组合机床外文文献

Int J Adv Manuf Technol (2006) 29: 178–183 DOI 10.1007/s00170-004-2493-9
ORIGINAL ARTICLE
Ferda C. C ? etinkaya
Unit sized transfer batch scheduling in an automated two-machine ?ow-line cell with one transport agent
Received: 26 July 2004 / Accepted: 22 November 2004 / Published online: 16 November 2005 ? Springer-Verlag London Limited 2005 Abstract The process of splitting a job lot comprised of several identical units into transfer batches (some portion of the lot), and permitting the transfer of processed transfer batches to downstream machines, allows the operations of a job lot to be overlapped. The essence of this idea is to increase the movement of work in the manufacturing environment. In this paper, the scheduling of multiple job lots with unit sized transfer batches is studied for a two-machine ?ow-line cell in which a single transport agent picks a completed unit from the ?rst machine, delivers it to the second machine, and returns to the ?rst machine. A completed unit on the ?rst machine blocks the machine if the transport agent is in transit. We examine this problem for both unit dependent and independent setups on each machine, and propose an optimal solution procedure similar to Johnson’s rule for solving the basic two-machine ?owshop scheduling problem. Keywords Automated guided vehicle · Lot streaming · Scheduling · Sequencing · Transfer batches entire lot to ?nish its processing on the current machine, while downstream machines may be idle. It should be obvious that processing the entire lot as a single object can lead to large workin-process inventories between the machines, and to an increase in the maximum completion time (makespan), which is the total elapsed time to complete the processing of all job lots. However, the splitting of an entire lot into transfer batches to be moved to downstream machines permits the overlapping of different operations on the same product while work proceeds, to complete the lot on the upstream machine. There are many ways to split a lot: transfer batches may be equal or unequal, with the number of splits ranging from one to the number of units in the job lot. For instance, consider a job lot consisting of 100 identical items to be processed in a three-stage manufacturing environment in which the ?ow of its operations is unidirectional from stage 1 through stage 3. Assume that the unit processing time at stages 1, 2, and 3 are 1, 3, 2 min, respectively. If we do not allow transfer batches, the throughput time is (100)(1+3+2) = 600 min (see Fig. 1a). However, if we create two equal sized transfer batches through all stages, the throughput time decreases to 450 min, a reduction of 25% (see Fig. 1b). It is clear that the throughput time decreases as the number of transfer batches increases. Flowshop problems have been studied extensively and reported in the literature without explicitly considering transfer batches. Johnson [1], in his pioneering work, proposed a polynomial time algorithm for determining the optimal makespan when several jobs are processed on a two-machine (two-stage) ?owshop with unlimited buffer. With three or more machines, the problem has been proven to be NP-hard (Garey et al. [2]). Besides the extension of this problem to the m -stage ?owshop problem, optimal solutions to some variations of the basic two-stage problem have been suggested. Mitten [3] considered arbitrary time lags, and optimal scheduling with setup times separated from processing was developed by Yoshida and Hitomi [4]. Separation of the setup, processing and removal times for each job on each machine was considered by Sule and Huang [5]. On the other hand, ?owshop scheduling problems with transfer batches have been examined by various researchers. Vickson
1 Introduction
Most classical shop scheduling models disregard the fact that products are often produced in lots, each lot (process batch) consisting of identical parts (items) to be produced. The size of a job lot (i.e., the number of items it consists of) typically ranges from a few items to several hundred. In any case, job lots are assumed to be indivisible single entities, although an entire job lot consists of many identical items. That is, partial transfer of completed items in a lot between machines on the processing routing of the job lot is impossible. But it is quite unreasonable to wait for the
F.C. ?etinkaya (u) Department of Industrial Engineering, Eastern Mediterranean University, Gazimagusa-T.R.N.C., Mersin Turkey E-mail: ferda.cetinkaya@https://www.360docs.net/doc/1b14475910.html,.tr Tel.: +90-392-6301052 Fax: +90-392-3654029

179 Fig. 1. Processing a without transfer batches and b with transfer batches
2 Problem description
