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Real-time order flowtime estimation methods for two-stage hybrid flowshops
Omega 42 (2014) 1–8
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Real-time order flowtime estimation methods for two-stage hybrid flowshops Geun-Cheol Lee n College of Business Administration, Konkuk University, 1 Hwayang-dong, Gwangjin-gu, Seoul 143-701, Korea
a r t i c l e i n f o
abstract
Article history: Received 26 June 2011 Accepted 7 February 2013 Processed by B. Lev Available online 20 February 2013
In this study, we consider a problem of estimating order flowtimes in two-stage hybrid flowshops, where orders arrive dynamically and various scheduling schemes can be used. To solve the problem, we devise several order flowtime estimation methods, and each method is specific to the scheduling scheme used in the shop. Whenever an order arrives, the flowtime of the order is estimated by using one of the proposed methods. In the methods, we consider not only the current workload but also the expected workload in the near future, the volume of which mainly depends on the scheduling scheme. To evaluate the performance of the proposed methods, we obtained the actual flowtimes of orders from simulation runs, and compared them with the estimated flowtimes of the orders. The results of a series of computational experiments show the superior performance of the proposed methods over the several existing methods. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Flowtime estimation Hybrid flowshop Dispatching rule Simulation
1. Introduction As customer satisfaction becomes one of the most prominent goals of many companies, the problems inherent to order promising or available-to-promise (ATP) are becoming increasingly more important. In particular, if a company produces make-to-order products, delivering orders on time is regarded as an essential capability that the company must possess. However, on-time delivery is difficult to achieve in today’s dynamic business environment. Even if two orders have the same number of products with the same type and due-date, the flowtimes of these orders can vary because of different inventory status, scheduling schemes, and so on. In this paper, we propose realtime methods for estimating order flowtimes in hybrid flowshops, where orders arrive dynamically and the various scheduling rules can be used. Here, the flowtime of an order is the time interval between the arrival and the completion of the order; this interval is also called the flow allowance in many due-date assignment studies. Order flowtime estimation, order promising, ATP, and duedate assignment have been popular issues that many researchers have been focusing on. Several survey studies have already been conducted in this regard. Recently, Framinan and Leisten [1] classified the order promising approach into six categories, according to the combinations of integration of its three
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constituent decisions: i.e., order selection, due-date assignment, and order scheduling. Keskinocak and Tayur [2] thoroughly summarized several topics relevant to due-date management: e.g., scheduling policies, off-line vs. on-line models, service constraints, and so on. Kaminsky and Hochbaum [3] primarily reviewed analytical approaches including queuing models, which are largely limited to simple models such as static, common duedates, and single or parallel machine shops. For earlier reviews on the related fields, we refer to Gordon et al. [4] and Cheng and Gupta [5]. Although a significant number of existing studies on order promising problems exists, several practical limitations remain in those studies. First, many studies consider a very simple machine system. Particularly in cases of analytical approaches, most of them consider only a single machine model. Second, many existing studies ignore the dynamic characteristics of order arrivals, and they assume that the jobs to be produced are given. However, in practice, it is common that orders arrive in the system dynamically. Third, in previous studies, the scheduling schemes of the systems have mostly been a naive dispatching rule, e.g., FIFO (First In First Out), and this is especially common if the system involves complex processes. In this study, these limitations are overcome toward a more practical flowtime estimation. Note that, in this study, we focus on estimating the flowtimes of orders. Many previous studies on due-date assignment or order promising problems consider the estimation of flowtimes as a pre-step to assign or quote due-dates. Basically, they consider total work (TWK), jobs in queue (JIQ), or jobs in bottleneck (JIBQ)
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for estimating flowtimes or flow allowances [5]. While TWK only considers the workload of a job itself, the inventory workload is taken into account in JIQ and JIBQ. Unlike these methods, Lawrence [6] estimates the flowtime of an order using the actual flowtimes of the latest completed orders. Recently, the same decision problem was addressed by Lee [7], whose proposed methods were devised for long hybrid flowshops and were mainly focused on the inventory workload at the bottleneck stage. In the later section, these methods are compared with the proposed methods. The contribution of this study is that we develop methods which can estimate the flowtimes of orders more accurately than the existing methods under practical environments. Furthermore, four dispatching rules are incorporated in the proposed methods, which, to the best of our knowledge, is the largest number of scheduling schemes considered in this field. The rest of this paper is structured as follows: Section 2 presents a detailed description of the considered problem. In Section 3, the proposed methods are introduced. Using the proposed methods, we conducted a series of computational experiments, the results of which are presented in Section 4. Finally, we conclude the paper with some discussions and the future directions for this study.
2. Problem description In this study, we consider two-stage hybrid flowshops with dynamic order arrival. The hybrid flowshop consists of two stages that are connected serially and each stage has its own operation for each product type. Each operation can be handled by one of the parallel machines at the stage. A product is completed after processed at stages 1 and 2 sequentially. In this study, we assume that parallel machines in the same stage have the same capabilities; hence, each stage can be referred to as an identical parallel machine system. Although the two-stage is the simplest layout of hybrid flowshops, the scheduling problem of the two-stage hybrid flowshop is well known to be NP-complete [8]. Due to its difficulty, in the earlier studies on this area, case-oriented studies were mainly performed, such as Tsubone et al. [9] and Kim et al. [10]. Nevertheless, two-stage hybrid flowshops have received much attention because of their applications, which can be easily noticed in recent survey papers [11,12]. Even very recent studies on hybrid flowshop scheduling problems continue to consider two-stage layouts [13–15]. In the considered shop, orders arrive dynamically, which means we have no information about a certain order prior to its arrival. When orders arrive, they contain information regarding which and how many products should be produced until what time. Once an order arrives in the system, it is released as multiple lots into the first stage. In this study, a lot is regarded as the minimum production unit so that the processing time of one lot of each product type is provided. The released lots immediately become part of the inventory at stage 1, if no machine is available. All the waiting lots at each stage are sequenced according to a pre-specified dispatching rule. In this study, we could use several different dispatching rules; however, only one rule is selected and used at both stages during the single production run. We assume that a lot cannot be interrupted by another lot, once a machine starts processing it; the inventory buffer size in front of the machines at each stage is infinite; and no machine failures occur. In this study, we estimate the flowtime of each order at the time the order arrives. We anticipate how the order will flow through the two-stage hybrid flowshop using the proposed estimation methods, in which the current workload, future orders, and the dispatching rule used in the system are mainly considered.
3. Proposed methods In the considered system, not only FIFO but also EDD (Earliest Due-Date), SLACK, and SPT (Shortest Processing Time) can be used as the scheduling scheme; hence, an appropriate flowtime estimation method for each scheduling scheme is necessary. In this study, we devise different flowtime estimation methods for different scheduling schemes. However, they analyze the flowtime of an order in the same manner. Fig. 1 shows the schematic breakdown of an order flowtime. The flowtime can be decomposed into two parts: flowtimes caused by processing the order itself, and workloads with higher priorities than that of the order. Workloads with higher priorities could come from the current inventory and from future orders, and those workloads should be processed prior to the order just arrived. The proposed estimation methods use different logics in obtaining the flowtimes caused by processing these workloads with higher priorities, depending on which dispatching rule is used in the shop. In the proposed methods, we employ list scheduling procedures to achieve more accurate estimation results. In each method, we select the stage with the higher inventory level as the target stage to which a list scheduling procedure is applied in order to construct a virtual schedule. Applying a list scheduling procedure to a stage is similar to solving a parallel machine scheduling problem using a dispatching rule, which can be done in a very short time.
3.1. Notation and formula We devise equations to obtain the flowtimes of orders in the proposed methods. The notations used in the flowtime estimation formula are as follows: i k
d aik pik
Lik LSdk ðOÞ
wdik
d f^ ik
d r^ i2
index of order index of stage index of dispatching rule arrival time point of order i at stage k in the shop processing time of a lot from order i at stage k set of orders waiting at stage k when order i arrives in the shop function that returns the completion time of the virtual schedule which is constructed by list scheduling on order set O at stage k under dispatching rule d partial flowtime of order i through stage k incurred by the current workload under rule d, which can be obtained by LSd1 ðfig \ Li1 Þai1 , if k¼1; LSd2 ðfig \ Li1 \ Li2 Þai1 , o/w estimated partial flowtime of order i through stage k expected to be incurred by future orders under dispatching rule d estimated ready time point of order i at stage 2 under dispatching rule d
Flowtime of an order just arrived Flowtime caused by processing workloads with higher priorities than that of the order Flowtime caused by current inventory with higher priority
Flowtime caused by future orders with higher priorities
Flowtime caused by processing the order itself
Fig. 1. Schematic breakdown of an order flowtime.
G.-C. Lee / Omega 42 (2014) 1–8
adi2
virtual arrival time point of order i at stage 2 considering only the current workload, which can be obtained while processing LSd1 ðfig \ Li1 Þ estimated arrival time point of order i at stage 2, which
a^ i2
d
can be obtained by adi2 þ f^ i1 Q(O) function that returns the number of lots in order set O max(x,y) function that returns the largest value between values x and y In the notation, much of the information is real-time data which can be obtained while running a simulation. Once an order (e.g., order i) arrives in the shop, the flowtime of order i needs to be estimated at time ai1. First, the inventory status values of both stages at time ai1, i.e., Li1 and Li2, are obtained: The function, LSdk ðOÞ, virtually performs a list scheduling on order set O at stage k with dispatching rule d (i.e., every lot included in order set O is selected according to d and then loaded and processed on one of the machines at stage k) and returns the completion time of the resulting virtual schedule. wdik can be obtained by using the virtual schedule result and the arrival time d information. The procedures for obtaining f^ ik are all different depending on which dispatching rule is used. With the inventory informad
tion, f^ i2 can be estimated in different ways according to dispatching d
d rule d. The detail procedures for f^ ik and r^ i2 are presented in the next subsection. The estimated arrival time point of order i at stage 2 ða^ i2 Þ d
consists of f^ i1 and adi2 , and the latter can be obtained while applying a list scheduling on the current workload at stage 1. Using the above notation, we could estimate the flowtime of order i under dispatching d rule d ðF^ i Þ using the following equation: 8 d > < wd þ f^ i1 þ maxðr^ di2 a^ i2 ,0Þ þpi2 , if Q ðLi1 Þ ZQ ðLi2 Þ d i1 ^F ¼ i d > d ^ : otherwise wi2 þ f i2 ,
ð1Þ
The flowtime of order i is estimated with either of the two equations in (1), depending on the inventory status of the two stages. If stage 1 has more inventory load than stage 2, the flowtime at stage 1 and the tail time at stage 2 are combined. The flowtime at stage 1 is the estimated time to process the current workload at stage 1 and future orders on machines at stage 1, and the tail time at stage 2 can be estimated by comparing the d difference between r^ i2 and a^ i2 . If there is more inventory load at stage 2, only the flowtime at stage 2 is used for the flowtime of order i in the shop. The flowtime at stage 2 is the estimated time to process the current workload at both stages and future orders on machines at stage 2. Note that, when we estimate the flowtimes, virtual schedulings are applied to the workloads with higher priorities than that of order i. Eq. (1) is used to estimate the flowtimes of orders under all considered dispatching rules. The only difference among the proposed methods is that they use d d different procedures for obtaining the estimators f^ ik and r^ i2 .
the orders arrived are dispatched in ascending order of their arrival times, which means that no lot can be processed before another lot that arrived earlier. Therefore, future orders do not FIFO affect the flowtime of an order (i) just arrived, i.e., f^ ik ¼ 0, for all FIFO values of k. Similarly, when estimating r^ i2 , only the current inventory, i.e., lots waiting at stages 1 and 2, needs to be P FIFO considered. Hence, r^ i2 can be calculated from j A ðLi1 \Li2 Þ pj2 = m2 , where mk is the number of machines at stage k.
