Download pdf - Single machine scheduling with randomly compressible processing times

This article was downloaded by: [Nova Southeastern University]On: 07 October 2014, At: 14:27Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

Stochastic Analysis and ApplicationsPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lsaa20

Single machine scheduling with randomly compressibleprocessing timesX. D. Qi , G. Yin a & J. R. Birge ba Department of Mathematics , Wayne State University , Detroit, MI, 48202, U.S.A.b McCormick School of Engineering and Applied Science , Northwestern University ,Evanston, IL, 60208, U.S.A.Published online: 15 Feb 2007.

To cite this article: X. D. Qi , G. Yin & J. R. Birge (2002) Single machine scheduling with randomly compressible processingtimes, Stochastic Analysis and Applications, 20:3, 591-613

To link to this article: http://dx.doi.org/10.1081/SAP-120004116

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/loi/lsaa20

http://dx.doi.org/10.1081/SAP-120004116

http://www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/page/terms-and-conditions

SINGLE MACHINE SCHEDULING WITHRANDOMLY COMPRESSIBLE

PROCESSING TIMES

X. D. Qi,1 G. Yin,2,* and J. R. Birge3

1TAC Automotive Group, 500 Town Center Dr., Suite 100,

Dearborn, MI 481262Department of Mathematics, Wayne State University,

Detroit, MI 482023McCormick School of Engineering and Applied Science,

Northwestern University, Evanston, IL 60208

ABSTRACT

This work is concerned with single machine scheduling with

random compression of processing times. The objective is to find

the optimal sequence to minimize the cost based on earliness,

tardiness and compression. The analysis is carried out under a

common due date. Both absolute derivation cost and squared

derivation cost are considered. For both constrained problems and

unconstrained problems, it is shown that an optimal schedule must

be V-shaped. Remarks on common slack model is also provided.

Key Words: Scheduling; Random compression; Earliness;

Tardiness; Due date; V-shaped sequence

591

Copyright q 2002 by Marcel Dekker, Inc. www.dekker.com

*Corresponding author. E-mail: [email protected]

STOCHASTIC ANALYSIS AND APPLICATIONS, 20(3), 591–613 (2002)

Dow

nloa

ded

by [

Nov

a So

uthe

aste

rn U

nive

rsity

] at

14:

27 0

7 O

ctob

er 2

014

1. INTRODUCTION

The sweeping changes produced by computer-based technologies have a

significant impact on management of manufacturing systems; see for example,[28] and the references therein. Owing to these changes, due-date assignment and

scheduling problems in which the jobs have compressible processing times has

received much attention lately.

The concept of compressible processing time was originated from the area of

project planning and control; see, for example [15], among others. The motivation in

the field of scheduling is of the same spirit. Increasing manufacturing flexibility is a

key to improving responsiveness. In particular, production volume flexibility has

long been a corporate goal in cyclical industries. Nowadays, it is well recognized that

the ability to use trial-and-error to tune the performance of a system is virtually

useless on an environment in which changes occur faster than the lessons can be

learned. Optimal scheduling and rescheduling after changes in parameters or

constraints are of critical importance. This is a greater need for understanding on

compressible processing times, and their effect. In many manufacturing systems,

jobs can be accomplished in shorter or longer durations by increasing or decreasing

additional resources. Not only the use of compressible processing times is justified,

but also it provides more realistic models. Studies on problems with compressible

processing times was initiated by [25,26]. A survey along the line of scheduling is

in [19]. To mention some of the recent progress on this problem, we cite the work[11,13,18,24,27]. More recently, [1], [8], and [20] extend the problem to include the due-

date aspect. To be more specific, in [20], the authors consider the common due-date

assignment and single-machine scheduling problem in which the objective function

is the sum of penalties based on earliness, tardiness and processing time

compressions. The main idea is to reduce the underlying problem to an assignment

problem. The authors of [1] extend the results of [20] to the parallel-machine

scheduling case. The reference [8] further generalizes the result to the case where the

cost of compression is a general convex function of the amount of compression.

In [12], we consider a due-date assignment and single-machine scheduling problem in

which a penalty for due-dates is added to the objective function which includes the

penalties for earliness, tardiness and processing time compressions. The objective is

to determine jointly the optimal due-dates, the optimal sequence, and the optimal

processing time compressions to minimize the total penalty. Again, the key point is

to reduce the underlying problem to an assignment problem. The paper [9] presents a

single-machine scheduling model in which the job processing times are controllable

with linear costs. Solutions of a dynamic programming algorithm and a fully

approximation scheme for the problem are obtained.

All of the aforementioned research to date has been concerned with

deterministic compressible times. To the best of our knowledge, the case of

random compressible times have not received much attention. However, more

QI, YIN, AND BIRGE592

Dow

nloa

ded

by [

Nov

a So

uthe

aste

rn U

nive

rsity

] at

14:

27 0

7 O

ctob

er 2

014

often than not, the deterministic formulation alone does not reflect the reality

well. Job processing time compressions are usually unknown in advance, are

unavoidably to include uncertainty, and cannot be predicted before hand. In this

work, we formulate the single-machine scheduling problem as one that job

processing times are randomly compressible. The objective is to find the optimal

job sequence or the optimal due dates to minimize the total penalty based on the

job earliness/tardiness and compressions. Common due-date-assignment method

and common slack-due-date-assignment method are used to assign due dates to

jobs. The analysis is carried out under the assumption that the compressions are

independent and identically distributed random variables.

The rest of the paper is arranged as follows. Section 2 presents the problem

description and modeling. Section 3 discusses the relationship between V-shaped

optimal sequence and the monotone-about-T property of the cost function.

Section 4 deals with the absolute derivation objective function. Both

continuously distributed random compressions and discrete ones are considered.

Section 5 treats the squared derivation objective function. Some remarks on the

common slack model are given in section 6. Conclusion and topics for further

study are discussed in section 7.