The two-machine ?ow-line cell considered in this paper was ?rst introduced by Panwalker [15] (without explicitly considering the transfer batches and the setup times), and can be described as follows. Suppose that there are several job lots (process batches), all consisting of a given number of identical items (units), are available simultaneously at time zero to be processed on a ?ow-line cell with two machines which are continuously available, and ready at time zero. Each job lot is ?rst processed on machine 1 then on 2. The loading of a new piece (unit) on the ?rst machine and the unloading of a ?nished piece from the second machine are performed in a negligibly small time by a dedicated automated mechanism. Furthermore, before processing each unit, a setup is required for each machine. Setups can be either unit dependent (also known as attached or inseparable) or independent (also known as detached or separable). When setup is unit dependent, it cannot be performed on machine 2 in anticipation of the arriving unit. In this case, setup time may include the time required to set the unit in jigs and ?xtures and to adjust the tools. On the other hand, when setup is independent of the unit to be processed, it can be performed on a machine as soon as the previous unit is ?nished on the same machine. In this case, setup may consist of activities such as acquiring the required jigs and ?xtures, and setting them on the machine. There is also one transport agent in the system (such as an automated guided vehicle, a robotic arm or a crane), which picks up a completed unit from the ?rst machine, delivers it to the second machine, and returns to the ?rst machine. If the automated guided vehicle (AGV) is not available at the time when the processing of a unit has been completed on the ?rst machine, then machine 1 becomes blocked until the unit is removed by the AGV. That is, a unit, having completed processing on the ?rst machine, remains on the machine until the AGV becomes available to remove it. When the AGV reaches the second machine, it delivers the unit to the second machine if machine 2 is free (i.e., machine 2 is ready to receive the unit); otherwise, the AGV places the unit in a storage space and immediately starts its return to machine 1. The time to load and unload the AGV is included in the travel time, and the time to remove a unit from machine 1 is considered to be negligibly small. Note that the transport agent is assumed to be automated for convenience only; the actual unit need not be automated. Furthermore, it is continuously available and ready at time zero, unless the ?ow-line is in a busy state. Figure 2 illustrates an example of the above behaviour for two cases in which unit dependent setups and unit independent setups are assumed in Fig. 2a and Fig. 2b, respectively, and there are two job lots, each having two units. Note that the AGV is initially available for transferring the completed units at machine 1, and therefore the ?rst unit transfer batch does not wait for AGV. Machine 1 will be blocked after the second transfer batch of the ?rst job in the sequence, since the AGV is not available at the time when the processing of second transfer batch has been completed on the ?rst machine.
and Alfredson [6] examined two- and three-machine ?owshops assuming there is no limit on the number of transfer batches so that it is optimal to use unit sized transfer batches. They showed that the two-machine problem with unit sized transfer batches is easily solved by a slight modi?cation of Johnson’s well known algorithm, and also proved that pre-emption of job lots is not necessary in the optimal solution. That is, a job is split into unit sized transfer batches that are processed consecutively (not intermingled with the transfer batches of other job lots). Cetinkaya and Kayaligil [7] and Baker [8] extended this study to obtain a uni?ed solution procedure, which handles separable and inseparable setups, respectively. Cetinkaya [9] and Vickson [10] independently showed that the scheduling problem with either equal or unequal transfer batches decomposes into an easily identi?able sequence of single job problems, even with setup times and transfer times. Sriskandarajah and Wagneur [11] studied the lot streaming and scheduling multiple products in two-machine, no-wait ?owshops. Kim et al. [12] proposed a scheduling rule for a two-stage ?owshop with identical parallel machines at each stage. On the other hand, studies on job lot scheduling with transfer batches for different shop structures are very limited. Examples of these studies include Dauzere-Peres and Lasserre [13] and Sen and Benli [14], which examine the job shops and open shops, respectively. The rest of this paper is organised as follows. In Sect. 2, we describe our ?ow-line cell in detail. The properties of the optimal sequence that minimises the makespan of all job lots with unit sized transfer batches are developed in Sect. 3. In Sect. 4, we propose a polynomial time algorithm to ?nd the optimal sequence. Next, in Sect. 5 a numerical example of the algorithm is provided to illustrate the proposed algorithm. Finally, the paper concludes with a summary and directions for future work.