EDD SLACK EDD SLACK 3.2.2. Estimating f^ ik , f^ ik , r^ i2 , and r^ i2 Under EDD and SLACK, a lot can be passed by other lots that arrive later in the shop. That is, some future orders would affect the flowtime of the newly arrived order, because such future
d orders have higher priorities. The estimator f^ ik represents the time to process such future orders, and this time needs to be included in the estimated flowtime of the newly arrived order. In d this study, we devise an equation to obtain the value of f^ ik . Before the equation is presented, we introduce notations used in the EDD SLACK : equation for f^ ik and f^ ik
l Q pk
rdik
d f^ ik ¼ wdik l ðQ =2Þ ðpk =mk Þ rdik
ð2Þ
Estimating the partial flowtime of future orders requires the multiplication of three terms: (1) the average number of future orders that arrive while processing the current workload ðwdik lÞ; (2) the average time required to process a certain order ððQ =2Þ ðpk =mk ÞÞ; and (3) the expected ratio of future orders having higher priorities than that of order i during wdik at stage k ðrdik Þ. The third EDD SLACK , can be obtained by another term, which differentiates f^ ik and f^ ik equation, where the following notations are used.
di Ii Imax Imin Si Smax
FIFO FIFO 3.2.1. Estimating f^ ik and r^ i2 Under FIFO, the arrival times of orders are the most important factor when estimating the flowtimes of the orders. All the lots in
mean arrival rate of orders maximum order size mean processing time of a lot at stage k expected ratio of future orders having higher priorities than that of order i during wdik at stage k
Using the above notations, under EDD or SLACK, the estimated partial flowtime of order i through stage k incurred by future orders under dispatching rule d can be estimated via the following equation:
3.2. Estimators To obtain the flowtime estimates using Eq. (1), we need to find d d the right values for the estimators (i.e., f^ ik and r^ i2 ) in the equation, and these values are mainly affected by the dispatching rule used in the shop. In the following subsections, the procedures to obtain d d f^ ik and r^ i2 for four dispatching rules are presented, respectively.
3
Smin vdi
udik
due-date of order i interval between arrival time and due-date of order i in the shop, i.e., di ai1 maximum interval between arrival times and due-dates among the orders currently in the shop minimum interval between arrival times and due-dates among the orders currently in the shop initial slack of order i, which is calculated from di ðai1 þwSLACK Þ in this study i2 maximum initial slack among the orders currently in the shop minimum initial slack among the orders currently in the shop relative urgency of order i at time ai1, which is calculated from (Ii Imin)/(Imax Imin), if d ¼EDD or (Si Smin)/ (Smax Smin), if d ¼SLACK relative urgency of order i at stage k at time ai1 þ wdik , min Þ=ðImax Imin Þ, if which is calculated from ððIi wEDD ik ÞI
d ¼EDD or ððSi wSLACK ÞSmin Þ=ðSmax Smin Þ, if d ¼SLACK ik
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G.-C. Lee / Omega 42 (2014) 1–8
Using the above notation, under EDD or SLACK, rdik can be estimated using the following equation: 8 d d v þu > if udik Z 0 < i 2 ik , d ð3Þ rik ¼ ðvdi Þ2 > otherwise : 2ðvd ud Þ , i
ik
The estimation methods for SLACK and EDD are similar because both methods consider the relative urgencies of orders. From a dispatching point of view, they are different because the priority values of orders under EDD do not change throughout the production, while those of orders under SLACK keep changing over time. However, when we obtain the relative urgencies of orders under SLACK, we use initial slack values, which can be EDD SLACK obtained when the orders arrive. Note that r^ i2 and r^ i2 can be obtained from the sum of the second stage processing times of all the lots waiting at both stages with earlier due-dates and tighter slack values than order i, respectively.
SPT SPT 3.2.3. Estimating f^ ik and r^ i2 Like in EDD and SLACK, a lot can be passed by other lots that arrive later under SPT. However, this phenomenon occurs more often under SPT, because the priority criteria, i.e., processing times, have nothing to do with the arrival times of orders. Note that, in EDD and SLACK, due-dates and slack values of orders are somewhat affected by the arrival times of the orders, which means that lots which arrive later tend to have tighter duedates and smaller slacks. For example, a lot is rarely passed by other lots which arrive much later under EDD and SLACK; however, this can happen commonly under SPT. In the worst case, some lots can be even deadlocked if other lots with shorter processing times keep arriving. Therefore, we need to overestimate the impact of future orders SPT EDD SLACK when obtaining f^ ik , compared to obtaining f^ ik and f^ ik Eq. (2) considers the volume of the (first) future load occurring while processing the current workload, but that is not enough under SPT. In addition to the first future load, we also need to consider the additional volume of the (second) future load occurring while processing the first future load. Based on Eq. (2), the estimated partial flowtime caused by the second future load can be
calculated
from
SPT
½f^ ik l ðQ =2Þ ðpk =mk ÞrSPT ik ,
i.e.,
½wSPT ik
2
flðQ =2Þ ðpk =mk ÞrSPT ik g . Likewise, we could add the amount of 3
SPT the third future load with ½wSPT ik flðQ =2Þ ðpk =mk Þrik g , and so SPT on. Theoretically, f^ ik could be obtained by summing all of the 2
3
following future loads, i.e., fwSPT ik ðf þ f þ f þ. . .Þg, where
f ¼ l ðQ =2Þ ðpk =mk ÞrSPT ik . However, that is too excessive deliberation, therefore, in this study, we chose to consider the next SPT three future loads. Hence, the final equation for f^ ik is as follows: SPT 2 3 f^ ik ¼ wSPT ik ðf þ f þ f Þ
ð4Þ
SPT where f ¼ lðQ =2Þ ðpk =mk ÞrSPT is the ik . In this equation, rik expected ratio of the future orders having shorter processing at stage k. rSPT can be obtained times than order i during wSPT ik ik from (cik 1)/P, where cik is the rank of the processing time of a lot in order i at stage k in ascending order, and P is the number of product types, respectively. Now, we could estimate the flowtime of an order using Eq. (1) with the relevant method for the partial flowtime caused by future orders in two-stage hybrid flowshops. The next section shows the performances of the proposed methods by computational experiments of comparisons with the existing methods.
4. Computational experiments 4.1. Simulation model To see how the proposed methods work in two-stage hybrid flowshops, we developed a simulation model representing several configurations of the shop. In many studies on hybrid flowshop scheduling, simulation experiments were commonly used to evaluate the performances of the proposed methods [6,7,16–18]. The model simulates order arrivals and lot flows, such as waiting, processing, and so on. The simulation model employed in this study was developed using the computer language C. In the model, some aspects can be pre-specified by the user, and others have to be randomly generated in a certain manner. The pre-specified items are the number of product types (P), the shop type (H), the maximum size of orders (Q), and the due-date tightness of orders (g). To control the duration of each simulation run, the warm-up period and simulation end time are also prespecified. These are the parameters to be established by the user prior to the simulation run. Some randomly generated configurations are also predetermined prior to the simulation run. The processing time of one lot in order i at stage k (pik) is generated randomly from DUð1,100Þ ðmk =mÞ þDUð0,50Þ, where DU(LB,UB) is the discrete uniform distribution with range [LB,UB], and m is the average number of machines at one stage in the shop. Once the shop configuration is determined, we can simulate the arrivals and productions of orders in the shop. To simulate the arrivals of orders, the interarrival times of orders are generated exponentially according to a certain arrival rate, which is determined in accordance with the shop capability. Additionally, the product type, size, and due-date of an order are determined randomly upon its arrival. The product type of the order is arbitrarily selected as one of the P product types and the size of order i (Qi) is generated from DU(1,Q). To generate the due-date of FIFO order i, we employ the function ai þ R F^ i , where ai is the FIFO arrival time of order i in the shop and F^ i is the estimated flowtime of order i under FIFO. The due-date range of an order (R) is generated from DU(1/2g,3/2g) in this study. The developed model is a typical discrete-event simulation model, in which the simulation clock progresses according to the time of the next earliest event. Three types of events occur in the model: the order arrival, lot arrival, and processing completion. When an order arrives, the arrival time of the next order is determined. Once a lot is released to the shop, the lot becomes inventory at the first stage of the shop if no machine is available. If a machine is available, the lot is loaded onto that machine. Then the lot completion time is determined, which leads to the processing completion of the lot, i.e., the third event. The machine that becomes available finds the next lot to be processed by using the dispatching rules. In this manner, all the lots continue to be processed until the simulation clock reaches the end of the simulation. In the model, not only the shop configurations, but also the dispatching rules and the order flowtime estimation methods are implemented. Once the simulation begins, the estimation method is executed whenever a new order arrives at the shop.
4.2. Benchmark methods In this study, we used seven existing flowtime estimation methods as benchmarks to investigate the performance of the proposed methods. The first three existing methods are from Lee [7], in which estimation methods for three different dispatching rules were developed for long multi-stage (kZ30) hybrid flowshops. Those methods are similar to our proposed methods, so
G.-C. Lee / Omega 42 (2014) 1–8
that they can also be applied to the two-stage hybrid flowshop problem easily. However, the proposed methods are much more systematic and sophisticated than the existing methods. The estimation methods for FIFO, EDD, and SPT in Lee [7] are denoted as EM-F, EM-E, and EM-S, respectively. In addition to these methods, three parametric methods and one method based on exponential smoothing were also used as benchmarks. The first parametric method is TWK, which is one of the most basic and simple methods in estimating the flowtime of an order. TWK considers the total processing time of the order just arrived and determines the relationship between the flowtime and the total processing time by using linear regression. Unlike TWK, the inventory status is considered in the other parametric methods, JIQ and JIBQ. The number of all waiting lots throughout the shop is counted in JIQ, while the inventory only at the bottleneck stage is considered in JIBQ. Unlike these three parametric methods, the method based on exponentially smoothing (Exponential Smoothed Flowtime, ESF) uses the actual flowtimes of the last two completed orders. For details regarding TWK, JIQ, JIBQ and ESF, please refer to the studies of Sha and Hsu [17] and Lawrence [6].
