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International Journal of Production Research
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The burn-in test scheduling problem with batch
dependent processing time and sequence dependent
setup time
W.L. Pearn a , J.S. Hong a & Y.T. Tai b
a
Department of Industrial Engineering & Management , National Chiao Tung University , Taiwan
b
Department of Information Management , Kainan University , Taoyuan , Taiwan Published online: 26 Jun 2012.
To cite this article: W.L. Pearn , J.S. Hong & Y.T. Tai (2013) The burn-in test scheduling problem with batch dependent processing time and sequence dependent setup time, International Journal of Production Research, 51:6, 1694-1706, DOI: 10.1080/00207543.2012.694488
To link to this article: http://dx.doi.org/10.1080/00207543.2012.694488
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Vol. 51, No. 6, 15 March 2013, 1694–1706
The burn-in test scheduling problem with batch dependent processing time and sequence
dependent setup time
W.L. Pearna, J.S. Hongaand Y.T. Taib*
a
Department of Industrial Engineering & Management, National Chiao Tung University, Taiwan; b
Department of Information Management, Kainan University, Taoyuan, Taiwan (Received 7 November 2011; final version received 17 April 2012)
The burn-in test scheduling problem (BTSP) is a variation of the complex batch processing machine scheduling problem, which is also a generalisation of the liquid crystal injection scheduling problem with incompatible product families and classical identical parallel machine problem. In the case we investigated on the BTSP, the jobs are clustered by their product families. The product families can be clustered by different product groups. In the same product group, jobs with different product families can be processed as a batch. The batch processing time is dependent on the longest processing time of those jobs in that batch. Setup times between two consecutive batches of different product groups on the same batch machine are sequentially dependent. In addition, the unequal ready times are considered in the BTSP which involves the decisions of batch formation and batch scheduling in order to minimise the total machine workload without violating due dates and the limited machine capacity restrictions. Since the BTSP involves constraints on unequal ready time, batch dependent processing time, and sequence dependent setup times, it is more difficult to solve than the classical parallel batch processing machine scheduling problem with compatible product families or incompatible product families. These restrictions mean that the existing methods cannot be applied into real-world factories directly. Consequently, this paper proposes a mixed integer programming model to solve the BTSP exactly. In addition, two efficient solution procedures which solve the BTSP are also presented. Keywords: batch; incompatible product family; mixed integer programming; burn-in test
1. Introduction
In order to enhance efficiency of production, batch processing machines are very often involved in factory applications, in particular in the semiconductor (Lee et al. 1992, Lee and Uzsoy 1999, Pearn et al. 2004a) and thin film transistor liquid crystal display (TFT-LCD) manufacturing processes (Chung et al. 2009, Tai and Lai 2011). The burn-in test scheduling problem is an essential batch processing machine scheduling problem, which was first proposed by Lee et al. (1992). In an integrated circuit (IC) final test process of the semiconductor manufacturing process, burn-in test operation includes parallel batch processing machines which are often used to detect and screen out those infant defects. This process is essential, especially on those newly developed process or new designed devices to ensure their reliability to avoid ‘dead on arrival’. In a TFT-LCD manufacturing process, the ageing test operation (also referred to as the burn-in test operation) is used to detect the early failures before jobs are delivered (Chung et al. 2009). Lee and Uzsoy (1999) considered the batch processing machine scheduling problem with compatible product families in which the processing times are dependent on the longest processing time of all the jobs in the batch. However, the variety of burn-in specifications raised from customers’ requirements, such as temperatures and processing times in the burn-in operation, are increasing. Setup times between two consecutive batches of different product groups on the same batch machine are sequentially dependent. Consequently, in this paper, we consider a burn-in test scheduling problem (BTSP) which is a multi-dimensional parallel batch processing machines scheduling problem (see Figure 1) involving the constraints of batch dependent processing time, sequence dependent setup time, unequal ready time, due dates and limited machine capacity. The BTSP is more complicated than other classical parallel batch processing machine problems proposed by Chung et al. (2009) and Tai and Lai (2011).
Batch processing machine scheduling problems are commonly categorised into two types: compatible product family and incompatible product family. It should be noted that the BTSP considers the essential problem
*Corresponding author. Email: yttai@mail.knu.edu.tw
ISSN 0020–7543 print/ISSN 1366–588X online ß 2013 Taylor & Francis
http://dx.doi.org/10.1080/00207543.2012.694488 http://www.tandfonline.com
characteristics involving the two essential problem types simultaneously. In the first batch processing machine scheduling problem with compatible product family, it is assumed that jobs belonging to different product families may be simultaneously processed. The batch processing time is determined by the longest job processing time in that batch. The first researchers to address the problem arising in a burn-in oven of the final test in the semiconductor industry are Lee et al. (1992). In addition, the existing research works on the batch processing machine scheduling problems with compatible product family characteristics involve the single batch processing machine scheduling problem (Uzsoy 1994, Erramilli and Mason 2006, Chou et al. 2006, Kashan et al. 2006, Wang et al. 2007) and the parallel batch processing machine scheduling problem (Chang et al. 2004, Van Der Zee 2004, 2007, Chung et al. 2009). However, the existing research works do not take unequal ready times and setup time into considerations.