2. PROBLEM FORMULATION

This work considers a single-machine scheduling problem involving n jobs

with the due-dates dj, for j ¼ 1; 2; . . .; n: A job j has a deterministic processing

time pj, which can be compressed by an amount zj, 1 # j # n: Here zj is a random

variable and 0 # zj # �zj: So the actual processing time of job j is pj 2 zj: We

assume that �zj , pj; 1 # j # n: This condition is necessary because if there is a

job j such that �zj ¼ pj; then it may be compressed to have zero processing time

which is not reasonable in practice.

We assume that all the jobs are ready at time zero and no pre-emption is

allowed. Let Rj, Ej and Tj denote the actual completion time, the earliness and the

tardiness of job j, respectively. Then

Ej ¼ maxð0; dj 2 RjÞ and Tj ¼ maxð0;Rj 2 djÞ:

Let S denote the job sequence and d the vector of due dates, that is,~d ¼ ðd1; d2; . . .; dnÞ: We are interested in the following two models:

(1) Model 1: The objective function is of the form

JðS; ~dÞ ¼ EXn

j¼1

ðaEj þ bTj þ bzjÞ; ð1Þ

SINGLE MACHINE SCHEDULING 593

Dow

nloa

ded

by [

Nov

a So

uthe

aste

rn U

nive

rsity

] at

14:

27 0

7 O

ctob

er 2

014

where a, b and b are positive real numbers representing unit penalties

for earliness, tardiness and compression, respectively.

(2) Model 2: The objective function has the following form.

JðS; ~dÞ ¼ EXn

j¼1

ðaðRj 2 djÞ2 þ bzjÞ; ð2Þ

where a and b are positive real numbers representing unit penalties

for earliness/tardiness and compression, respectively.

The problem is to determine the optimal sequence S*, or the optimal

due-date vector d* or both to minimize the function J(S, d ). If the due

dates are given and fixed, the problem is called constrained. If the due dates

are decision variables and to be determined, then the problem is called

unconstrained.

Let Cj denote the deterministic completion time of job j, which is sequence

dependent. Define C0 ¼ 0: Without loss of generality, we assume that Cj is the

completion time of job j for a fixed sequence which can be taken as S ¼

{1; 2; . . .; n}: Then,

Cj ¼Xj

i¼1

pi:

Consequently, we have

Rj ¼Xj

i¼1

ðpi 2 ziÞ ¼ Cj 2 Zj;

where

Zj ¼Xj

i¼1

zi;

which is the total compression of the first j jobs in the sequence. In this work we

treat the case where the compressions zj; j ¼ 1; 2; . . .; n; are independent and

identically distributed variables. Let �zj ¼ �z; j ¼ 1; 2; . . .; n; then

0 # Zj # j�z

Remark 2.1. We assume that 0 # zj # �zj; that is, zj is a bounded random

variable. This condition is satisfied for example, for random variables following

uniform distribution or Beta distribution.


Dow

nloa

ded

by [

Nov

a So

uthe

aste

rn U

nive

rsity

] at

14:

27 0

7 O

ctob

er 2

014

3. PRELIMINARY

We begin with the following deterministic problem. Consider the single

machine scheduling problem where the objective function is given as

JðSÞ ¼Xn

j¼1

ðgðCjÞ þ vjÞ; ð3Þ

where Cj is the completion time of the jth job in the sequence S, g(t ) is a

continuous function and vj is a constant which does not depend on Cj. So

gðCjÞ þ vj is the contribution of the jth job. Note that gðCjÞ þ vj also depends on

the due dates for the problems with early-tardy penalties.

A typical result for problems with early-tardy costs is the V-shaped

property of the optimal schedule. The concept of V-shaped property was first

introduced in [14]. To proceed, we recall the definition first.

Definition 3.1. A sequence is said to be V-shaped with respect to processing

times if in the sequence the jobs before (resp. after) the job with the shortest

processing times are arranged in nonincreasing (resp. nondecreasing) order of

processing times.

Definition 3.2. A function f(t ) is said to be monotone about T if it is

nonincreasing for t # T and nondecreasing for t $ T (see [2]).

Lemma 3.3. If g(t ) is a monotone function about T, then there exists an V-

shaped sequence that minimizes the cost function (3).

Proof. The main steps are outlined below. A similar proof may also be found

in [2].

Without loss of generality, we may assume the optimal sequence is S ¼

{1; 2; . . .; n}: Let Sˆ

denote the sequence obtained by interchanging job j and job

j þ 1 in S. Then

JðSÞ2 JðSÞ ¼ gðCj21 þ pjþ1Þ2 gðCj21 þ pjÞ:

Consider two consecutive jobs j and j þ 1 such that pj , pjþ1 and Cj21 þ pjþ1 #

T: Note that Cj21 þ pj , Cj21 þ pjþ1 and since the function g(t ) is nonincreasing

for t # T ;

JðSÞ2 JðSÞ ¼ gðCj21 þ pjþ1Þ2 gðCj21 þ pjÞ , 0:


Dow

nloa

ded

by [

Nov

a So

uthe

aste

rn U

nive

rsity

] at

14:

27 0

7 O

ctob

er 2

014

Thus, jobs j and j þ 1 can be interchanged without increasing the objective

function. Repeating the above procedure for all such consecutive jobs results in a

sequence in which all jobs completing by T are ordered in LPT (largest

processing time) sequence. The same argument can be applied to jobs starting

after T to show that an SPT (smallest processing time) sequence minimizes the

contribution to the objective function J(S).

Note that the resulting sequence need not be V-shaped because the processing

time of the job that starts before the due-date T and completes after T may be

longer than that of each of its adjacent jobs. In what follows we show this can not

be the case, adn conclude the desired V-shaped property.

Now consider the situation where pj . pj21; pj . pjþ1 and Cj21 , T ,

Cj21 þ pj: There are three cases.

