Open shop scheduling with maximal machines

DISCRETE APPLIED MATHEMATICS EISEVIER

Discrete Applied

Mathematics

78 (1997)

175--l 87

Open shop scheduling with maximal machines George

J. Kyparisis*,

Christos

Koulamas

Department of Decision Sciences and Information Systems, College of Business Administration, Florida International Universit,y, Miami, FL 33199, USA Received

11 June 1996; revised November

1996

Abstract The objective of this paper is to develop polynomial algorithms for the open shop makespan problem. It is shown that when a machine majorizes all other machines and the ith largest processing time on that machine is at least as large as the processing times of all operations on machines i through m, the problem becomes polynomially solvable. The utilization of the duality property between jobs and machines leads to a similar polynomial algorithm when a job majorizes all other jobs and the ith largest processing time of this job is at least as large as the processing times of all operations for jobs i through n. Kevwords:

Scheduling;

Open shop

1. Introduction

The open-shop minimum schedule length (makespan) problem is as follows: IZjobs Jl,Jl,. ,J, are processed on m machines Ml,Mz,. .,M,,,. Each job Ji has m operations, O,, with operation j of Ji processed on Mj for tij time units. The processing order of job operations is immaterial. Two operations of the same job cannot be processed simultaneously, nor can any machine process more than one job at a time. We assume that preemption is not allowed, i.e. any commenced operation has to be completed without interruptions. Open-shop schedules differ from flow-shop and job-shop schedules in that no restrictions are placed on the processing order for any job in the open shop. The objective is to find, for each machine, the schedule which obeys the constraints and minimizes the schedule length (makespan). Using the notation of Graham et al. [4], we can denote a scheduling problem by r/,0lr, where a specifies the operating environment; ,IY,restrictions in problem parameters; and y, the objective to be optimized. In this paper, ME {0,03, Om}, where 0 * Corresponding

author. E-mail: [email protected]

0166-218X/97/$17.00 Copyright PZI SOl66-218X(97)00018-8

0 1997 Elsevier

Science B.V. All rights reserved

176

G. J. Kyparisis,

denotes

C. Koulamas I Discrete

an open shop with an arbitrary

an open shop with exactly m (three) the problem

makespan.

maximal job, dominated They will be denoted

Restrictions machine,

Applied

Mathematics

number

machines, in problem

78 (1997)

175-187

of machines,

and Om (03)

and y = C,,,,

where C,,,

parameters

ordered machines,

include

maximal

denotes represents machine,

ordered jobs, and ordered problem.

by Ml 3 Ml, J1 2 Jr, A4,
Mj 2 Ml, Jj 3 Jl, ord. The

definitions are (see [6, I]): Machine Ml is maximal (majorizes all other machines) (A41 3 MI): An open shop has maximal machine Ml if til > til for i = 1,. . . , n, and 1 = 2,. . . , m. Job J1 is maximal (majorizes all other jobs) (Jl 2 JI): An open shop has maximal job J1 if tlj 3 tlj for 1=2,. . .,n, and j = 1,. . . ,m. Machine IV, is dominated by &I, (MY Ml): An open shop has ordered

machines

if tij 2 til for

i=l,..., n, and j,l=l,..., m, j Jk): An open shop has ordered jobs if for j = 1,. , m, and i,k=l,..., n, i
and ordered jobs. Notice that machine domination

is the strongest assumption

3

tkj

with respect to restricting

machine loads followed by ordered machines and then by machine maximal open shops. The general O//C,,, problem is strongly NP-hard [4]. Gonzalez and Sahni [3] showed is binary NP-hard; the complexity of 03//C,,, is open with respect that 03//C,,,,, to a unary encoding of the data. Gonzalez and Sahni [3] developed an elegant O(n) algorithm for 02//C,,,,, Since the Gonzalez and Sahni paper, numerous researchers attempted to identify Adiri and Aizikowitz [l] developed an O(n) polynomially solvable cases of O//C,,,. when one of the machines is dominated by any of the other algorithm for 03//C,,, two machines. Liu and Bulfin [6] developed an O(n) algorithm for 03/‘Mj 3 MI/C,,,,, (ordered machines) with the additional constraint that the job with the longest processing time on Ml is not the job with the longest processing time on M2. Fiala [2] used is solvable by an 0(n2m3) algorithm results from graph theory to prove that O//C,,, whenever n

m:xxtij

> (16m’log,m’+

5m’)mytij,

j=l

where m’ = 2n”sz ml. Sevast’janov algorithm whenever n

max i

c ( tij

j=l

2

1 m2 - 1 + m-l

[8] showed that O//C,,,

max >

is solvable

by an O(n2m2)

tij

i,j

by reducing the O//C,,, problem to the compact vector summation problem. Fiala and Sevast’janov showed that when the above condition holds C,,, = maxi ~~=, tij.