5
4.3. Test results To test the proposed methods, we generated the problem instances with a variety of parameter combinations. We considered three levels of the product type (P), 10, 50, and 100; three levels of the shop type (H), (m1 ¼2, m2 ¼4), (5, 5), and (4, 2); three levels of the maximum order size (Q), 30, 60, and 90; and three levels of the customer due-date tightness (g), 6, 8, and 10. Five replications for each case added up to a total of 405 instances ( ¼3 3 3 3 5). Under each dispatching scheme, four proposed methods and seven previous methods were tested, which means we tested all the proposed flowtime estimation methods under each dispatching rule, even though each proposed method is devised for a specific dispatching rule. In the tables, the proposed estimation methods for FIFO, EDD, SLACK, and SPT are denoted as EEM-F, EEM-E, EEM-L, and EEM-S, respectively. Table 1 shows the overall comparison results between the proposed order flowtime estimation methods and the existing methods. To assess the performance of the flowtime estimation methods, we compared the actual flowtime, which was obtained
Table 1 Overall performance of the proposed order flowtime estimation methods under pre-specified dispatching rules. Rules
Proposed methods EEM-F
Existing methods
EEM-E
EEM-L
EEM-S
EM-F
EM-E
EM-S
TWK
JIQ
JIBQ
ESF
FIFO
ac
0.01 (0.999)bc
0.31 (0.80)
0.28 (0.82)
0.73 (0.40)
0.08 (0.994)
0.36 (0.82)
0.64 (0.41)
0.79 (0.01)
0.24 (0.97)
0.63 (0.69)
0.26 (0.83)
EDD
2.33 (0.75)
0.08 (0.98)
0.19 (0.97)
1.34 (0.52)
2.41 (0.78)
1.25 (0.95)
1.17 (0.49)
1.53 (0.24)
3.25 ( 0.33)
4.35 ( 0.38)
1.50 (0.60)
SLACK
1.85 (0.78)
0.12 (0.975)
0.07 (0.980)
1.22 (0.51)
1.92 (0.81)
0.98 (0.96)
1.06 (0.48)
1.43 (0.22)
2.83 ( 0.35)
3.86 ( 0.37)
1.24 (0.63)
SPT
27.63 ( 0.03)
23.51 (0.06)
23.92 (0.05)
0.44 (0.77)
28.05 ( 0.01)
24.30 (0.11)
0.62 (0.78)
3.15 (0.44)
29.69 (0.04)
43.94 (0.06)
11.68 ( 0.03)
a b c
Average of mean relative error ratio (RER). Average of correlation coefficient (CC) between the actual and the estimated flowtimes. Best result among 11 methods is shown in bold.
Fig. 2. Averages of mean relative error ratio of all methods under a specific dispatching rule.
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G.-C. Lee / Omega 42 (2014) 1–8
after the corresponding order was completed, and the estimated flowtime, which was obtained by applying the proposed or existing methods immediately after the order arrived. Two measures were employed for the comparison. One is referred to as the relative error ratio (RER), representing the absolute deviation between the estimated and actual flowtimes, divided by the actual flowtime, as introduced by Moses et al. [18]. The smaller RER value refers to the more accurate estimation. The test results on RER values are also depicted as bar charts in Fig. 2, to show the clear differences in performances between the tested methods.
Table 2 Results of ANOVA on differences across the problem configuration changes.
Estimation method Dispatching rule Shop type Number of product types Maximum order size Due-date tightness Residuals a b
df
Sum of squares
Mean square
F-value
10 3 2 2 2 2 17,798
234,941.7 1,148,082.9 5976.5 2208.7 17,446.8 685.0 3,737,552.7
23,494.2 382,694.3 2988.2 1104.3 8723.4 342.5 210.0
111.9a 1822.4a 14.2a 5.3b 41.5a 1.6
Statistically different at the significance level of 0.001. Statistically different at the significance level of 0.01.
Table 3 Results of the paired t-test on difference in performance between the best and the second best methods. Rules
Best performing method
Second best performing method
Results
FIFO EDD SLACK SPT
EEM-F EEM-E EEM-L EEM-S
EM-F EEM-L EEM-E EM-S
a
a
a a a
Statistically different at the significance level of 0.001.
Another measure is the correlation coefficient (CC) between the actual and the estimated flowtimes. If the CC value is close to one, it means the estimation is close to the actual value. CC was used in the study of Sha and Hsu [17] for the same comparison. As can be seen in Table 1, the proposed methods outperformed the previous methods in all dispatching rules, except the CC criteria under SPT. The numbers in bold are the best results for each case. In terms of RER, the performances of EEM-F under FIFO (around 0.01), EEM-E under EDD, and EEM-L under SLACK (around 0.1) are relatively very good compared to other methods. The performance of EEM-S under SPT is not that outstanding, however 44% of RER and 77% of CC can be considered as fairly good results compared to other methods under SPT. To see any statistical differences across the problem configuration change, we performed the analysis of variance (ANOVA) on the mean RER values, and the results are summarized in Table 2. The table shows that, except for the due-date tightness, changes in all other configurations, (i.e., the shop type, the number of product types, and the maximum order size) affected the performances of the estimation methods. Additionally, we performed the paired t-test to find out whether the performance gap between the best and the second best estimation methods for each dispatching rule is statistically significant. In terms of mean RER, the best performing method for each dispatching rule is the corresponding proposed method. As shown in Table 1, the second best performing methods for FIFO, EDD, SLACK, and SPT, are EM-F, EEM-L, EEM-E, and EM-S, respectively. Table 3 shows that all four pairs have statistically significant differences, which means that the proposed estimation method for each dispatching rule outperforms the second-place method at a statistically significant level. To see the detail results of the test based on changes in the problem configuration, test results were summarized for each configuration type. Table 4 shows the performance results as the number of product types varies. For the proposed methods, except SPT, the performance does not significantly change depending on the number of product types. Under SPT, as the
Table 4 Performance of the proposed methods by the different number of product types. Rules
FIFO
No. of product types
10 50 100
EDD
10 50 100
SLACK
10 50 100
SPT
10 50 100
Proposed methods
Existing methods
EEM-F
EEM-E
EEM-L
EEM-S
EM-F
EM-E
EM-S
TWK
JIQ
JIBQ
ESF
0.008 (0.999) 0.010 (0.999) 0.010 (0.999)
0.29 (0.81) 0.32 (0.79) 0.31 (0.79)
0.27 (0.83) 0.30 (0.40) 0.29 (0.81)
0.73 (0.41) 0.72 (0.81) 0.74 (0.40)
0.07 (0.995) 0.08 (0.993) 0.08 (0.994)
0.33 (0.83) 0.38 (0.81) 0.37 (0.81)
0.69 (0.39) 0.61 (0.42) 0.60 (0.42)
0.74 (0.01) 0.82 (0.01) 0.82 (0.01)
0.22 (0.98) 0.24 (0.986) 0.25 (0.96)
0.56 (0.71) 0.81 (0.57) 0.52 (0.80)
0.25 (0.83) 0.27 (0.84) 0.27 (0.81)
2.03 (0.77) 2.60 (0.74) 2.36 (0.74)
0.07 (0.98) 0.08 (0.98) 0.08 (0.98)
0.17 (0.97) 0.21 (0.97) 0.20 (0.97)
1.30 (0.52) 1.38 (0.52) 1.34 (0.52)
2.10 (0.80) 2.68 (0.77) 2.44 (0.77)
1.09 (0.96) 1.39 (0.95) 1.27 (0.95)
1.25 (0.46) 1.17 (0.50) 1.09 (0.50)
1.41 (0.21) 1.67 (0.25) 1.50 (0.25)
3.07 ( 0.38) 3.58 ( 0.32) 3.10 ( 0.32)
3.76 ( 0.43) 5.56 ( 0.36) 3.75 ( 0.36)
1.33 (0.63) 1.65 (0.59) 1.51 (0.59)
1.58 (0.80) 2.08 (0.76) 1.88 (0.77)
0.11 (0.978) 0.12 (0.97) 0.12 (0.97)
0.06 (0.983) 0.08 (0.98) 0.07 (0.98)
1.19 (0.50) 1.26 (0.51) 1.21 (0.51)
1.64 (0.82) 2.16 (0.79) 1.96 (0.80)
0.84 (0.96) 1.10 (0.95) 1.00 (0.95)
1.13 (0.45) 1.07 (0.49) 0.98 (0.50)
1.29 (0.20) 1.58 (0.23) 1.42 (0.22)
2.64 ( 0.36) 3.10 ( 0.32) 2.75 ( 0.36)
3.43 ( 0.41) 4.81 ( 0.33) 3.35 ( 0.38)
1.10 (0.66) 1.39 (0.63) 1.25 (0.62)
