One-operator–two-machine flowshop scheduling with setup and dismounting times

*Corresponding author. Tel.: 00852 2766 5216; fax: 00852 2356 2682; e-mail: [email protected].

Computers & Operations Research 26 (1999) 715—730

One-operator—two-machine flowshop scheduling with setupand dismounting times

T.C. Edwin Cheng!,*, Guoqing Wang", Chelliah Sriskandarajah#

!Office of the Vice-President (Research and Postgraduate Studies), The Hong Kong Polytechnic University,Kowloon, Hong Kong

"Department of Business Administration, Jinan University, Guangzhou, People+s Republic of China#Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario, Canada

Received November 1997; received in revised form June 1998

Abstract

In this paper we study the problem of scheduling n jobs in a one-operator—two-machine flowshop. In sucha flowshop, before a machine begins processing a job, the operator has to set up the machine, and then themachine can process the job on its own. After a machine finishes processing a job, the operator needs toperform a dismounting operation before setting up the machine for another job. The setup and dismountingoperations are either separable or nonseparable. The objective is to minimize the makespan. Confining ourstudy to cyclic-movement schedules which require the operator to move between the two machines accordingto some cyclic pattern, we show that both the cyclic-movement separable and nonseparable setup anddismounting problems are NP-complete in the strong sense. We then propose some heuristics and analyzetheir worst-case error bounds. ( 1999 Elsevier Science Ltd. All rights reserved.

Scope and purpose

The one-worker—multiple-machine (OWMM) concept is widely applied in just-in-time (JIT) manufactur-ing systems. In an OWMM system, a worker tends several machines simultaneously where each of themachines performs a different operation, and the different machines make up a flow line. In this paper, westudy a scheduling problem in the one-operator—two-machine system where the operator moves between twomachines according to some cyclic pattern, and performs the setup and dismounting operations on bothmachines. Both separable and nonseparable setup and dismounting problems are considered.

Keywords: One-operator—two-machine; Flowshop scheduling; Complexity; Algorithms

0305-0548/99/$— see front matter ( 1999 Elsevier Science Ltd. All rights reserved.PII: S 0 3 0 5 - 0 5 4 8 ( 9 8 ) 0 0 0 8 7 - 2

1. Introduction

In this paper, we study a scheduling problem in a one-operator—two-machine flowshop whichcan be stated as follows: there are n jobs to be processed by a two-machine flowshop system whichis tended by one operator. Each job must be processed first on machine 1 and then on machine 2.All jobs are available for processing on the first machine at time zero. Before a machine beginsprocessing a job, the operator has to set up the machine for the job, then the machine can processthe job automatically. After a machine finishes processing a job, the operator needs to dismount thejigs and fixtures on it before setting it up for another job. The setup, processing and dismountingoperations are non-preemptive and sequence independent. For simplicity, we assume that thetransportation times and the walking times of the operator between the machines are negligible.The objective is to schedule the jobs such that the maximum job completion time, i.e., themakespan, is minimized.

There are two types of problems associated with the setup and dismounting operations, namelythe separable setup and dismounting problem and the nonseparable setup and dismountingproblem. For the separable setup and dismounting problem, the setup and dismounting operationsof a job can be performed without the involvement of the job, such as setting up jigs and fixtures ona machine, and dismounting them from a machine. Thus, the setup operation of a job on machine 2can be performed in anticipation of its arrival from machine 1, and so the job is immediatelyavailable for processing on machine 2 upon finishing its processing operation on machine 1. Forthe nonseparable setup and dismounting problem, all setup and dismounting operations can onlybe performed with the involvement of the job, such as job loading and unloading. Thus, the setupoperation on machine 2 of a job can only be started after its dismounting is finished on machine 1.It should be pointed out that some OWMM environment may involve both separableand nonseparable setup and dismounting operations. However, since the nonseparable setupand dismounting times are typically much smaller than the separable setup and dismountingtimes, we may assume that the nonseparable setup and dismounting times are negligible in thissituation.

The general case of our problem is to sequence the jobs and schedule the operator movesbetween the machines concurrently. But for practical reasons and theoretical convenience, we willonly consider cyclic-movement schedules in this paper. A cyclic-movement schedule requires theoperator to move between the two machines according to some cyclic pattern. With this restriction,we can identify the relevant cyclic-movement schedules for each problem as follows:

¹he cyclic-movement separable setup and dismounting problem. In a regular cycle (as shown inFig. 1a), the operator first performs the dismounting operation of job J

iand the setup operation

of job Ji`1

on machine 1, then performs the same operations on machine 2. In the initial cycle (asshown in Fig. 1b), the operator only performs the setup operations on both machines for the firstjob, and in the final cycle (as shown in Fig. 1c), the operator only performs the dismountingoperations on both machines for the last job.