G.J. Kyparisis. C. Koulamasl Discrete Applied Mathematics 78 (1997)

Notice that the results of Fiala and Sevast’janov with the maximum

of the number

of machines.

111

require that the load on the machine

load exceed the overall maximum

which is a function

175-187

operation

time by a certain factor

Also, since

n n max t,, 3 max ITi

i

c

tij3

J=I

Sevast’janov’s formula implies that n >, m *. This shows that the polynomial solvability is a type of limiting property which holds when the number of jobs, n, is of OIL,, at least an order of magnitude greater than the number of machines, m. We close our literature survey by mentioning the survey paper of Kubiak et al. [5] on the complexity of open-shop problems. In this paper we propose O(n) algorithms for both the 03//C,,,,, and the general O//C,,, problems when certain conditions are imposed on the job processing times. Our algorithms require that n 3 m and that a machine majorizes all other machines (since we are dealing with an open-shop problem we can renumber the machines so the majorizing machine is defined as Mt ; equivalently, our problem is Mt -maximal). In addition, we require that the ith largest processing time on A41 is at least as large as the processing times of all operations on machines i through m. By observing a duality property between jobs and machines we obtain a similar O(m) algorithm when IZ < m under the assumption that a job majorizes all other jobs and the ith largest processing time of this job is at least as large as the processing times of all operations for jobs i through n. We also derive an O(n) algorithm for the 03//C,,, problem under slightly different assumptions. Specifically, we require ordered machines in the 03//C,,, case but at the same time we place less restrictive assumptions on the individual job processing times. The rest of the paper is organized as follows. Section 2 is devoted to the 03//C,,, case. Our algorithm for the general O//Cmas is presented in Section 3. The conclusions of this research

2. Scheduling

are summarized

in Section 4.

the three machine open shop

In this section we prove the following result which shows that the NP-hard problem 03//C,,,,, can be solved in polynomial time under certain conditions. Theorem 1. 03/1’v4 2 MI/C,,, f the jobs cun be arranged

( or d ere d machines)

can be solved in polynomiul

time

so that

Proof. We shall prove that a particular schedule is an optimal solution to 03/M, 2 under (Cl). It will be convenient to represent this schedule by a directed MIIC,,, graph as follows (see [7]). We define a vertex 0, with an associated weight t,, for

G.J. Kyparisis, C. Koulamasl Discrete Applied Mathematics 78 (1997)

178

175-187

(Ml) (‘442)

Fig. 1. Graph for an optimal schedule

for 03//C,,,,.

each operation Oij of job Ji on machine Mj. Also, we define arcs directed from a given vertex 0, towards specified other vertices of the graph (see Fig. 1). In addition, we define two special vertices, both with zero weight, the initial vertex 00 and the final vertex 0,. The graph is shown in Fig. 1 where each row corresponds to a different machine specified in parentheses. Given this graph, the completion time Cij of operation 0, of job Ji on machine Mj in the above schedule is equal to the maximum-weight directed path from 00 to Oij. Therefore, the time at which all jobs are completed is given by C,,, = C, where we define C, as the completion time of 0,. Any path from 0, to 0, which attains this maximum weight is called a critical path (see [7]). Observe that the optimal value of C,, for 03//C,,, is bounded below by

LB=max l Furthermore,

.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ i=l

i=l

for 03/n/j

LB = max

e

2 M//C,,

tii, 2

i=l

i=l

tij, 2

j=l

j=l

j=l

j=l

under assumption

tzj

i

(Cl)

,

j=l

since, by Mj 3 Ml, j < 1, one has ti2 < tii and tis < til) for all i, and, by (Cl ) and Mj 2 Ml, j < I, one has t31+ t32+ t33 d tl~ + tzl + t31. We shall prove in the sequel that any directed path ?rk from 00 to 0, has a weight W(q) which satisfies W(r&) < LB. This will imply that also C,, = maxk w(r&) < LB and since LB < C,,, we conclude that Cmax = LB = max

2 i=l

til, &

tlj, e

j=l

j=l

t2j

.

