra(i) 2 C&) and min [ra(i) + a.(i)I 2 q(k-l) , then sequence k s i l n k l i l n

(~(1) , . . . , s (k - 1)) is an optimal initial sequence. We apply the decomposition pr*

cedure for the jobs sequenced in nondecreasing r order, then ul = (s(1) , . . . , s(k - 1))

for the largest k value found. After we have h e d 01, we determine the dominance

order 4 on the remaining jobs in S = J\Sl. Note that the dominance order is cal-

culated oniy once at the root node. This eiiminates the potentiaiiy large overhead of

having to calculate and update dynamically changing dominance conditions at every

node of the search tree.


3.4.1 Test problems

For each problem with n jobs 372 integer data (r i , u+, b*) were generated. The proces-

ing t imes a. and b, were bot h uniformly distributed between [l ,1001. Release t imes ri

were uniformly distributed in the range [O, n- 101- R] for R E {0.2,0.4,0.5,0.6,0.8,1.0),

following the technique used by Hariri and Potts [16]. For each R and n combination,

50 problems were generated. We used the random number generator of Taillard [42]

to generate these problems. This means that a l l our test problems are reproducible

by running the problem generation procedure fkom the same seeds. (The seeds used

have b e n saved and are available from the author on request.)

The branch and bound algorithm was coded in Sun Pascal 4.2 and run on a Sun

Sparc5 workstation. Three separate versions of the algorithm were m, testing the

effectiveness of the dominance order and the different tie-breakers Tl and T2. Alg*

rithm Al has the dominance order i 'turned on' and uses tie-breaker Tl. Algorithm

A:! ais0 has the dominance order 'turneci on', but it uses Tz to break ties. Finaliy, al-

gorithm A3 has the dominance order 'turned off' with tiebreaker T2. Each version

of the algorithm was run until either it obtained the optimal solution or the number

of nodes branched from in the tree reached one million for the problem. In the latter

case, the problem was declareci unsolved.

Table 3.4 contains the results of the computational experiment for the different

R values. For each group of 50 problems we report: the total CPU time requird

for the 50 problems (denoted by total CPU); the average CPU tirne for the solved

problems in the group (denoted by avg CPW); the total number of nodes in all of the

treg (denoted by total nodes); the number of problems solved (denoted by solved);

and for each of the unsolved problems we calculate the gap between the best solution

obtained and the smallest lower bound among the left-over nodes in the tree, the

maximum gap (denoted maz gap) is the largest of these over al1 the problems in the

grou P-

Our results indicate that the difficulty in solving a problem depends much

more on the value of R than on the number of jobs. The actual size of the problem

for the branch and bound algorithm is determineci by how many jobs are fixeci by

the decomposition procedure. In general, as R increases the percentage of the jobs

fixeci increases. Recall, in the mode1 n 101 is the expected total processing time

of the jobs, and the range of the release times is [O, n 101 - RI. For small R values

(R < 0.4), the decomposition procedure is ineffective and ticg l e s than 2 percent

of the jobs. However, since a l l the j o b are ready relatively early, release times play

l e s of a role and the problem is easily solved in just a few nodes. For large R values

(R > 0.6), the opposite is true, the jobs do not become ready simultaneously, there

may be gaps in the schedule, and the decomposition procedure fixes over 90 percent

of the jobs on average. The difficult problem are those with R values concentrated

close to R = 0.5, in the range R E [0.4,0.6], with the m a t diflicult being R = 0.5

itself. For R = 0.4 the decomposition procedure fixes only 3 percent of the jobs, whiIe

for R = 0.6 it fixes a much larger 70 percent of the jobs on average. In spite of this

large Merence in the dectiveness of the decomposition procedure for R = 0.4 and

0.6, we are still able to solve at least 90 percent of these problems, even those with

500 jobs. Whereas, for R = 0.5 the decompaition procedure fixes around 20 percent

of the jobs, and the problems appear to be more d.ifEcult with a clear decreasing trend

in the number of problems solved as n increases.

Recall, a problem was 'unsolved' when the number of nodes branched fiom

reached a million. Thus the total number of nodes for a group of 50 problems,

contains a million nodes for each unsolved problem. As can be seen, the algorithm

typically generated much fewer nodes for the solved problems. In those groups where

there were unsolved problems, the solution obtained was nearly optimal as measuted

by the maximum gap. The large& such gap was on the order of 1 to 2 percent,

rneaning that in the worst case we were withh this range of the optimal solution.

The average gap was much m e r , less than 0.5 percent.

Comparing the dinerent versions of the aigonthm, we found that versions Al

and A2 (with the dominance order included) together always solved as many problems

as version Ag (without the dominance order included), and in some cases a few more.

The main ben& of using the dominance order is the substantial reduction in the

CPU t h e required to solve a group of problems. This reduction in CPU time is

approximately 80 percent over d the groups, aithough sometimes it is more than

this. Consider for example the group of problems with R = 0.4 and n = 200: here

Al and A2 both took approximately 37 minutes to complete, while A3 took over

12.5 hours to do so. We also found that this speed-up factor tends to increase as n

increases. This explains why we did not run algorithm Ag for groups with n = 500

jobs, as it would have required excessively long times. The use of the Merent tie-

breaking d e s in Al and A2 did not result in any clearly identifiable Merences in

the relative performance.

3.5 Summary

This Chapter considered a new branch and bound algorithm for the problem

F2 Ir j / C-. The main features of the proposed algorithm are: an adaptive branch-

ing d e that fixes jobs at both ends of the schedule, and a new dominance order that

subst antidy reduces branching. In general, computationai results indicated that the

algorithm performed well in solving a large percentage of test problems, some with

up to 500 jobs. We found that the dificult problems were those where the range of

release times was approximately 1/2 the expected total procgsing time of the jobs

(i.e., R = 0.5 in our model). Even when we failed to solve a problem, we were always

very close to the optimal solution. We aiso found that the use of the dominance or-

der resulted in a reduction in computation tirne of roughly 80 percent. In summary,

we feel that the p r o p d aigorithm is relatively more effective than previous solution

methods for the problem F 2 / r j / Cm, in that it found fewer hard problems and it

was &O able to solve much larger problems than previous algorithms. The fact that

most problems were solved to optimum within a few seconds, means that the dg*

nthm has the potential of being used as a callable subroutine for F2 I r j / Cm,-type

subproblems generated during the solution of more cornplex scheduling problems, for

exampie problem Fm Irj/ Cm--

Table 3.4: Results of computational experiment for problem F2 / r j / Cmu

n d g total CPU avg CPU total nodes solved m m gap

Ai 0.28 0.0056 107 50 - 40 A2 0.24 0.0048 107 50 -

A3 0.24 0.0048 107 50 - A1 1.28 0.0256 1787 50 -

60 A;! 1.25 0.0250 1787 50 - A3 2.76 0.0552 1789 50 - Al 1.08 0.0216 89 50 -

80 A3 1.09 0.0218 89 50 - A3 0.61 0.0122 89 50 -

AI 1.86 0.0372 111 50 - 100 A2 1.83 0.0366 111 50 -

A3 1 .O2 0.0204 129 50 - Ai 1 1 -95 0.2390 218 50 -

200 Az 12.13 0.2426 218 50 - A3 5.37 O. 1074 218 50 - Ai 5:54.83 7.0966 1384 50 -

500 A2 4:41.84 5.6368 1384 50 - R = 0.4

Al 1:28.11 0.0050 1,001,861 49 O. 14 40 A2 1:M. 16 0.0046 1 ,001,861 49 0.38

Table 3.4: (continued) . n d g total CPU avg CPU total d e s solved max gap

(hrs:min:sec) (min:sec) (%> R = 0.5

A l 3:30.76 0.0853 2,027,081 48 0.62 40 A2 5:59.66 0.0926 2,029,231 48 0.62

A3 10:08.70 0.4476 2,051,022 48 0.62 AI 18: 15.17 0.7275 5,189,188 45 1.37

60 A2 2024.29 0.7349 5,189,000 45 1.55 A3 38:43.69 13.2302 5,733,770 45 1.37 Al 1: 14: 10.25 1.3020 8,211,107 42 0.79

80 A2 1:18:09.24 1.4863 8,210,998 42 0.79 AS 2:36: 10.73 8.3622 8,438,701 42 O. 79 Al 1:59:06.36 5.3600 9,540,374 41 1.26