¹he cyclic-movement nonseparable setup and dismounting problem. In a regular cycle (as shown inFig. 2a), the operator performs the dismounting peration of job J

iand the setup operation of job

Ji`1

on machine 1, then performs the dismounting operation of job Ji~1

and the setup operationof J

ion machine 2. The first and the last cycles are shown in Fig. 2b and c, respectively.

716 T.C.E. Cheng et al. / Computers & Operations Research 26 (1999) 715—730

Fig. 1. A cyclic-movement schedule for the separable setup and dismounting problem: \ — setup, K — processing,Y — dismounting.

Fig. 2. A cyclic-movement schedule for the nonseparable setup and dismounting problem: \ — setup, K — processing,Y — dismounting.

It is clear that the cyclic-movement schedules may be sub-optimal. However, apart from yieldinganalytical simplicity, the cyclic movement assumption is also reasonable in the real-world due tothe following two reasons:

(i) A natural way of scheduling jobs in a one-operator—two-machine flowshop is to follow thecyclic movement as the operator and the machine operations tend to overlap, thus the cyclicmovement will reduce the interference between the operator and machine operations.

(ii) It is evident that, for the cyclic-movement separable setup and dismounting problem, there isno intermediate storage between the two machines and, for the cyclic-movement nonsepar-able setup and dismounting problem, there are at most two jobs waiting before machine 2 ata time. These observations are in line with the concept of just-in-time manufacturing wherebuffer inventories are strictly restricted [1].

The model under study is indeed motivated by the one-worker—multiple-machine (OWMM)concept, which is widely applied in just-in-time (JIT) manufacturing systems [2]. In an OWMMsystem, a worker tends several machines simultaneously where each of the machines performsa different operation. Although it is common to have a worker tend several identical machinessimultaneously, the different machines in an OWMM system make up a flow line. In such a system,the machine layout is organized in a logical sequence according to the order of the operations to beperformed on the jobs. The worker tends and rotates from machine to machine upon completion ofeach operation. Thus, the worker follows a cyclic movement. Such a system configuration offers thebenefit of reducing labor requirements as well as work-in-process inventory.

T.C.E. Cheng et al. / Computers & Operations Research 26 (1999) 715—730 717

Another area of application of the models under study is the scheduling of robotic cells in whicha robot tending several machines uses double grippers, one to remove a finished part froma machine while the other to load another part. Such robotic cells have long been used inmanufacturing [3]. Since the robot can carry two parts at a time, it usually follows the cyclicmovement. In fact, Su and Chen [4] proved the optimality of the cyclic movement for thecyclic-movement nonseparable setup and dismounting problem when all jobs are identical, andprovided some analytical results showing the effectiveness of a double-gripper robot cell overa single-gripper robot. They also noted the lack of literature on the scheduling of robotic cellswhere a robot uses double grippers.

Due to their theoretical challenges and practical significance, many different versions of theflowshop makespan problem have received considerable attention from researchers for decades.For general information on the flowshop scheduling problems, the reader is referred to the works ofGraham et al. [5], Dudek et al. [6], Lawler et al. [7], Hall and Sriskandarajah [8]. Particularly, wenote that much attention has recently been devoted to similar problems in which an m-machineflowshop is served with a single intermediate agent (e.g., an AGV or a robot) which performscertain operations such as transportation, loading and unloading. These problems have beenanalyzed under two scenarios: (i) in single part type production — finding the best policy of the agentmoves [9, 10], and (ii) in multi-product type production — for a fixed sequence of agent moves,finding the best processing sequence for the parts [11—14].

This paper deals with problems in the second scenario. But there are two major differencesbetween our problems and most of the problems studied under this scenario: (i) Most studiedproblems only deal with the nonseparable setup and dismounting (load and unload) operations.The only exception is the problem studied by Levner et al. [15]. Although the separable setupoperations are considered in [15], the operations are not performed by the intermediate agent intheir model: (ii) Most studied problems implicitly assume that the robot uses a single gripper,thus the unload and load operations for adjacent parts on a machine have to be performedseparately. The problem studied by Agnetis et al. [16] is an exception. In fact, the dynamics of thesystem in [16] is actually the same as the cyclic-movement nonseparable setup and dismountingproblem considered in this paper. But their study only concerns the situation in which the parts areproduced in lots, where each lot consists of several identical parts to be produced consecutively.

The rest of the paper is organized as follows. In the next section, we will introduce the notationto be used. Then the cyclic-movement separable setup and dismounting problem and thecyclic-movement nonseparable setup and dismounting problem are considered in Sections 3 and 4,respectively. Finally, we present our conclusions in the last section.

2. Notation

To avoid confusion, we will first make a distinction between the terms schedule and sequence asfollows:

A sequence is a complete ordering of the set of jobs, whereas a schedule is a sequence togetherwith the starting time arrangements for the jobs.