29.80 ( 0.04) 26.35 ( 0.02) 26.73 ( 0.02)
25.54 (0.05) 22.28 (0.06) 22.71 (0.06)
25.98 (0.04) 22.68 (0.05) 23.11 (0.05)
0.40 (0.85) 0.46 (0.73) 0.45 (0.723)
30.24 ( 0.02) 26.75 (0.00) 27.16 (0.00)
26.60 (0.09) 22.98 (0.12) 23.31 (0.12)
0.59 (0.88) 0.63 (0.75) 0.63 (0.725)
3.36 (0.48) 3.25 (0.43) 2.83 (0.40)
34.16 (0.05) 31.39 (0.03) 23.52 (0.04)
48.09 (0.07) 45.50 (0.05) 38.22 (0.05)
14.21 ( 0.04) 10.57 ( 0.03) 10.26 ( 0.03)
G.-C. Lee / Omega 42 (2014) 1–8
7
Table 5 Performance of the proposed methods by the different shop types. Rules
FIFO
Shop types
(2,4) (4,2) (5,5)
EDD
(2,4) (4,2) (5,5)
SLACK
(2,4) (4,2) (5,5)
SPT
(2,4) (4,2) (5,5)
Proposed methods
Existing methods
EEM-F
EEM-E
EEM-L
EEM-S
EM-F
EM-E
EM-S
TWK
JIQ
JIBQ
ESF
0.007 (0.999) 0.007 (0.999) 0.014 (0.999)
0.32 (0.77) 0.30 (0.80) 0.30 (0.81)
0.29 (0.80) 0.28 (0.82) 0.28 (0.83)
0.77 (0.39) 0.73 (0.42) 0.68 (0.40)
0.07 (0.997) 0.07 (0.996) 0.10 (0.989)
0.37 (0.81) 0.33 (0.83) 0.38 (0.81)
0.65 (0.40) 0.63 (0.41) 0.63 (0.42)
0.78 (0.01) 0.79 (0.01) 0.81 (0.01)
0.19 (0.98) 0.34 (0.95) 0.17 (0.990)
0.18 (1.00) 1.14 (0.43) 0.57 (0.65)
0.28 (0.81) 0.26 (0.83) 0.25 (0.84)
2.70 (0.73) 2.28 (0.76) 2.01 (0.76)
0.07 (0.98) 0.08 (0.98) 0.08 (0.98)
0.21 (0.97) 0.18 (0.97) 0.18 (0.97)
1.52 (0.51) 1.38 (0.53) 1.13 (0.52)
2.81 (0.76) 2.38 (0.79) 2.03 (0.79)
1.48 (0.95) 1.23 (0.95) 1.04 (0.95)
1.26 (0.48) 1.16 (0.49) 1.08 (0.50)
1.61 (0.25) 1.49 (0.20) 1.48 (0.26)
3.54 ( 0.39) 3.19 ( 0.28) 3.02 ( 0.33)
3.62 ( 0.41) 6.36 ( 0.46) 3.08 ( 0.26)
1.71 (0.58) 1.45 (0.61) 1.34 (0.62)
2.08 (0.76) 1.84 (0.78) 1.62 (0.79)
0.11 (0.97) 0.12 (0.976) 0.12 (0.975)
0.06 (0.98) 0.07 (0.98) 0.07 (0.98)
1.36 (0.49) 1.25 (0.52) 1.05 (0.51)
2.18 (0.79) 1.93 (0.82) 1.66 (0.81)
1.14 (0.95) 0.97 (0.96) 0.84 (0.96)
1.13 (0.47) 1.06 (0.48) 0.99 (0.49)
1.47 (0.23) 1.41 (0.17) 1.41 (0.24)
3.03 ( 0.39) 2.82 ( 0.33) 2.64 ( 0.33)
3.09 ( 0.40) 5.79 ( 0.47) 2.71 ( 0.25)
1.37 (0.61) 1.23 (0.64) 1.13 (0.65)
31.83 ( 0.03) 29.42 ( 0.05) 21.63 (0.00)
26.94 (0.05) 25.04 (0.03) 18.54 (0.08)
27.44 (0.04) 25.48 (0.02) 18.85 (0.07)
0.40 (0.81) 0.51 (0.73) 0.40 (0.77)
32.31 ( 0.02) 29.88 ( 0.02) 21.95 (0.02)
27.83 (0.10) 26.00 (0.09) 19.06 (0.14)
0.63 (0.80) 0.65 (0.75) 0.58 (0.80)
3.33 (0.39) 2.99 (0.46) 3.12 (0.46)
33.80 (0.05) 27.83 (0.04) 27.44 (0.03)
35.00 (0.05) 53.59 (0.07) 43.22 (0.05)
12.09 ( 0.04) 11.42 ( 0.04) 11.54 ( 0.02)
Table 6 Performance of the proposed methods by the different maximum order sizes. Rules
FIFO
Max. order sizes
30 60 90
EDD
30 60 90
SLACK
30 60 90
SPT
30 60 90
Proposed methods
Existing methods
EEM-F
EEM-E
EEM-L
EEM-S
EM-F
EM-E
EM-S
TWK
JIQ
JIBQ
ESF
0.006 (0.999) 0.010 (0.999) 0.013 (0.999)
0.22 (0.89) 0.32 (0.78) 0.38 (0.71)
0.20 (0.91) 0.30 (0.80) 0.35 (0.75)
0.73 (0.40) 0.73 (0.41) 0.73 (0.41)
0.04 (0.997) 0.08 (0.993) 0.10 (0.992)
0.22 (0.90) 0.37 (0.80) 0.49 (0.75)
0.64 (0.39) 0.64 (0.41) 0.64 (0.42)
0.74 (0.01) 0.80 (0.00) 0.84 (0.02)
0.14 (0.993) 0.22 (0.987) 0.34 (0.94)
0.32 (0.76) 0.67 (0.71) 0.89 (0.60)
0.19 (0.89) 0.26 (0.81) 0.34 (0.77)
0.89 (0.88) 2.47 (0.73) 3.62 (0.64)
0.05 (0.99) 0.08 (0.98) 0.10 (0.97)
0.09 (0.988) 0.21 (0.97) 0.27 (0.95)
0.89 (0.52) 1.35 (0.52) 1.78 (0.52)
0.93 (0.89) 2.54 (0.77) 3.75 (0.69)
0.51 (0.98) 1.30 (0.95) 1.94 (0.93)
0.79 (0.50) 1.17 (0.49) 1.54 (0.48)
1.04 (0.21) 1.56 (0.23) 1.99 (0.28)
1.88 ( 0.46) 3.22 ( 0.31) 4.64 ( 0.23)
2.22 ( 0.46) 3.64 ( 0.37) 7.20 ( 0.31)
0.63 (0.77) 1.58 (0.58) 2.29 (0.46)
0.72 (0.89) 1.92 (0.76) 2.89 (0.67)
0.07 (0.990) 0.13 (0.97) 0.15 (0.96)
0.04 (0.99) 0.07 (0.98) 0.09 (0.97)
0.86 (0.50) 1.22 (0.51) 1.57 (0.51)
0.76 (0.90) 1.99 (0.79) 3.01 (0.72)
0.42 (0.98) 1.00 (0.95) 1.54 (0.93)
0.76 (0.48) 1.06 (0.48) 1.36 (0.47)
1.01 (0.19) 1.43 (0.20) 1.84 (0.26)
1.71 ( 0.46) 2.74 ( 0.37) 4.04 ( 0.21)
2.00 ( 0.44) 3.09 ( 0.37) 6.50 ( 0.31)
0.54 (0.79) 1.28 (0.61) 1.91 (0.50)
36.79 ( 0.05) 25.47 ( 0.02) 20.61 (0.00)
33.33 (0.00) 21.21 (0.06) 15.98 (0.11)
33.68 ( 0.01) 21.62 (0.05) 16.46 (0.09)
0.43 (0.78) 0.44 (0.77) 0.45 (0.76)
37.24 ( 0.05) 25.88 (0.00) 21.03 (0.03)
34.08 (0.04) 21.92 (0.12) 16.90 (0.17)
0.55 (0.80) 0.63 (0.79) 0.67 (0.77)
3.12 (0.40) 2.82 (0.43) 3.50 (0.48)
42.64 (0.05) 24.65 (0.04) 21.78 (0.03)
61.55 (0.07) 32.73 (0.05) 37.53 (0.05)
13.03 ( 0.03) 11.39 ( 0.03) 10.62 ( 0.04)
number of product types increases, the performance slightly gets worse. Tables 5 and 6 show the performance results as the shop configuration and the maximum order size vary, respectively. As you can see from Table 5, the proposed methods do not differ much across the shop types. The performance worsens slightly in the balanced case (5,5) of EEM-F under FIFO and the unbalanced
case (4,2) of EEM-S under SPT. Among the problem configurations, the maximum order size is the one that most strongly affects the performance of the estimation methods. Table 6 clearly shows the performance worsening for the proposed methods as the maximum order size increases. The larger the order size, the longer the stay time of the order in the shop, which could cause more errors in estimating the flowtime of the order.
8
G.-C. Lee / Omega 42 (2014) 1–8
Table 7 Summary of the main contributions. Aspect
This study
Remarks
Layout of production system Arrival of orders Scheduling schemes
Two-stage hybrid flowshop
This study considers the production system that has many applications, while many previous studies on this topic still consider simple systems, such as a single machine model
Dynamic FIFO, EDD, SLACK, SPT
This study considers more practical environments than the previous studies, where order arrivals are mostly static This study considers four different dispatching rules, to the best of our knowledge, which is the largest number of scheduling schemes considered on this topic
5. Conclusions In this study, we proposed real-time order flowtime estimation methods for two-stage hybrid flowshops, where the orders arrive dynamically and the scheduling is carried out using one of four dispatching rules, FIFO, EDD, SLACK, and SPT. For each dispatching rule, a specific flowtime estimation method was devised, considering the characteristics of the dispatching rule. However, the estimation methods have the same framework in that they each consider three types of workloads, i.e., the order that just arrived, the current inventory, and future orders. In order to validate the performance of the proposed methods, computational experiments were conducted. The results of the computational experiments demonstrate that the proposed flowtime estimation methods outperform other existing methods. The main contributions of this study are summarized in Table 7. This study can be extended in several ways. First, we need to consider estimation methods under other dispatching rules and even under batch scheduling methods. Second, new methods need to be developed for more complex production systems, such as k-stage (k Z3) hybrid flowshops, production systems with reentrant flows, and so on. In addition, the proposed methods eventually need to be employed in due-date quotation or order promising methods, which could lead to an integrated method that solves the problem of order promising and internal scheduling simultaneously.
Acknowledgment This work was supported by the Korea Research Foundation Grant funded by the Korean Government (No. KRF-2009-327B00181). References [1] Framinan JM, Leisten R. Available-to-promise (ATP) systems: a classification and framework for analysis. International Journal of Production Research 2009;48(11):3079–103.