100 A2 1:52:27.48 5.3471 9,340,357 41 1 -26 A3 4:19:29.52 8.3293 10,143,663 40 1.26 Al 3:33: 16.72 1.3308 13,019,825 37 0.97

200 A2 6:15:50.78 1.2841 13,020,073 37 0.97

Table 3.4: (cont inued) . n d g total CPU avg CPU total nodes solved maz: gap

Chapter 4 Minirnizing maximum lat eness in the two-machine flow-shop with release tirnes

4.1 Introduction

We consider the two-machine flow-shop problem with re1ease times where the objec-

tive is to minimk the maximum latengs for permutation schedules. This problem

denoted by F2 /rjYperm/ L,, is a well known strongly NP-hard scheduling prob

lem [27]. Much of the underlying interest in the problem F2 Irj, petml L;, stems

hom the fact that it arises aaturdy in the solution of more complicated scheduling

problems, such as the problem FrnIIC,, [24]. The complexity resdt above suggests

that in order to optimally solve the problem F2/rj,perm/ L,, it is necessary to

resort to some efficient enmerative technique, such as branch and bound.

Branch and bound algorithms are characterized by various components: the

branching scheme employed, the different types of upper and lower bounds used,

along with other featurg. Grabowski [14] and Grabowski et al. [15] presented branch

and b o ~ d algorithms for the problems F2 Irjy petml L;, and F m /rj, Perm/ L,,

respectively. These algorithms used a branching scheme that exploited certain dom-

inance properties of a critical path. Tadei et al. [41] tried this type of branchuig

scheme for the problem F2 Irj/ Cm=, but found it to be less effective than their tra-

ditional n-ary branching scheme, that fixes jobs only at the beguining of the schedule.

Potts [34] considered an adaptive brandiing scheme that fkes jobs at both ends of

the schedule, and found this branching scheme to be quite eflFective for the problem

Fm//Cma. Dominance rules are another cornmon feature of branch and bound dg*

rithms, they are tsrpicaily used to eliminate nodes (More their bouods are calculated)

in order to reduce computation time and storage requirements. Cheng et al. [4] a p

ply approximate dominance des, in the context of fuzzy inference, for scheduling

and searching in flow-shop problems such as F3//C,. and F m /perm/ Cm,. The

authors were able to find a nearly optimal initial schedule by repeatedly applying

the fuzzy dominance nile to schedule the jobs. This fuzzy schedule also proved very

usefil for tie-breaking purposes, to decide which way to branch among several nodes

with the same lower bound.

In this Chapter, we consider a new branch and bound algorithm for solving the

problem F 2 / r j , perml L,, with the following main features. We use an adaptive

n - q branching rule, a variant of the one used by Potts. We derive a new dominance

order on the job set, using the proof technique of subset-restricted interchange that

employs a new interchange operator which we introduce. In addition, we make use of

fuzzy dominance properties for initial scheduling and tie-breaking. We also incorpo

rate a simple decomposition procedure that reduces the problem size by fixing jobs

at the beginning of the schedule. We use six very quickly computable lower bouncis

at each node of the tree. The algorithm represents an effective tool for solving large

instances of the strongly NP-hard problem F2 /ri, perm/ LI-. In a large scale com-

putational experiment, the algorithm has solved in a matter of a few seconds 4,384

of the 4,500 randomly generated test problems with up to 200 jobs. Even for the un-

solved problems, the best solution found by the algorithm was on average within l e s

than 0.5% of the optimal value.

The rest of the Chapter is organized as foliows. In the next section, we derive

several new dominance results for the problem F2 Irj, p e m / L;,. In section 3, we

present the details of our branch and bound algorithm, a pseudocode for it is found

in Appendix B. In section 4, we discuss the results of a large scale computational

experiment . In the last section, we summarize our results.

4.2 Dominance results

In this section, we present dominance results for the problem F2 /ri, p e m / L',, .

First we derive a new dominance order 4, using subset-restkcted intenhange for a

new interchange operator which we introduce. Secondly, we consider a fuzzy approx-

imation of dominance that is used in both initial scheduling and searching.

4.2.1 Subset-restricted interchange

Recail that we foilow Monma [3û] in definhg our interchange operators. Let sl be a

sequence with job m preceding job k. In general, sl is of the form sl = (XmYkZ),

where X, Y and Z are subsequences of J, and let U and V be disjoint subsequences

of Y. Three types of interchanges of jobs k and m that leave k preceding m in the

resulting sequence s 2 , are the foilowhg:

To define more precisely the new SI operator, consider again sequence sl = (XmYkZ)

and a partition of Y into Y = U U V, where U = (u1u2u3) and V = (v1w2v3v4), and Y

is of the form Y = ( U ~ U ~ V ~ V ~ U ~ V ~ V ~ ) . Then SI applied to sequence SI, for this particu-

lar choice of U and V, gives Sequence s2 = (Xu1u2u3 kmv1v2u3v4Z). We c d it s h d e

interchange because the interchange of jobs has the resulting net effect of 'shufllhg'

sequence Y into subsequences U and V, and placing jobs k and rn between them.

Notice that SI may change the relative order of some u's and v's after interchange,

but it never changes the relative order of two il's or two u's. Thus SI depends on the

choice of subsequences U and V. SI generalizes BI and FI: If we let V and U be

the sets of jobs in sequences V and U (for the sake of brevity, we do not distinguish

between sets and sequences) when V = 0 or U = 0, then SI reduces to BI or FI,

respectively. Fùrther, these interchanges all reduce to adjacent pairwise interchange

in the case when Y = 0.

Recall that the problem F2//C,, is solved by the Johnson onier: £kt order

the job6 with aj 5 b, in nondec+easing a order foiiowed by the jobs with aj > b, in

n o n z n ~ n g 6 order [21]. To emphasize the partitioning of the jobs in the Johnson

order we define Jab = { j laj > 6, ). T b ordering of the jobs

is an adjacent interchange order for the problem F2//Cm,, and since it also com-

pletely orders every pair of jobs, it is an optimal ordering (i-e., an optimal sequence) .

An adjacent interchange order for the problem F2 /rj, perm/ L:, is the intersection

of the nondecreasing r order, the nonincreasing q order and the Johnson order, as d e

fined forrnally below. Note that this order is no longer a complete order, rather it is

ody a partial order.

Definition 4.1 Adjacent Interchange Order a : k a m if r k 5 r,,,, qk 2 q,,, and

(2) k , rn E Jas( and a& 5 4, or, (ii) k E Ja>b OT,

(iii) k, rn E c.Ja>b and bk 2 b,,, .