Now, we introduce the notation to be used throughout this paper.


2.1. Notation

J"MJ1,2, J

nN: a job set

siand s@

i: setup times of J

ion machines 1 and 2, respectively

riand r@

i: dismounting times on machines 1 and 2, respectively

piand q

i: processing times of J

ion machines 1 and 2, respectively

p"SJ*1+

,2 , J*n+

T: a job sequence, where the subscript [i] denotes the ith entity in the sequencen: a scheduleC.!9

(n): makespan of the schedule n.

Adopting the three-field notation proposed by Graham et al. [5] to describe schedulingproblems, we denote the general problem as F2(O1)/cyclic/C

.!9. The cyclic-movement separable

setup and dismounting problem and the cyclic-movement nonseparable setup and dismountingproblem will be denoted as F2(O1)/cyclic, ssd/C

.!9and F2(O1)/cyclic, nsd/C

.!9, respectively.

3. Cyclic-movement separable setup and dismounting problem

We consider F2(O1)/cyclic, ssd/C.!9

in this section. First, we formulate the problem to minimizethe makespan.

Given a feasible schedule n with the job sequence p"SJ*1+

,2 , J*n+

T for F2(O1)/cyclic, ssd/C.!9

,let ¹

*i+, i"0, 1,2 , n, be the duration between the start time of the dismounting operations of

J*i+

and J*i`1+

on machine 1 (as shown in Fig. 1). Let

¹*0+"maxMs

*1+#p

*1+, s

*1+#s@

*1+N,

¹*n+"maxMr

*n+#r@

*n+, q

*n+#r@

*n+N.

The values of ¹*i+

, i"1,2 , n!1, can be easily computed as follows:

¹*i+"maxMr

*i+#s

*i`1+#p

*i`1+, q

*i+#r@

*i+#s@

*i`1+, r

*i+#s

*i`1+#r@

*i+#s@

*i`1+N.

Then, we have

C.!9

(n)"n+i/0

¹*i+

"maxMs*1+#p

*1+, s

*1+#s@

*1+N#maxMr

*n+#r@

*n+, q

*n+#r@

*n+N

#

n~1+i/1

maxMr*i+#s

*i`1+#p

*i`1+, q

*i+#r@

*i+#s@

*i`1+, r

*i+#s

*i`1+#r@

*i+#s@

*i`1+N

"

n+i/1

(ri#r@

i#s

i#s@

i)#maxMp

*1+!s@

*1+, 0N#maxM0, q

*n+!r

*n+N

#

n~1+i/1

maxMp*i`1+

!r@*i+!s@

*i`1+, q

*i+!r

*i+!s

*i`1+, 0N. (1)


With this formulation, we immediately obtain two important results. First, we establish thecomputational complexity for the problem with the following theorem.

Theorem 1. ¹he problem F2(O1)/cyclic, ssd/C.!9

is NP-complete in the strong sense.

Proof. This is achieved by a reduction from the classical three-machine no-wait schedulingproblem, denoted as F3/no-wait/C

.!9, which has been shown to be NP-complete in the strong

senses by Rock [17]. For an instance on the F3/no-wait/C.!9

problem with the job setJ@"MJ@

1,2 , J@

nN, let x

i, y

iand z

idenote the processing times of J@

ion the first, second and third

machine, respectively. Then the makespan of the schedule n with the job sequencep"SJ@

*1+,2 , J@

*n+T, Z(n), can be written (see [8]) as

Z(n)"n+i/1

yi#x

*1+#z

*n+#

n~1+i/1

maxMx*i`1+

!y*i+

, z*i+!y

*i`1+, 0N.

To construct an instance for F2(O1)/cyclic, ssd/C.!9

, we can set pi"x

i, s

i"r@

i"y

i, s@

i"r

i"0

and qi"z

i. It is immediately clear that minimizing Z(n) is equivalent to minimizing C

.!9(n). j

Next, we identify two polynomially solvable cases using Theorem 2 below.

Theorem 2. ¼hen si"s or r@

i"r, i"1,2, n, F2(O1)/cyclic, ssd/C

.!9can be solved in O(n2 ) time.

Proof. When si"s, i"1,2 , n, from Eq. (1), we have

C.!9

(n)"n+i/1

(ri#r@

i#s

i#s@

i)#maxMp

*1+!s@

*1+, 0N#maxM0, q

*n+!r

*n+N

#

n~1+i/1

maxMp*i`1+

!r@*i+!s@

*i`1+, q

*i+!r

*i+!s

*i`1+, 0N

"

n+i/1

(ri#s#s@

i)#maxMp

*1+!s@

*1+, 0N#maxMr@

*n+, q

*n+!r

*n+#r@

*n+N

#

n~1+i/1

maxMp*i`1+

!s@*i`1+

, q*i+!r

*i+#r@

*i+!s, r@

*i+N. (2)

When the last job is fixed to be Jl, i.e., J

*n+"J

l, l"1,2, n, this special case reduces to the

F2/no-wait/C.!9

problem with the job set J@"MJ@1,2, J@

nN such that

p@i"maxMp

i!s@

i, 0N, i"1,2 , n,

q@l"0,

q@i"maxMq

i!r

i!s#r@

i, r@

iN, i"1,2 , n, iOl.