In the second batch processing machine scheduling problem with incompatible product family, some literature has investigated the solution procedures for the single batch processing machine to obtain exact or approximated solutions. Uzsoy (1994) is the first one considered the single batch processing machine scheduling problem with incompatible job families to minimise total completion time and makespan. Mehta and Uzsoy (1998), Dobson and Nambimadom (2001), Perez et al. (2005) and Tangudu and Kurz (2006) investigated the single batch processing machine with the incompatible product family characteristic to obtain exact or approximated solutions. For the parallel batch processing machines scheduling problems, Uzsoy (1994, 1995), Koh et al. (2004, 2005), Mo¨nch et al. (2005, 2006), Malve and Uzsoy (2007), Castro and Novais (2009) and Tai and Lai (2011) presented the solution procedures to solve the scheduling problem in which one batch only contain the jobs clustered in the same product family. For the problems, the batch processing times are dependent on their jobs product family in that batch.
In this paper, the BTSP with the essential problem characteristics involves compatible and incompatible product family simultaneously. In the classical batch scheduling problems with compatible product families, the sequence dependence setup time is not considered due to fixed burn-in temperature. In the classical batch scheduling problems with incompatible product families, constant processing time is considered in one batch. However, the BTSP involves constraints of unequal ready times, limited machine capacity, batch dependent processing times and sequence dependence setup time, and is a variation of the classical parallel batch processing machine scheduling problem considered by Chung et al. (2009) and Tai and Lai (2011). The longest processing time dependent on the batch formation is the ultimate processing time. Once batches containing multiple jobs are formed and are processed on parallel batch processing machines successively, setup times between two consecutive batches of different product groups on the same burn-in test machine are incurred. In this paper, the objective of the BTSP is to schedule jobs without violating the constraints of unequal ready times, due dates, batch depended processing time and sequence dependent setup times, while the total machine workload is minimised. To the best of the authors’ knowledge, the parallel batch processing machine scheduling problem with constraints that we investigated has not been considered by other researchers.
In this paper, we first formulate the BTSP as a mixed integer linear programming (MILP) model in order to obtain the exact solutions. If the computation time is a primary concern, two heuristic algorithms are presented and therefore could be an effective way of scheduling the jobs in the burn-in test operation with consideration of the batch dependent processing time, unequal ready time, due date, sequence dependent setup time and limited
Unequal ready time
Limited machine capacity
Due date Batch sequence Number of batches Batch dependent processing times Total workload Sequence dependent setup time
Figure 1. The multiple dimensions of the BTSP.
machine capability. This paper is organised as follows: Section 2 describes the burn-in test scheduling problem and Section 3 presents the MILP model for the BTSP. The practical illustration is shown in Section 4. Two heuristic algorithms are provided in Section 5. The test problem design and computation tests are presented in Section 6. Finally, Section 7 includes the conclusions.
2. The burn-in test scheduling problem in an IC final test process
An IC final testing process is the final stage of semiconductor manufacturing. In the burn-in test operation, many thousands of IC chips are arrayed in the oven with various settings of temperature and voltage to detect infant defects (Pearn et al. 2004a). Notably, there are several classifications (or grades) for the semiconductor products: military grade, industrial grade and consumer grade. Different devices have their own life time and infant mortality period. Consequently, the burn-in test time and temperature may vary accordingly. In the BTSP we investigated the jobs clustered by their product families. Product families are clustered by their product groups. In the same product group, jobs with different product families can be processed as a batch. The batch processing time is dependent on the longest processing times of those jobs in the batch. In a final testing house, the jobs in the burn-in test operation come from the upstream operations which is called ‘final test’ (FT) (Pearn et al. 2004a); thus, the ready times of jobs are usually unequal, which depend on the completion time of the final test operation. The formed batches must be processed on any of the parallel batch processing machines and be completed before their due dates. Setup times between two consecutive batches of different product groups on the same burn-in test machine are sequentially dependent. Machines are arranged in parallel and jobs to be processed are typically in hundreds of product families.
2.1 The scheduling problem description for the BTSP
The BTSP involves the batch formation and batch sequence can be stated as follows. The job set C ¼ fcijji ¼1, 2, 3, . . . , I, j ¼ JðiÞg including
PI
i¼1JðiÞ jobs and each job cij clustered by product family i, and I
product families was clustered into G product groups G ¼ fgyjy ¼0, 1, 2, . . . , Yg, where y ¼ 0 represents a pseudo
product group. The jobs are formed into batches Hyb (bth batch associated with product group y) in which that
product groups of jobs are mutually incompatible to be processed on the batch machine set K ¼ fkmjm ¼1, 2, . . . , Mg containing M identical parallel batch machines. Let W represent the predetermined
machine capacity and express the same time unit as that of processing time and setup time. Let the number of batches be denoted by B. Due to the considerations of real-world production restrictions in the burn-in test, job cij
has a ready time denoted by rij, a processing time denoted by pi, and a latest starting time denoted by eij which
relates to the due date dij, where eij¼dijpi. It should be noted that the processing time of batch Hybin the BTSP is
depend on the longest processing time of the composite jobs with product family (i) in that batch; therefore, ptyb¼maxfpiji 2 gyg. The batch ready time is the latest ready times of those composite jobs; the batch latest starting
time is the smallest starting time of those composite jobs. Consequently, the combination of jobs in a batch can directly determine the tightness of the time window in that batch.
