Int J Adv Manuf Technol (2010) 50:789–792DOI 10.1007/s00170-010-2522-9

ORIGINAL ARTICLE

Notes on “minimizing the earliness/tardiness costson parallel machine with learning effects and deterioratingjobs: a mixed nonlinear integer programming approach”

Yunqiang Yin · Dehua Xu

Received: 27 October 2009 / Accepted: 4 January 2010 / Published online: 9 February 2010© Springer-Verlag London Limited 2010

Abstract The aim of this paper is to point out thatthe mixed nonlinear integer programming model pro-posed by Toksarı and Guner (Int J Adv Manuf Technol38:801–808, 2008) is incorrect. We present a mixed non-linear 0–1 programming model for the same schedulingproblem based on their model.

Keywords Scheduling · Learning effect ·Deterioration jobs · Common due date

1 Introduction

As we observe, the model proposed in Toksarı andGuner [1] is incorrect. The aim of this note is to pointout the errors in the model and to provide a modifiedmodel for the problem.

We shall follow the notations and terminologiesgiven in Toksarı and Guner [1]. There are n jobs to bescheduled on m parallel machines. Each machine canhandle at most one job at a time. If job i is scheduledin position r in a sequence on a machine, its actualprocessing time pr is defined as:

pr = [pr + (α × tr)

]ra,

where pr is the basic processing time of job i, α (α > 0)

is deterioration effect, which is the amount of increase

Y. Yin · D. Xu (B)School of Mathematics and Information Sciences,East China Institute of Technology, Fuzhou,Jiangxi 344000, Chinae-mail: schedulingresearch@gmail.com

Y. Yine-mail: yinyunqiang@126.com

in the processing time of a job per unit delay in itsstarting time, a (a ≤ 0) is the learning index, and tr isthe starting time of the job scheduled in position r.

Toksarı and Güner [1] proposed the following mixednonlinear integer programming model for the consid-ered scheduling problem.

Objective function:

minn∑

r=1

m∑

j=1

(

T jr

(n∑

i=1

ti X jir

)

+ E jr

(n∑

i=1

ei X jir

))

Subjective to:

p jr =n∑

i=1

(

pi +r−1∑

k=1

(

α

(n∑

v=1

pv X jvr

)

ka

×r−1∏

z=k+1

(1 + αza)

))

ra X jir

( j = 1, · · · , M, r = 1, · · · , N) (1)

C j1 = t j + p j1, ( j = 1, · · · , N) (2)

C jr = C j,r−1 +n∑

I=1

n∑

i=1

sIi X jIr X ji,r+1 + p jr(I �= i),

(r = 2, · · · , N; j = 1, · · · , M) (3)m∑

j=1

n∑

r=1

X jir = 1, (i = 1, · · · , N) (4)

X jIr +n∑

i=1

X ji,r+1 ≤ 2 (I �= i),

(I = 1, · · · , N; r = 1, · · · , N − 1; j = 1, · · · , M)

(5)

790 Int J Adv Manuf Technol (2010) 50:789–792

X jI,r+1 ≤n∑

i=1

X jir (I �= i),

(I = 1, · · · , N; j = 1, · · · , M; r = 1, · · · , N − 1) (6)

C jr

n∑

i=1

X jir − T jr + E jr = dn∑

i=1

X jir,

(r = 1, · · · , N; j = 1, · · · , M) (7)

P jr, C jr, T jr, E jr, t j ≥ 0,

(i = 1, · · · , N; j = 1, · · · , M; r = 1, · · · , N) (8)

X jir{0, 1} (9)

The meanings of the variables (see [1]) in the modelare as follows:

M number of machines availableN number of jobs to be scheduledd common due date for all jobs

T jr length of time job scheduled in position r onmachine j is tardy

E jr length of time job scheduled in position r onmachine j is early

ti tardiness cost per period for job iei earliness cost per period for job i

p jr actual processing time of job scheduled in posi-tion r on machine j

C jr actual complete time of job scheduled in positionr on machine j

pi basic processing time of job iα deterioration effecta learning effectt j starting time of the first scheduled job on ma-

chine jX jir 1 if job i is done in position r on machine j, 0

otherwisesIi setup time for job I when it immediately follows

job i

2 Comments

In this section, we point out some errors in the abovemodel. Before proceeding, we first provide a lemma.