As F2/no-wait/C.!9

is solvable by the Gilmore and Gomory algorithm in O(n log n) time [8, 18],one evident way to solve the special case is to obtain an optimal solution n*

lwith J

l, l"1,2 , n, as


the last job by solving the corresponding no-wait problem, and then choose the optimal schedulen* such that C

.!9(n*)"minMC

.!9(n*

l) D l"1,2, nN. This straightforward scheme solves the case

in O(n2 log n) time. In fact, this O(n2 log n) scheme may be modified to be performed in essentiallyquadratic time according to Levner et al. [16].

Similarly, when r@i"r@, i"1,2 , n, we have

C.!9

(n)"n+i/1

(ri#r@

i#s

i#s@

i)#maxMp

*1+!s@

*1+, 0N#maxM0, q

*n+!r

*n+N

#

n~1+i/1

maxMp*i`1+

!r@*i`1+

!s@*i`1+

, q*i+!r

*i+!s

*i`1+, 0N

"

n+i/1

(ri#r@#s@

i)#maxMp

*1+!s@

*1+#s

*1+, s

*1+N#maxM0, q

*n+!r

*n+N

#

n~1+i/1

maxMp*i`1+

!s@*i`1+

#s*i`1+

!r@, q*i+!r

*i+, s

*i`1+N. (3)

When the first job is fixed to be Jk, i.e., J

*1+"J

k, k"1,2 , n, this special case reduces to the

F2/no-wait/C.!9

problem with the job set J@"MJ@1,2 , J@

nN such that

p@k"0,

p@i"maxMp

i!r@!s@

i#s

i, s

iN, i"1,2 , n, iOk,

q@i"maxMq

i!r

i, 0N, i"1,2 , n.

Following the same earlier argument, we can show that this case can also be solved in quadratictime. The proof is complete. j

Furthermore, we note that when si"0 or r@

i"0, i"1,2, n, it is evident that F2(O1)/

cyclic, ssd/C.!9

directly reduces to the F2/no-wait/C.!9

problem, thus can be solved by the Gilmoreand Gomory algorithm in O (n log n) time. Based on these two simpler solvable special cases, we canconstruct a heuristic, SSD, for F2(O1)/cyclic, ssd/C

.!9.

Heuristic SSDStep 1. For a given instance of F2/(O1)/cyclic, ssd/C

.!9, if +n

i/1si*+n

i/1r@i, set r@

i"0,

i"1,2 , n; otherwise, set si"0, i"1,2 , n.

Step 2. Apply the Gilmore and Gomory algorithm to find the optimal schedule n@ with the jobsequence p"SJ

*1+,2, J

*n+T for the corresponding special case.

Step 3: Construct a feasible schedule nSSD

for F2(O1)/cyclic, ssd/C.!9

with the same job sequence p.

It is clear that the time complexity of SSD is O (n log n). Let n* be an optimal schedule forF2(O1)/cyclic, ssd/C

.!9, *"minM+n

i/1si, +n

i/1r@iN, S"+n

i/1(si#s@

i) and R"+n

i/1(ri#r@

i). In

the following theorem, we establish the worst-case error bound for the heuristic SSD.

Theorem 3. C.!9

(nSSD

)/C.!9

(n*))5/4, if *)minMS, RN/2; C.!9

(nSSD

)/C.!9

(n*))4/3, if *)

maxMS, RN/2; C.!9

(nSSD

) /C.!9

(n*))3/2, otherwise.


Proof. Let C.!9

(n@) be the makespan of the no-wait problem considered in Step 2. Clearly, wehave

C.!9

(nSSD

))C.!9

(n@ )#*.

Also, it is obvious that

C.!9

(n* )*C.!9

(n@),

and

C.!9

(n* )*S#R.

So we get

C.!9

(nSSD

)C

.!9(n* )

)

C.!9

(n@ )#*C

.!9(n*)

)

C.!9

(n@)C

.!9(n*)

#

*S#R

)1#*

S#R.

In what follows, we shall obtain three worst-case error bounds for SSD for three possiblecases.