(1) Cj21 þ pjþ1 . T : Interchanging job j and job j þ 1; we have

T , Cj21 þ pjþ1 , Cj21 þ pj:

It follows that

JðSÞ2 JðSÞ ¼ gðCj21 þ pjþ1Þ2 gðCj21 þ pjÞ , 0;

since g(t ) is a nondecreasing function for t . T : Hence we can

interchange job j and job j þ 1 without increasing the objective

function.

(2) Cj22 þ pj , T : Let ~S denote the sequence obtained by interchanging

jobs j and j 2 1: Then

Cj22 þ pj21 , Cj22 þ pj , T ;

which leads to

Jð ~SÞ2 JðSÞ ¼ gðCj22 þ pjÞ2 gðCj22 þ pj21Þ , 0

since g(t ) is a nonincreasing function for t , T : Hence we can

interchange job j and job j 2 1 without increasing the objective

function.

(3) Consider the case where neither case (1) nor case (2) is satisfied, i.e.,

Cj21 þ pjþ1 , T and Cj22 þ pj . T : First, we have

gðCj21 þ pjþ1Þ2 gðCj22 þ pj21Þ , 0

since Cj22 þ pj21 ¼ Cj21 , Cj21 þ pjþ1 , T and g(t ) is non-

increasing for t , T : Similarly,

gðCj22 þ pjÞ2 gðCj21 þ pjÞ , 0

since T , Cj22 þ pj , Cj21 þ pj and g(t ) is nondecreasing for t . T :


Dow

nloa

ded

by [

Nov

a So

uthe

aste

rn U

nive

rsity

] at

14:

27 0

7 O

ctob

er 2

014

Thus we have

JðSÞ2 JðSÞ þ Jð ~SÞ2 JðSÞ

¼ gðCj21 þ pjþ1Þ2 gðCj21 þ pjÞ þ gðCj22 þ pjÞ

2 gðCj22 þ pj21Þ

¼ ½gðCj21 þ pjþ1Þ2 gðCj22 þ pj21Þ�

þ ½gðCj22 þ pjÞ2 gðCj21 þ pjÞ�

, 0:

Therefore, JðSÞ2 JðSÞ , 0; or Jð ~SÞ2 JðSÞ , 0 or both.

Hence, there exists a V-shaped optimal schedule. The proof is complete. A

Corollary 3.4. If the function g(t ) is convex, then there exists an V-shaped

sequence that minimize the cost function (3).

Proof. The conclusion holds because a convex function must be monotone

about T for some T. A

4. MODEL 1

Throughout Sections 4 and 5, we use the common due-date-assignment

method to assign due-dates to the jobs, that is,

dj ¼ d; j ¼ 1; 2; . . .; n:

In this case the objective function can be denoted as J(S, d ).

4.1. Continuous Processing Time Compressions

In this subsection, we consider the case where zj; j ¼ 1; 2; . . .; n; have a

continuous distribution with mean m. Recall that 0 # Zj # j�z: Let Hj(x ), 0 #

x # j�z; denote the distribution function of Zj. Then Hj(x ) is sequence independent

and Hjð0Þ ¼ 0: The objective function can be expressed as

JðS; dÞ ¼ EXn

j¼1

f 1jðCjÞ

!;


Dow

nloa

ded

by [

Nov

a So

uthe

aste

rn U

nive

rsity

] at

14:

27 0

7 O

ctob

er 2

014

where

f 1jðCjÞ ¼ aEj þ bTj þ bzj

¼ amaxð0; d2 RjÞ þ bmaxð0;Rj 2 dÞ þ bzj

¼ amaxð0; d2 Cj þ ZjÞ þ bmaxð0;Cj 2 d2 ZjÞ þ bzj:

If Cj # d; we have

Eðf 1jðCjÞÞ ¼ aðd2 Cj þ EðZjÞÞ þ bEðzjÞ ¼ aðd2 Cj þ jmÞ þ bm:

If Cj $ dþ j�z; we have

Eðf 1jðCjÞÞ ¼ bðCj 2 dþ EðZjÞÞ þ bEðzjÞ ¼ bðCj 2 dþ jmÞ þ bm:

If d , Cj , dþ j�z; we have

Eðf 1jðCjÞÞ ¼ a

Z j�z

Cj2d

ðd2 Cj þ xÞdHjðxÞ

þ b

Z Cj2d

0

ðCj 2 d2 xÞdHjðxÞ þ bm

¼ a

Z j�z

Cj2d

ðd2 Cj þ xÞdHjðxÞ þ a

Z Cj2d

0


2 a

Z Cj2d

0


þ b

Z Cj2d

0

ðCj 2 d2 xÞdHjðxÞ þ bm

¼ a

Z jz

0

ðd2 Cj þ xÞdHjðxÞ þ ðaþ bÞ

Z Cj2d

0

ðCj 2 d2 xÞdHjðxÞ

þ bm

¼ aðd2 CjÞ

Z jz

0

dHjðxÞ þ a

Z j�z

0

xdHjðxÞ

þ ðaþ bÞðCj 2 d2 xÞHjðxÞjCj2d

0

þ ðaþ bÞ

Z Cj2d

0

HjðxÞdx þ bm

¼ aðd2 CjÞ þ aEðZjÞ þ ðaþ bÞ

Z Cj2d

0

HjðxÞdx þ bm

¼ aðd2 CjÞ þ ðaþ bÞ

Z Cj2d

0

HjðxÞdx þ ajmþ bm:


Dow

nloa

ded

by [

Nov

a So

uthe

aste

rn U

nive

rsity

] at

14:

27 0

7 O

ctob

er 2

014

Define

g1ðtÞ ¼

aðd2 tÞ þ ajm; for t # d;

bðt 2 dÞ þ bjm; for t $ dþ j�z;

aðd2 tÞ þ ðaþ bÞR t2d

0HjðxÞdx þ ajm; for d , t , dþ j�z:

8>>><>>>:

Then

Ef 1jðCjÞ ¼ g1ðCjÞ þ bm:

To proceed, we show the following lemma holds.

Lemma 4.1. g1(t) is a convex function.