In general, if an adjacent interchange order is o d y a partial order (and not a

complete order), it need not be a dominance order. This is the case for e defined

above, if we consider the instance of F 2 /rj, p a / L;, in Example 4. Here we have

3 4 (since r3 = ri = 10, q3 = 10 2 O = ql, and 3 , l E J,>o with 4 = 25 > 15 = bl):

however, the unique optimal sequence is (1,2,3,4) with L;, = 125. Thus we see

that 6 is not a dominance order since there is no optimal sequence that is a linear

extension of r. We will use subset-restricted interchange to find a suborder of r that

Example 4 A 4 job pmblem to illustrate that 4 is not necessarily a dominance

Recdl, that the SI operator above interchanges k and n 'around' sequence Y,

for some particular choice of U and V. Intuitively, whether or not such an interchange

leads to a reduction in cust (for a given sequencing function f and adjacent interchange

order r ) , should depend on the composition of Y, as well as on the choice of U and

V. We consider interchanges that are restricteci by conditions on Y and define the

subset-restricted interchange condition for SI .

Definition 4.2 An adjacent interchange order 4 together with the collection of m b -

sets {u,+, U Vk+, Ik+m} satisfes the Shufle Interchange Condition for a sapent-

ing function f ij for al1 jobs k , m and sequences X , Y , Z them exist a partition U

and V into disjoint subsequenees of Y such that kem, U C U k f , , and V C V+,

imp$ that f (XUkmVZ) j f (XrnYkZ).

Next we examine the stmcture of Ukf and VkG, for the different types of

pairs k+ rn for the problem F2 Irj, p e m / L,.

Theorem 4.1 If k E Joc6 - and ak 5 a,,,, then k+m together with the sets U k f m =

{ u lu E J&, ru 5 > ; n , ~ 5 a,,,) and Vkfm = { V Iqv 5 qk) satisfy the S h u . e Inter-

change Condition.

Proof. Given a sequence s we constmct the directeci graph G(s) to evaluate &,(s).

Each job s ( j ) is represented by four nodes with weights r .~), n,b), bsCi), and q . ~ ) ,

respectively. L ~ , ( S ) is the length of the longest 'node-weighted' path fkom the

start node to the finish node. These paths can be uniquely identifieci by the triples

(s(i),s(j),s(k)), for 1 < i 5 j 5 k I: n, representing the end points of the three

horizontal segments on the path (see Figure 4.1). By definition,

and we can evaluate L;, ( s ) as the maximum over all such paths in G ( s ) .

Let sl = (XmYkZ) be a sequence for pair k e m with k E Ja<b - and ak 5 u,,,. Let U C Uke, and V C Vk4, be disjoint subsequences of Y, with Y = U u V. We apply shutae interchange to sl, with this U and V, and obtain sequence s2 =

(X UkmVZ) . (We use lower case letters x, u, v , and z to refer to arbitrary generic

elements of subsequencg X, U, V, or Z, respectively.) We can dernonstrate that s;!

is not worse than SI, by exhibithg for every path in sa a dominating path in sl (see

Figure 4.2). For example, mnsider the pat h (u, k, v ) in s2 , then the correspondhg

Figure 4.1: Directed graph G(s) for problem F2 /rj, perm/ L;,.

dominating path in sl is (m, rn, k), which can be proved as follows. (Note that we

use UI,~.~ to represent the subsequence of U starting with u and ending in the last job

in U; and l+vl to represent the subsequence of V from its beginning to job u.)

To complete the proof, we present Table 4.1 which gives the dominating paths

in sl for each path in sp. The last column contains the argument why each path is a

dominating path. I

Theorem 4.2 If k E JaCb - and m E Ja>*, then kem together with the sets Ukf =

{U 1" E Ja<bi - ru 5 Tm} and V&, = {v lu E Ja>br qv 5 q k ) satisfies the Shufle Inter-

change Condition.

Proof. Similarly, Table 4.2 contains the appropriate dominating paths. .

Table 4.1 : Dominat ing paths for Theorem 4.1.

Table 4.1: (continued).

Table 4.2: Dominating paths for Theorem 4.2.

Table 4.2: (continued) .

Figure 4.2: Directed graphs G ( s z ) and G(sl) for S I .

Theorem 4.3 If m E Ja>b and bk 2 bm7 then k e n together with the sets Ukf ,,, =

{u Ir, 5 r, ) and hem = ( V IV E J.,), qv 5 qk,b,, 5 bi )satisfes the ShuBe Inter-

change Condition.

Proof. Symmetric to Theorem 4.1. . 4.2.2 New dominance order for ~ 2 / r ~ , ~ e r m / ~ k ,

In this section, we use subset-restricted interchange to derive a new dominance order

on the jobs, denoted by i , for the problem F2 /ri, perrnl L:,. We define the follow-

ing sets of jobs for every pair krm. (Note that a pair kam may satisfy more than

one of the foliowing three conditions, so that more than one rnay be applicable to a

pair- >

1. I fak<bka.nda&%, thenlet u:+,, = { u l ~ E Jo<biru 5 r m , ~ 5 > qk) VAm = { ~ I ~ V ~ P L } .

2. Euk 5 bk and% 1 bm, thenlet Ut*, = {uIuE Jasb7ru < rm}

%Lm = {u IV E Ja>br qv 5 qk}. 3- If a > 6 , and bk 2 bm, then let Uiem = {u Ifu 5 r, )

% L m = {v IV E Ja>b, qv 5 q k , h 5 bk, r u > rrn )- These are just the sets ukf, and Vk4m fkom the previous three Theorems with

additional conditions in cases 1 and 3, to ensure that UL,mnVlem = 0 and maintain

that vi # for E Ui,m and vi E V&. In case I, q, > qk has b e n added, and

T, > Tm has been added in case 3. For a given pair k ~ m , then krm is included in the

dominance order 4 exactly when any of the applicable sets Uif, U VAm is the entire

job set. Thus, the dominance order 4 is defined as the suborder of r consisting of

pairs kem, that can be interchangeci around any intermediate sequence using the SI

operator for the appropriate choice of U and V.

Definition 4.3 If k f r n and Lli,m u Viem = J for sorne i applicable to k and rn,

then k 4 m.

To illustrate the definition of 4 , consider again Example 4. The adjacent

interchange order r consists of the pairs 264, 3r4, and (as we saw previously)

3 e 1. For pair 2 4 4 both Theorems 4.1 and 4.2 apply, and checking the conditions

for these we see that both U : f 4 U %le, and UZf, U V&, are equal to J , thus we

have that 2 4 4. For pairs 3 r 4 and 3 4 1 o d y Theorem 4.3 applies. We also have

U : e 4 ~ ~ f 4 = J1 thus 3 4 4. However, u:+, U%\l # J because job 2 $ s\l since 2 E Jasb, and 2 $ U$, since r2 > rl , thus 3 # 1. Therefore, we have that the

dominance order 4 is the suborder of econsisting of the pairs 2 4 4 and 3 4 4, and

we see that the unique optimal sequence (1,2,3,4) is indeed a iineax extension of 4,

as we would expect.

We define the foltowing suborders of 4 for the dinerent types of pairs in 4

where A x B represents all pairs with first element from set A and second element fkom

set B. These suborders will be used fùst to demonstrate that + is indeed a partial

Lemma 4.4 Suborder 4 is a partial ~ n i e r .