[2] Keskinocak P, Tayur S. Due-date management policies, in handbook of quantitative supply chain analysis: modeling in the e-business era. Norwell, MA: Kluwer Academic Publishers; 2004 485–553. [3] Kaminsky P, Hochbaum D. Due-date quotation models and algorithms, handbook of scheduling: algorithms, models, and performance analysis. Boca Raton: CRC Press Inc.; 2004. [4] Gordon V, Proth JM, Chu C. A survey of the state-of-the-art of common duedate assignment and scheduling research. European Journal of Operational Research 2002;139(1):1–25. [5] Cheng TCE, Gupta MC. Survey of scheduling research involving due-date determination decisions. European Journal of Operational Research 1989;38(2):156–66. [6] Lawrence SR. Estimating flowtimes and setting due-dates in complex production systems. IIE Transactions 1995;27(5):657–68. [7] Lee GC. Estimating order lead times in hybrid flowshops with different scheduling rules. Computers and Industrial Engineering 2009;56(4): 1668–74. [8] JND Gupta. Two-stage, hybrid flowshop scheduling problem. Journal of the Operational Research Society 1988;39(4):359–64. [9] Tsubone H, Ohba M, Takamuki H, Miyake Y. A production scheduling system for a hybrid flow shop: a case study. Omega-International Journal of Management Science 1993;21(2):205–14. [10] Kim JS, Kang SH, Lee SM. Transfer batch scheduling for a two-stage flowshop with identical parallel machines at each stage. Omega-International Journal of Management Science 1997;25(5):547–55. [11] Ruiz R, Antonio J, Rodriguez V. The hybrid flow shop scheduling problem. European Journal of Operations Research 2011;205:1–18. [12] Ribas I, Leisten R, Framina JM. Review and classification of hybrid flow shop scheduling problems form a production system and a solutions procedure perspective. Computers and Operations Research 2010;37:1439–54. [13] Yao FS, Zhao M, Zhang H. Two-stage hybrid flow shop scheduling with dynamic job arrivals. Computers and Operations Research 2012;39(7): 1701–2. [14] Carpov S, Carlier J, Nace D, Sirdey R. Two-stage flow shop with precedence constraints and parallel machines at second stage. Computers and Operations Research 2012;39(3):736–45. [15] Rossi A, Puppato A, Lanzetta M. Heuristics for scheduling a two-stage hybrid flow shop with parallel batching machines: application at a hospital sterilisation plant. International Journal of Production Research 2012. http:// dx.doi.org/10.1080/00207543.2012.737942. [16] Hunsucker JL, Shah JR. Performance of priority rules in a due date flow-shop. Omega-International Journal of Management Science 1992;20(1):73–89. [17] Sha DT, Hsu SY. Due-date assignment in wafer fabrication using artificial neural networks. International Journal of Advanced Manufacturing Technology 2004;23(9–10):768–75. [18] Moses S, Grant H, Gruenwald L, Pulat P. Real-time due-date promising by build-to-order environments. International Journal of Production Research 2004;42(20):4353–75.
Contents lists available at SciVerse ScienceDirect
Omega journal homepage: www.elsevier.com/locate/omega
Real-time order flowtime estimation methods for two-stage hybrid flowshops Geun-Cheol Lee n College of Business Administration, Konkuk University, 1 Hwayang-dong, Gwangjin-gu, Seoul 143-701, Korea
a r t i c l e i n f o
abstract
Article history: Received 26 June 2011 Accepted 7 February 2013 Processed by B. Lev Available online 20 February 2013
In this study, we consider a problem of estimating order flowtimes in two-stage hybrid flowshops, where orders arrive dynamically and various scheduling schemes can be used. To solve the problem, we devise several order flowtime estimation methods, and each method is specific to the scheduling scheme used in the shop. Whenever an order arrives, the flowtime of the order is estimated by using one of the proposed methods. In the methods, we consider not only the current workload but also the expected workload in the near future, the volume of which mainly depends on the scheduling scheme. To evaluate the performance of the proposed methods, we obtained the actual flowtimes of orders from simulation runs, and compared them with the estimated flowtimes of the orders. The results of a series of computational experiments show the superior performance of the proposed methods over the several existing methods. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Flowtime estimation Hybrid flowshop Dispatching rule Simulation
1. Introduction As customer satisfaction becomes one of the most prominent goals of many companies, the problems inherent to order promising or available-to-promise (ATP) are becoming increasingly more important. In particular, if a company produces make-to-order products, delivering orders on time is regarded as an essential capability that the company must possess. However, on-time delivery is difficult to achieve in today’s dynamic business environment. Even if two orders have the same number of products with the same type and due-date, the flowtimes of these orders can vary because of different inventory status, scheduling schemes, and so on. In this paper, we propose realtime methods for estimating order flowtimes in hybrid flowshops, where orders arrive dynamically and the various scheduling rules can be used. Here, the flowtime of an order is the time interval between the arrival and the completion of the order; this interval is also called the flow allowance in many due-date assignment studies. Order flowtime estimation, order promising, ATP, and duedate assignment have been popular issues that many researchers have been focusing on. Several survey studies have already been conducted in this regard. Recently, Framinan and Leisten [1] classified the order promising approach into six categories, according to the combinations of integration of its three
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0305-0483/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.omega.2013.02.004
constituent decisions: i.e., order selection, due-date assignment, and order scheduling. Keskinocak and Tayur [2] thoroughly summarized several topics relevant to due-date management: e.g., scheduling policies, off-line vs. on-line models, service constraints, and so on. Kaminsky and Hochbaum [3] primarily reviewed analytical approaches including queuing models, which are largely limited to simple models such as static, common duedates, and single or parallel machine shops. For earlier reviews on the related fields, we refer to Gordon et al. [4] and Cheng and Gupta [5]. Although a significant number of existing studies on order promising problems exists, several practical limitations remain in those studies. First, many studies consider a very simple machine system. Particularly in cases of analytical approaches, most of them consider only a single machine model. Second, many existing studies ignore the dynamic characteristics of order arrivals, and they assume that the jobs to be produced are given. However, in practice, it is common that orders arrive in the system dynamically. Third, in previous studies, the scheduling schemes of the systems have mostly been a naive dispatching rule, e.g., FIFO (First In First Out), and this is especially common if the system involves complex processes. In this study, these limitations are overcome toward a more practical flowtime estimation. Note that, in this study, we focus on estimating the flowtimes of orders. Many previous studies on due-date assignment or order promising problems consider the estimation of flowtimes as a pre-step to assign or quote due-dates. Basically, they consider total work (TWK), jobs in queue (JIQ), or jobs in bottleneck (JIBQ)
2
G.-C. Lee / Omega 42 (2014) 1–8
for estimating flowtimes or flow allowances [5]. While TWK only considers the workload of a job itself, the inventory workload is taken into account in JIQ and JIBQ. Unlike these methods, Lawrence [6] estimates the flowtime of an order using the actual flowtimes of the latest completed orders. Recently, the same decision problem was addressed by Lee [7], whose proposed methods were devised for long hybrid flowshops and were mainly focused on the inventory workload at the bottleneck stage. In the later section, these methods are compared with the proposed methods. The contribution of this study is that we develop methods which can estimate the flowtimes of orders more accurately than the existing methods under practical environments. Furthermore, four dispatching rules are incorporated in the proposed methods, which, to the best of our knowledge, is the largest number of scheduling schemes considered in this field. The rest of this paper is structured as follows: Section 2 presents a detailed description of the considered problem. In Section 3, the proposed methods are introduced. Using the proposed methods, we conducted a series of computational experiments, the results of which are presented in Section 4. Finally, we conclude the paper with some discussions and the future directions for this study.
2. Problem description In this study, we consider two-stage hybrid flowshops with dynamic order arrival. The hybrid flowshop consists of two stages that are connected serially and each stage has its own operation for each product type. Each operation can be handled by one of the parallel machines at the stage. A product is completed after processed at stages 1 and 2 sequentially. In this study, we assume that parallel machines in the same stage have the same capabilities; hence, each stage can be referred to as an identical parallel machine system. Although the two-stage is the simplest layout of hybrid flowshops, the scheduling problem of the two-stage hybrid flowshop is well known to be NP-complete [8]. Due to its difficulty, in the earlier studies on this area, case-oriented studies were mainly performed, such as Tsubone et al. [9] and Kim et al. [10]. Nevertheless, two-stage hybrid flowshops have received much attention because of their applications, which can be easily noticed in recent survey papers [11,12]. Even very recent studies on hybrid flowshop scheduling problems continue to consider two-stage layouts [13–15]. In the considered shop, orders arrive dynamically, which means we have no information about a certain order prior to its arrival. When orders arrive, they contain information regarding which and how many products should be produced until what time. Once an order arrives in the system, it is released as multiple lots into the first stage. In this study, a lot is regarded as the minimum production unit so that the processing time of one lot of each product type is provided. The released lots immediately become part of the inventory at stage 1, if no machine is available. All the waiting lots at each stage are sequenced according to a pre-specified dispatching rule. In this study, we could use several different dispatching rules; however, only one rule is selected and used at both stages during the single production run. We assume that a lot cannot be interrupted by another lot, once a machine starts processing it; the inventory buffer size in front of the machines at each stage is infinite; and no machine failures occur. In this study, we estimate the flowtime of each order at the time the order arrives. We anticipate how the order will flow through the two-stage hybrid flowshop using the proposed estimation methods, in which the current workload, future orders, and the dispatching rule used in the system are mainly considered.
3. Proposed methods In the considered system, not only FIFO but also EDD (Earliest Due-Date), SLACK, and SPT (Shortest Processing Time) can be used as the scheduling scheme; hence, an appropriate flowtime estimation method for each scheduling scheme is necessary. In this study, we devise different flowtime estimation methods for different scheduling schemes. However, they analyze the flowtime of an order in the same manner. Fig. 1 shows the schematic breakdown of an order flowtime. The flowtime can be decomposed into two parts: flowtimes caused by processing the order itself, and workloads with higher priorities than that of the order. Workloads with higher priorities could come from the current inventory and from future orders, and those workloads should be processed prior to the order just arrived. The proposed estimation methods use different logics in obtaining the flowtimes caused by processing these workloads with higher priorities, depending on which dispatching rule is used in the shop. In the proposed methods, we employ list scheduling procedures to achieve more accurate estimation results. In each method, we select the stage with the higher inventory level as the target stage to which a list scheduling procedure is applied in order to construct a virtual schedule. Applying a list scheduling procedure to a stage is similar to solving a parallel machine scheduling problem using a dispatching rule, which can be done in a very short time.
3.1. Notation and formula We devise equations to obtain the flowtimes of orders in the proposed methods. The notations used in the flowtime estimation formula are as follows: i k
d aik pik
Lik LSdk ðOÞ
wdik
d f^ ik
d r^ i2
index of order index of stage index of dispatching rule arrival time point of order i at stage k in the shop processing time of a lot from order i at stage k set of orders waiting at stage k when order i arrives in the shop function that returns the completion time of the virtual schedule which is constructed by list scheduling on order set O at stage k under dispatching rule d partial flowtime of order i through stage k incurred by the current workload under rule d, which can be obtained by LSd1 ðfig \ Li1 Þai1 , if k¼1; LSd2 ðfig \ Li1 \ Li2 Þai1 , o/w estimated partial flowtime of order i through stage k expected to be incurred by future orders under dispatching rule d estimated ready time point of order i at stage 2 under dispatching rule d
Flowtime of an order just arrived Flowtime caused by processing workloads with higher priorities than that of the order Flowtime caused by current inventory with higher priority
Flowtime caused by future orders with higher priorities
Flowtime caused by processing the order itself
Fig. 1. Schematic breakdown of an order flowtime.