180 Fig. 2. Unit sized transfer batch processing in the automated ?ow-line cell
5. No unit sized transfer batch may be pre-empted once an operation has begun. 6. Job lots can be split, i.e., pre-emption of job lots is allowed. This means that all transfer batches of a job lot may not necessarily be processed consecutively on each machine. They can be intermingled with the unit transfer batches of other job lots. 7. Machine processing times, setup times and travel/return times of the transport agent are independent of sequence. 8. No storage space exists after the ?rst machine for storing the processed transfer batches, but the buffer preceding the second machine is suf?ciently large to store all transfer batches ?nished on machine 1 and carried to the second machine by the transport agent. 9. The loading/unloading times on the machines and the times for placing transfer batches into the storage space before the second machine are considered to be negligibly small, and thus are taken as zero. 10. Unit processing times, setup times, travel/return times, and number of unit sized transfer batches for each job lot are known and deterministic. 2.1 Notation The following notation will be used throughout the paper so as to develop the proposed optimal scheduling rule: n j m uj p j ,m S j ,m t r = number of job lots to be scheduled = index for job lots, j = 1, 2, ..., n = index for machines, m = 1, 2 = size (number of unit sized transfer batches) of job lot j = unit processing time of job lot j on machine m = setup (unit dependent or independent) time on machine m to be performed before processing each unit sized transfer batch of job lot j = travel time of the transport agent carrying a unit from machine 1 to machine 2 = return time of the empty transport agent from machine 2 back to machine 1
Levler et al. [16] extended the scheduling problem studied by Panwalker for the two-machine ?ow-line cell without unit sized transfer batch con?gurations to the case where loading and unloading times are assumed to be non-negligible. Stevens and Gemmill [17] studied Panwalker’s work with the objective of minimising maximum lateness for the set of jobs. The problem we deal with in this paper is another extension of the problem studied by Panwalker, and can be stated as follows: ? Given a set of job lots whose unit sized transfer batches are carried by a transport agent between the machines of a twomachine ?ow-line cell where setups (unit dependent or independent) are required before processing, it is desired to ?nd a schedule of the process batches along with their transfer batches so that the makespan is minimised. We are unaware of any previous research that addresses the two-machine ?ow-line cell with one transport agent, where unit independent setups and the possibility of splitting the jobs into unit sized transfer batches exist simultaneously. That is, transfer batch idea is discussed in neither Panwalker’s paper, or the study by Levler et al. [16]. Similarly, unit dependent setups are not considered either paper. When the loading times on both machines are considered as unit dependent setup times and each unit sized transfer batch are assumed to be independent individual job, our proposed procedure gives results equivalent to those of Levler et al. [16]. The following assumptions are necessary to further describe the characteristics of the ?ow-line: 1. Both machines 1 and 2 are continuously available and ready at time zero. 2. The transport agent is continuously available and ready at time zero, unless the ?ow-line is in a busy state. 3. Both machines 1 and 2, as well as the transport agent, may be idle. 4. Each machine can process only one unit sized transfer batch of a job lot at any given time.