Proof. For t , d; we have

›g1ðtÞ

›t¼ 2a:

For t . dþ j�z; we have

›g1ðtÞ

›t¼ b:

For d , t , dþ j�z; taking the derivative, we arrive at

›g1ðtÞ

›t¼ 2aþ ðaþ bÞHjðCj 2 dÞ:

Furthermore, taking the second derivative of g1(t ) leads to

›2g1ðtÞ

›t 2¼ ðaþ bÞ

d

dxHjðCj 2 dÞ . 0:

Note that Hjð0Þ ¼ 0: Then

›g1ðd2Þ

›t¼ 2a;


Dow

nloa

ded

by [

Nov

a So

uthe

aste

rn U

nive

rsity

] at

14:

27 0

7 O

ctob

er 2

014

and

›g1ððdþ j�zÞþÞ

›t¼ b:

Thus, ›g1(t )/(›t ) is continuous at points d and dþ j�z: Therefore, g1(t ) is a convex

function. A

Theorem 4.2. For Model 1 with the continuous job processing time

compressions, there exists a V-shaped sequence that minimizes its objective

function.

Proof. The proof follows immediately from Corollary 3.4 and

Lemma 4.1. A

4.2. Discrete Processing Time Compressions

In this subsection, we consider the case where each job compression has a

discrete distribution. Assume that for each job j, zj ¼ �z with probability p and

zj ¼ 0 with probability 1 2 p: Then EðzjÞ ¼ p�z and Eðz2j Þ ¼ p�z2: That is, the

random compression of each job follows a Bernoulli distribution. As a result, Zj is

a binormial random variable with mean jz and variance jpð1 2 pÞ�z2:Assume that pj and d are all integers. Then Cj is also an integer. For

simplicity, assume that �z ¼ 1: Similar to the continuous compression case, the

objective function can be expressed as

JðS; dÞ ¼ EXn

j¼1

f 2jðCjÞ

!;

where

f 2jðtÞ ¼ amaxð0; d2 t þ ZjÞ þ bmaxð0; t 2 d2 ZjÞ þ bzj:

If Cj # d; we have

Eðf 2jðCjÞÞ ¼ aðd2 Cj þ EðZjÞÞ þ bEðzjÞ ¼ aðd2 Cj þ jpÞ þ bp:

If Cj $ dþ j�z; we have

Eðf 2jðCjÞÞ ¼ bðCj 2 dþ EðZjÞÞ þ bEðzjÞ ¼ bðCj 2 dþ jpÞ þ bp:


Dow

nloa

ded

by [

Nov

a So

uthe

aste

rn U

nive

rsity

] at

14:

27 0

7 O

ctob

er 2

014

If d , Cj , dþ j�z; we have

Eðf 2jðCjÞÞ ¼ aXj

l¼Cj2d

ðd2 Cj þ lÞj

l

!plð1 2 pÞj2l

þ bXCj2d

l¼0

ðCj 2 d2 lÞj

l

!p lð1 2 pÞj2l þ bp

¼ aXj

l¼Cj2d

ðd2 Cj þ lÞj

l

!plð1 2 pÞj2l

þ aXCj2d

l¼0

ðd2 Cj þ lÞj

l

!plð1 2 pÞj2l

2 aXCj2d

l¼0

ðd2 Cj þ lÞj

l

!p lð1 2 pÞj2l

þ bXCj2d

l¼0

ðCj 2 d2 lÞj

l

!p lð1 2 pÞj2l þ bp

¼ aXj

l¼0

ðd2 Cj þ lÞj

l

!plð1 2 pÞj2l

þ ðaþ bÞXCj2d

l¼0

ðCj 2 d2 lÞj

l

!plð1 2 pÞj2l þ bp

¼ aðd2 CjÞXj

l¼0

j

l

!plð1 2 pÞj2l

þ aXj

l¼0

ll

j

!plð1 2 pÞj2l

þ ðaþ bÞXCj2d

l¼0

ðCj 2 d2 lÞj

l


¼ aðd2 CjÞ þ aEðZjÞ

þ ðaþ bÞXCj2d

l¼0

ðCj 2 d2 lÞj

l


¼ aðd2 CjÞ þ ðaþ bÞXCj2d

l¼0

ðCj 2 d2 lÞj

l

!p lð1 2 pÞj2l

þ ajp þ bp:


Dow

nloa

ded

by [

Nov

a So

uthe

aste

rn U

nive

rsity

] at

14:

27 0

7 O

ctob

er 2

014

Define

g2ðtÞ ¼

aðd2 tÞ þ ajp; t # d;

bðt 2 dÞ þ bjp; t $ dþ j�z;

aðd2 tÞ þ ðaþ bÞPt2d

l¼0 ðt 2 d2 lÞj

l

!plð1 2 pÞj2l þ ajp; else:

8>>>>><>>>>>:

Then Ef 2jðlÞ ¼ g2ðCjÞ þ bp:

Lemma 4.3. g2(t) is a monotone about T function for some T.

Proof. Obviously, g2(t ) is a continuous function. For d , t , dþ j�z; we have

g2ðCj þ 1Þ2 g2ðCjÞ ¼ 2aþ ðaþ bÞ

£ ½XCjþ12d

l¼0

ðCj þ 1 2 d2 lÞ

j

l

0@1Aplð1 2 pÞj2l

2XCj2d

l¼0

ðCj 2 d2 lÞ

j

l

0@1Aplð1 2 pÞj2l�

¼ 2aþ ðaþ bÞ

£ ½XCj2d

l¼0

ðCj þ 1 2 d2 lÞ

j

l

0@1Aplð1 2 pÞj2l

2XCj2d

l¼0

ðCj 2 d2 lÞ

j

l

0@1Aplð1 2 pÞj2l�

¼ 2aþ ðaþ bÞXCj2d

l¼0

j

l

0@1Aplð1 2 pÞj2l:

Let T be the maximal value of Cj such that

g2ðCj þ 1Þ2 g2ðCjÞ # 0:


Dow

nloa

ded

by [

Nov

a So

uthe

aste

rn U

nive

rsity

] at

14:

27 0

7 O

ctob

er 2

014

Then g2(t ) is nonincreasing for l # T and nondecreasing for t $ T : Therefore,

g2(t ) is monotone about T function. A

Theorem 4.4. For Model 1 with discrete compressions, there exists a V-shaped

sequence that minimizes its objective function.