G.-C. Lee / Omega 42 (2014) 1–8
adi2
virtual arrival time point of order i at stage 2 considering only the current workload, which can be obtained while processing LSd1 ðfig \ Li1 Þ estimated arrival time point of order i at stage 2, which
a^ i2
d
can be obtained by adi2 þ f^ i1 Q(O) function that returns the number of lots in order set O max(x,y) function that returns the largest value between values x and y In the notation, much of the information is real-time data which can be obtained while running a simulation. Once an order (e.g., order i) arrives in the shop, the flowtime of order i needs to be estimated at time ai1. First, the inventory status values of both stages at time ai1, i.e., Li1 and Li2, are obtained: The function, LSdk ðOÞ, virtually performs a list scheduling on order set O at stage k with dispatching rule d (i.e., every lot included in order set O is selected according to d and then loaded and processed on one of the machines at stage k) and returns the completion time of the resulting virtual schedule. wdik can be obtained by using the virtual schedule result and the arrival time d information. The procedures for obtaining f^ ik are all different depending on which dispatching rule is used. With the inventory informad
tion, f^ i2 can be estimated in different ways according to dispatching d
d rule d. The detail procedures for f^ ik and r^ i2 are presented in the next subsection. The estimated arrival time point of order i at stage 2 ða^ i2 Þ d
consists of f^ i1 and adi2 , and the latter can be obtained while applying a list scheduling on the current workload at stage 1. Using the above notation, we could estimate the flowtime of order i under dispatching d rule d ðF^ i Þ using the following equation: 8 d > < wd þ f^ i1 þ maxðr^ di2 a^ i2 ,0Þ þpi2 , if Q ðLi1 Þ ZQ ðLi2 Þ d i1 ^F ¼ i d > d ^ : otherwise wi2 þ f i2 ,
ð1Þ
The flowtime of order i is estimated with either of the two equations in (1), depending on the inventory status of the two stages. If stage 1 has more inventory load than stage 2, the flowtime at stage 1 and the tail time at stage 2 are combined. The flowtime at stage 1 is the estimated time to process the current workload at stage 1 and future orders on machines at stage 1, and the tail time at stage 2 can be estimated by comparing the d difference between r^ i2 and a^ i2 . If there is more inventory load at stage 2, only the flowtime at stage 2 is used for the flowtime of order i in the shop. The flowtime at stage 2 is the estimated time to process the current workload at both stages and future orders on machines at stage 2. Note that, when we estimate the flowtimes, virtual schedulings are applied to the workloads with higher priorities than that of order i. Eq. (1) is used to estimate the flowtimes of orders under all considered dispatching rules. The only difference among the proposed methods is that they use d d different procedures for obtaining the estimators f^ ik and r^ i2 .
the orders arrived are dispatched in ascending order of their arrival times, which means that no lot can be processed before another lot that arrived earlier. Therefore, future orders do not FIFO affect the flowtime of an order (i) just arrived, i.e., f^ ik ¼ 0, for all FIFO values of k. Similarly, when estimating r^ i2 , only the current inventory, i.e., lots waiting at stages 1 and 2, needs to be P FIFO considered. Hence, r^ i2 can be calculated from j A ðLi1 \Li2 Þ pj2 = m2 , where mk is the number of machines at stage k.
EDD SLACK EDD SLACK 3.2.2. Estimating f^ ik , f^ ik , r^ i2 , and r^ i2 Under EDD and SLACK, a lot can be passed by other lots that arrive later in the shop. That is, some future orders would affect the flowtime of the newly arrived order, because such future
d orders have higher priorities. The estimator f^ ik represents the time to process such future orders, and this time needs to be included in the estimated flowtime of the newly arrived order. In d this study, we devise an equation to obtain the value of f^ ik . Before the equation is presented, we introduce notations used in the EDD SLACK : equation for f^ ik and f^ ik
l Q pk
rdik
d f^ ik ¼ wdik l ðQ =2Þ ðpk =mk Þ rdik
ð2Þ
Estimating the partial flowtime of future orders requires the multiplication of three terms: (1) the average number of future orders that arrive while processing the current workload ðwdik lÞ; (2) the average time required to process a certain order ððQ =2Þ ðpk =mk ÞÞ; and (3) the expected ratio of future orders having higher priorities than that of order i during wdik at stage k ðrdik Þ. The third EDD SLACK , can be obtained by another term, which differentiates f^ ik and f^ ik equation, where the following notations are used.
di Ii Imax Imin Si Smax
FIFO FIFO 3.2.1. Estimating f^ ik and r^ i2 Under FIFO, the arrival times of orders are the most important factor when estimating the flowtimes of the orders. All the lots in
mean arrival rate of orders maximum order size mean processing time of a lot at stage k expected ratio of future orders having higher priorities than that of order i during wdik at stage k
Using the above notations, under EDD or SLACK, the estimated partial flowtime of order i through stage k incurred by future orders under dispatching rule d can be estimated via the following equation:
3.2. Estimators To obtain the flowtime estimates using Eq. (1), we need to find d d the right values for the estimators (i.e., f^ ik and r^ i2 ) in the equation, and these values are mainly affected by the dispatching rule used in the shop. In the following subsections, the procedures to obtain d d f^ ik and r^ i2 for four dispatching rules are presented, respectively.
3
Smin vdi
udik
due-date of order i interval between arrival time and due-date of order i in the shop, i.e., di ai1 maximum interval between arrival times and due-dates among the orders currently in the shop minimum interval between arrival times and due-dates among the orders currently in the shop initial slack of order i, which is calculated from di ðai1 þwSLACK Þ in this study i2 maximum initial slack among the orders currently in the shop minimum initial slack among the orders currently in the shop relative urgency of order i at time ai1, which is calculated from (Ii Imin)/(Imax Imin), if d ¼EDD or (Si Smin)/ (Smax Smin), if d ¼SLACK relative urgency of order i at stage k at time ai1 þ wdik , min Þ=ðImax Imin Þ, if which is calculated from ððIi wEDD ik ÞI
d ¼EDD or ððSi wSLACK ÞSmin Þ=ðSmax Smin Þ, if d ¼SLACK ik
4
G.-C. Lee / Omega 42 (2014) 1–8
Using the above notation, under EDD or SLACK, rdik can be estimated using the following equation: 8 d d v þu > if udik Z 0 < i 2 ik , d ð3Þ rik ¼ ðvdi Þ2 > otherwise : 2ðvd ud Þ , i
ik
The estimation methods for SLACK and EDD are similar because both methods consider the relative urgencies of orders. From a dispatching point of view, they are different because the priority values of orders under EDD do not change throughout the production, while those of orders under SLACK keep changing over time. However, when we obtain the relative urgencies of orders under SLACK, we use initial slack values, which can be EDD SLACK obtained when the orders arrive. Note that r^ i2 and r^ i2 can be obtained from the sum of the second stage processing times of all the lots waiting at both stages with earlier due-dates and tighter slack values than order i, respectively.
SPT SPT 3.2.3. Estimating f^ ik and r^ i2 Like in EDD and SLACK, a lot can be passed by other lots that arrive later under SPT. However, this phenomenon occurs more often under SPT, because the priority criteria, i.e., processing times, have nothing to do with the arrival times of orders. Note that, in EDD and SLACK, due-dates and slack values of orders are somewhat affected by the arrival times of the orders, which means that lots which arrive later tend to have tighter duedates and smaller slacks. For example, a lot is rarely passed by other lots which arrive much later under EDD and SLACK; however, this can happen commonly under SPT. In the worst case, some lots can be even deadlocked if other lots with shorter processing times keep arriving. Therefore, we need to overestimate the impact of future orders SPT EDD SLACK when obtaining f^ ik , compared to obtaining f^ ik and f^ ik Eq. (2) considers the volume of the (first) future load occurring while processing the current workload, but that is not enough under SPT. In addition to the first future load, we also need to consider the additional volume of the (second) future load occurring while processing the first future load. Based on Eq. (2), the estimated partial flowtime caused by the second future load can be
calculated
from
SPT
½f^ ik l ðQ =2Þ ðpk =mk ÞrSPT ik ,
i.e.,
½wSPT ik
2
flðQ =2Þ ðpk =mk ÞrSPT ik g . Likewise, we could add the amount of 3
SPT the third future load with ½wSPT ik flðQ =2Þ ðpk =mk Þrik g , and so SPT on. Theoretically, f^ ik could be obtained by summing all of the 2
3
following future loads, i.e., fwSPT ik ðf þ f þ f þ. . .Þg, where
f ¼ l ðQ =2Þ ðpk =mk ÞrSPT ik . However, that is too excessive deliberation, therefore, in this study, we chose to consider the next SPT three future loads. Hence, the final equation for f^ ik is as follows: SPT 2 3 f^ ik ¼ wSPT ik ðf þ f þ f Þ
ð4Þ
SPT where f ¼ lðQ =2Þ ðpk =mk ÞrSPT is the ik . In this equation, rik expected ratio of the future orders having shorter processing at stage k. rSPT can be obtained times than order i during wSPT ik ik from (cik 1)/P, where cik is the rank of the processing time of a lot in order i at stage k in ascending order, and P is the number of product types, respectively. Now, we could estimate the flowtime of an order using Eq. (1) with the relevant method for the partial flowtime caused by future orders in two-stage hybrid flowshops. The next section shows the performances of the proposed methods by computational experiments of comparisons with the existing methods.
4. Computational experiments 4.1. Simulation model To see how the proposed methods work in two-stage hybrid flowshops, we developed a simulation model representing several configurations of the shop. In many studies on hybrid flowshop scheduling, simulation experiments were commonly used to evaluate the performances of the proposed methods [6,7,16–18]. The model simulates order arrivals and lot flows, such as waiting, processing, and so on. The simulation model employed in this study was developed using the computer language C. In the model, some aspects can be pre-specified by the user, and others have to be randomly generated in a certain manner. The pre-specified items are the number of product types (P), the shop type (H), the maximum size of orders (Q), and the due-date tightness of orders (g). To control the duration of each simulation run, the warm-up period and simulation end time are also prespecified. These are the parameters to be established by the user prior to the simulation run. Some randomly generated configurations are also predetermined prior to the simulation run. The processing time of one lot in order i at stage k (pik) is generated randomly from DUð1,100Þ ðmk =mÞ þDUð0,50Þ, where DU(LB,UB) is the discrete uniform distribution with range [LB,UB], and m is the average number of machines at one stage in the shop. Once the shop configuration is determined, we can simulate the arrivals and productions of orders in the shop. To simulate the arrivals of orders, the interarrival times of orders are generated exponentially according to a certain arrival rate, which is determined in accordance with the shop capability. Additionally, the product type, size, and due-date of an order are determined randomly upon its arrival. The product type of the order is arbitrarily selected as one of the P product types and the size of order i (Qi) is generated from DU(1,Q). To generate the due-date of FIFO order i, we employ the function ai þ R F^ i , where ai is the FIFO arrival time of order i in the shop and F^ i is the estimated flowtime of order i under FIFO. The due-date range of an order (R) is generated from DU(1/2g,3/2g) in this study. The developed model is a typical discrete-event simulation model, in which the simulation clock progresses according to the time of the next earliest event. Three types of events occur in the model: the order arrival, lot arrival, and processing completion. When an order arrives, the arrival time of the next order is determined. Once a lot is released to the shop, the lot becomes inventory at the first stage of the shop if no machine is available. If a machine is available, the lot is loaded onto that machine. Then the lot completion time is determined, which leads to the processing completion of the lot, i.e., the third event. The machine that becomes available finds the next lot to be processed by using the dispatching rules. In this manner, all the lots continue to be processed until the simulation clock reaches the end of the simulation. In the model, not only the shop configurations, but also the dispatching rules and the order flowtime estimation methods are implemented. Once the simulation begins, the estimation method is executed whenever a new order arrives at the shop.