Proof. Since 4 is a suborder of + , it only remains to show that i is transitive, i.e.,

we want to show that for any jobs k, 1, and m if k 4 1 and 1 4 ml then k + m. First we consider the case where ail of the jobs k, 1, and rn are in Ja<b. - In this case,

we have that k 41 I and 1 m, and we want to demonstrate that k m. By

Definition 4.3 k 41 rn if U V:f, = J . Since k 4 r m and k, 1, and m are in

Jocb, - we have that rk 5 rl 5 rm and ak 5 al 5 h. These imply that ULf, > uLfl, and since VAm = Vif , we have that ULfm U VAm contains LI&., U V A l . However,

k + 1 1 implies that LI:+, U Vie, = J , which by the above observation and Definition

4.3 implies that k 41 m. Next we examine the case where k, 1 E Jacb - and rn E J&,,

in which k +1 1 and 1 +2 m. We consider separately the different cases of Definition

4.3 that may apply for pair 1 4 2 m. If case 1 applies for 1 4 2 rn, then (a* 5) al 5 a,

and the previous argument carries over exactly as above. For case 2, we assume that

u & ~ u = J. Since k ~ l f r n , we have that r k 5 ri I. r, and qk 2 q~ > q,,,. These imply, however, that U$,, = Ufemand VLm 2 V;:,, so we also have that

w k ~ u V,fm = J . Thus, by Definition 4.3 we have that k 4 2 m. Finally, if

case 3 appiies to 1 4 2 rn, then br 2 6, and U,+, U C l f , = J. We notice that

(Uf,, u V;f ,) n Jas* equsls W,fm n Jasb, which equals W:+, . Since n Ja6 , whjch equ& V&,. Since U i f 1 n Ja>b = 0 and Vie, u VA, = J ,

this implies that V&, contains ali of the jobs in Ja,b- Combining these two results

we have that qem U &*+, contains all of the jobs in J, fiom which it follows by

Definition 4.3 that k 4 2 m. The remaining cases, where k, 1, and m are in Ja,b k 4 3 1

and 1 4 3 m, or k E Ja* with k 4 2 1 and 1 +3 rn, follow by symmetry.

Thus we see that 4 is a transitive order indeed. I

Theorem 4.5 Partial order 4 is a dominance onlerforprvblem F2 /ri, perrn/ L,.

Proof. Let s be an optimal sequence. We find an optimal iinear extension of + by

repeatedly applying the SI operator to sequence s without increasing the length of

the longest path. We demonstrate the comparabilities in the order 4 3 , 4 1 and 4 2 ,

respect ively.

Assume that s is not a Linear extension of 4 3 . Let m E J,,* be the last job

in s with some job j f J,,, with j 4 3 m, such that j is after rn in S. Let k be the

first such job j in s, then s is of the form s = ( X m Y k Z ) . Since k 4 rn, k and m

can be interchanged around any intermediate sequence using SI, in particular around

subsequence Y. Since k +3 m, Uke, u Vkf, = Ulem U Vtem = J. Applying SI to

s with U = (r,3,m 0 Y and V = Vtem n Y, we obtain sequence ( X U k m V Z ) . This

sequence now orders k and rn properly, and it does not introduce m y new violations

of 4 3 : To see this, consider any jobs u E U and v E V. Since u E Utern and

v E V:+m, V # U, so v # u, which impiies that v 7t3 u. Also, m #3 u follows by the

choice of rn, and v k follows since r, > r, 2 r k . By repeatedly applying the above

procedure until there is no such job rn, we obtain an optimal sequence s' which is a

linear extension of 4 3 .

For +1, we proceed in exactly the same way. Assume that sequence s' is not a

linear extension of +. Let rn E JaCb - be the Last job in s' with some job j E J.<* - with

j m, su& that j is after m in S. Let k be the first such job j in s', then s' îs of the

form s' = (XrnYkZ) . Applying SI to s' with LI = U:+,, n Y and V = VArn n Y ?

we obtain sequence (XUkrnVZ). This sequence now orders k and rn properly, and it

does not introduce any new violations of +1 or 4 3 : For il, once again consider any

jobs v E U and v E V. Since, v f u, then v pl u. Also rn $1 u since q, > qk 2 qm,

and v +l k follows by the choice of k. To see that this doesn't introduce any new

violations of 4 3 eitber, note that a.il jobs in Y n Jo>b must be in V, and for these

jobs their relative order is unchangeci. Repeatedly applying SI we obtain an optimai

sequence s" that is a linear extension of i l and 4 3 .

Finally, assume that sequence s is not a h e a r extension of 4 2 . Let rn E Ja>b

be the last job in s with some job j E Ja<b - with j 4 2 m, such that j is after m in

S. Let k be the first such job j in sl', then sr' is of the form sl' = ( X m Y k Z ) . Since

k 4, rn, we know for some i that J = UiGm U Vieml applying SI for this choice of

i with U = n Y and V = Viem n Y, we obtain sequence (XUkmVZ). This

sequence now orders k and rn properly, and it does not introduce any new violations

of 4: As above, for any jobs u f U and v E V we have that v # u which implies

that v f i u. 111 addition, we have rn #3 u (=+ m # u) and v #l k (=+ v # k) as

above. Thus, we see that this interchange does not introduce any new violations of 4.

Continuing to apply SI until there is no such job m, we obtain an optimal sequence

s* that is &O a linear extension of 4. I

4.2.3 Fuzzy approximation of dominance

Dominance d e s on sequences are usuaily used to speciS whether a node can be

elirninated before its Iower bound is calculated in a branch and bound algorithni-

However, they also can be used heuristically in hding a good initial solution, or

directing the search in case of ties [4].

Theorern 4.6 Consider s partial sequence u for the pmblem ~ 2 / r ~ / t , . If them

are unseqzlenced jobs i and j such that

d(aij) 5 c f ( u j i ) , 15 15 2 (1)

Lm, W j ) 5 L : ~ W ) (2)

then the partial sequence ai j dominates oji, i. e., any wmpletion of o i j h a an L,

which is not larger than the L, for the same wmpletion of u j i .

Proof. Consider an arbitrary complet ion sequence w for the remaining uasequenced

jobs. If Eq.(l) holds, then every job in w will be able to start no later on both

machines, in the sequence (uijw) than in the sequence (ojiw). This implies that

no job iri w can have a larger L' value in the sequence (ui jw) than in the sequence

(o j iw) . Combining this with Eq. (2) complets the proof.

If there exists a job i for which Eq.(l) and Eq.(2) hold for al1 unsequenced

jobs j, then sequence a has an optimal completion with job i sequenced next. Such a

job i, very rarely exists, however. This suggests that if they only upproximately hold,

i-e., they hold for job i with cdmost all unsequenced jobs j , then job i may precede

another job j in an optimal completion of sequence a with high probability. We

measure the closeness of this a p p r h a t i o n by a fuzzy membership function. We use

this fwzy inference to find a good initial sequence, and to break ties between nodes

with the same lower bound when branching. These techniques proved very useful for

the problem Frn//C,, [4]. Let

~ ' ( o i j ) = ~ ( a i j ) - ( j ) , 151 5 2

o3 (o i j ) = L;=(U~ j ) - ~ k ~ ( u j 2 ) ) -

The fun( membership fvnction t hat represents the likelihood t hat job i precede. job

j in an optimal completion of a is given by

p&, j) = 0.5 -

where D(oi j) = CL, a1D1(ui j ) , D-(a)

ai, al, as 5 1 and Cf=, al = 1 ) are real numbers. (Note that this definition ensures

that O 5 pu@, j ) 5 1.) Let S be the set of jobs not scheduled in o. Then, the

likelihood of job i E S dominating the remaining jobs after partial sequence a is

measured by

and job i* satisfjing

pz (i') = max pz ( i ) 1ES

is identifieci as the job that immeàiately foliows a .