This will mean that the minimum makespan is achieved by either of the three feasible paths: the path composed of all operations Oil, or the path composed of all operations Oij, or the path composed of all operations Ozj, thus proving the theorem. By systematic inspection of the graph, we can identify several classes of directed paths, 71k, from 00 to 0, and compute their path lengths W(7ck). Using the convention

c

tij := 0

i=i,

if il >iz,

j = l,...,

3,

G. J. Kyparisis, C. Koulamas I Discrete Applied Mathematics 78 (I 997) 175-187

these path lengths

are computed

179

below as follows:

n ~(7~1) = C tit < LB, 1-I l-l

I

w(n2)c~t~l+~ti2<

Ctil+tl,+~f~]

r-2

w(n3) =

i=l

C

i=2

=ktl]


l=&...,n,

(1)

z-2

i=i

LB,

t2j d

(2) (3)

w(n6)

=

C

tlj

d

LB,

/=I

47c7)

=

t]2

+

t,3

+

==c

til d

t23

<

t12

+

t]3

+

t,,

<

(4)

LB,

LB,

(5)

i=l

I w(n9>

n

ES3

=

+

I-

Ctil

+ t]]

d

=

+

tl3

+

tll

d

i=3 n

t2] + Ctj]

+ t]] = Cti]

I=1

c


(6)

z=l n

&I +

t21 +

tll

=

i=3

GI d

c

LB,

(7)

i=l

n

w(nll)=Cti3< Ctil I=1

+

n

n ti3

C

n

i=3

n

w(7110)

Cti]

i=l

i=3

I


(8)

i=l

where (1) is implied by MI 3 A42 and by the inequality

t/l < tll ; (2) by MI > ~4~; (3)

by t12 < t21; (4) by t23 d t21 d tll; (5) by MI 2 A42 b M3, t13 d t/2 < t21, and t,.l d tll; (6) and (7) follow similarly to (5); and (8) follows from the inequality Mt >, Mj. Since the classes of paths zk listed above exhaust all possible paths through the network and they all satisfy the inequality w(zk) < LB, the theorem is proved in view of the earlier remarks. 0

180

G. J. Kyparisis,

Theorem

C. Koulamas I Discrete

1 generalizes

Theorem

tli = max tii,

(C3)

78 (1997)

time if the jobs can be arranged

so that

i=l,...,n

condition

tii = max ti,, i=l,...,n

to (C2) under which Theorem t12 = max ti2, i=

1 also holds is:

t12 < t21.

1,..., n

Observe also that the optimal schedule in the proof of Theorem of the schedules in the proof of Theorem 3 in [6].

3. Scheduling

175-187

t22 = max tj2.

i=l,...,n

A complementary

Mathematics

3 in Liu and Bulfin [6] who proved that 03/Mi >

MI/C,,,~~ can be solved in polynomial (C2)

Applied

1 corresponds

to one

the m machine open shop

In this section we prove our main theorem which shows that the NP-hard can be solved in polynomial time under certain conditions.

problem

OllCln,,

Theorem 2. O//Cmax with m < n can be solved in polynomial

time if the jobs can be

arranged so that (C4) til

2

Ml 3 Ml, i=l

tjk,

I = 2,. . . , m

(Ml -maximal)

,..., m, j=l,...,

n, i
In order to prove the theorem we first that condition (C4) states that machine of the ith operation on machine Mi, til, operations on machines Mi, . ,A4,. To holds one should order the machines max til 2 max ti2 3 . . 3 max i

i

and subsequently t,,

3

tz,

3

i

need to prove the following lemma. Observe A41 is maximal and that the processing time is at least as large as processing times of all maximize the likelihood that condition (C4)

so that tim

order the jobs so that ...

2

tm,.