Lemma 1 For a given schedule with n jobs, let sr,r−1(r =2, · · · , n) be the setup time of the job in position r whenit immediately follows the job in position r − 1 in theprocessing sequence and s1,0 = 0. If the f irst job starts

at time t0 ≥ 0, then the actual processing time and thecomplete time of the job scheduled in the position r are

pr =(

pr +r−1∑

k=1

(

αpkkar−1∏

z=k+1

(1 + αza)

)

+r∑

k=1

(

sk,k−1α

r−1∏

z=k

(1 + αza)

)

+t0αr−1∏

k=1

(1+αka)

)

ra

and

Cr = prra +r−1∑

k=1

(

pkkar∏

z=k+1

(1 + αza)

)

+r∑

k=1

(

sk,k−1

r∏

z=k

(1 + αza)

)

+ t0r∏

k=1

(1 + αka),

respectively, wherej∏

k=i(1 + αza) = 1 if i < j.

Proof (by induction)

p1 = (p1 + αt0)1a,

C1 = p1 + t0 = (p1 + t0(1 + α1a))1a

p2 = (p2 + α(C1 + s2,1))2a

= (p2 + αp1 + t0α(1 + α1a) + αs2,1)2a,

C2 = C1 + s2,1 + p2 = C1 + s2,1 + (p2 + α(C1 + s2,1))2a

= p22a + (1 + α2a)C1 + (1 + α2a)s2,1

= p22a + p1(1 + α2a) + t0(1 + α1a)(1 + α2a)

+ (1 + α2a)s2,1.

Suppose Lemma 1 holds for r = i, i.e.,

pi =(

pi +i−1∑

k=1

(

αpkkai−1∏

z=k+1

(1 + αza)

)

+i∑

k=1

(

sk,k−1α

i−1∏

z=k

(1 + αza)

)

+t0αi−1∏

k=1

(1+αka)

)

ia

and

Ci = piia +i−1∑

k=1

(

pkkai∏

z=k+1

(1 + αza)

)

+i∑

k=1

(

sk,k−1

i∏

z=k

(1 + αza)

)

+ t0i∏

k=1

(1 + αka).

Int J Adv Manuf Technol (2010) 50:789–792 791

Then

pi+1 = (pi+1 + α(Ci + si+1,i))(i + 1)a

=(

pi+1+αpiia+αsi+1,i+i−1∑

k=1

(

αpkkai∏

z=k+1

(1+αza)

)

+i∑

k=1

(

sk,k−1α

i∏

z=k

(1 + αza)

)

+ t0αi∏

k=1

(1 + αka)

)

(i + 1)a

=(

pi+1 +i∑

k=1

(

αpkkai∏

z=k+1

(1 + αza)

)

+i+1∑

k=1

(

sk,k−1α

i∏

z=k

(1 + αza)

)

+ t0αi∏

k=1

(1 + αka)

)

(i + 1)a

and

Ci+1 = Ci + si+1,i + pi+1

= Ci + si+1,i + (pi+1 + α(Ci + si+1,i))(i + 1)a

= pi+1(i + 1)a + (1 + α(i + 1)a)Ci

+ (1 + α(i + 1)a)si+1,i

= pi+1(i + 1)a +i∑

k=1

(

pkkai+1∏

z=k+1

(1 + αza)

)

+i+1∑

k=1

(

sk,k−1

i+1∏

z=k

(1 + αza)

)

+ t0i+1∏

k=1

(1 + αka).

This completes the proof. ��

What follows are some comments on the above pro-gramming model.

Comment 1 Note that in the notations part ofSection 2 in Toksarı and Guner [1], the authors denotedby ti and t j the tardiness cost per period for job i and thestarting time of the first scheduled job on machine j,respectively, which may cause confusion. In the sequel,we denote by τ j the starting time of the first scheduledjob on machine j.