To simplify the modelling of BTSP, we use information of product group G to represent the information of product family directly, C ¼ fcyjjy ¼0, 1, 2, . . . , Y, j ¼ Jð yÞg. Thus, variables rij, pi, eij, and dijcan be transformed
into variables ryj, pyj, eyjand dyj respectively. Let tybmrepresent a starting time of batch Hyb processed on machine
km. The scheduled batch starting time (tybm) should not be less than the batch ready time and not be greater than the
batch latest starting time. Furthermore, one batch can be processed on a machine on the condition that the accumulated number of jobs in that batch does not exceed the machine’s capacity (maximum number of jobs can be processed simultaneously on a machine) (bmax). Each batch is processed without pre-emption on one machine. No job can co-exist in more than one batch. In addition, let syy0 be the sequence dependent setup time between any two
consecutive batches associated with different product groups (y and y0) on the same machine. The BTSP can be
solved to form batches appropriately as well as to find a schedule for those batches that satisfies the ready time and due date restrictions without violating the machine capacity constraints, while achieving the objective of minimising total workload.
3. A mixed integer programming model formulation
The mixed integer linear programming (MILP) model, with considerations of the real-world production environment, is proposed to obtain the detailed schedule for the BTSP. The objective of this MILP model is to minimise the total workload of burn-in test on the premise of satisfying the practical constraints. Before the mathematical model is presented, the decision variables used in the formulation are listed below.
xyjb¼
1 if job cyjis assigned to batch Hyb,
0 otherwise:
hyjm¼
1 if job cyjis assigned to machine km,
0 otherwise:
fybm¼
1, if the batch Hybassociated with product group y is processed on machine km;
0, otherwise:
zyby0b0m¼
1 if batch Hybassociated with product group y is scheduled immediately
following batch Hy0b0associated with product group y0on machine km,
0 otherwise: 8 > < > :
The mathematical formulation of the BTSP is shown as follows.
MinimiseX M m¼1 XY y¼0 XB b¼1 ptybmþ XY y¼0 XB b¼1 XY y0¼0 XB b0¼1 zyby0b0msyy0 ! ( ) ð1Þ subject to XB b¼1
xyjb¼1, for all y, j, ð2Þ
XM
m¼1
hyjm¼1, for all y, j, ð3Þ
XJi
j¼1
xyjbbMAX, for all y, b, ð4Þ
f0mm¼1, for all m, ð5Þ XM m¼1 fybm XJi j¼1
xyjb for all y, b, ð6Þ
XM
m¼1
fybm1, for all y, b, ð7Þ
hyjm1 þ Q1ðxyjbþfybm2Þ, for all y, j, b, m, ð8Þ
ptybmpyjþQ1ðxyjbþhyjm2Þ, for all y, j, b, m, ð9Þ
XB b¼1 ptybmþ XY y¼0 XB b¼1 XY y0¼0 XB b0¼1 zyby0b0msyy0 ! W, for all m, b06¼b, ð10Þ
Q1fybmtybm, for all y, b, m, ð11Þ
XM m¼1
tybmryjxyjb, for all y, j, b, ð12Þ
XM m¼1 tybmdyj XB b¼1 xyjb XM m¼1
ptybm, for all y, j, b, ð13Þ
tybmþptybmþsyy0ty0b0mQ2ðzyby0b0m1Þ 0, for all y, y0, b, b0, m, b06¼b, ð14Þ
ðzyby0b0mþzy0b0ibmÞ þQ2ðfybmþfy0b0m2Þ 1, for all y, y0, b, b0, m, b06¼b, ð15Þ
ðzyby0b0mþzy0b0ibmÞ Q2ðfybmþfy0b0mÞ 0, for all y, y0, b, b0, m, b06¼b, ð16Þ
ðzyby0b0mþzy0b0ybmÞ Q2ðfy0b0mfybmþ1Þ 0, for all y, y0, b, b0, m, b06¼b, ð17Þ
XY y¼0 XB b¼1 zyby0b0mþ1 ¼ XY y¼0 XB b¼1
fybm, for all m, b, b0¼1, . . . , B, b 6¼ b0, ð18Þ
X
Hyb6¼Hy0 b0
zyby0b0m1, for all y, y0, b, m, b0¼1, . . . , B, b 6¼ b0, ð19Þ
X
Hyb6¼Hy0 b0
zy0b0ybm1, for all y, y0, b, m, b0¼1, . . . , B, b 6¼ b0, ð20Þ
xyjb2 f0, 1g, for all y, j, b, ð21Þ
zyby0b0m2 f0, 1g, for all y, y0, b, b0, m, b06¼b, ð22Þ
fybm2 f0, 1g, for all y, b, m: ð23Þ
The objective function (1) states that the total setup times and processing times in the burn-in test operation is minimal in order to minimise total workload. Constraints in (2) and (3) ensure that each job is exactly assigned to one batch and processed on one machine. Constraints in (4) are the batch size constraint, which requires that the sum of number of jobs contained in one batch processed simultaneously should not exceed the maximal batch size. Constraints in (5) guarantee that the only pseudo-job c0jis scheduled on one machine. Constraints in (6) states that
if a product group y assigned to batch Hyb, then there must have job(s) in product group y assigned to batch Hyb.