Lemma 1. O//C,,,, where m = n, can be solved in polynomial arranged so that (C4) holds.

time if the jobs can be

Proof. We shall prove that a particular schedule is an optimal solution to O//C,,, under (C4). As in the proof of Theorem 1, we will represent this schedule by a directed graph. The graph is shown in Fig. 2 where each row corresponds to a different machine specified in parentheses. The order of operations on machines Mj is as follows: Ml : (021,. . , O,, ,011), : (012,. . . , Om2),Mk:(Okk,Ok+l,k,...,Omk,Olk,02k,...,0k-l.k), der of operations for jobs Ji is as follows : J1 : (0

k=$...,m.

M2

12, Ol,,

O,,m--1,.

The

Or-

. . ,013,011>, J2 : (021,

C J. Kyparisis. C. Koulamas I Discrete Applied Mathematics 78 (1997)

Fig. 2. Graph

022,02m,02,m-I,...,023),

for an optimal

schedule

for O//c‘,,,

in Lemma

53:(033,031,032,03m,03,m-1,....034),

Ok3,Okl,OkZ,Okm,Ok.m-l,...,Ok,kfl),

k==d,...,m

-

175-187

IRI

I (n-m).

Jk::okk,Ok.k-I...-.

1,

O,,,,m--l,. . , 0m3, O,,,,, 0,~). As in the proof of Theorem 1, the completion Jm: ~Qnim time C,i of operation

0,

is equal to the maximum-weight

directed

path from 00 to

O,,. Therefore, the time at which all jobs are completed is given by C,,, = C, C, is the completion time of 0,. Observe that the optimal value of C,,,,, for Ol/Cmax is bounded below by

LB =

max

j.k=l.__.,m

Furthermore,

{z

tii’ g

condition

since

MI > M,, j =

since

tk, <

t,l,

i, k =

LB = g tjl i=l

1,.

tki}’

(C4) implies

,m

and

1,. . . , m, so we obtain for O/lCmax under (C4) that

where

182

G.J. Kyparisis, C. KoulamaslDiscrete

Applied Mathematics 78 (1997)

175-187

We shall prove in the sequel that any directed path rtk from 0s to 0, w(zk) which satisfies W(71k) < LB. This will imply that also C,,, and since LB < C,,,

has a weight

= maxk W(zk) < LB

we conclude that

C,,=LB=ktil. i=l

This means that the minimum

makespan

is achieved by the feasible path composed

of

all operations Oil, thus proving the theorem. By systematic inspection of the graph, we can identify several classes of directed and compute their path lengths w(zk). We consider the paths, 7Lk, from 0s to 0, following

cases.

Case 1. Paths passing through operation 021. Any directed path ret from 00 to 0, passing through 021 is composed of operations on different machines in the following order (Mt,M~,M,,M,,_t ,..., Ms). A close observation of Fig. 2 indicates that the order results in a path ret composed of operations on the Wl,~Z,M?l,M-1,. . . ,M3) first s (1 < s d m) of these machines. The length of such a path ret, w(rc~ ), is given by

=

$

$ +$ ti2

til +

$

Cm +

km-l +

.. +

i=$, tiJs,

P

where we define 2 < It < 12 d ...

(h,..., k,)=(2,3 As before,

1, = ly,

l,=kl,...,

6 1,-t < ly,

,..., m,l),

s=3 ,..., m, jt=l,

j,=l,“““+l,

s= l,,..,

m,

jz=2.

CFzi, tv = 0 if il > i2, 1 < j < m, so that if

we adopt the convention

czs_,

km+l_s, ly=k,,,+l_s,

ti,js is the last term in the expression

w(rc1). Using condition

(C4)

we obtain I”“” w(nlIz~l,l+

2

i=2

+

tQ+

5

i=[l+l

tim+

i=il+l

t11,2 + ti,,, +

t19,m-l

+

5

ti,m_*+“‘+

2

i=I,+l

. . . +

ti,jf

i=l,_,+l

ti,_,,j~

pax &

hl i=2

+

t11,2

+

tiz,m

+

t13,m-l

+

. . . +

tl,_,,js

(9)

G. J. Kyparisis,

C. Koulamas I Discrete

Applied

Mathematics

78 (1997)

175-187

183

I”“’

c =c <

'il +

'II +

t,l

+

tm_l,l

+

.