3 Optimal scheduling rule
In this section we examine the problem for both unit dependent and independent setups on each machine. The following cases need to be investigated: Case 1: t + r S j,1 ? d j + p j,1 for 1 j n , where d j = 0 if setups are dependent and d j = S j,2 if setups are independent. Case 2: t + r > S j,1 ? d j + p j,1 for 1 j n , where d j = 0 if setups are dependent and d j = S j,2 if setups are independent. The following theorems characterise the optimal solution of the problem. Theorem 1. If t + r S j,1 ? d j + p j,1 for 1 j n, where d j = 0 for dependent setups case and d j = S j,2 for independent setups case, all transfer batches of the same job lot are processed consequently, i.e., no transfer batch of every job lot is intermingled

181
with the transfer batches of other job lots, and job lot h precedes job lot i if min{ Si,1 ? di + pi,1 , Sh ,2 ? dh + ph ,2 } min{ Sh ,1 ? dh + ph ,1 , Si,2 ? di + pi,2 }. Proof. Assume that we allow every transfer batch of all jobs to be scheduled independently, one by one. In this case, the problem involves the scheduling of NT = n j =1 u j distinct objects, where u j is the number of unit sized transfer batches of job lot j . Let (k) be a transfer batch, which is sequenced in the k th position for a sequence of NT transfer batches. Furthermore, let C(k),m denote the completion time of transfer batch (k) on machine m (m = 1, 2). Then, the completion time of transfer batch (k) on machine 1 is given by:
k k
? either no transfer batch of every job lot is intermingled with the transfer batches of other jobs (i.e., all job lots are processed consecutively), or ? at most one job lot is pre-empted and one transfer batch of this job lot intermingles with the other jobs. Proof. The proof is based on a known result. If we consider the classical two-machine ?owshop scheduling problem without transfer batches, then it is known that for any speci?ed presequence of one or more jobs in the job set, it is optimal to order the remaining jobs according to Johnson’s rule. Let T = {1, 2, ..., NT } be the set of all transfer batches and σ1 be the sequence of transfer batches obtained by Johnson’s rule, with the modi?ed processing times (denoted by primes) p j,1 = max{t + r, S j,1 ? d j + p j,1 } and p j,2 = S j,2 ? d j + p j,2 . That is, in sequence σ1 , transfer batch h precedes transfer batch i if min{ pi,1 , ph ,2 } min{ ph ,1 , pi,2 }. Suppose that we move the transfer batch in the k th position (2 k NT ) of the sequence σ1 to the ?rst position in σ1 and create a new sequence σk . In this manner, the modi?ed processing times of the remaining NT ? 1 transfer batches are not affected. This implies that the processing order of transfer batches in σk will be same as in σ1 . If t + r > S(k),1 ? d(k) + p(k),1 , then p(k),1 is reduced from t + r to S(k),1 ? d(k) + p(k),1 . Hence, according to Johnson’s rule with modi?ed processing times, Cmax (σk ) Cmax (σ1 ) where Cmax (σk ) and Cmax (σ1 ) are the corresponding makespans of sequences σk and σ1 , respectively. On the other hand, if t + r S(k),1 ? d(k) + p(k),1 , then the modi?ed processing time of the transfer batch in the k th position remains the same, i.e., p j,1 = S j,1 ? d j + p j,1 . But, in this case, moving the transfer batch in the k th position to the ?rst position contradicts Johnson’s rule with modi?ed processing times. That is, Cmax (σk ) > Cmax (σ1 ). Thus, the only way to improve the makespan is to move a transfer batch in the k th position (2 k NT ) of the sequence σ1 to the ?rst position in σ1 , if t + r > S(k),1 ? d(k) + p(k),1 . In this manner, we create at most NT ? 1 different sequences in addition to σ1 . However, it is suf?cient to create at most n ? 1 different sequences by bringing only one transfer batch of each job lot j with t + r > S j,1 ? d j + p j,1 , since the processing times of the transfer batches of job lot j are equal. This implies that the remaining u j ? 1 transfer batches of job lot j are sequenced in an unbroken string, i.e. they are processed consecutively. Thus, one of these n sequences (at most) gives the optimal sequence with minimum makespan.