Case 1: *)minMS, RN/2.Since S#R*4* in this case, we have C

.!9(n

SSD)/C

.!9(n*))5/4. To show the bound is tight,

consider the instance with J"MJ1,2 , J

4N such that s

i"¸, s@

i"0, i"1,2, 4, and

p1"1, q

1"¸#2, r

1"¸, r@

1"0,

p2"1, q

2"¸#2, r

2"¸, r@

2"0,

p3"2¸, q

3"2, r

3"0, r@

3"0,

p4"1, q

4"1, r

4"0, r@

4"2¸,

where ¸A1.For this instance, we have S"4¸, R"4¸ and *"+4

i/1r@i"2¸, and so *"minMS, RN/2. The

schedule obtained by SSD and the optimal schedule are shown in Fig. 3. We have

C.!9

(nSSD

)C

.!9(n* )

"

10¸#48¸#5

.

It is clear that C.!9

(nSSD

)/C.!9

(n*)P5/4, as ¸PR.

Case 2. *)maxMS, RN/2.Since S#R*2*#minMS,RN*2*#minM+n

i/1si, +n

i/1r@iN" 3*, we have C

.!9(n

SSD)/C

.!9(n* )

)4/3. To show the bound is tight, consider the instance with J"MJ1, J

2N such that s@

i"0 and

ri"0, i"1, 2, and

s1"¸, p

1"¸, q

1"2, r@

1"0,

s2"¸#1, p

2"1, q

2"1, r@

2"¸,

where ¸A1.


Fig. 3. Schedules nSSD

and n*: \ — setup, K — processing, Y — dismounting.



For this instance, we have S"2¸#1, R"¸ and *"+4i/1

r@i"¸, and so *)maxMS, RN/2.

The schedule obtained by SSD and the optimal schedule are shown in Fig. 4. We have

C.!9

(nSSD

)

.!9(n*)

"

4¸#33¸#4

.


(nSSD

)/C.!9

(n*)P4/3, as ¸PR.

Case 3. *'maxMS, RN/2.Since S#R*2* in this case, we have C

.!9(n

SSD) /C

.!9(n*))3/2. To show the bound is tight,

consider the instance with J"MJ1, J

2N such that s@

i"0 and r

i"0, i"1, 2, and

s1"¸, p

1"¸, q

1"2, r@

1"0,

s2"1, p

2"1, q

2"1, r@

2"¸,

where ¸A1.For this instance, we have S"¸#1, R"¸ and *"+4

i/1r@i"¸. The schedule obtained by

SSD and the optimal schedule are shown in Fig. 5. We have

C.!9

(nSSD

)C

.!9(n* )

"

3¸#32¸#4

. (4)


(nSSD

)/C.!9

(n*)P3/2, as ¸PR. The proof is complete. j

Note that we can also set r@"+ni/1

r@i/n or s"+n

i/1si/n in Step 1 of SSD. But in this case, the

time complexity of the algorithm will be O(n2) according to Theorem 2. Furthermore, it is not clearwhether these settings would result in any improvement in the worst-case performance of theheuristic.




4. Cyclic-movement nonseparable setup and dismounting problem

In this section we consider F2(O1)/cyclic, nsd/C.!9

. We first give a simple proof for the strongNP-completeness for the problem by showing that it reduces to the traditional two-machineflowshop with a limited buffer size. Clearly, when all setup and dismounting times approach zero,the cyclic-movement constraint of F2(O1)/cyclic, nsd/C

.!9simply reduces to a job starting time

constraint, and whenever there are two jobs waiting for processing before machine 2, machine 1 willbe blocked. So F2(O1)/cyclic, nsd/C

.!9is equivalent to the traditional two-machine flowshop with

limited buffer size b"1, which is denoted as F2/b"1/C.!9

. Since F2/b"1/C.!9

is NP-completein the strong sense [16], we have the following theorem.

Theorem 4. F2(O1)/cyclic, nsd/C.!9

is NP-complete in the strong sense even when si"s@

i"

ri"r@

i"0, i"1,2 , n.

In the remainder of this section, we will provide two heuristics for the problem and analyze theirworst-case error bounds.

The first heuristic, NSD1, is based on a solvable case of the problem in which qi"0 and

r@i"0, i"1,2 , n. Let n be a feasible schedule with the job sequence p"SJ

*1+,2 , J

*n+T for the

special case. Referring to Fig. 2, we see that

C.!9

(n)"n+i/1

¹i

"s*1+#p

*1+#

n~1+i/2

(r*i~1+

#s*i+#maxMp

*i+, s@

*i~1+N)#r

*n+#s@

*n+

"

n+i/1

(si#r

i)#p*1+#s@

*n+#

n~1+i/2

maxMp*i+

, s@*i~1+

N. (5)

Thus the problem reduces to the F2/no-wait/C.!9

problem with the job set J@"MJ@1,2 , J@

nN such

thatp@i"p

i, q@

i"s@

i, i"1,2 , n.

With this result, we construct the following heuristic:

Heuristic NSD1Step 1. For a given instance of F2(O1)/cyclic, nsd/C

.!9with job set J"MJ

1,2 , J

nN, construct an

instance of the F2/no-wait/C.!9

problem with job set J@"MJ@1,2, J@

nN such that

p@i"p

i, q@

i"s@

i, i"1,2 , n.