The BTSP could process different product families with the same product group simultaneously. Constraints in (7) ensure that each batch Hyb consisted by one product group. The term Q1is a constant as it is sufficiently large in
value to satisfy Q1ðxyjbþfybm2Þ 0 and Q1ðxyjbþhyjm2Þ 0. Constraints in (8) are contingent constraints, if
job cyj with product group y assigned to batch Hyb and Hyb being processed on machine km( fybm¼1), then job cyj
processed by machine km(hyjm¼1). Constraints in (9) states that the processing time of batch Hyb is the longest
processing time of the jobs in batch Hyb. Constraint (10) is a capacity constraint, for each machine km, the machine
workload does not exceed the machine capacity. Constraints in (11) are also contingent constraints. That is, if batch Hybassociated with product group y is not assigned to machine km( fybm¼0), then the starting time of batch Hybon
machine km(tybm) should be not greater than 0. Constraints in (12) and (13) indicate that the starting times of each
batch are greater than or equal to the ready times and are not greater than the latest starting times of those, respectively. The batch ready time is the latest ready time of all the jobs clustered in one batch. The batch latest starting time is determined by the due dates of each job in the batch minus the processing time of the batch. Term Q2
is the chosen constant as it is sufficiently large in value to satisfy the zyby0b0m¼0 or 1 which is required for constraints
in (14)–(17). Constraints in (14) ensure the satisfaction of the inequality tybmþptybmþsyy0ty0b0m, where batch Hyb
immediately precedes batch Hy0b0(zyby0b0m¼1). Constraints in (15)–(17) are the precedence constraints provided and
their applicability has been demonstrated by Pearn et al. (2002). These constraints state their sequence relations. Constraints in (18) indicate that the number of batches processed on machine kmis equal to one more of the number
of the combination of two batches that are processed consecutively. Constraints in (19) ensure that at most one batch Hyb associated with product group y can be scheduled behind batch Hy0b0 associated with product group y0
directly for all the batches, which are scheduled on the same machine km. Similarly, constraints in (20) ensure that at
most one batch Hy0b0 associated with product group y0can be scheduled behind batch Hyb associated with product
group y directly for all the batches, which are scheduled on the same machine km. Constraints in (21)–(23) indicate
that xyjb, zyby0b0mand fybmare binary integer variables. In the MILP model, the total number of variables is I2B2Mþ
2IBM þ NIB þ NIM and the total number of constraint equations is ð9=2ÞI2BðB 1ÞM þ IBM þ 3IBþ
2NIBM þ2NIB þ2NIþ3M, where Gi is the job number in product group i, I is the total number of product
groups, NI¼G0þG1þG2þ þGI1 and B is the number of batches.
4. Solutions for the BTSP
To demonstrate the applicability of the scheduling mechanism for the BTSP, we consider a case in the burn-in test operation of a testing house for illustration purposes. In the following example, there are two parallel batch machines (k1and k2) and 12 independent jobs clustered into seven product families and their family processing times
are 180, 150, 160, 160, 145, 155 and 150 minutes, respectively. These seven product families are categorised into three product groups as shown in Table 1. The ready times and due dates for the 12 jobs is shown in Table 2. Furthermore, the setup times between different product groups are sequence dependent and presented in Table 3. In this illustrative example, the maximal number of jobs in one batch is two. Initially, it is assumed that each job will be contained in an individual batch (B ¼ 12). These 12 jobs should be clustered into appropriate batch numbers while the MILP model is performed; then, those batches are scheduled on the two identical parallel batch machines. The batch processing time is not affected by the machine processing it, but it is associated with job families composed in that batch. Finally, the machine capacity is set to 1200 minutes.