Comment 2 Lemma 1 indicates that constraint set(1) in the model is incorrect, the authors ignored thestarting time of the sequence in machine j and the setup

times of the jobs. Constraint set (1) should be changedto

p j1 =N∑

i=1

(pi + ατ j)X ji1 ( j = 1, · · · , M)

p jr =N∑

i=1

⎛

⎜⎝pi+

r−1∑

k=1

(

α

(N∑

v=1

pv X jvk

)

kar−1∏

z=k+1

(1 + αza)

)

+r∑

k=1

⎛

⎝

⎛

⎝N∑

i=1

N∑

k=1,k�=i

sik X jir X jk,r−1

⎞

⎠α

r−1∏

z=k

(1+αza)

⎞

⎠

+ τ jα

r−1∏

k=1

(1 + αka)

⎞

⎟⎠ra X jir,

( j = 1, · · · , M, r = 2, · · · , N)

or

p j1 =N∑

i=1

(pi + ατ j)X ji1 ( j = 1, · · · , M)

p jr =N∑

i=1

⎛

⎝pi+α

⎛

⎝C j,r−1+N∑

I=1

N∑

i=1,i �=I

sIi X jIr X ji,r−1

⎞

⎠

⎞

⎠raX jir,

(r = 2, · · · , N; j = 1, · · · , M)

Comment 3 Note that C jr = C j,r−1 + sr,r−1 + p jr (r =2, · · · , N; j = 1, · · · , M). Hence constraint set (3) inthe model is incorrect and should be changed as

C jr = C j,r−1 +n∑

I=1

n∑

i=1,i �=I

sIi X jIr X ji,r−1 + p jr,

(r = 2, · · · , N; j = 1, · · · , M)

Comment 4 One may have noted that the constraintsin the model are insufficient. We observe that morethan one job may be processed in the first position ofthe processing sequence on one machine according tothe schedule produced by the model. For example, con-sider a six-job problem with the same basic processingtime 2 and a common due date d = 2 on two machines1 and 2. It can be checked that the objective valueproduced by the model is 0, which implies that each ofthe six jobs is the first job processed by machine 1 or 2.This is impossible since that a machine cannot processmore than one job at a time. The missing constraint setof the model is as follows:

N∑

i=1

X jir ≤ 1, (r = 1, · · · , N; j = 1, · · · , M),

which ensures that at most one job can be assigned toeach position of a machine.

792 Int J Adv Manuf Technol (2010) 50:789–792

The modified mixed nonlinear 0–1 programmingmodel for the scheduling problem under considerationis as follows:

Objective function:

minN∑

r=1

M∑

j=1

(

T jr

(N∑

i=1

ti X jir

)

+ E jr

(M∑

i=1

ei X jir

))

Subjective to:

p j1 =N∑

i=1

(pi + ατ j)X ji1 ( j = 1, · · · , M) (10)

p jr =N∑

i=1

⎛

⎝pi+α

⎛

⎝C j,r−1+N∑

I=1

N∑

i=1,i �=I

sIi X jIr X ji,r−1

⎞

⎠

⎞

⎠raX jir,

(r = 2, · · · , N; j = 1, · · · , M) (11)

C j1 = τ j + p j1, ( j = 1, · · · , N) (12)

C jr = C j,r−1 +N∑

I=1

N∑

i=1,i �=I

sIi X jIr X ji,r−1 + p jr,

(r = 2, · · · , N; j = 1, · · · , M) (13)

M∑

j=1

N∑

r=1

X jir = 1, (i = 1, · · · , N) (14)

N∑

i=1

X jir ≤ 1, (r = 1, · · · , N; j = 1, · · · , M) (15)

X jIr +N∑

i=1,i �=I

X ji,r+1 ≤ 2,

(I = 1, · · · , N; r = 1, · · · , N − 1; j = 1, · · · , M)

(16)

X jI,r+1 ≤N∑

i=1,i �=I

X jir,

(I = 1, · · · , N; j = 1, · · · , M; r = 1, · · · , N − 1)

(17)

C jr

N∑

i=1

X jir − T jr + E jr = dN∑

i=1

X jir,

(r = 1, · · · , N; j = 1, · · · , M) (18)

p jr, C jr, T jr, E jr, t j ≥ 0,

(i = 1, · · · , N; j = 1, · · · , M; r = 1, · · · , N) (19)

X jir ∈ {0, 1},(i = 1, · · · , N; j = 1, · · · , M; r = 1, · · · , N) (20)

Reference

1. Toksarı MT, Guner E (2008) Minimizing the earliness/tardiness costs on parallel machine with learning effects anddeteriorating jobs: a mixed nonlinear integer programmingapproach. Int J Adv Manuf Technol 38:801–808