The d e determining i* in this way is referred to as the nile and the

schedule obtained by repeatedly applying the fuzzy d e is referred to as the furzy

schedule (or fuzy sequence). To obtain our fuzy sequence we apply the fuzy nile

with al = 1/3 for I = 1,2,3. This requires 0(n2) t h e .

4.3 Branch and bound

In this section, we outline the basic components of our p r o p d branch and bound

algorithm to solve the problem F2 I r j , perm/ L;,. A pseudocode for it is &en in

Appendix B.

4.3.1 Branching rde

In order to exploit the symmetry of the problem, we consider a variant of Potts'

adaptive branching d e that fixes jobs at both ends of the schedule [34] . More

precisely, each node of the search tree is represented by a pair (al, oz), where al and

a2 are the initial and finai partial sequences, respectively. Let Si denote the set of

jobs in Oi for i = 1, 2, then the set of iinfixed jobs is S = J\(Sl U S2). We use 41,

to refer to the restriction of 4 to the set S. An immediate successor of (al, oz) in

the tree is eit her of the form ( q i , 02) for a type 1 branching; or (al, iaz ) for a type

2 branching, where i is a minimal or maximal job in +I,, respectively. The types of

the branchings are aii the same within a level of the tree. The type for a given level

k is fixed on the very k t visit to level k according to the following d e : branch in

the direction of the fewest number of ties at the minimum lower bound. Let nl and

n2 be the number of ties at the minimum lower bound for potential type 1 and type

2 branchings, at level k. If nl < n2 the next branching is of type 1, while if nz < nl

then the branching is of type 2. If nl = nz then the branching is the same type as at

the previous level.

The search strategy is to branch to the ne- active node with the smallest

lower bound, breaking t i s by the appropriate h z y rule. For type 1 branchings

we use hizzy sequence 1 to break ties, this is the sequence obtained by repeatedly

applying the fuzzy nile to the forward problem. For type 2 branchings we use fuzzy

sequence 2 to break ties, this is the analogous sequence for the reverse problem-

Upper bounds are calculated only for the nIst n nodes, afterward the upper bound is

evaluated only at leaf nodes. For the first n noda we calculate four separate upper

bounds. The first two upper bounds are obtained by sequencing the unfixed jobs in

the fuzzy sequence for the forward and reverse problems, respectively. Each of these

requires only O(n) time working fiom the previously established fwzy sequences. The

two remaining upper bounds are obtained by sequencing the iinfixed jobs in ready

q + b order for the forward problem, and ready r + a order for the reverse problem.

Each of these upper bounds requires O(n log n) time to cornpute. For leaf nodes, there

are no unfixeci jobs and the upper bound is just the length of the sequence obtained

by concatenating al and 0 2 .

Since the branch and bound tree may require the computation of lower bounds

for a potentially very large number of nodes, it is important that we use lower bounds

which require only O(n) time per bound. (We have also experïmented with some

potentidy better lower bounds, requiring O(n log n) time, but they have noticeably

slowed down the algorithm without any substantial increase in the number of problems

solved.) We consider six lower bounds for each node (al , az). We calculate lower

bounds on the lengths of different paths for the iinfixed jobs S, and we combine these

in various ways with the actual Iengths of the fixeci sequences al and 0 2 . For al

we look at the forward problem and we let C'(a l ) and @(al) be the completion

times on machines 1 and 2, respectively. For S we consider the forward problem, and

assume that these jobs cannot start on machines 1 and 2 before C 1 ( o l ) and C(o l ) ,

respectively. Ignoring release times for the jobs in S , let L1 ( S ) be the completion

time on machine 2 of the Johnson sequence on S. Let L2(S) be the completion time

on machine 1 of the jobs in S sequenced in nondecreasing r order, that is, L2(S) is

the earliest time at which the jobs in S c m be completed on machine 1. Similarly, if

we relax the capacity constraint on machine 1 and sequence the jobs in nondecreasing

r + a order, then the length of this schedule on machine 2, LJ(S), is a lower bound

on when the jobs in S can complete on machine 2. For a2 we define the completion

times C1 (02) and e (a2) on machines 1 and 2, respectively, for the reverse problem.

Once again, we assume that the jobs in S cannot start before these t h . We let

&(S) be the completion tirne on machine 1 of the reverse Johnson sequence. As for

the forward problem, the earliest that the jobs in S can complete on machine 2 in the

reverse problem is to sequence them in nondecreasing q order, we denote the length

of this schedule on machine 2 by LS(S) . Relaxhg the capacity constraint on machine

2, and sequencing the jobs on machine 1 in nondecreasing q + b order, the length of

this scheduie denoted by Ls(S), is a lower bound on when the jobs in S W h on

machine 1 in the reverse problem. We compute the foilowing for the node (al, 02):

LBi = LI(S)+C2(02)

LB2 = Lz(S) +C1(uz)

LB3 = L3(S) + C2(u2)

LB4 = C'(ai) + L4(S) LB5 = c2(o1) +L5(S)

LBs = C'(ai) + Ls(S).

Findy, the lower bound is the maximum of the above six lower bounds and the lengths

of the h e d sequences al and a 2 (in the forward and reverse problems), L:,(O~) and

L:, (u2 ) , respectively.

At the mot node we evaluate two additional lower bounds, these are single

machine preemptive bounds obtained by relaxhg the capacity constraints on each of

machines 1 and 2 in the forward problem, respectively. It is well known that these

are solved by the preemptive ready Jackson rule, which reguires O (n log n) t ime to

compute.

4.3.3 Decomposition and dominance

We find a starting sequence al by applying the following simple decomposition prcl

cedure, which is a generalization of the one used in [41] for the problem F2/ri/C-.

Given a sequence s, then partial sequence sk-' = (s(l), . . . , s(k - 1)) is an opti-

mal initial sequence if there exists a k E [2, n] such that min r,(i) 2 Ci(k-l), mjn k l i l n k<rsn

+as(il] 2 c(k(,_l), and we have LB(J\(s~-')) 1 L,(sk-'), where LB(J\(S~-'))

is computed as foliows. We apply the decompmition procedure for the jobs sequenced

in nondecreasing r order, and we use the two preemptive singie machine bounds men-

tioned in the last paragraph of the previous section to determine LB( J\(sk-')). Then

the initial partial sequence al is sk-' for the largest k value found.

After we have fixed 01, we determine the dominance order 4 on the remaining

jobs in S = J\&. In addition, it is possible to dynamidy update the dominance

order 4 at each node (al, OZ), to see whether king jobs in al and CQ can acld new

comparabilities, thus further reducing the amount of branching.


4.4.1 Test problerns

For each problem with n jobs 472 integer data (ri, ai, bi, q,) were generated. The

processing times and bi were bot h uniformly distributed between [l ,100]. Release

times ri and delivery times qi were udorrnly distributed in the range [O, n - 101 R]

and [O, n -101 QI, respectively, following the technique used by Hariri and Potts [16].

Different R and Q values were tested for values Q 5 R and R E (0.2,0.4,0.6,0.8,1 .O}.

For each individual R, Q, and n combination, 50 probiems were generated. We used

the random number generator of Taillard [42] to generate these problems. This means

that ali of our test problems are reproducible by running the problem generation

procedure fiom the same seeds. (The seeds used have b e n saved and are available

from the author on request.)