One optimal solution solved by the MILP for the BTSP with incompatible product groups is described as follows. Seven batches associated with different product groups are formed, namely HA1, HA4, HB4, HB5, HC1, HC2,
and HC3. Batch HA1 is the first batch associated with product group A and is composed of c11 and c12 of
productfamily 1. Therefore, the ready time and due date for HA1 are 200 and 920, respectively. In the example we
investigated, three batches of group C (HC1, HC2, and HC3) are processed on machine k1; four batches with two
product groups, group A (HA1 and HA4) and B (HB4 and HB5) are processed on machine k2with one setup time
because batches with different product groups are scheduled on this machine. The machine schedules of the optimal solution are depicted in Figure 2, where the total workload is 1155 minutes and the computation time on PC (Intel T2300 1.66 GHz with 512MB RAM) is 2671 CPU seconds.
To explore the run time growth with example problem size increasing, we ran eight example problems and show the computation results in Table 4, which contains various settings of number of jobs and product groups. It is noted that the computation time increases with the example problem size rapidly. In industrial applications, there
Table 1. The category of product family. Product group Product family Number of jobs Product group Product family Number of jobs A 1 2 C 5 2 A 2 2 C 6 1 B 3 1 C 7 2 B 4 2
Table 2. Ready times and due dates of the 12 jobs.
Job Product group Ready time Due date Job Product group Ready time Due date c11 A 30 950 c42 B 200 700 c12 A 200 920 c51 C 22 742 c21 A 230 950 c52 C 129 749 c22 A 23 943 c61 C 140 960 c31 B 150 870 c71 C 200 920 c41 B 100 820 c72 C 70 790
are hundreds of jobs which need to be scheduled for the daily production plan. Therefore, for large-scale BTSP, the optimal solutions may not be obtained easily within reasonable amount of computation time.
5. Heuristic algorithms
For the large-scale BTSP, the mixed integer linear programming model is computationally inefficient. Therefore, two heuristic algorithms are proposed to solve the BTSP, if the computation time is the primary concern. In this paper, we presented two two-phase heuristic algorithms for the BTSP, referred to as delay window-time parallel
HC1 HC2 HC3 HB5 HB4 HA1 400 200 0 100 300 500 600 700 800 80 100 260 420 615 180 200 355 505 650
Setup Time Process Time
Batch machine 1 Batch machine 2 HA1: c11, c12 HC1: c61, c71 HB5: c41 HB4: c31, c42 HA 4: c21, c22 HC2 : c72 HC3: c51, c52 435 HA4 765
Figure 2. The Gantt chart of the illustrative example for the BTSP.
Table 4. Summary of example problems and computation times. Number of jobs Number of product groups Number of machines CPU seconds 9 3 2 356 12 3 2 2671 12 3 3 6210 16 4 4 8735 18 4 4 11501 18 4 5 17851 21 4 5 * 24 5 5 *
Remarks: *No solutions can be found within reasonable amount of compu-tation time.
Table 3. Setup times of the three product groups. To From U A B C U – 20 20 20 A 0 – 60 150 B 0 15 – 60 C 0 35 90 –
saving algorithm (DWPSA) and delay window-time generalised saving algorithm (DWGSA), in order to generate efficient solutions for the large-scale BTSP.
Notably, the batch processing machine scheduling problem can be decomposed into two parts, batch forming and batch scheduling, which is suggested by Mehta and Uzsoy (1998) and Mo¨nch et al. (2005). In this paper, the two heuristic algorithms essentially consist of two phases each. Phase I forms appropriate batches to accommodate unequal ready times on the parallel burn-in test machines. In this phase, we incorporate the merits of the DELAY heuristic solution procedure proposed by Lee and Uzsoy (1999) and further modified by Chung et al. (2009) in order to accommodate the parallel processing batch machine with incompatible product families in the paper. The ready time of the selected jobs to be put in the forming batch need to be no later than the formed ready time of batch plus the processing time multiplied by a parameter. The main purpose is to avoid the jobs to cause excessive delays and to avoid jobs already waiting to be processed to cause the violation of due date constraint. However, this constraint might limit only few jobs could be put on one batch for some cases when variation of ready time of jobs is large. In such cases, the total processing time will increase dramatically since more batches are formed in this phase.
In the BTSP, we modify the constraint to limit the ready times of the selected jobs that should be no later than the latest starting time of the forming batch. In Phase II, it applies the idea of network transformation of wafer probing scheduling problem (WPSP) and liquid crystal injection scheduling problem, provided by Pearn et al. (2004b) and Tai and Lai (2011), respectively. In this phase, it incorporates the modified saving based algorithms to schedule those batches formed in Phase I in order to reduce the total workload. In addition, we incorporate and modify the saving function to allow the batch with shorter process window time to be processed earlier. The two proposed heuristic algorithms are described as follows.
5.1 The delay window-time parallel saving algorithm (DWPSA) Phase I (batch formation)
Step 0: Let the available set ({AS}) be the set of jobs which are available to be selected as a batch. Sort all jobs associated with processing times in descending order of magnitude as an unscheduled-job list (US). Set b ¼ 1. Step 1: Choose the first job (cyj) on the unscheduled-job list into the available set, {AS}. Set the decision time point
(tp) as the ready time of the first job (ryj). Let Eyis the smallest latest starting time of the jobs in the available set in
which product group is y.