Step 2. Apply the Gilmore and Gomory algorithm to find the optimal schedule n@ with jobsequence p@"SJ@

*1+,2 , J@

*n+T for the no-wait problem.

Step 3. Construct a feasible schedule nNSD1

with the corresponding job sequence p"SJ

*1+,2 , J

*n+T for F2(O1)/cyclic, nsd/C

.!9.

The time complexity of NSD1 is O(n log n). Let n* be an optimal schedule for F2(O1)/cyclic, nsd/C

.!9, S@"+n

i/1s@iand Q@"+n

i/1(q

i#r@

i). In the following theorem, we establish the

worst-case error bound for the heuristic NSD1.

Theorem 5. If S@*Q@, C.!9

(nNSD1

)/C.!9

(n*))3/2; otherwise, C.!9

(nNSD1

)/C.!9

(n*))2.

Proof. Let C.!9

(n*NW

) be the makespan obtained for the problem in Step 2 of NSD1. Followingthe proof of Theorem 3, we can easily show that

C.!9

(nNSD1

))C.!9

(n@)#Q@,

C.!9

(n* )*C.!9

(n@),

C.!9

(n* )*S@#Q@,

and so

C.!9

(nNSD1

)C

.!9(n*)

)

C.!9

(n@)#Q@C

.!9(n*)

)1#Q@

S@#Q@.

When S@*Q@, it is obvious that C.!9

(nNSD1

)/C.!9

(n* ))3/2. To show this bound is tight,consider the instance J"MJ

1, J

2, J

3N such that r

i"0 and r@

i"0, i"1, 2, 3, and

s1"1, s@

1"¸, p

1"1, q

1"1,

s2"1, s@

2"¸#1, p

2"2, q

2"2¸,

s3"2¸, s@

3"1, p

3"¸#1, q

3"1,

where ¸A1.For this instance, we have S@"2¸#2 and Q@"2¸#2, and so S@"Q@. The schedule n

NSD1obtained by NSD1 and the optimal schedule are shown in Fig. 6. We have

C.!9

(nNSD1

)C

.!9(n*)

"

6¸#64¸#8

.


(nNSD1

)/C.!9

(n*)P3/2, as ¸PR.When S@(Q@, it is obvious that C

.!9(n

NSD1)/C

.!9(n*))2. To show this bound is tight, consider

the instance J"MJ1, J

2N such that s

i"1, r@

1"0, i"1, 2, and

s@1"2, p

1"1, q

1"1, r

1"¸,

s@2"1, p

2"2, q

1"¸, r

2"0,

where ¸A1.


Fig. 6. Schedules nNSD1

and n*: \ — setup, K — processing.

Fig. 7. Schedules nNSD1


For this instance, we have S@"3 and Q@"¸#1, and so S@(Q@. The schedule nNSD1

obtainedby NSD1 and the optimal schedule are shown in Fig. 7. We have

C.!9

(nNSD1

)C

.!9(n*)

"

2¸#7¸#8

.


(nNSD1

)/C.!9

(n*)P2, as ¸PR. The proof is complete. j

To construct another heuristic for F2(O1)/cyclic, nsd/C.!9

, let us consider a restricted version ofthe problem, namely N¼, in which the dimounting and setup operations on machine 2 in a cyclemust immediately follow the dismounting and setup operations on machine in the previous cycle(Fig. 8). Now we consider a special case where the setup and dismounting operations are jobindependent for N¼, i.e., s

i"s, s@

i"s@, r

i"r and r@

i"r@, i"1,2 , n . As shown in Fig. 8, it is not

difficult to see that the makespan of the problem N¼ can be expressed as follows:

C.!9

(nNW

)"n+i/0

¹i

"2s#r#p*1+#maxMs#r#p

*2+, s@#q

*1+, s#r#s@N

#

n~1+i/3

maxMs#r#p*i+

, s@#r@#q*i~1+

, s#r#s@#r@N

#maxMr#p*n+

, s@#r@#q*n~1+

, r#s@#r@N#s@#2r@#q*n+

. (6)

Following a similar argument as that for the proof of Theorem 2, we can show that, when thesetup and dismounting operations are job independent, N¼ can be solved in O (n3) time. Hence, bysolving this special case for N¼, we may get a near-optimal solution for F2(O1)/cyclic, nsd/C

.!9.

But instead of directly dealing with N¼, we introduce a simplified problem, namely SN¼, which


Fig. 8. A schedule for problem NW: \ — setup, K — processing, Y — dismounting.

seeks to minimize the following function:

C.!9

(nSNW

)"s#r#p*1+

#s@#r@#q*n+

#

n+i/2

Ms#r#p*i~1+

, s@#r@#q*i+

, s#r#s@#r@N. (7)

Following the discussion of the special cases in Section 3, we can easily show that SN¼ can besolved in O (n2) time.