+

tj~,,

(10)

1=2

m

(11)

t,l = LB,

IFI

where (9) follows from tii < tll; (10) from ti

1

3

tjk,

k 3 i; and ( 11) from j, = 11,’ + I,

s = 3,. . , m (the cases jr = 1, j2 = 2 are proved similarly). Cclse 2. Puths passing through operation

012. Any directed path 7~ from 00 to 0,

passing through 012 is composed of operations on different machines in the following order (M2,Mm,Mm_t, . . . ,M3,h4, ). A close observation of Fig. 2 indicates that the (M2,M,,,,M,,~-t,. . . ,M~,MI) order results in a path 7r2 composed of operations on the first s (1 < s d m) of these machines. The length of such a path 7~2, w(rc2), is given by I W(X2)

=

e

=

ti2

+

5

ti,

+

5

ti,m-l

+

”

+

‘2

tj3

(=I

1=1,

i=12

r-l,,_1

[I

lz

I3

in’“*

C

ti2

+

C

i=l

tim +

C

I=/,

tj,m-I

+

’

+ C

1’12

+

5

til

I= I,,

I

ti,j<,

I:/,-,

where we define 1 G 1, 612 < ... (kl,...,

k,)=(1,2

Using condition

W(7T2)=

dl,_,


1,=x- ,,...,

,..., m), j,=I~““+l,

k,+,_,s,

l~=k,+,_,,

s=2 ,..., m-

1, jr=2,

s=l,._.,

m,

j,=l.

(C4) we obtain

2

5 ti,+ 2

t,2 +

[=/,+I

i-l

t/,.m+ t/:,m-I

+

ti,m-1 + . .

i=l?+l

+

”

. +

+

fj

t,,,>

l=l,-,+I t/ _,,,>

I”‘”

&

fil +

t[,,, +

t,l +

'*I +

t/z,m-]

+

'

-t tl,_,,j5

(12)

!=I /:,,’

<

c

tm-l,l

+

'"

+

tj~,l

(13)

t=I

t,l = LB, where (12) follows from tij < t,l; (13) from til 3 tjk, k 3 i; and (14) from j, = ly s = 2,. . . , m - 1 (the cases jt = 2, j, = 1 are proved similarly).

(14) + 1,

G. J. Kyparisis, C. Koulamas I Discrete Applied Mathematics 78 (1997)

184

175-187

Case 3. Paths passing through operation 033, Any directed path I-Q from 00 to 0, passing through

03s is composed

(~3,~1,~2,&,~m-l

order

of operations on different machines in the following A close observation of Fig. 2 indicates that the order results in a path 7-c3composed of operations on

, . . . ,A&). . . ,I&)

(~3,~1,~2,&dfm-l,

the first s (1 d s < m) of these machines.

The length

of such a path 7r3, w(rca), is

given by

4713)

= 2

ti3

+

2

til

+

5

i=l,

i=3

tiz

+

2

id2

tim +

5

’ ’ f

ti4

I

id,_

iA3

where we define

3 < 11d I2 d ...
d IF,

l,=kl,...,

,..., m,1,2),

j,=Zy+l,

k,,,+,_$, l~=k,+l_,,s=1,..., ,..., m, j2=1,

s=1,4

m, j3=2.

(C4) we obtain Ima”

w(n3

) =

5

ti3 +

t[,,l

+

2

til

+

2

i=[l+l

i=3

ti2

+

2

i=[z+l

+ t[*,2 + ti,,*

+

"

tim +

..’

i=lj+l

' +

+

j_:

ti,jA

i=l,_l+l

t[,_,,j,

P"

61 + t!,,l

&

+ t12,2 + tls,m +

. . -k tlS_,,jS

(15)

i=3 P” s <

c i==3

til + tll

+

t21

+

tml

+

' . . +

tjs,]

(16)

m =

c

til

=LB,

(17)

i=l

where (15) follows from tij < til; (16) fromfir 2 tj,+,k 3 i; and(17) from jS=ZF”+l, s=1,4,..., m (the cases j2 = 1, j3 = 2 are proved similarly). Finally, similarly to Case 3 we prove the general case k for k = 4,. . . , m.