The branch and bound algorithm was coded in Sun Pascal 4.2 and run on a Sun

Sparc5. Three separate versions of the algorithm were run. Algorithm Al has the

dominance order 4 'turned off'. Algorithm A2 has the dominance order 'tumeci

on', and it dynamically updates 4 at each node. Findy, algorithm A3 alSO has the

dominance order 'turned on', but 4 is only d d a t e d once at the root node. Each

version of the algorithm was run until either it obtained the optimal solution or the

number of nodes branched from in the tree reached one million for the problem. In

the latter case, the problem was declareci unsolved.

Tables 4.3, 4.4, and 4.5 contain the results of the computational experiment for

the different R and Q values. For each group of problems we report: the &action of

problems solved (denoted by solved); for each of the unsolved problems we calculate

the gap between the best solution obtained and the smalleest lower bound among the

left-over nodes in the tree, and the maximum gap (denoted maz gap) is the largest of

these over all the problems in the group; the total number of nodes in all of the trees

(denoted by total nodes); the average CPU t h e for the solved problems in the group

(denoted by avg CPU); and the total CPU tirne required for al l of the problems in

the group (denoted by total CPU).

Recall, a problem was 'unsolved' when the number of nodes branched from

reached a million. Thus the total number of nodes for a group of 50 problems,

contains a million nodes for each unsolved problem. As can be seen, the a l g o r i t h

typicdy generated mu& fewer nodes for the solved problems. In thme groups where

there were unsolved problems, the solution obtained was always nearly optimal as

measured by the maximum gap. The larggt such gap was on the order of 3 percent,

meaning that, in the worst case, we were within this range of the optimal solution.

The average gap was much smailer, less than 0.5 percent.

As the results indicate, ail three versions of the new branch and bound dg*

rithm proved very effective in solving the test problems. In total, each of the three

solved 4,384 of the 4500 randornly generated test problems with up t o 200 jobs. There

were clifferences in the tirne-performance of the three versions, however. In general,

algorithms Az and A3 (with the dominance order) required less time than version

Al (without the dominance order). Algorithms Al and A3 needed roughly 5 percent

and 15 percent l e s total CPU time, respectively, than Al for the 4500 test problems.

For the 4384 solved problems, the savings in total CPU t h e amounted to 15 percent

and 20 percent, respectively, when compareci to Al. Thus, both A2 and Ag made the

search through the tree faster, but the substantiaily larger overhead of A2 (i.e., of u p

dating 4 at every node of the tree) o&t a large part of the speed-up. In srimmary,

aigorithm A3 was the most effective method, it was extremely fast on all of the solved

problems, never requiring more than a few seconds.

The very extensive computational experiment has led to further insights. Our

results suggest that the difficulty in solving a particular problem depends much m m

on the valu= of R and Q, than it does on the number of jobs. Given this observation,

the diicult problems are those with intermediate R values, with the most àifiicult

being the problems with R = 0.4. We have also found that for a given R, the

problems with smaU Q are m a t nifficult;. Recaii, that the actual sue of the problem

for the branch and bound algorithm is determineci by how many jobs are fixed by

the decomposition procedure. We noticed some trends regarding the effectiveness

of the decompmition procedure. First, and most importantly, we noticed that as

R increases the percentage of jobs fixeci tends to increase. We also noticed that

for a given R value, the percentage of jobs fixed decreases for increasing Q values,

where recall that Q 5 R. For smail R values with R < 0.4, the decomposition

procedwe is ineffective and fixes at most 1 and 6 percent of the jobs for R = 0.2

and 0.4, respectively. Moreover , for these values the decomposition procedure tends

to become les effective as the number of jobs inaeases. For larger R values with

R 2 0.6, the decomposition p r o d u r e is quite effective, and it performs even better

as the number of jobs increases. For R = 0.6, the decomposition procedure fixes

around 40 percent of the jobs for the problems with n = 20, this increases to around

80 percent for n = 200. For R = 1 .O, the percentage of jobs fixeci increases to around

75 percent and 90 percent for problems with n = 20 and n = 200, respectively.

This Chapter considered a new branch and bound algorithm for the problem

F 2 /ri, perm/ L,. The main features of the proposed algorithm are: an adaptive

branching rde that fixes jobs at both ends of the schedule, a new dominance order

to reduce branching together with new fuzzy dominance properties that are used for

scheduling and tie-breaking, and a simple decomposition procedure that reduces the

problem size by bcing jobs at the beginning of the scheduie. in general, cornputa-

tional results indicated that the aigotitbm perfomed very weli. It has solved more

than 97% of the test problems with up to 200 jobs aithin a few seconds. Even when

we failed to solve a problem, we were always very close to the optimal solution. We

found that the use of the dominance order resulted in a reduction in computation

time of 15 to 20 percent. We also found that the use of fuzzy dominance properties

aided the search by breaking ties and by often finding a nearly optimal initial solu-

tion. In summary, we feel that the proposed algorithm is more effective than previous

solution methods for the problem F2/rj,perm/ L;,, in that i t found fewer hard

problems and it was aiso able to solve much larger problems. The fact that most

problems were solved to optimum within a few seconds, means that the algorithm

has the potential of being used as a callable subroutine for F 2 / r j , perm/ ~ k , - t ~ ~ e

subproblems generated during the solution of more complex scheduling problems.

Table 4.3: Results for (0.2,O.Z) and (0.4,0.2).

n alg solved max gap total nodes avq CPU total CPU (%) (sec) (hrs:min:sec)

R = 0.2, Q = 0.2 Al 50/50 - 24,666 0.0840 4.20

20 AZ 50150 - 24,593 0.0958 4.79 A3 50150 - 24,666 0.0828 4.14 AL 50/50 - 3,894 O. 1146 5.73

40 A2 50/50 - 3,893 O. 1190 5.95 A3 50150 - 3,894 O. 1098 5.49 Al 48/50 1.1401 2,001,789 0.2225 30:57.50

60 Az 48/50 1.1401 2,001,791 0.2056 33: 17.86 A3 48/50 1.1401 2,001,791 0.1954 2933.53 AL 49/50 0.4968 1,009,292 1.0969 20: 16.54

80 A2 49/50 0.4968 1,009,278 1.1967 22:48.84 A3 49/50 0.4968 1,009,288 1.0549 19:53.81 Al 48/50 0.6178 2,029,119 4.6633 2:40: 15.34

100 A2 48/50 0.6178 2,048,242 6.9856 2: 16:42.26 A3 48/50 0.6178 2,048,242 5.9825 2:01:24.72 Al 49/50 0.0655 1 ,O63,65O 38.7327 8:02:25.27

200 A2 49/50 0.0655 1,063,647 24.1063 7:06:01.35 A3 49/50 0.0655 1,063,647 22.0294 6: 11: 14-99

R = 0.4, Q = 0.2 Al 48/50 1.6575 2,947,498 2.9265 7:32.65

20 A2 48/50 1.6575 2,900,048 3.1988 8:11.50 A3 48/50 1.6575 2,903,012 2.7614 7: 14.97 Al 38/50 1.4881 13,365,909 7.3640 49:02.53

40 A2 38/50 1.4881 13,364,572 7.0429 4446.62

Table4.4: Results for (0.4,0.4) and R = 0.6.

n alg solued m a x g a p totalnodes avgCPU totalCPU (W (sec) (hrs:min:sec)

R = 0.4, Q = 0.4 Al 45/50 3.2119 5,505,470 1.9667 13:31.30

20 A2 45/50 3.2119 5,474,584 1.6951 14:21.46 A3 45/50 3.2119 5,468.818 1.5591 12S.13

Table4.5: Rsults for R = 0.8 and R = 1.0.