Proof. The proof follows immediately from Lemma 3.3 and Lemma 4.3. A

4.3. Due Date Assignment

In this section, we consider the problems where the common due date d is a

decision variable. The problem is to find an optimal d for a given sequence.

Proposition 4.5. For Model 1 with the continuous job compressions, for a fixed

job sequence, the optimal common due date d* must satisfy

Xn

j¼1

HjðCj 2 d* Þ ¼na

aþ b:

Proof. Taking derivative of J(S, d ) with respect to d, we have

›JðS; dÞ

›d¼Xn

j¼1

ða2 ðaþ bÞHjðCj 2 dÞÞ:

Setting ð›=›dÞJðS; dÞ ¼ 0 leads to

Xn

j¼1

HjðCj 2 dÞ ¼na

aþ b:

The proposition is thus proved. A

Proposition 4.6. For Model 1 with the discrete job compressions, for a fixed

job sequence, the optimal common due date d* is the maximal value of d such that

Xn

j¼1

XCj2d

l¼0

l

j

!plð1 2 pÞj2l #

na

aþ b:


Dow

nloa

ded

by [

Nov

a So

uthe

aste

rn U

nive

rsity

] at

14:

27 0

7 O

ctob

er 2

014

Proof. It is obvious that J(S, d ) is a continuous function of d.

JðS; dþ 1Þ2 JðS; dÞ

¼Xn

j¼1

0@aþ ðaþ bÞ

XCj2d21

l¼0

ðCj 2 d2 1 2 lÞ

l

j

0@1Aplð1 2 pÞj2l

0@

2XCj2d

l¼0

ðCj 2 d2 lÞ

l

j

0@1Aplð1 2 pÞj2l

1A1A

¼Xn

j¼1

aþ ðaþ bÞXCj2d21

l¼0

ðCj 2 d2 1 2 lÞ

l

j

0@1Aplð1 2 pÞj2l

0@

0@

2XCj2d21

l¼0

ðCj 2 d2 lÞ

l

j

0@1Aplð1 2 pÞj2l

1A1A

¼Xn

j¼1

a2 ðaþ bÞXCj2d21

l¼0

l

j

0@1Aplð1 2 pÞj2l

0@

1A:

It is easily seen that J(S, d ) is a monotone about T (for some T ) function of d. The

conclusion is immediate by setting JðS; dþ 1Þ2 JðS; dÞ # 0: A

5. MODEL 2

5.1. Objective Function Development

Recall that zj; j ¼ 1; 2; . . .; n; are independent and identically distributed

random variables. Assume that EðzjÞ ¼ m and Eðz2j Þ ¼ m: For Model 2, the

objective function can be expressed as

JðS; dÞ ¼ EXn

j¼1

ðaðCj 2 Zj 2 dÞ2 þ bzjÞ

!¼Xn

j¼1

Eðf 3jðCjÞÞ;


Dow

nloa

ded

by [

Nov

a So

uthe

aste

rn U

nive

rsity

] at

14:

27 0

7 O

ctob

er 2

014

where

f 3jðtÞ ¼ aðt 2 d2 ZjÞ2 þ bzj:

Thus we have

f 3jðCjÞ ¼ aðCj 2 dÞ2 2 2aðCj 2 dÞZj þ aZ2j þ bzj:

So

Eðf 3jðCjÞÞ ¼ aðCj 2 dÞ2 2 2aðCj 2 dÞEðZjÞ þ aEðZ2j Þ þ bEðzjÞ

¼ aðCj 2 dÞ2 2 2aðCj 2 dÞjmþ aEðZ2j Þ þ bm:

But

EðZ2j Þ ¼ E

Xj

i¼1

zi

!20@

1A ¼ E

Xj

i¼1

z2i þ

Xj

i–k

zizk

!¼ jm þ jðj 2 1Þm2:

Thus,

Eðf 3jðCjÞÞ ¼ aðCj 2 dÞ2 2 2ajmðCj 2 dÞ þ aðjm þ jðj 2 1Þm2Þ þ bm:

In what follows, we will discuss the V-shaped optimal schedules for both

constrained problems and unconstrained problems. However, Lemma 3.3 does

not apply to this model because in E( f3j(Cj)), the term 22a(Cj 2 d )jm depends

both on Cj and j. We will develop the V-shaped schedule using a job

interchanging argument.

5.2. Constrained Problem

Theorem 5.1. For a given and fixed common d, the optimal sequence that

minimizes its objective function must be V-shaped.

Proof. Without loss of generality, we assume that the optimal sequence is

S ¼ {1; 2; . . .; n}: Suppose S is not V-shaped. Then there are three consecutive

jobs j 2 1; j, j þ 1 in S such that pj . pj21 and pj . pjþ1: Let Sˆ

be the sequence


Dow

nloa

ded

by [

Nov

a So

uthe

aste

rn U

nive

rsity

] at

14:

27 0

7 O

ctob

er 2

014

obtained by interchanging jobs j and j þ 1; and S the sequence obtained by

interchanging jobs j and j 2 1: Let Dj;jþ1 ¼ JðS; dÞ2 JðS; dÞ and Dj21;j ¼

Jð ~S; dÞ2 JðS; dÞ: Then

Dj;jþ1 ¼ aðCj 2 pj þ pjþ1 2 dÞ2 2 2ajmðCj 2 pj þ pjþ1 2 dÞ

2 aðCj 2 dÞ2 þ 2ajmðCj 2 dÞ

¼ aðpjþ1 2 pjÞ2 þ 2aðpjþ1 2 pjÞðCj 2 dÞ2 2ajmðpjþ1 2 pjÞ

¼ aðpjþ1 2 pjÞ½pjþ1 2 pj þ 2ðCj 2 dÞ2 2jm�:

and

Dj21;j ¼ aðCj21 2 pj21 þ pj 2 dÞ2 2 2aðj 2 1ÞmðCj21 2 pj21 þ pj 2 dÞ

2 aðCj21 2 dÞ2 þ 2aðj 2 1ÞmCj21

¼ aðpj 2 pj21Þ2 þ 2aðpj 2 pj21ÞðCj 2 dÞ2 2ajmðpj 2 pj21Þ

¼ aðpj 2 pj21Þ½pj 2 pj21 þ 2ðCj21 2 dÞ2 2ðj 2 1Þm�:

Now we can compare Dj;jþ1 and Dj21;j: If Dj;jþ1 , 0; then jobs j and j þ 1 can be

interchanged with the objective function decreasing. This contradicts the

assumption that S is optimal.

Suppose Dj;jþ1 $ 0: Since pj 2 pjþ1 . 0; by the expression of Dj;jþ1;

ðpjþ1 2 pjÞ þ 2ðCj 2 dÞ2 2jm # 0: ð4Þ

In this case, we will show that Dj21;j , 0: Suppose Dj21;j $ 0: Since pj21 2 pj ,

0; by the expression of Dj21;j;

ðpj 2 pj21Þ þ 2ðCj21 2 dÞ2 2ðj 2 1Þm $ 0: ð5Þ

Subtracting (5) from (4), we obtain

ðpjþ1 þ pj21 2 2pjÞ þ 2pj 2 2m # 0:

Since pj . m;

pjþ1 þ pj21 2 2m , 0:

This is a contradiction because m , pj; j ¼ 1; 2; . . .; n: Thus, if Dj;jþ1 $ 0; then it

must be Dj21;j , 0: Therefore, jobs j and j 2 1 can be interchanged with the

objective function decreasing. Again, this contradicts the assumption that S is

optimal. Therefore, the optimal sequence must be V-shaped. A


Dow

nloa

ded

by [

Nov

a So

uthe

aste

rn U

nive

rsity

] at

14:

27 0

7 O

ctob

er 2

014

5.3. Unconstrained Problem

Taking derivative of J(S, d ) with respect to d, we get

›JðS; dÞ

›d¼Xn

j¼1

ð22aðCj 2 dÞ þ 2ajmÞ:

Setting ð›=›dÞJðS; dÞ ¼ 0; we arrive at (since a . 0)

d ¼ �C 21

n

Xn

j¼1

jm ¼ �C 2ðn þ 1Þ

2m;

where

�C ¼1

n

Xn

j¼1

Cj:

Thus, C is the mean completion time of schedule S. Note that C is sequence

dependent.

Theorem 5.2. For the unconstrained problem, the optimal sequence that

minimizes its objective function must be V-shaped.

Proof. The proof is similar to that of the constrained case. Assume that the

optimal sequence is S ¼ {1; 2; . . .; n}: Suppose S is not V-shaped. Then there are

three consecutive jobs j 2 1; j, j þ 1 in S such that pj . pj21 and pj . pjþ1: Let Sˆ

be the sequence obtained by interchanging jobs j and j þ 1; and S the sequence

obtained by interchanging jobs j and j 2 1: Let Dj;jþ1 ¼ JðS; dÞ2 JðS; dÞ and

Dj21;j ¼ Jð ~S; dÞ2 JðS; dÞ: Note that the mean completion time of schedule Sˆ

is

�C þ1

nðpjþ1 2 pjÞ;

and the mean completion time of schedule S is

�C þ1

nðpj 2 pj21Þ:


Dow

nloa

ded

by [

Nov

a So

uthe

aste

rn U

nive

rsity

] at

14:

27 0

7 O

ctob

er 2

014

Then we have

Dj;jþ1 ¼ a ðCj 2 pj þ pjþ1Þ2 �C þ1

nðpjþ1 2 pjÞ

� �þ

ðn þ 1Þ

2m

� �2

2 2ajm Cj 2 pj þ pjþ1 2 �C þ1

nðpjþ1 2 pjÞ

� �þ

ðn þ 1Þ

2m

� �

2 a Cj 2 �C þðn þ 1Þ

2m

� �2

þ2ajm Cj 2 �C þðn þ 1Þ

2m

� �

¼ a Cj 2 �C þðn þ 1Þ

2mþ

ðn 2 1Þ

nðpjþ1 2 pjÞ

� �2

2 2ajm Cj 2 �C þðn þ 1Þ

2mþ

ðn 2 1Þ

nðpjþ1 2 pjÞ

� �

2 a Cj 2 �C þðn þ 1Þ

2m

� �2

þ2ajm Cj 2 �C þðn þ 1Þ

2m

� �

¼ aðn 2 1Þ2

n2ðpjþ1 2 pjÞ

2

þ 2ðn 2 1Þ

nðpjþ1 2 pjÞ Cj 2 �C þ

ðn þ 1Þ

2m

� �

2 2ajmðn 2 1Þ

nðpjþ1 2 pjÞ

¼aðn 2 1Þ

nðpjþ1 2 pjÞ

ðn 2 1Þ

nðpjþ1 2 pjÞ

�

þ 2ðCj 2 �CÞ þðn þ 1Þ

2m2 2jm

�:

Similarly,

Dj21;j ¼aðn 2 1Þ

nðpj 2 pj21Þ

ðn 2 1Þ

nðpj 2 pj21Þ

�

þ2ðCj21 2 �CÞ þðn þ 1Þ

2m2 2ðj 2 1Þm

�:

Now we compare Dj;jþ1 and Dj21;j: If Dj;jþ1 , 0; then jobs j and j þ 1 can be

interchanged with the objective function decreasing. This contradicts the

assumption that S is optimal.