Case k. Paths passing through operation 0~. Any directed path z,k from 00 to 0, passing

through

Ok is composed

of operations

on different machines in the . . . ,Mk+l). A close observation order @fk&fk-1,. . .,M3,Ml,MwW,,,W,,-1, indicates that the (Mk,Mk_ 1,. . . , M3, Ml, M2, M,,,,M,,,- I,. . . , Mk+l ) order results xk composed of operations on the first s (1 < s < m) of these machines. The

following of Fig. 2 in a path length of

G.J. Kyparisis, C. Koulamas I Discrete Applied Mathematics 78 (1997)

185

175-187

such a path 7ck, W(zk), is given by

where we define k d

11 d

12 f

...

d

1,-l

d

(kl, . . ..k.)=(k,k+l,...,

Using condition

=

km+l-sr

m, j&,=1,

ly=k,+l_,,

s=

I,...,

m,

j,=ly+l, jk=2.

(C4) we obtain 11

w(nk)

l,=kl,...,

m,1,2 ,..., k-l),

k-2,k+l,...,

S=l,...,

IF,

[k--2

12 tik +

c

c

i=k

fi,k-I +

f

C

tim +

cc

lr-I tj3 $_

II.

c

61

$

r=li-2+1

c

t,2

i=lk_i+l

p'

“’

+

i=lk+l +tl,_,,l

c 1=/k-3+1

i=[l+l Ii+1

+

'.

c’

ti,j,

+ tl,,khl

f

.

. + t11_?,3

Gl,_,fl +

61 +

t11_,,2 +

tl~,k-1

+

tlk,m +

. . . + h_,,JT

+ tLL3

+ tL.1

+ Q_,,2

+

t/r,m +

”

+ t,,_,,,\

i=k

(18) PX

&

61 +

tk-l,l

+

i=k m =c

til=

LB,

i=l

where (18) follows from tq < til; (19) from til >, tjk,k > i; and (20) from j,=ly+l, s=l ,..,, k-2,k+l,..., m(thecasesjk_r=l,jk=2areprovedsimilarly). Since the classes of paths nk, k = 1,. . . , m, listed above exhaust all possible paths through the network and they all satisfy the inequality w(nk) < LB, the lemma is q proved in view of the earlier remarks.

186

G. J. Kyparisis,

The preceding

C. Koulamas I Discrete

lemma provides

Applied Mathematics

the foundation

78 (1997)

175-187

for the proof of Theorem

2.

Proof of Theorem 2. In order to prove the result we need to show that Lemma to the case where m < n. Define the operations

be extended as having

times equal to zero, i.e. tij = 0 for i = 1,. . . ,n, j = m + 1,. . . ,n.

processing

Observe that (C4) still holds for the enlarged case the result follows from Lemma

1.

so that in this

of Theorem

2 holds for O//Cmax with

1= 2,. . . , n, (J1-maximal) i
tliatkj,

with n machines

problem

0

It can be similarly proved that a counterpart m > n under a dual condition

JI 3 JI,

1 can

on the last n - m machines

For the problem

03//C,,,,

n, j=l,..., the following

m, ifman. corollary

follows directly from Theorem 2.

(M 1-maximal) can be solved in polynomial time if the jobs can be arranged so that Corollary

(C5)

2 MI/&,~

1. 03/M,

tll = max i=l,...,n

tjl,

t21

3

max

i=l,...,n

tj2,

t2i

3

max

i=l,...,n

ti3,

t3l

3

max

i=l,...,n

tj3.

Corollary 1 provides another condition under which 03//C,,, can be solved in polynomial time. Observe that, in contrast with Theorem 1, the assumption of ordered machines (Mj > Ml) is replaced by a less restrictive condition that machine Ml is maximal.

However,

condition

(Cl)

has to be strengthened

to (C5).

4. Conclusions We expanded the frontier of polynomially solvable O//C,,,,, problems by developing O(n) algorithms for machine maximal and/or job maximal O//C,, problems subject to some additional constraints on the job processing times. Our algorithms apply to problems with an arbitrary number of machines. Moreover, unlike Sevast’janov’s results, our results hold when the number of magnitude.

of jobs and machines

are of the same order

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