Step 2: Check whether there is a job (cyj0) with the same product group y on the unscheduled-job list which satisfies
constraints ryj0Ey and number of batches in the available set is less than the maximum number of jobs in one
batch (bMAX). If a job (c
yj0) satisfies the conditions, then put the job (cyj0) into the available set. Update the value of Ey
and go to Step 2, where Ey¼ feyjjcyj2 fASgg. Otherwise go to Step 3.
Step 3: Form the batch. Put the jobs in available set into batch Hybwhere Hyb¼ fASg. Remove those jobs which
are formed in batch Hyb from the unscheduled-job list. Select the largest ready times of all the jobs in the formed
batches as the batch ready time and select the smallest latest starting times of all the jobs in the formed batch as the batch latest starting time. Set b ¼ b þ 1.
Step 4: Check the unscheduled-job list, if the list still had unscheduled jobs. Clear all the jobs in the available set fASg ¼ , Ey¼0, and go to Step 2. Otherwise, the phase of batch formation is completed.
Phase II (batch schedule)
Phase II is to schedule those batches formed in Phase I. In the DWPSA algorithm, we apply the parallel savings algorithm, which proposed by Golden (1977) and modified by Pearn et al. (2004b). The DWPSA initially calculates the savings consisting of three terms: the setup time, processing time and the time window restrictions. For the time window restriction, we proposed a new calculation considering the ‘processing time window’ (eyjryj) and force the
batches with shorter processing time window be processed first. The DWPSA initially calculates the savings of all pairs of jobs and sorts those savings in the descending order of magnitude. The DWPSA creates a multiple of M batch processing machines schedules simultaneously. In constructing the schedule, the M pairs of the largest saving values are assigned to the M available machines. Notably, the inserted job looks most promising for maximal waiting time, setup time reduction, and machine capacity (W) constraints. The parallel savings algorithm then
searches downward from the savings list, for a batch which could be merged into the endpoints of current constructing schedules (at the front endpoint or rear endpoint) with the largest saving. The assigned procedure based on the saving procedure is repeated until all batches formed in Phase I are scheduled. The saving function in the DWPSA can be expressed as Equation (24).
WPSAHybHy0 b0 ¼ 0 if WPSAHybHy0 b05 0 or y ¼ y 0, b ¼ b0, 1 sUyþsy0Usyy0 þ0:01 1ptyb þ1 ðey0 b0ry0 b0Þ eyb ðeybrybÞ ey0 b0 otherwise: 8 > > < > > : ð24Þ
In the saving calculation of the DWPSA, three parameters, 1, 1 and 1, and three the ranges 0 11,
0 11 and 0 11, are added to the savings function as the weight of the ‘savings term’, ‘processing time
term’ and ‘time windows restrictions term’, respectively. Parameter 1represents weight of setup time savings, which
results from consecutively processing two batches Hyb and Hy0b0 as a batch pair. It can help to avoid a long setup
time being incurred. Parameter 1can help that the batches with longer processing times are forced to be processed
earlier than the others with the shorter ones. Finally, parameter 1 is used in the time windows restrictions term to
prevent the batches being processed after their due dates in order to enhance customer satisfaction. In addition, the scaled values of 0.01 are added for the second term in Equation (24); it is mainly used to make the appropriate measurements because batch processing times are longer in the BTSP, adding this scaled value to make clear distinction of each saving value. Furthermore, syy0 represents the setup time between any two consecutive batches
Hyband Hy0b0associated with different product groups, respectively; eyband rybrepresent the latest starting time and
the ready time of batch Hyb with product group y, respectively; U denotes the idle status of machine.
5.2 The delay window-time generalised saving algorithm (DWGSA)
Phase I of the DWGSA, which is referred to as the batch forming phase, also applies the modified DELAY algorithm as DWPSA. In Phase II, we incorporate the merit of DGSA heuristic solution procedure proposed by Tai and Lai (2011) for the LCISP. The inserting batch (Hyb) is chosen for scheduling, maximising the savings while
minimising the insertion cost. This inserting batch selection criterion can avoid this algorithm to create a new schedule on another machine with a high setup time (Pearn et al. 2004b). In addition, unlike the insertion strategy of the DWPSA, the DWGSA considers only the end of points, but also those positions between two consecutive batches. Let upos1, Hyb, upos
be the additional setup cost when batch Hyb is inserted between position (pos 1)
and pos in schedule PS. Let u
pos1, Hyb, upos
be the minimal insertion cost value. upos1, Hyb, upos
¼supos1yþsyupos1supos1upos 1 12 ð25Þ
u
pos1, Hyb, upos
¼min upos1, Hyb, upos
ð26Þ Batch Hyb is chosen, which maximises the savings upos1, Hyb, upos
while minimising the insertion cost u
pos1, Hyb, upos
and which avoids the algorithm to create a new schedule on another machine with a high setup time 2sUy. In addition to taking into account machine capacity, the latest starting time constraints of all batches
must also be examined for violation before a batch is inserted. The procedure is repeated until all batches are scheduled or all schedules are full and cannot be expanded.