n alg solved max gap total nudes avg CPU total CPU (%) (sec) (hrs:min:sec)

R = 0.8, Q = 0.2,0.4,0.6,0.8 Al 200/200 - 1164 0.0026 0.51

20 A2 200/200 - 1149 0.0029 0.58 A3 200/200 - 1149 0.0027 0.54 Al 200/200 - 914 0.0056 1.11

40 A2 200/200 - 912 0.0056 1.11 A3 200/200 - 912 0.0053 1 .O6 Al 199/200 0.2625 1,001,163 0.0098 2:32.97

60 A2 199/200 0.2625 1,001,167 0.0097 6:53.71 A3 199/200 0.2625 1,001,167 0.0095 2:33.83 Al 198/200 0.1513 2,000,359 0.0013 6: 10.13

8G A2 198/200 0.1513 2,000,357 0.0012 3:35.69 A3 198/200 0.1513 2,000,357 0.0012 3:30.68 Al 200/200 - 257 0.0181 3.61

100 Ag 200/2(10 - 261 0.0183 3.65 As 200/200 - 261 0.0180 3.59 Al 200/200 - 259 0.0571 11.42

200 A2 200/200 - 267 0.0569 11.37 A3 200/200 - 267 0.0568 11.36

Chapter 5 Thesis summary Ln this thesis, we have introduced a new technique, subset-restricted pairwise inter-

change, that generalizes well-known methodç used to derive dominance results for

scheduling problems. Subset-restricted pairwise interchange is a generalization of

pairwise job interchange t hat incorporates subset-restrictions on intermediate jobs.

The traditional methods of pairwise job interchange can be viewed as special cases,

where t hese subsets are all udormly either the empty set (adjacent pairwise inter-

change) or the entire job set (nonadjacent pairwise interchange).

We have used this technique to derive dominance orders for the one- and

two-machine scheduling problems with release times 1 / r j / fma, F2 / r j / Cm,, and

F2 / r j , perm/ L;,. The dominance order we derive is a suborder of the adjacent

interchange order for the problem. We have shown that in general the adjacent in-

terchange order is not a dominance order, but that there is a subset of this order

that is a dominance order. These results generalize the dominance results of Erschler

et al. [8], [9], and [IO], for the one-machine scheduling problems 1 /rj,G/ Cm, and

1 / r j / Lrnax-

We have tested the effectiveness of the dominance orders in efficient branch and

bound dgorithms. We found that the dominance orders were &mtive at reducing

the number of branches in the search tree, and that this led to a sizeable reduction

in the total computation times. Our experiments indicated that the dificulty in

solving these problems depends much more on the range of release (and delivery)

times than on the number of jobs. We have classifieci problem instances as either

'easy' or 'hard' on this basis. Our test problems were generated using the random

number generator of Taillard [42], thus they are easily reproducible, and can serve as

benchmark problems for future researcchers.

There are several directions for future research. A natural extension would be

to consider problems with more than two machines (i.e. m > 2) such as F m Irj/ Cm=,

FrnIIC,,, and Fm / r j / L,. These techniques could be appiied directly to derive

dorninance orders, or the proposeci aigorithms could be used as subroutines to solve

F 2 /r,/ Cm,- or F2 I r j , p e n l ~ , - t ~ ~ e subproblerns generated during the solution

process. Another extension would be to apply these techniques to problems where the

objective is to minimize the total cost, instead of the maximum cost. Continuhg in

this vein, still another extension would be to consider multiple criteria combinations

of the above two types of objectives. These directions will be the subject of future

research .

Appendix A Algorit hm for F2/rj/Cmm Calculations for root node

a apply decomposition procedure with nondecreasing r sequence to fix al;

a let S := J\S1, 0 2 := 0; nodes := 0;

a construct the dominance order 4;

a caiculate initial UB and LB, and let BestMakespan := UB;

a call recursive procedure Opt that facilitates branching;

procedure Opt (al, S, 02, Best Makespan) ;

begin

if nodes 5 1000000 then

begin

nodes := nodes + l;{increment the number of nodes brancheci fkorn)

if (nodes 5 n) or (ISI = 1) then calculate UB and (possibly) update

Best Makespan

if (LB < BestMakespan) and ([SI # 1) then

begin

find lists of minimal and mazimal jobs in 41,

if type = 1 then calculate lower bounds for mÊn2mal and sort in

nondecreasing order

else if type = 2 then calculate lower bounds for maximal and sort in

nondecreasïng order

else (type not set)

begin

must calculate and sort lower bounds for both minimal and

maximal to find nl and nz to determine type for node

if nl < no then type := 1

else if nl > nz then t ype := 2

else type := previous-t ype;

end;

if t ype = 1 tben list := minimal

else if type = 2 then list := maximal ;

while ( l i s t # 0) and (Bes tMakespan > lower bound fkom head of List)

do

begin

remove head of List

let job be the job that is fixed and let LB be its lower bound

remove job £rom 4

S := S\{job);

if type = 1 then add job to al

else if t ype = 2 then add job to oz;

Opt (ol , S, 02 , Best Makespan) ; {recursive d l )

if type = 1 then remove job from a1

else if type = 2 then remove job fiom 02;

S := S u (joû); restore job to 4

end; {while)

end;{if (LB < BestMakespan) and (ISl # l)}

end;{if nodes 5 1000000)

end;

Appendix B Algorithm for F 2 / r j , p e r m / ~ m m Calculations for root node

apply decomposition procedure with nondecreasing r sequence to fbc al;

0 let S := J\Sl, 0, := 0; nodes := 0;

0 construct the dominance order 4;

a find fuzzy sequence land k z y sequence 2;

a cdculate initial U B and LB, and let BestU B := U B;

c d recursive procedure Opt that facilitates branching;

procedure Opt (ul , S, 0 2 , BestU B);

begin

if nodes 5 1000000 then

begin

nodes := nodes + l;{increment the number of nodes branched from)

if (nodes 5 n) or ((SI = 1) then calculate UB and (possibly) update BestUB

if (LB < BestUB) and (1st # 1) then

begin

find lists of minimal and maximal jobs in +ls if type = 1 then calculate lower bounds for minimal and sort in

nondecreasing lower bound order breaking ties using fuzzy sequence 1

else if type = 2 then calculate lower bounds for maximal and sort in

nondecreasing lower bound order breaking ties using fuzzy sequence 2

else (type not set)

begin 90

calculate and sort lower bounds for both minimal and maximal

breaking t ies accordingly

find nl and n~ to detennine type for node

if nl < n2 then type := 1

else if nl > n;! then t ype := 2

else type := previoztsAype;

end;

if type = 1 then list := minimal

else if type = 2 then list := maximal;

while (list # 0) and (BestU B > lower bound boom head of l ist) do

begin

remove head of list

let job be the job that is fixed and let LB be its lower bound

remove job fiom + S := S\{job);

if type = 1 then add job to al

else if type = 2 then add job to as;

Opt (ol , S, 02 , BestU B ) ; {recursive c d )

if type = 1 then remove job from al

else if type = 2 then remove job from oz;

S := S u {job); restore job to 4

end; {while)

end;{if (LB < BestUB) and (ISI # 1)) end;{if nodes 5 1000000}

end;

Bibliography [l] Baker, K.R., and Z.S. Su (1974). Sequencing with due dates and early start tirnes to

minimize maximum tardines. Naval Res. Logist. Quart. 21, 171-176.