4.2. Benchmark methods In this study, we used seven existing flowtime estimation methods as benchmarks to investigate the performance of the proposed methods. The first three existing methods are from Lee [7], in which estimation methods for three different dispatching rules were developed for long multi-stage (kZ30) hybrid flowshops. Those methods are similar to our proposed methods, so
G.-C. Lee / Omega 42 (2014) 1–8
that they can also be applied to the two-stage hybrid flowshop problem easily. However, the proposed methods are much more systematic and sophisticated than the existing methods. The estimation methods for FIFO, EDD, and SPT in Lee [7] are denoted as EM-F, EM-E, and EM-S, respectively. In addition to these methods, three parametric methods and one method based on exponential smoothing were also used as benchmarks. The first parametric method is TWK, which is one of the most basic and simple methods in estimating the flowtime of an order. TWK considers the total processing time of the order just arrived and determines the relationship between the flowtime and the total processing time by using linear regression. Unlike TWK, the inventory status is considered in the other parametric methods, JIQ and JIBQ. The number of all waiting lots throughout the shop is counted in JIQ, while the inventory only at the bottleneck stage is considered in JIBQ. Unlike these three parametric methods, the method based on exponentially smoothing (Exponential Smoothed Flowtime, ESF) uses the actual flowtimes of the last two completed orders. For details regarding TWK, JIQ, JIBQ and ESF, please refer to the studies of Sha and Hsu [17] and Lawrence [6].
5
4.3. Test results To test the proposed methods, we generated the problem instances with a variety of parameter combinations. We considered three levels of the product type (P), 10, 50, and 100; three levels of the shop type (H), (m1 ¼2, m2 ¼4), (5, 5), and (4, 2); three levels of the maximum order size (Q), 30, 60, and 90; and three levels of the customer due-date tightness (g), 6, 8, and 10. Five replications for each case added up to a total of 405 instances ( ¼3 3 3 3 5). Under each dispatching scheme, four proposed methods and seven previous methods were tested, which means we tested all the proposed flowtime estimation methods under each dispatching rule, even though each proposed method is devised for a specific dispatching rule. In the tables, the proposed estimation methods for FIFO, EDD, SLACK, and SPT are denoted as EEM-F, EEM-E, EEM-L, and EEM-S, respectively. Table 1 shows the overall comparison results between the proposed order flowtime estimation methods and the existing methods. To assess the performance of the flowtime estimation methods, we compared the actual flowtime, which was obtained
Table 1 Overall performance of the proposed order flowtime estimation methods under pre-specified dispatching rules. Rules
Proposed methods EEM-F
Existing methods
EEM-E
EEM-L
EEM-S
EM-F
EM-E
EM-S
TWK
JIQ
JIBQ
ESF
FIFO
ac
0.01 (0.999)bc
0.31 (0.80)
0.28 (0.82)
0.73 (0.40)
0.08 (0.994)
0.36 (0.82)
0.64 (0.41)
0.79 (0.01)
0.24 (0.97)
0.63 (0.69)
0.26 (0.83)
EDD
2.33 (0.75)
0.08 (0.98)
0.19 (0.97)
1.34 (0.52)
2.41 (0.78)
1.25 (0.95)
1.17 (0.49)
1.53 (0.24)
3.25 ( 0.33)
4.35 ( 0.38)
1.50 (0.60)
SLACK
1.85 (0.78)
0.12 (0.975)
0.07 (0.980)
1.22 (0.51)
1.92 (0.81)
0.98 (0.96)
1.06 (0.48)
1.43 (0.22)
2.83 ( 0.35)
3.86 ( 0.37)
1.24 (0.63)
SPT
27.63 ( 0.03)
23.51 (0.06)
23.92 (0.05)
0.44 (0.77)
28.05 ( 0.01)
24.30 (0.11)
0.62 (0.78)
3.15 (0.44)
29.69 (0.04)
43.94 (0.06)
11.68 ( 0.03)
a b c
Average of mean relative error ratio (RER). Average of correlation coefficient (CC) between the actual and the estimated flowtimes. Best result among 11 methods is shown in bold.
Fig. 2. Averages of mean relative error ratio of all methods under a specific dispatching rule.
6
G.-C. Lee / Omega 42 (2014) 1–8
after the corresponding order was completed, and the estimated flowtime, which was obtained by applying the proposed or existing methods immediately after the order arrived. Two measures were employed for the comparison. One is referred to as the relative error ratio (RER), representing the absolute deviation between the estimated and actual flowtimes, divided by the actual flowtime, as introduced by Moses et al. [18]. The smaller RER value refers to the more accurate estimation. The test results on RER values are also depicted as bar charts in Fig. 2, to show the clear differences in performances between the tested methods.
Table 2 Results of ANOVA on differences across the problem configuration changes.
Estimation method Dispatching rule Shop type Number of product types Maximum order size Due-date tightness Residuals a b
df
Sum of squares
Mean square
F-value
10 3 2 2 2 2 17,798
234,941.7 1,148,082.9 5976.5 2208.7 17,446.8 685.0 3,737,552.7
23,494.2 382,694.3 2988.2 1104.3 8723.4 342.5 210.0
111.9a 1822.4a 14.2a 5.3b 41.5a 1.6
Statistically different at the significance level of 0.001. Statistically different at the significance level of 0.01.
Table 3 Results of the paired t-test on difference in performance between the best and the second best methods. Rules
Best performing method
Second best performing method
Results
FIFO EDD SLACK SPT
EEM-F EEM-E EEM-L EEM-S
EM-F EEM-L EEM-E EM-S
a
a
a a a
Statistically different at the significance level of 0.001.
Another measure is the correlation coefficient (CC) between the actual and the estimated flowtimes. If the CC value is close to one, it means the estimation is close to the actual value. CC was used in the study of Sha and Hsu [17] for the same comparison. As can be seen in Table 1, the proposed methods outperformed the previous methods in all dispatching rules, except the CC criteria under SPT. The numbers in bold are the best results for each case. In terms of RER, the performances of EEM-F under FIFO (around 0.01), EEM-E under EDD, and EEM-L under SLACK (around 0.1) are relatively very good compared to other methods. The performance of EEM-S under SPT is not that outstanding, however 44% of RER and 77% of CC can be considered as fairly good results compared to other methods under SPT. To see any statistical differences across the problem configuration change, we performed the analysis of variance (ANOVA) on the mean RER values, and the results are summarized in Table 2. The table shows that, except for the due-date tightness, changes in all other configurations, (i.e., the shop type, the number of product types, and the maximum order size) affected the performances of the estimation methods. Additionally, we performed the paired t-test to find out whether the performance gap between the best and the second best estimation methods for each dispatching rule is statistically significant. In terms of mean RER, the best performing method for each dispatching rule is the corresponding proposed method. As shown in Table 1, the second best performing methods for FIFO, EDD, SLACK, and SPT, are EM-F, EEM-L, EEM-E, and EM-S, respectively. Table 3 shows that all four pairs have statistically significant differences, which means that the proposed estimation method for each dispatching rule outperforms the second-place method at a statistically significant level. To see the detail results of the test based on changes in the problem configuration, test results were summarized for each configuration type. Table 4 shows the performance results as the number of product types varies. For the proposed methods, except SPT, the performance does not significantly change depending on the number of product types. Under SPT, as the
Table 4 Performance of the proposed methods by the different number of product types. Rules
FIFO
No. of product types
10 50 100
EDD
10 50 100
SLACK
10 50 100
SPT
10 50 100
Proposed methods
Existing methods
EEM-F
EEM-E
EEM-L
EEM-S
EM-F
EM-E
EM-S
TWK
JIQ
JIBQ
ESF
0.008 (0.999) 0.010 (0.999) 0.010 (0.999)
0.29 (0.81) 0.32 (0.79) 0.31 (0.79)
0.27 (0.83) 0.30 (0.40) 0.29 (0.81)
0.73 (0.41) 0.72 (0.81) 0.74 (0.40)
0.07 (0.995) 0.08 (0.993) 0.08 (0.994)
0.33 (0.83) 0.38 (0.81) 0.37 (0.81)
0.69 (0.39) 0.61 (0.42) 0.60 (0.42)
0.74 (0.01) 0.82 (0.01) 0.82 (0.01)