Now let nNW

and nSNW

be feasible schedules with the same job sequence for N¼ and SN¼,respectively. By comparing Eqs. (6) and (7), we have

C.!9

(nNW

)!s!r@)C.!9

(nSNW

))C.!9

(nNW

)#s#r@. (8)

Based on these results, we can now construct a heuristic for the general problem as follows:

Heuristic NSD2Step 1. For a given instance of F2(O1)/cyclic, nsd/C

.!9, let s"+n

i/1si/n, r"+n

i/1ri/n,

s@" +ni/1

s@i/n and r@"+n

i/1r@i/n.

Step 2. Construct an instance of SN¼ and find the optimal schedule n*SNW

with the job sequencep"SJ

*1+,2 , J

*n+T.

Step 3. Construct a feasible schedule nNSD2

with the same job sequence p for the instance ofF2(O1)/cyclic, nsd/C

.!9.

It is easy to see that the completely of NSD2 is O (n2). While it is clear thatC

.!9(n

NSD2)/C

.!9(n* ))2 for the general problem, the following theorem shows that the worst-

case performance of the heuristic may be better when the setup and dismounting times arejob-independent.

Theorem 6. If si"s, s@

i"s@, r

i"r and r@

i"r@, i"1,2 , n, C

.!9(n

NSD2)/C

.!9(n* ))3/2.

Proof. As si"s, s@

i"s@, r

i"r and r@

i"r@, i"1,2 , n, it is clear that for a feasible schedule n with

the same job sequence corresponding to a feasible schedule nNW

for problem N¼, we have

C.!9

(n))C.!9

(nNW

).


Hence, from Eq. (8), we have

C.!9

(nNSD2

))C.!9

(n*SNW

)#s#r@. (9)

Now assume that, without loss of generality, n* is an optimal schedule with the sequencep"SJ

1,2 , J

nT for F2(O1)/cyclic, nsd/C

.!9. We can assume that n* is a saturated schedule by

which we mean that, except for the initial idle time I*"2s#r#p

1on machine 2 and the final idle

time I&"s@#2r@#q

non machine 1, there is no idle time on both machines. So we have

C.!9

(n* )"(2s#p1#r)#

n~1+i/2

(pi#s)#(s@#2r@#q

n). (10)

In such a saturated schedule, we have

qi*s#r, i"1,2, n!1. (11)

Now, construct a job set JM "MJM1,2 , JM

nN with

pN1"0, qN

1"s@#q

1, (12)

pNi"s#r#p

i, qN

i"s@#r@#q, i"2,2, n!1, (13)

pNn"r#p

n, qN

n"0, (14)

where pNiand qN

idenote the processing times of JM

ion machines 1 and 2, respectively. It is easy to see

that the schedule n* with the sequence p6 "SJM1,2, JM

nT is indeed optimal for JM and the intermediate

inventories are limited by a buffer size b"1, and

C.!9

(n* )"C.!9

(nb/1

)#2s#r#s@#2r@#p1#q

n. (15)

Then, following the argument due to Papadimitriou and Kanellakis [19], we can easily showthat there exists a feasible scdhedule n

b/0with the sequence p8 "SJM

1, JM

*2+,2, JM

*n~1+, JM

nT for

F2/no-wait/C.!9

with the job set JM such that

C.!9

(nb/0

)C

.!9(n

b/1))

32

. (16)

Now construct a feasible schedule n1,n

with the corresponding sequence p@"SJ1, J

*2+,2,

J*n~1+

, JnT for SN¼. From Eqs. (7) and (11)— (14), we see that

C.!9

(n1,n

))C.!9

(nb/0

)#s#r@#p@1#q@

n

"C.!9

(nb/0

)#2s#r#s@#2r@#p1#p

n. (17)

From Eqs. (10) and (15)— (17), we have

C.!9

(nNSD2

)C

.!9(n*)

)

C.!9

(n*SNW

)#s#r@C

.!9(n* )

)

C.!9

(n1,n

)#s#r@C

.!9(n*)

)

C.!9

(nb/0

)#3s#r#s@#3r@#p1#p

nC

.!9(n

b/1)#2s#r#s@#2r@#p

1#q

n


)

32!

r#s@#p1#p

n2(C

.!9(n

b/1)#2s#r#s@#2r@#p

1#q

n)

(

32.

This completes the proof. j

However, we are yet to find an instance which shows that the bound is attainable. But, accordingto Papadimitriou and Kanellakis [19], we know that the bound C

.!9(n*

SNW)/C

.!9(n*))3/2 is tight

when si"s@

i"r

i"r@

i"0, i"1,2 , n.