[2] Baker, K.R., E.L. Lawler, J.K. Lenstra, and A.H.G. h o o y Kan (1983). Preemptive

scheduling of a single machine tu minimize maximum cost subject to release dates

and precedence constraints. Oper. Res. 3 1,381-386.

[3] Carlier, J. (1982). The onemachine sequencing problem. Eumpean J. Oper. Res. 1 1

42-47.

[4] Cheng, J., H. Kise, and H. Matsumoto (1997). A branch and bound algorithm with

fuzzy inference for a permutation flowshop scheduiing problem. Evmpeun J. Oper.

Res. 96, 578-590.

[5] Della Croce, F. (1995). Generalized pairwise interchanges and machine scheduling.

European J. Oper. Res. 83, 310-319.

(61 Dushnik, B., and E.W. Miller (1941). Par t idy ordered sets. Amer. J. Math. 63,

600-610.

[7] Emrnons, H. (1969). One machine sequencing to rninimize certain hinctions of job

tardiness. Oper. Res. 17, 701-715.

[8] Erschler, J., G. Fontan, C . Merce, and F. Raubellat (1982). Applying new dominance

concepts to job schedule optimization. Eumpean J. Oper. Res. 11, 6@66.

[9] Erschler, J., G. Fontan, C. Merce, and F. Roubellat (1983). A new dominance concept

in schedding n jobs on a single machine with ready tùnes and due dates. Oper. Res.

31, 114-127.

[IO] Erschler, J., G. Fontan, and C. Merce (1985). Un nouveau concept de dominance pour

l'ordonnancement de travaux sur une machine. R. A. I. R. O. Recherche op&mtionnelle

19, 15-26.

[Il] Fishburn, P.C. (1985) Interval M e r s and Intemal Gmphs, J. Wdey, New York, N.Y.

[12] Fontan, G. (1980). Notion de dominance et son application à l'étude de certains prob

lèmes d'ordonnancement, Thèse de Doctorat ès Sciences, Université Paul Sabatier,

Toulouse, Rance.

[13] Garey, M.R., and D.S. Johnson (1979) Cornputers and Intmctabzlzty: a Guide to the

Theory of NP-Completeness, F'reeman, San Fkancisco.

[14] Grabowski, J. (1980). On twemachine scheduling with release and due dates to min-

imize maximum Iateness. Opsearch 17, 13M54.

[15] Grabowski, J., E. SkubaIska, and C. Smutnickî (1983). On flow shop scheduiing with

release and due dates to mbimize maximum lateness. J. Oper. Res. Soc. 34,615-620.

[16] Hariri, A.M., and C.N. Potts (1983). An algorithm for single machine sequencing

with release dates to mhimïze total weighted completion tirne. Discrete Appl. Math.

5, 99-109.

[17] Hall, L.A. (1993). A note on generalizing the maximum latengs criterion for schedd-

ing. Discrete Appl. Math. 47, 129-137.

[18] Hall, L.A. (1994). A polynomial approximation scheme for a constrained flow shop

scheduling problem. Math. Opns. Res. 19, 6885.

[19] Hall, L.A., and D.B. Shmoys (1992). Jackson's rule for single machine scheduling

making a good heuristic better. Math. Opns. Res. 17, 22-35.

1201 Jackson, J.R (1955). Scheduling a production line to mhimke maximum tardines.

Research Report 43, Management Science Research Project, Univ. of California, Los

Angeles.

[2 11 Johnson, S.M. (1954). Optimal tw* and three-stage production schedules with setup

times included. Naval Res. Logist. Quart. 1, 61-68.

[22] Kovalyov, M.Y., and F. Werner (1997). A polynomial approximation scheme for prob-

lem F2 I r j / Cm,. Oper. Res. Lett. 20, 75-79.

[23] Lageweg, B. J., J.K. Lenstra, and A.H.G. Rhnooy Kan (1976). Minimizing maximum

lateness on one machine: comput uat ional experience and some applications. Stast.

Neerlandzca 30, 25-41.

(241 Lageweg, B. J., J.K. Lenstra, and A.H.G. Rinnooy Kan (1978). A general bounding

scheme for the permutation Bowshop problem. Oper. Res. 26, 53-67.

[25] Lawler, E.L. (1973) Optimal sequencing of a single machine subject to precedence

constraints. Management Science 19, 544346.

[26] Lawler, E.L., J.K. Lenstra, A.H.G. Rinnooy Kan, and D.B. Shmoys (1989). Sequenc-

ing and scheduling: Algorithms and complexity. Report BSR8909, CWI Amsterdam.

[27] Lenstra, J.K., A.H.G. Rinnooy Kan, and P. Brucker (1977). Complexity of machine

scheduling problems. Annals of Disc. Math. 1, 343-362.

[28] McMahon, G.B., and M. Florian (1975). On schedulïng with ready times and due

dates to minimize maximum lateness. Oper. Res. 23, 475-482.

[29] Merce, C . (1979). Etude de l'existence de solutions pour certains problémes d'ordon-

nancement, Thése de Doctorat Ingénieur, Université Paul Sabatier, Toulouse, France.

[30] Moma, C.L. (1978). Properties and efficient algonthms for certain classes of sequenc-

ing problems, Ph. D. Thesis, Cornell University, Ithaca, New York.

[3 11 Papadimitriu, C. H., and M. Yannakakk (1979). Scheduling intenal-or derd t asks.

SIAM J. Computzng 8, 405409.

[32] Pinedo, M. (1995) SCHEDULING: Thwry, Algorithms and Systems, Prentice Hall,

Englewood Clif&, N.J.

[33] Potts, C.N. (1980). Analysis of a heuristic for one machine sequencing with release

d a t a and delivery times. Oper. Res. 28, 14361441.

[34] Potts, C.N. (1980). An adaptive branching d e for the permutation flow-shop prob-

lem. Eumpean J. ûper. Res. 5, 19-25.

[35] Pot ts , C. N. (1985). Analysis of heurist ics for twemachine flow-s hop sequencing s u b

ject to release dates. Math. Opns. Res. 10, 576-584.

[36] Rinnooy Kan, A.H.G.,B.J. Lageweg, and J.K. Lenstra (1975). Minimizing total costs

in one machine scheduling. Oper. Res. 23, 908927.

[37] Smith, W .E. (1956). Various optimizers for single-stage production. Naval Res. h g i s t .

Quart. 3, 59-66.

[38] Szwarc, W. (1990). Parametric precedence relations in single machine scheduling.

Oper. Res. Lett. 9, 133-140.

[39] Szwarc, W., M.E. Paner, and J.J. Liu (1988). The single machine problem with a

quadratic cost function of completion times. Management Sci. 34, 1480-1488.

[40] Szwarc, W., and J.J. Liu (1993). Weighted tardiness single machine scheduling with

proportional weights. Management Sci. 39, 626-632.

[4 11 Tadei, R , J.N.D. Gupta, F. Della Croce, and M. Cortesi (1998). Minimising makespan

in the twemachine flow-shop with release times. J. Qper. Res. Soc. 49, 77-85.

[42] Taiilard, E. (1993). Benchmarks for basic scheduling problems. Eumpeun J. Oper.

Res. 64, 278-285.

(431 Zdralka, S., and J. Grabowski (1989). An algorithm for single machine sequencing

with release dates to minjmize maximum cost. Discmte Appl. Math. 23, 73-89.

paul a. stephenson, m.b.a., m.sc.,collectionscanada.gc.ca/obj/s4/f2/dsk1/tape9/pqdd... · as either...

Documents