upos1, Hyb, upos ¼2sUy upos1, Hyb, upos 1 22 ð27Þ u pos1, Hyb, upos
¼max upos1, Hyb, upos
ð28Þ
6. Test problem design and computation results comparisons
To analyse and compare the performance of those algorithms on various BTSP with different characteristics, we randomly generate 72 problems with different characteristics. The experimental design involves five essential characteristics: product group ratio, tightness of due date, processing time variation, setup time variation and
maximal number of jobs allowed in one batch. In these 72 test problems, they all have 80 job numbers. For the BTSP, the concentration of product group has great impact on the total incurred setup time in machine schedule. If a product group contains a large number of families (jobs), the total setup time may be decreased in the machine schedule. On the other hand, if a product group contains a small number of families (jobs), this product group will contribute a larger value of setup time. In this paper, we define an index, called product group ratio, representing the concentration of jobs in a product group, which is the division of the number of product families by the number of product groups. The larger the product group ratio the stronger the family concentration.
In these 72 test problems, we set the product group ratio at 4 and 6. Since the tightness of due date may impact on the solutions of BTSP, for those jobs with large time windows, they could be processed as a batch in jobs with the same product group and later ready time. In addition, for batches with later due dates (larger time windows), they allow postponement of scheduling to minimise the setup time. On the test problems, we set two levels of the tightness of the due dates, four and six times the processing time. In the factor of processing time variation, it can be characterised by three magnitudes, large (L), medium (M), and small (S). In the factor of setup time variation, two magnitudes involving large (L) and small (S) are characterised. Accordingly, the processing times in a problem instance are generated from uniform distributions in [150,440], [190,390] and [150,430] for large, medium and small variations, respectively. The large setup time variation is 2860 and the small setup time variation is 1361. Machine number is set as five and maximal number of job allowed in one batch is 5, 6 and 7 jobs in order to get 72 problem configurations. The five different experimental factors are listed in Table 5. In addition, the ready times are generated from uniform distributions in [0, 1440] and the machine capacity is 3200 minutes. In those problem instances, once the batch processing begins, it is non-pre-emptive until the batch is completely processed. Processing and ready times are measured in minutes. All jobs should be formed as batches and be processed completely by the minimal total workload.
In order to solve those BTSP problem instances with distinct experimental factors using the two heuristic algorithms, the program codes of the two heuristic algorithms are written in Visual Basic 6.0. In the two heuristic algorithms, the number of formed batches is determined by Phase I, which is referred to as batch formation. It may be affected by the factors, including product group ratio, processing time variation and maximal number of job allowed in one batch. Tables 6 and 7 present the solutions of the 72 problem configurations generated by the proposed DWPSA and DWGSA algorithms with five parallel batch machines in each.
In Tables 6 and 7, the ‘6LTS’ represents the batches with large processing time variation, tight due date tightness, and small setup time variations, which 80 jobs are clustered in six product groups. In addition, term B0indicates the
number of formed batches obtained from Phase I of the two heuristic algorithms for various problem instances. In our testing, the run times for problems are very fast. Tables 6 and 7 display the run time required for the two algorithms in CPU seconds. We have found that none of them required more than 0.13 CPU seconds. In comparing the two developed algorithms, the test results showed that the DWGSA receives 41 best solutions and seven tight solutions (out of 72) than those of the DWPSA. We particularly note that in those problems with loose capacity loading, the DWGSA significantly outperformed the DWPSA. In the 72 problem instances, the product group ratio has great impact on the capacity loading. For example, the capacity loading is tighter when product group ratio is set to six rather than set to four in the 80 job problems due to the difference of the number of formed batch. In Tables 6 and 7, the results indicated that DWGSA significantly outperformed DWPSA for problems with loose capacity loading (product group ratio being four), as DWGSA received 29 best solutions and four tight solutions out of 36 test problems.
Table 5. Experimental factors for the BTSP problems.
Factor Value considered Number of values
Product group ratio 4, 6 2
Tightness of due date T, S 2
Processing time variation L, M, S 3
Setup time variation L, S 2
Maximal number of job allowed in one batch 5, 6, 7 3
Total problem instances 72
Table 7. Run times and total workload results for the problem instances using the DWGSA heuristic algorithm with 1¼1 and 2¼1.