C(k),1 =
v =1
C(v?1),1 + S(k),1 + p(k),1 =
v =1
( S(v),1 + p(v),1 ) . (1)
since t + r S j,1 ? d j + p j,1 for 1 k NT . In other words, machine 1 is never blocked since the unit processing time of all job lots on machine 1 are at least equal to t + r . By the same reasoning, the completion time of transfer batch (k) on machine 2 is expressed as: C(k),2 = max{C(k?1),2 + d(k) , C(k),1 + t } + S(k),2 ? d(k) + p(k),2 . (2) By successive application of the expression (Eq. 2), substituting C(k),1 given in (Eq. 1), we obtain the makespan of NT transfer batches as:
z
C( NT ),2 = max
0 z NT
( S(v),1 ? d(v) + p(v),1 )
v =1 z ?1
(3)
?
v =1 NT
( S(v),2 ? d(v) + p(v),2 ) + t ( S(k),2 + p(k),2 ) .
k =1
+
Since the second and third terms in the expression above are constants that are independent of the sequence, it is suf?cient to minimise the ?rst term. Thus, in an optimal sequence of the transfer batches, transfer batch h is processed before transfer batch i if: min{ Si,1 ? di + pi,1 , Sh ,2 ? dh + ph ,2 } min{ Sh ,1 ? dh + ph ,1 , Si,2 ? di + pi,2 } . If transfer batches h and i belongs to the same job lot, then the inequality in (Eq. 4) becomes an equality, which implies that all transfer batches of the same job lot are processed consecutively. That is, in the optimal sequence, no transfer batch of all job lots intermingles with the transfer batches of other job lots. Theorem 2. If t + r > S j,1 ? d j + p j,1 for 1 j n, where d j = 0 for dependent setups case and d j = S j,2 for independent setups case, then in the optimal sequence, (4)
4 Proposed algorithm
Based on Theorems 1 and 2, an optimal sequencing algorithm running in O(n log n ) time is presented as follows: Step 1: Let d j = 0 for j ∈ J = {1, 2, ..., n } if setups are unit dependent, and d j = S j,2 for j ∈ J = {1, 2, ..., n } if setups are unit independent. Also, let p j,1 = max{t + r, S j,1 ? d j + p j,1} and p j,2 = S j,2 ? d j + p j,2 for j ∈ J = {1, 2, ..., n };

182
Step 2: 1. Decompose the job lot set J into two parts: J A = { j p j,1 p j,2 } and J B = { j p j,1 > p j,2 }. 2. Sort the job lot numbers of set J A in non-decreasing order of p j,1 , and denote the resulting list as J A . 3. Sort the job lot numbers of set J B in non-increasing order of p j,2 , and denote the resulting list bas J B . 4. Let σ1 = { J A , J B }. Calculate the corresponding makespan Cmax (σ1 ). 5. If p j,1 t + r for every job lot j ∈ J , then if σ1 is an optimal sequence, stop; otherwise, go to Step 3. Step 3: 1. Set k = 2. 2. If k > n , go to Step 4; otherwise, let the job lot in the k th position of σ1 be job lot z . 3. If pz ,1 t + r , set Cmax (σk ) = +∞, and go to Step 3(4); otherwise move a transfer batch of job lot z to the ?rst position in σ1 to obtain a new sequence σk . Calculate Cmax (σk ) and go to Step 3(4). 4. Set k = k + 1. Return to Step 3(2). Step 4: σ j is an optimal sequence if Cmax (σ j ) = min {Cmax (σk )}, and Cmax (σ j ) is the corresponding minimal makespan. Notice that the sequence σ1 obtained in Step 2 of the algorithm above is optimal for the case where the pre-emption of job lots is not allowed. Thus, there is no need to apply Steps 3 and 4.
1 k n
Fig. 3. Candidate sequences in the example problem
The sequences obtained in Step 3 and their corresponding makespans are as follows: σ2 = {31 , 11 , 12 , 13 , 32 , 33 , 21 , 22 , 23 } , σ3 = {21 , 11 , 12 , 13 , 31 , 32 , 33 , 22 , 23 } , Cmax (σ2 ) = 46 Cmax (σ3 ) = 49 (5) (6)
1
Sequence σ2 is optimal since: min {Cmax (σ j )} = min{48, 46, 49} = 46 (see Fig. 3).
