*Corresponding author. Tel.: 852-2766-5215; fax: 852-2356-2682.
E-mail address:[email protected] (T.C.E. Cheng)
Solvable cases of permutation
#owshop scheduling
with dominating machines
S. Xiang
!
, G. Tang
"
, T.C.E. Cheng
#,
*
!Department of Economics, Anhui College of Architectural Industry, Hefei, Anhui 230022, People's Republic of China
"Department of Management, Shanghai Second Polytechnic University, Shanghai 200041, People's Republic of China
#Ozce of the Vice-President (Research&Postgraduate Studies), The Hong Kong Polytechnic, University Hung Hom, Kowloon, Hong Kong Received 25 June 1998; accepted 22 July 1999
Abstract
In this paper we study the permutation#owshop scheduling problem with an increasing and decreasing series of dominating machines. The objective is to minimize one of the"ve regular performance criteria, namely, total weighted completion time, maximum lateness, maximum tardiness, number of tardy jobs and makespan. We establish that these
"ve cases are solvable by presenting a polymonial-time solution algorithm for each case. ( 2000 Elsevier Science B.V. All rights reserved.
Keywords: Scheduling; Permutation#owshop; Dominating machines
1. Introduction
It is common to observe that in many manufac-turing plants, the production facilities are arranged in series through which the jobs are processed. Often, the jobs have to be processed by each of the serial facilities in the same order. Such a production plant con"guration is referred to as a#owshop. If there are technological or other constraints that demand the jobs to go through the facilities in the same order, the environment is called a permuta-tion#owshop.
A permutation#owshop scheduling problem can be stated as follows. There arenjobsJ
1,J2,2,Jn
to be processed successively on m machines
M
1,M2,2,Mm in that order. Each job can be
processed on no more than one machine at any time, while each machine can handle only one job and the processing of a job may not be interrupted. Moreover, we assume that the same job order is chosen on each machine. Hence, the set of feasible solutions is given by the setSof all permutations of (1, 2,2,n).
We de"ne the following notation:
p
i,j "processing time of jobJj on machineMi;
d
j "due date of jobJj;
w
j "weight of jobJj;
C
j "completion time of jobmutation; Jj in a given
per-¸
.!9"ness of a given permutation;maxMCj!djDj"1,2,nN, maximum
late-¹
.!9"maxM0,¸.!9N, maximum tardiness of a given permutation;
C
c(s) "the objective function value for s"(s(1), s(2),2,s(n)).
In abbreviated notation, it is denoted asM
k'Mr.
De5nition 2. The machines form an increasing series of dominating machines (idm) i!
M
1(M2(2(Mm.
De5nition 3. The machines form a decreasing series of dominating machines (ddm) i!
M
1'M2'2'Mm.
De5nition 4. The machines form an increasing} de-creasing series of dominating machines (idm}ddm) i!
M
1(M2(2(Mh'2'Mm,
where 1)h)m.
De5nition 5. The machines form a decreasing} in-creasing series of dominating machines (ddm}idm) i!
M1'M2'2'M
h(2(Mm,
where 1)h)m.
De5nition 6. Jobs are processed at theearliest pos-sible time if any job is not allowed to wait between two successive machines unless its predecessor job has not"nished processing on the same machine.
For previous work on#owshop scheduling in an environment of a series of dominating machines,
the reader is referred to Refs. [1}4,6,7]. Using the three-"eld notation for problem classi"cation, a permutation #owshop with an increasing} de-creasing and a dede-creasing}increasing series of dominating machines to minimize a performance criterion c is denoted as FDidm}ddmDc and
Observation. For a given permutation s"(s(1), s(2),2,s(n)) for the problem FDidm}ddmDc, the
completion timeC
s(j) of job Js(j), if jobs are pro-cessed at the earliest possible time, is as follows:
C where the"rst item is only dependent on jobJ
s(1), the second term is the weighted completion time of all the jobs on machineM
hand is minimized by the
weighted shortest processing time (WSPT) order-ing, and the last term l0"+n
j/1ws(j)+mi/h`1pi,s(j)
"+n
j/1wj+mi/h`1pi,j is independent of
Algorithm 1.
Step 1: Construct a WSPT ordering s"(s(1), s(2),2,s(n)) of processing timesph,j,j"1,2,n, of
all the jobs on machineM
h. That is,
p
h,s(1)/ws(1))ph,s(2)/ws(2))2)ph,s(n)/ws(n). Step 2: Lets
j be thes(j) forward-shift permutation
ofs, i.e.
s
j"(s(j),s(1),2,s(j!1),s(j#1),2,s(n)).
Calculate its objective function valuec(s
j).
Step 3: Letc(s
jH)"minMc(s
j)Dj"1,2,nN. An
opti-mal solution is
s
jH"(s(jH),s(1),2,s(jH!1),s(jH#1),2,s(n)), i.e. thes
jHforward-shift permutation ofs.
Theorem 1. For the problem FDidm}ddmD+w
jCj,
Algorithm 1 generates an optimal solution.
Proof. For any feasible solution s@"(s@(1), s@(2),2,s@(n)), let s@(1)"s(k), where s(k) is the kth
element of s"(s(1),s(2),2,s(n)) obtained in Step
1 of Algorithm 1. From the Observation and using the method of adjacent pairwise interchange [8], we knowc(s@)*c(s
k)*c(sjH). h
Algorithm 1 requires O(n2) time to obtain an optimal solution since the computational complex-ity of WSPT is O(nlogn) and the computation of
c(s
jH) needs O(n2) time.