Suppose Dj;jþ1 $ 0: Since pj 2 pjþ1 . 0; by the expression of Dj;jþ1;

ðn 2 1Þ

nðpjþ1 2 pjÞ þ 2ðCj 2 �CÞ þ

ðn þ 1Þ

2m2 2jm # 0: ð6Þ


Dow

nloa

ded

by [

Nov

a So

uthe

aste

rn U

nive

rsity

] at

14:

27 0

7 O

ctob

er 2

014

In this case, we will show that Dj21;j , 0: Suppose Dj21;j $ 0: Since pj21 2 pj ,

0; by the expression of Dj21;j;

ðn 2 1Þ

nðpj 2 pj21Þ þ 2ðCj21 2 �CÞ þ

ðn þ 1Þ

2m2 2ðj 2 1Þm $ 0: ð7Þ

Subtracting (7) from (6), we obtain

ðn 2 1Þ

nðpjþ1 þ pj21 2 2pjÞ þ 2pj 2 2m # 0:

So

ðpjþ1 þ pj21 2 2pjÞ þ 2pj 2 2mþ1

nð2pj 2 pjþ1 þ pj21Þ # 0:

Thus,

pjþ1 þ pj21 2 2mþ1

nð2pj 2 pjþ1 þ pj21Þ # 0:

This is a contradiction because pj . pjþ1; pj . pjþ1 and m , pj; j ¼ 1; 2; . . .; n:Thus, if Dj;jþ1 $ 0; then it must be Dj21;j , 0: Therefore, jobs j and j 2 1 can

be interchanged with the objective function decreasing. Again, this contradicts

the assumption that S is optimal. Therefore, the optimal sequence must be

V-shaped. A

6. REMARKS ON COMMON SLACK CASE

In this section, we consider the situation where the job due dates are

assigned using common slack methods, that is,

dj ¼ k þ pj; j ¼ 1; 2; . . .; n:

Then the objective function can be denoted by J(S, k ).

6.1. Objective Function Development

For Model 1 with continuous job compressions, J(S, k ) can be written as

JðS; kÞ ¼Xn

j¼1

aðdj 2 CjÞ þ ðaþ bÞ

Z Cj2dj

0

HjðxÞdx þ ajmþ bm

� �

¼Xn

j¼1

aðk þ pj 2 CjÞ þ ðaþ bÞ

Z Cj2k2pj

0


� �

¼Xn

j¼1

aðk 2 Cj21Þ þ ðaþ bÞ

Z Cj212k

0


� �:


Dow

nloa

ded

by [

Nov

a So

uthe

aste

rn U

nive

rsity

] at

14:

27 0

7 O

ctob

er 2

014

For Model 1 with discrete job compressions, J(S, k ) can be written as

JðS; kÞ ¼Xn

j¼1

24aðk 2 Cj21Þ þ ðaþ bÞ

XCj212k

l¼0

Cj21 2 k 2 lÞj

l

!plð1 2 pÞj2l þ ajp þ bp

!#:

For Model 2, J(S, k ) can be written as

JðS; kÞ ¼Xn

j¼1

ðaðCj 2 djÞ2 2 2aðCj 2 djÞjmþ aðjm þ jðj 2 1ÞmÞ þ bmÞ

¼Xn

j¼1

ðaðCj 2 k 2 pjÞ2 2 2aðCj 2 k 2 pjÞjm

þ aðjm þ jðj 2 1ÞmÞ þ bmÞ

¼Xn

j¼1

ðaðCj21 2 kÞ2 2 2aðCj21 2 kÞjmþ aðjm þ jðj 2 1ÞmÞ þ bmÞ:

6.2. Job Scheduling

Remark 6.1. The results on optimal V-shaped sequence for the common due-date

case continue to hold for the common-slack case, that is, for each model, there exists

a V-shaped sequence that minimizes its objective function. The proofs of these

results are similar to that of the common-due-date case, which we omitted here.

6.3. Due Date Assignment

Again, the results on due date assignment for the common due date case

still hold for the common slack case. We provide the results below without

proofs. The proofs are similar to the common due-date case.

Remark 6.2. For Model 1 with the continuous job compressions, for a fixed job

sequence, the optimal common slack k* must satisfy

Xn

j¼1

HjðCj21 2 k* Þ ¼na

aþ b:


Dow

nloa

ded

by [

Nov

a So

uthe

aste

rn U

nive

rsity

] at

14:

27 0

7 O

ctob

er 2

014

Remark 6.3. For Model 1 with the discrete job compressions, given a fixed job

sequence, the optimal common slack k* is the maximal integer such that

Xn

j¼1

XCj212k

l¼0

l

j

!plð1 2 pÞj2l #

na

aþ b:

Remark 6.4. For Model 2, for a fixed job sequence, the optimal common due

date k* is given by

k* ¼1

n

Xn

j¼1

Cj21 2ðn þ 1Þ

2m:

7. CONCLUDING REMARKS

In this work, we have considered the single-machine scheduling problem

with randomly compressible processing times. The objective is to find the

optimal job sequence or the optimal due dates to minimize the total penalty based

on the job earliness/tardiness and compressions. Common due-date-assignment

method and common slack-due-date-assignment method are used to assign due

dates to jobs. Under the assumption that the job compressions are independent

and identically distributed variables, it is shown that there exists V-shaped

optimal schedules for both constrained problems and the unconstrained

problems.

Further study includes the investigation of problems where the jobs are

assigned distinct due dates and/or the job compressions are not identically

distributed. In a recent paper [23], we treated random processing time with objective

function under expected earliness/tardiness costs. It is interesting to continue

further work in this direction and examine a combination of random processing

times and random compressible times. It is also interesting to obtain bounds in the

same spirit of [4].

ACKNOWLEDGMENTS

Research of G. Yin was supported in part by the National Science

Foundation under grant DMS-9877090. Research of J. R. Birge was supported in

part by the National Science Foundation under grant DMII-9523275.