j 3
5 Numerical example
In order to illustrate the above solution procedure, let us consider a three-job problem in which the travel time t is equal to 2 units of time from machine 1 to machine 2, and r is equal to 1 unit of time for the return from machine 2 to machine 1. We assume that setups are unit independent, and job lots can be pre-empted. The modi?ed processing times p j,1 and p j,2 , which are obtained in Step 1 of the proposed algorithm, are already included in the data shown in Table 1:
Table 1. Data for the three job lot problem Job lot Setup time Unit processing time size Machine 1 Machine 2 Machine 1 Machine 2 uj Sj,1 Sj,2 pj,1 pj,2 pj,1 3 3 3 2 2 1 2 1 2 3 2 2 4 2 3 4 4 4
6 Conclusion
This study extends previous work [15] in two-machine ?owshop scheduling by considering the possibility of splitting the job lots into unit sized transfer batches to be moved to the downstream machine. The problem is examined for both unit dependent and independent setups on each machine, and a polynomial-time algorithm is established to ?nd the optimal schedule for both cases. Potential areas for future research involve modifying the problem itself. Studying the problem with more than two machines and more than one transport agent is one of the possible extensions, although it appears to be dif?cult. Another direction pertains to cases in which setup times are sequence dependent. One issue of importance is the examination of the same problem that relaxes assumption (9). Individual job attributes could also be changed so that some ready times are non-zero.
JobLot j 1 2 3
pj,2 4 2 3
References
1. Johnson SM (1954) Optimal two and three-stage production schedules with setup times included. Naval Res Logist Q 1:61–68 2. Garey MR, Johnson DS, Sethi R (1976) The complexity of ?owshop and jobshop scheduling. Math Oper Res 1:117–129 3. Mitten LG (1959) Sequencing n jobs on two machines with arbitrary time lags. Manage Sci 5: 293–298
Step 2 gives us the sequence σ1 = {11 , 12 , 13 , 31 , 32 , 33 , 21 , 22 , 23 }, in which ji denote the i th transfer batch of job lot j . The makespan Cmax (σ1 ) for this sequence is 48.

183 4. Yoshida T, Hitomi K (1979) Optimal two-stage production scheduling with setup times separated. AIIE Trans 11:261–263 5. Sule DR, Huang KY (1983) Sequency on two and three machines with setup, processing and removal times separated. Int J Prod Res 21: 723–732 6. Vickson RG, Alfredsson BE (1992) Two-and three-machine ?ow shop scheduling problems with equal sized transfer batches. Int J Prod Res 30:1551–1574 7. Cetinkaya FC, Kayaligil MS (1992) Unit sized transfer batch scheduling with setup times. Comput Ind Eng 22:177–183 8. Baker KR (1995) Lot streaming in the two-machine ?ow shop with setup times. Ann Oper Res 57:1–11 9. Cetinkaya FC (1994) Lot streaming in a two-stage ?ow shop with setup, processing and removal times separated. J Opl Res Soc 45:1445–1455 10. Vickson RG (1995) Optimal lot streaming for multiple products in a two-machine ?ow shop. Eu J Opl Res 85:556–575 11. Sriskandarajah C, Wagneur E (1999) Lot streaming and scheduling multiple products in two-machine no-wait ?owshops. IIE Trans 31: 695–707 12. Kim J-S, Kang S-H, Lee SM (1997) Transfer batch scheduling for a two-stage ?owshop with identical parallel machines at each stage. Omega 25:547–555 13. Dauzere-Peres S, Lasserre JR (1997) Lot streaming in job shop scheduling. Oper Res 45:584–595 14. Sen A, Benli OS (1999) Lot streaming in open shops. Oper Res Lett 23:135–142 15. Panwalker SS (1991) Scheduling of a two-machine ?owshop with travel time between machines. J Opl Res Soc 42:609–613 16. Levler E, Kogan K, Levin I (1995) Scheduling a two-machine robotic cell: a solvable case. Ann Oper Res 57:217–232 17. Stevens JW, Gemmill DD (1997) Scheduling a two-machine ?owshop with travel times to minimize maximum lateness. Int J Prod Res 35:1–15