3. ProblemsFDidm}ddmDLmaxandFDidm}ddmDTmax
Note thatc(s)"¸
.!9or¹.!9in this case. For the problemFDidm}ddmD¸
.!9, Ho and Gupta [4] pre-sented an optimal solution algorithm. Algorithm 2 below solves the problem more simply.
Algorithm 2.
Step 1: Construct an earliest due date (EDD) order-ing s"(s(1),s(2),2,s(n)) of due dates dj,j"
1,2,n, of all the jobs. That is, ds(1))ds(2)
)2)d
s(n).
Step 2: Lets
j be thes(j) forward-shift permutation
ofs, i.e.
s
j"(s(j),s(1),2, ,s(j!1),s(j#1),2,s(n)).
Calculate its objective function valuec(s
j).
Step 3: Letc(s
jH)"minMc(s
j)Dj"1,2,nN. An
opti-mal solution is
s
jH"(s(jH),s(1),2,s(jH!1),s(jH#1),2,s(n)), i.e. thes(jH) forward-shift permutation ofs.
Theorem 2. For the problems FDidmD¸
.!9 and FDidmD¹
.!9, Algorithm 2 generates an optimal solu-tion.
Proof. For any feasible solution s@"(s@(1), s@(2),2,s@(n)), let s@(1)"s(k), where s(k) is the kth
element ofs"(s@(1),s@(2),2,s@(n)) in Step 1 of
Al-gorithm 2. From the observation (note h"m in this case) and using the method of adjacent pair-wise interchange, we knowc(s@)*c(s
k)*c(sjH). Similarly, Algorithm 2 requires O(n2) time to obtain an optimal solution since the computational complexity of EDD is O(nlogn) and the computa-tion ofc(s
jH) needs O(n2) time.
Algorithm 3 below generates an optimal solution to the problems FDidm}ddmD¸
.!9 and FDidm}ddmD¹
.!9.
Algorithm 3.
Step 1: Let d@j"d
j!+mk/h`1pk,j for j"1,2,n.
Construct an EDD orderings"(s(1),s(2),2,s(n))
of due datesd@
j,j"1,2,n, of all the jobs. That is,
d@
s(1))d@s(2))2)d@s(n). Step 2: Lets
j be the s(j) forward-shift permutation
ofs, i.e.
s
j"(s(j),s(1),2,s(j!1),s(j#1),2,s(n)).
Calculate its objective function valuec(s
j).
Step 3: Letc(s
jH)"minMc(s
j)Dj"1,2,nN. An
opti-mal solution is
s
Theorem 3. For the problemsFDidm}ddmD¸
.!9 and FDidm}ddmD¹
.!9, Algorithm 3 generates an optimal solution.
Proof. By the Observation, we know thatC3 s(j) is independent of permutation. The problems FDidm}ddmD¸
Theorem 2 accordingly. h
Algorithm 3 again requires O(n2) time to obtain an optimal solution.
4. ProblemFDidm}ddmD+Uj Note that c(s)"+n
j/1;j in this case. Using
Moore's algorithm [9] ntimes in Algorithm 3 to calculate objective function values, Algorithm 4 be-low generates an optimal solution for the problem FDidm}ddmD+;
j be thes(j) forward-shift permutation
ofs, i.e.
s
j"(s(j),s(1),2,s(j!1),s(j#1),2,s(n)).
Calculate its objective function valuec(s
j) resulting
from"xing jobJ
s(j) in the"rst position and using Moore's algorithm to calculate the minimum num-ber of tardy jobs for the remaining (n!1) jobs, together with jobJ
s(j) if it is tardy.
i.e. thes(jH) forward-shift permutation ofs.
Theorem 4. For the problem FDidm}ddmD+;
j,
Algorithm 4 generates an optimal solution.
Proof. Same as Theorem 3. h
Algorithm 4 also requires O(n2) time to obtain an optimal solution.
Since the second term is independent of permuta-tion,C.!9 is minimized by placing in the"rst posi-tion the job which minimizes the "rst term and placing in the last position the job which minimizes the third term. Algorithm 5 below generates an optimal solution for the problemFDidm}ddmDC
.!9.
Algorithm 5.
Step 1: Let job J
v and jobJt satisfy the following
Eqs. (3) and (4), respectively:
h~1
v"t, which is unique in satisfying Eqs. (3) and (4). Let jobJ
Step 3: An optimal solution is one in which jobs J
position, respectively, and the other jobs are placed between them in any order.
Theorem 5. For the problem FDidm}ddmDC .!9, Algorithm 5 generates an optimal solution.
Proof. The result follows from (2) directly. h
Algorithm 5 only needs O(n) time to obtain an optimal solution.
6. Conclusions
We have studied the permutation #owshop scheduling problem with an increasing and de-creasing series of dominating machines to minimize one of the "ve regular performance criteria. We have shown that these cases are solvable and pre-sented their solution algorithms.
Acknowledgements
G. Tang was partially supported by the National Natural Science Foundation of China. T.C.E. Cheng was partially supported by The Hong
Kong Polytechnic University under grant number G-S201.
References
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[3] I. Adiri, D. Pohoryles, Flowshop/no-idle or no-wait sched-uling to minimize the sum of completion times, Naval Research Logistics Quarterly 29 (1982) 495}504.
[4] J.N.D. Gupta, M. Ikram, Optimal schedules for two special structure#owshops, 1977, unpublished paper.
[5] J.C. Ho, J.N.D. Gupta, Flowshop scheduling with domi-nant machines, Computers and Operations Research 22 (2) (1995) 237}246.
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