5. Conclusions

In this paper, we have considered the problem of scheduling n jobs in a one-operator—two-machine flowshop so as to minimize the makespan. Confining our attention to cyclic-movementschedules, we have first identified the relevant cyclic-movement schedules for the separable andnonseparable setup and dismounting problems, respectively. We have shown that both problemsare NP-complete in the strong sense. We also have proposed some heuristics and analyzed theirworst-case error bounds.

A multistage system is more realistic in practice, and so the one-operator—m-machine flowshopproblem is a worthy topic for further research. Since the makespan problem in such a system isnecessarily strongly NP-complete, research should focus on the design of efficient heuristics andanalysis of their performance.

Acknowledgements

This research was supported in part by The Hong Kong Polytechnic University under grantnumber 350/239. We are grateful to two anonymous referees for their helpful comments.

References

[1] Cheng TCE, Podolsky S. Just-in-time manufacturing: an introduction, 2/e. London: Chapman & Hall, 1996.[2] Krajeswski LJ, Ritzman LP. Operations management: strategy and analysis. Reading, MA: Addison-Wesley, 1987.[3] East N. A human guide to robots. UK: Alpin Press, 1983.[4] Su Q, Chen FE. Optimal sequencing of double-gripper gantry robot moves in tightly-coupled serial production

systems. IEEE Transactions on Robotics and Automation 1996;12:22—30.[5] Graham RL, Lawler EL, Lenstra JK, Rinnooy Kan AHG. Optimization and approximation in deterministic

sequencing and scheduling: a survey. Annals of Discrete Mathematics 1979;5:287—326.[6] Dudek RA, Panwalker SS, Smith ML. The lessons of flowshop scheduling research. Operations Research

1992;40:7—13.[7] Lawler EL, Lenstra JK, Rinnooy Kan AHG, Shmoys DB. Sequencing and scheduling: algorithms and complexity.

In: Nemhauser GL, Rinnooy Kan AHG (editors), Logistics of production and inventory. Amsterdam: North-Holland, 1993.


[8] Hall NG, Sriskandarajah C. A survey of machine scheduling problems with blocking and no-wait in process.Operations Ressearch 1996;44:510—24.

[9] Crama Y, van de Klundert J. Cyclic scheduling of identical parts in a robotic cell. Operations Research1997;45:952—6.

[10] Levner E, Kats V, Levit VE. An improved algorithm for the cyclic flowshop scheduling in a robotic cell. EuropeanJournal of Operational Research 1997;97:500—8.

[11] Kise H, Shioyama T, Ibaraki T. Automated two-machine flowshop scheduling: a solvable case. IIE Transactions1991;23:10—6.

[12] Sethi SP, Sriskandarajah C, Sorger G, Blazewicz J, Kubiak W. Sequencing of parts and robot moves in a roboticcell. International Journal of Flexible Manufacturing Systems 1992;4:331—58.

[13] Hall NG, Kamoun H, Sriskandarajah C. Scheduling in robotic cells: classification, two and three machine cells.Operations Research 1997;45:421—39.

[14] Hall NG, Kamoun H, Sriskandarajah C. Scheduling in robotic cells: complexity and steady state analysis.European Journal of Operational Research 1998;109:43—65.

[15] Levner E, Kogan K, Maimon O. Flowshop scheduling of robotic cells with job-dependent transportation andset-up effects. Journal of the Operational Research Society 1995;46:1447—55.

[16] Agnetis A, Pacciarelli D, Rossi F. Lot scheduling in a two-machine cell with swapping devices. IIE Transactions1996;28:911—7.

[17] Rock H. The three-machine no-wait flowshop problem is NP-complete. Journal of the Association for ComputingMachinery 1984;33:336—45.

[18] Gilmore PC, Gomory RE. Sequencing a one-state variable machine: a solvable case of the traveling salesmanproblem. Operations Research 1964;12:655—79.

[19] Papadimitriou CH, Kanellakis PC. Flowshop scheduling with limited temporary storage. Journal of the Associ-ation for Computing Machinery 1980;27:533—49.

T.C.E. Cheng is Vice-President (Research and Postgraduate Studies) and Chair of Management, The Hong KongPolytechnic University. He obtained a Ph.D. in Operations Research from the University of Cambridge. His expertise isin the fields of production and operations management and operations research. His publications have appeared ina wide variety of journals.

G. Wang is a lecturer in the Department of Business Administration, Jinan University. He received a B.Sc. inMechanical Engineering and an M.Sc. in Management Engineering from Harbin Institute of Technology, People’sRepublic of China. His research interests are in operations management and machine scheduling.

C. Sriskandarajah is a professor in the Department of Mechanical and Industrial Engineering, University of Toronto,Canada. He has published numerous articles on machine scheduling.


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One-operator–two-machine flowshop scheduling with setup and dismounting times