Dow

nloa

ded

by [

Nov

a So

uthe

aste

rn U

nive

rsity

] at

14:

27 0

7 O

ctob

er 2

014

REFERENCES

1. Alidaee, B.; Ahmadian, A. Two Parallel Machine Sequencing Problems

Involving Controllable Job Processing Times. Eur. J. Oper. Res. 1993, 70,

335–341.

2. Al-Turki, U.M.; Mittenthal, J.; Raghavachari, M. More informative V-

Shaped Optimal Schedule for a Class of Single Machine Problems;

Technical Report No. 37-93-377, Department of Decision and Engineering

Systems, Rensselaer Polytechnic Institute: Troy, New York, 1993.

3. Baker, K.R.; Scudder, G.D. Sequencing with Earliness and Tardiness

Penalties: A Review. Oper. Res. 1990, 38, 22–36.

4. Birge, J.R.; Glazebrook, K.D. Bounds on Optimal Values in Stochastic

Scheduling; Technical Report TR96-2, Department of Industrial and

Operations Engineering, University of Michigan, 1996.

5. Birge, J.R.; Louveaux, F. Introduction to Stochastic Programming;

Springer-Verlag: New York, 1997.

6. Cai, X.; Tu, F.S. Scheduling Jobs with Random Processing Times on a

Single Machine Subject to Stochastic Breakdowns to Minimize Early-

Tardy Penalties. Naval Res. Logistics J. 1996, 43, 1127–1146.

7. Cheng, T.C.E. Optimal Assignment of Slack Due-dates and Sequencing of

Jobs with Random Processing Times on a Single Machine. Eur. J. Oper.

Res. 1991, 51, 348–353.

8. Cheng, T.C.E.; Chen, Z.L.; Li, C.L. Parallel-Machine Scheduling with

Controllable Processing Times. IIE Trans. 2002, In press.

9. Cheng, T.C.E.; Chen, Z.L.; Li, C.L.; Lin, B.M.-T. Scheduling to Minimize

the Total Compression and Late Costs. Naval Res. Logistics J. 1998, 45,

67–82.

10. Cheng, T.C.E.; Gupta, M. Survey of Scheduling Research Involving Due-

Date Determination Decisions. Eur. J. Oper. Res. 1989, 38, 156–166.

11. Cheng, T.C.E.; Janiak, A. Resource Optimal Control in Some Single-

Machine Scheduling Problems. IEEE Trans. Automatic Control 1994.

12. Cheng, T.C.E.; Oguz, C.; Qi, X.D. Due-Date Assignment and Single

Machine Scheduling with Compressible Processing Times. Int. J. Prod.

Econ. 1996, 43, 107–113.

13. Daniels, R.L.; Sarin, R.K. Single Machine Scheduling with Controllable

Processing Times and Number of Jobs Tardy. Oper. Res. 1989, 37,

981–984.

14. Eilon, S.; Chowdhury, I.G. Minimizing Waiting Time Variance in the

Single Machine Problem. Manage. Sci. 1977, 23, 567–575.

15. Elmaghraby, S.E. Activity Networks; Wiley: New York, 1977.

16. Graves, S.C. A Review of Production Scheduling. Oper. Res. 1981, 29,

646–675.


Dow

nloa

ded

by [

Nov

a So

uthe

aste

rn U

nive

rsity

] at

14:

27 0

7 O

ctob

er 2

014

17. Maddox, M.J.; Birge, J.R. Using Second Moment Information in Stochastic

Scheduling. In Recent Advances in Control and Optimization of

Manufacturing Systems; Yin, G., Zhang, Q., Eds.; Springer-Verlag:

London, 1996; 99–120.

18. Nowicki, E.; Zdrzalka, S. A Two-Machine Flow Shop Scheduling Problem

with Controllable Job Processing Times. Eur. J. Oper. Res. 1988, 34,

208–220.

19. Nowicki, E.; Zdrzalka, S. A Survey of Results for Sequencing Problems

with Controllable Processing Times. Discrete Appl. Math. 1990, 26,

271–287.

20. Panwalkar, S.S.; Rajagopalan, R. A Single Machine Sequencing Problem

with Controllable Processing Times. Eur. J. Oper. Res. 1992, 59, 298–302.

21. Panwalkar, S.S.; Smith, M.L.; Seidmann, A. Common Due-Date Assign-

ment to Minimize Total Penalty for the One Machine Scheduling Problem.

Oper. Res. 1982, 30, 391–399.

22. Papadimitrous, C.H.; Stieglitz, K. Combinatorial Optimization: Algorithms

and Complexity; Prentice-Hall, Englewood Cliffs: New Jersey, 1982.

23. Qi, X.D.; Yin, G.; Birge, J. Scheduling Problems with Random Processing

Times Under Expected Earliness/Tardiness Costs. Stoch. Anal. Appl. 2000,

18, 453–473.

24. Van Wassenhove, L.N.; Baker, K.R. A Bicriterion Approach to Time/Cost

Trade-Offs in Sequencing. Eur. J. Oper. Res. 1982, 11, 48–52.

25. Vickson, R.G. Two Single-Machine Sequencing Problems Involving

Controllable Job Processing Times. AIIE Trans. 1980, 12, 258–262.

26. Vickson, R.G. Choosing the Job Sequence and Processing Times to

Minimize Total Processing Plus Flow Cost on a Single Machine. Oper. Res.

1980, 29, 1155–1167.

27. Zdrzalka, S. Scheduling Jobs on a Single Machine with Release Dates,

Delivery Times and Controllable Processing Times: Worst-Case Analysis.

Oper. Res. Lett. 1991, 10, 519–524.

28. Yin, G.; Zhang, Q., (Eds.) Mathematics of Stochastic Manufacturing

Systems. Proc. 1996 AMS-SIAM Summer Seminar in Applied Mathematics,

Lectures in Applied Mathematics; Amer. Math. Soc.: Providence, RI, 1997.


Dow

nloa

ded

by [

Nov

a So

uthe

aste

rn U

nive

rsity

] at

14:

27 0

7 O

ctob

er 2

014