0.22 (0.98) 0.24 (0.986) 0.25 (0.96)
0.56 (0.71) 0.81 (0.57) 0.52 (0.80)
0.25 (0.83) 0.27 (0.84) 0.27 (0.81)
2.03 (0.77) 2.60 (0.74) 2.36 (0.74)
0.07 (0.98) 0.08 (0.98) 0.08 (0.98)
0.17 (0.97) 0.21 (0.97) 0.20 (0.97)
1.30 (0.52) 1.38 (0.52) 1.34 (0.52)
2.10 (0.80) 2.68 (0.77) 2.44 (0.77)
1.09 (0.96) 1.39 (0.95) 1.27 (0.95)
1.25 (0.46) 1.17 (0.50) 1.09 (0.50)
1.41 (0.21) 1.67 (0.25) 1.50 (0.25)
3.07 ( 0.38) 3.58 ( 0.32) 3.10 ( 0.32)
3.76 ( 0.43) 5.56 ( 0.36) 3.75 ( 0.36)
1.33 (0.63) 1.65 (0.59) 1.51 (0.59)
1.58 (0.80) 2.08 (0.76) 1.88 (0.77)
0.11 (0.978) 0.12 (0.97) 0.12 (0.97)
0.06 (0.983) 0.08 (0.98) 0.07 (0.98)
1.19 (0.50) 1.26 (0.51) 1.21 (0.51)
1.64 (0.82) 2.16 (0.79) 1.96 (0.80)
0.84 (0.96) 1.10 (0.95) 1.00 (0.95)
1.13 (0.45) 1.07 (0.49) 0.98 (0.50)
1.29 (0.20) 1.58 (0.23) 1.42 (0.22)
2.64 ( 0.36) 3.10 ( 0.32) 2.75 ( 0.36)
3.43 ( 0.41) 4.81 ( 0.33) 3.35 ( 0.38)
1.10 (0.66) 1.39 (0.63) 1.25 (0.62)
29.80 ( 0.04) 26.35 ( 0.02) 26.73 ( 0.02)
25.54 (0.05) 22.28 (0.06) 22.71 (0.06)
25.98 (0.04) 22.68 (0.05) 23.11 (0.05)
0.40 (0.85) 0.46 (0.73) 0.45 (0.723)
30.24 ( 0.02) 26.75 (0.00) 27.16 (0.00)
26.60 (0.09) 22.98 (0.12) 23.31 (0.12)
0.59 (0.88) 0.63 (0.75) 0.63 (0.725)
3.36 (0.48) 3.25 (0.43) 2.83 (0.40)
34.16 (0.05) 31.39 (0.03) 23.52 (0.04)
48.09 (0.07) 45.50 (0.05) 38.22 (0.05)
14.21 ( 0.04) 10.57 ( 0.03) 10.26 ( 0.03)
G.-C. Lee / Omega 42 (2014) 1–8
7
Table 5 Performance of the proposed methods by the different shop types. Rules
FIFO
Shop types
(2,4) (4,2) (5,5)
EDD
(2,4) (4,2) (5,5)
SLACK
(2,4) (4,2) (5,5)
SPT
(2,4) (4,2) (5,5)
Proposed methods
Existing methods
EEM-F
EEM-E
EEM-L
EEM-S
EM-F
EM-E
EM-S
TWK
JIQ
JIBQ
ESF
0.007 (0.999) 0.007 (0.999) 0.014 (0.999)
0.32 (0.77) 0.30 (0.80) 0.30 (0.81)
0.29 (0.80) 0.28 (0.82) 0.28 (0.83)
0.77 (0.39) 0.73 (0.42) 0.68 (0.40)
0.07 (0.997) 0.07 (0.996) 0.10 (0.989)
0.37 (0.81) 0.33 (0.83) 0.38 (0.81)
0.65 (0.40) 0.63 (0.41) 0.63 (0.42)
0.78 (0.01) 0.79 (0.01) 0.81 (0.01)
0.19 (0.98) 0.34 (0.95) 0.17 (0.990)
0.18 (1.00) 1.14 (0.43) 0.57 (0.65)
0.28 (0.81) 0.26 (0.83) 0.25 (0.84)
2.70 (0.73) 2.28 (0.76) 2.01 (0.76)
0.07 (0.98) 0.08 (0.98) 0.08 (0.98)
0.21 (0.97) 0.18 (0.97) 0.18 (0.97)
1.52 (0.51) 1.38 (0.53) 1.13 (0.52)
2.81 (0.76) 2.38 (0.79) 2.03 (0.79)
1.48 (0.95) 1.23 (0.95) 1.04 (0.95)
1.26 (0.48) 1.16 (0.49) 1.08 (0.50)
1.61 (0.25) 1.49 (0.20) 1.48 (0.26)
3.54 ( 0.39) 3.19 ( 0.28) 3.02 ( 0.33)
3.62 ( 0.41) 6.36 ( 0.46) 3.08 ( 0.26)
1.71 (0.58) 1.45 (0.61) 1.34 (0.62)
2.08 (0.76) 1.84 (0.78) 1.62 (0.79)
0.11 (0.97) 0.12 (0.976) 0.12 (0.975)
0.06 (0.98) 0.07 (0.98) 0.07 (0.98)
1.36 (0.49) 1.25 (0.52) 1.05 (0.51)
2.18 (0.79) 1.93 (0.82) 1.66 (0.81)
1.14 (0.95) 0.97 (0.96) 0.84 (0.96)
1.13 (0.47) 1.06 (0.48) 0.99 (0.49)
1.47 (0.23) 1.41 (0.17) 1.41 (0.24)
3.03 ( 0.39) 2.82 ( 0.33) 2.64 ( 0.33)
3.09 ( 0.40) 5.79 ( 0.47) 2.71 ( 0.25)
1.37 (0.61) 1.23 (0.64) 1.13 (0.65)
31.83 ( 0.03) 29.42 ( 0.05) 21.63 (0.00)
26.94 (0.05) 25.04 (0.03) 18.54 (0.08)
27.44 (0.04) 25.48 (0.02) 18.85 (0.07)
0.40 (0.81) 0.51 (0.73) 0.40 (0.77)
32.31 ( 0.02) 29.88 ( 0.02) 21.95 (0.02)
27.83 (0.10) 26.00 (0.09) 19.06 (0.14)
0.63 (0.80) 0.65 (0.75) 0.58 (0.80)
3.33 (0.39) 2.99 (0.46) 3.12 (0.46)
33.80 (0.05) 27.83 (0.04) 27.44 (0.03)
35.00 (0.05) 53.59 (0.07) 43.22 (0.05)
12.09 ( 0.04) 11.42 ( 0.04) 11.54 ( 0.02)
Table 6 Performance of the proposed methods by the different maximum order sizes. Rules
FIFO
Max. order sizes
30 60 90
EDD
30 60 90
SLACK
30 60 90
SPT
30 60 90
Proposed methods
Existing methods
EEM-F
EEM-E
EEM-L
EEM-S
EM-F
EM-E
EM-S
TWK
JIQ
JIBQ
ESF
0.006 (0.999) 0.010 (0.999) 0.013 (0.999)
0.22 (0.89) 0.32 (0.78) 0.38 (0.71)
0.20 (0.91) 0.30 (0.80) 0.35 (0.75)
0.73 (0.40) 0.73 (0.41) 0.73 (0.41)
0.04 (0.997) 0.08 (0.993) 0.10 (0.992)
0.22 (0.90) 0.37 (0.80) 0.49 (0.75)
0.64 (0.39) 0.64 (0.41) 0.64 (0.42)
0.74 (0.01) 0.80 (0.00) 0.84 (0.02)
0.14 (0.993) 0.22 (0.987) 0.34 (0.94)
0.32 (0.76) 0.67 (0.71) 0.89 (0.60)
0.19 (0.89) 0.26 (0.81) 0.34 (0.77)
0.89 (0.88) 2.47 (0.73) 3.62 (0.64)
0.05 (0.99) 0.08 (0.98) 0.10 (0.97)
0.09 (0.988) 0.21 (0.97) 0.27 (0.95)
0.89 (0.52) 1.35 (0.52) 1.78 (0.52)
0.93 (0.89) 2.54 (0.77) 3.75 (0.69)
0.51 (0.98) 1.30 (0.95) 1.94 (0.93)
0.79 (0.50) 1.17 (0.49) 1.54 (0.48)
1.04 (0.21) 1.56 (0.23) 1.99 (0.28)
1.88 ( 0.46) 3.22 ( 0.31) 4.64 ( 0.23)
2.22 ( 0.46) 3.64 ( 0.37) 7.20 ( 0.31)
0.63 (0.77) 1.58 (0.58) 2.29 (0.46)
0.72 (0.89) 1.92 (0.76) 2.89 (0.67)
0.07 (0.990) 0.13 (0.97) 0.15 (0.96)
0.04 (0.99) 0.07 (0.98) 0.09 (0.97)
0.86 (0.50) 1.22 (0.51) 1.57 (0.51)
0.76 (0.90) 1.99 (0.79) 3.01 (0.72)
0.42 (0.98) 1.00 (0.95) 1.54 (0.93)
0.76 (0.48) 1.06 (0.48) 1.36 (0.47)
1.01 (0.19) 1.43 (0.20) 1.84 (0.26)
1.71 ( 0.46) 2.74 ( 0.37) 4.04 ( 0.21)
2.00 ( 0.44) 3.09 ( 0.37) 6.50 ( 0.31)
0.54 (0.79) 1.28 (0.61) 1.91 (0.50)
36.79 ( 0.05) 25.47 ( 0.02) 20.61 (0.00)
33.33 (0.00) 21.21 (0.06) 15.98 (0.11)
33.68 ( 0.01) 21.62 (0.05) 16.46 (0.09)
0.43 (0.78) 0.44 (0.77) 0.45 (0.76)
37.24 ( 0.05) 25.88 (0.00) 21.03 (0.03)
34.08 (0.04) 21.92 (0.12) 16.90 (0.17)
0.55 (0.80) 0.63 (0.79) 0.67 (0.77)
3.12 (0.40) 2.82 (0.43) 3.50 (0.48)
42.64 (0.05) 24.65 (0.04) 21.78 (0.03)
61.55 (0.07) 32.73 (0.05) 37.53 (0.05)
13.03 ( 0.03) 11.39 ( 0.03) 10.62 ( 0.04)
number of product types increases, the performance slightly gets worse. Tables 5 and 6 show the performance results as the shop configuration and the maximum order size vary, respectively. As you can see from Table 5, the proposed methods do not differ much across the shop types. The performance worsens slightly in the balanced case (5,5) of EEM-F under FIFO and the unbalanced
case (4,2) of EEM-S under SPT. Among the problem configurations, the maximum order size is the one that most strongly affects the performance of the estimation methods. Table 6 clearly shows the performance worsening for the proposed methods as the maximum order size increases. The larger the order size, the longer the stay time of the order in the shop, which could cause more errors in estimating the flowtime of the order.
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Table 7 Summary of the main contributions. Aspect
This study
Remarks
Layout of production system Arrival of orders Scheduling schemes
Two-stage hybrid flowshop
This study considers the production system that has many applications, while many previous studies on this topic still consider simple systems, such as a single machine model
Dynamic FIFO, EDD, SLACK, SPT
This study considers more practical environments than the previous studies, where order arrivals are mostly static This study considers four different dispatching rules, to the best of our knowledge, which is the largest number of scheduling schemes considered on this topic
5. Conclusions In this study, we proposed real-time order flowtime estimation methods for two-stage hybrid flowshops, where the orders arrive dynamically and the scheduling is carried out using one of four dispatching rules, FIFO, EDD, SLACK, and SPT. For each dispatching rule, a specific flowtime estimation method was devised, considering the characteristics of the dispatching rule. However, the estimation methods have the same framework in that they each consider three types of workloads, i.e., the order that just arrived, the current inventory, and future orders. In order to validate the performance of the proposed methods, computational experiments were conducted. The results of the computational experiments demonstrate that the proposed flowtime estimation methods outperform other existing methods. The main contributions of this study are summarized in Table 7. This study can be extended in several ways. First, we need to consider estimation methods under other dispatching rules and even under batch scheduling methods. Second, new methods need to be developed for more complex production systems, such as k-stage (k Z3) hybrid flowshops, production systems with reentrant flows, and so on. In addition, the proposed methods eventually need to be employed in due-date quotation or order promising methods, which could lead to an integrated method that solves the problem of order promising and internal scheduling simultaneously.
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