bmax 5 6 7
Problem configuration Total workload B0 Time (sec) Total workload B0 Time (sec) Total workload B0 Time (sec)
4MSL 5290 17 0.093 5155 16 0.103 3765 12 0.110 4MSS 5305 17 0.101 5170 16 0.111 3780 12 0.109 4MTL 5310 17 0.115 5055 16 0.105 3765 12 0.095 4MTS 5290 17 0.094 5090 16 0.099 3765 12 0.094 4SSL 5195 17 0.094 5070 16 0.095 3730 12 0.099 4SSS 5210 17 0.099 5085 16 0.099 3745 12 0.099 4STL 5215 17 0.104 5090 16 0.102 3730 12 0.114 4STS 5215 17 0.104 5180 16 0.093 3730 12 0.108 4LSL 5195 17 0.100 5070 16 0.110 3730 12 0.102 4LSS 5210 17 0.108 5085 16 0.109 3745 12 0.104 4LTL 5175 17 0.108 5255 16 0.108 3745 12 0.106 4LTS 5175 17 0.109 5280 16 0.097 3745 12 0.102 6MSL 5220 18 0.117 4790 16 0.117 4075 14 0.102 6MSS 5205 18 0.116 4800 16 0.116 4090 14 0.111 6MTL 5230 18 0.116 4770 16 0.116 4075 14 0.111 6MTS 5305 18 0.111 4785 16 0.111 4090 14 0.113 6SSL 5310 18 0.123 4885 16 0.123 4155 14 0.123 6SSS 5370 18 0.100 4870 16 0.100 4150 14 0.100 6STL 5250 18 0.115 4875 16 0.115 4215 14 0.125 6STS 5265 18 0.116 4890 16 0.116 4310 14 0.096 6LSL 5560 18 0.111 5135 16 0.111 4575 14 0.108 6LSS 5525 18 0.109 5220 16 0.109 4575 14 0.103 6LTL 5580 18 0.111 5265 16 0.111 4885 14 0.111 6LTS 5690 18 0.126 5280 16 0.126 4885 14 0.114
Table 6. Run times and total workload results for the problem instances using the DWPSA heuristic algorithm with 1¼0:6, 1¼0:5, and 1¼0:5.
bmax 5 6 7
Problem configuration Total workload B0 Time (sec) Total workload B0 Time (sec) Total workload B0 Time (sec)
4MSL 5310 17 0.144 5205 16 0.150 3805 12 0.154 4MSS 5310 17 0.149 5190 16 0.151 3800 12 0.139 4MTL 5350 17 0.145 5055 16 0.150 3835 12 0.139 4MTS 5345 17 0.144 5055 16 0.151 3800 12 0.144 4SSL 5215 17 0.144 5090 16 0.144 3770 12 0.144 4SSS 5215 17 0.159 5090 16 0.144 3765 12 0.147 4STL 5215 17 0.149 5120 16 0.132 3860 12 0.149 4STS 5215 17 0.139 5155 16 0.132 3765 12 0.144 4LSL 5215 17 0.139 5090 16 0.130 3765 12 0.144 4LSS 5215 17 0.160 5090 16 0.155 3765 12 0.155 4LTL 5195 17 0.169 5225 16 0.169 3785 12 0.157 4LTS 5195 17 0.149 5230 16 0.139 3810 12 0.142 6MSL 5240 18 0.179 4770 16 0.182 4095 14 0.169 6MSS 5190 18 0.176 4770 16 0.178 4095 14 0.172 6MTL 5190 18 0.188 4810 16 0.178 4095 14 0.178 6MTS 5190 18 0.180 4830 16 0.185 4095 14 0.182 6SSL 5270 18 0.191 4855 16 0.185 4155 14 0.165 6SSS 5280 18 0.186 4855 16 0.191 4155 14 0.199 6STL 5310 18 0.186 4855 16 0.191 4155 14 0.203 6STS 5265 18 0.186 4855 16 0.177 4155 14 0.190 6LSL 5550 18 0.186 5115 16 0.177 4615 14 0.189 6LSS 5525 18 0.191 5115 16 0.190 4635 14 0.194 6LTL 5590 18 0.186 5265 16 0.196 4705 14 0.196 6LTS 5525 18 0.186 5275 16 0.169 4705 14 0.177
Therefore, to confirm the results, we took an additional 24 problems with bmax¼6 and six parallel batch machines. These problems all have loose capacity loadings. The results of the experiments (see Table 8) indicated that the DWGSA algorithm perform remarkably well. The DWGSA approach achieved 16 best solutions and four tight solutions (out of 24) than those of the DWPSA algorithm in terms of total workload.
7. Conclusions
In this paper burn-in test scheduling problem (BTSP) has many real-world applications, involving the constraints of batch dependent processing time, sequence dependent setup time, unequal ready time, due dates and limited machine capacity. The BTSP is more complicated than other classical parallel batch processing machine problems with compatible or incompatible product families. In this paper, a mixed integer programming model was proposed to solve the BTSP exactly. If the computation time was a primary concern, two solution procedures based on the modified batch formation technologies and network algorithms were also developed to solve large-scale problems efficiently. In addition, a design on the test problems considering five factors involving product group ratio, tightness of due date, processing time variation, setup time variation, and maximal number of job allowed in one batch was conducted for the computation tests and comparisons. The computation results showed that the DWGSA performed well, particularly for loose capacity loading. In addition, all proposed algorithms solved the large-scale LCISP quite efficiently within a reasonable amount of computation time. In further research, some metaheuristic solution procedures, such as particle swarm optimisation and generic algorithm might be applied to solve the BTSP.
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