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Estimating order lead times in hybrid flowshops with different scheduling rules
Computers & Industrial Engineering 56 (2009) 1668–1674
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Estimating order lead times in hybrid flowshops with different scheduling rules Geun-Cheol Lee * College of Business Administration, Konkuk University, 1 Hwayang-dong Gwangjin-gu, Seoul 143-701, South Korea
a r t i c l e
i n f o
Article history: Received 24 June 2008 Received in revised form 30 October 2008 Accepted 30 October 2008 Available online 6 November 2008 Keywords: Lead time estimating Dispatching rules Hybrid flowshop Dynamic orders
a b s t r a c t In this study, we consider a problem of estimating order lead time in hybrid flowshops, where orders arrive dynamically. When an order arrives at the shop, the time duration between the arrival and completion (i.e., lead time) of the order is to be estimated. For good estimation, not only the order specification and the inventory status but also the scheduling method used in the shop should be taken into account. In this study, we consider three most common dispatching rules, i.e., FCFS, EDD, and SPT, for the shop scheduling, and develop three distinct estimation methods for the corresponding dispatching rules. A series of computational experiments were carried out and the results show that the proposed methods outperform the existing benchmarks in terms of two accuracy measures. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Today, as the customer satisfaction policy is strengthened in many companies, providing products and services on time becomes more important. In most practice environments, once a new order arrives, the due date of the order is determined after negotiation between the customer and a sales representative. When deciding the due date of the order, it is important that the promised date should be long enough for the capability of the production line as well as short enough for the customer’s expectation. This trade-off feature is already stated in several previous studies like Hopp and Sturgis (2001) and Spearman and Zhang (1999). In order to determine an appropriate due date considering the above feature, it is essential to use an accurately estimated lead time. In this study, we propose methods of estimating lead times of orders which arrive at the considered shop dynamically. A lot of studies have been done on order promising (or due date assignment) problems. Recently, the decision is also called Available-To-Promise (ATP) and many of production planning/scheduling solutions have modules that play the role of ATP. One point of view that distinguishes ATP from previous order promising is its short response time (Zhao, Ball, & Kotake 2005). Even though some manufacturing systems have products with complex process plans, order promising operations still need to be done in very short period of time. Because the order promising always requires the order lead time estimation (Moses, Grant, Gruenwald, & Pulat 2004), we need an estimation method which is not only accurate but also efficient. The scope of this study includes lead time estimating in com-
* Tel.: +82 2 450 4100; fax: +82 2 450 4141. E-mail address: [email protected]. 0360-8352/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2008.10.016
plex production environments not order promising which is left for the future research. In this study, we consider a lead time estimating problem in hybrid flowshops which are same with the traditional flowshops except there can be one more machines at each stage. Many researchers have taken into consideration in hybrid flowshop-type production systems due to their many applications. Especially the production lines of high-tech electronic products, such as semiconductor, TFT-LCD, PCB, can be considered as hybrid flowshops (Alisantoso, Khoo, & Jiang 2003; Quadt 2004). Actually, motivation of this study arises from the recent demand for good order promising methods from these industries. More studies on hybrid flowshoprelated scheduling can be found in the recent survey by Wang (2005). Although many studies related with the lead time estimating were surveyed by Gordon, Proth, and Chu (2002), the review mainly focused on single and parallel machines shops, because majority of the research dealt with those production shops. There is not much research of the lead time estimating (or due date assignment) on the complex processing systems like hybrid flowshops. Even though some researchers consider multi-resource shops for the due date assignments, most of them use FCFS (First Come First Served) as their scheduling schemes (Lawrence, 1995; Moses et al., 2004; Sha & Hsu, 2004; Spearman & Zhang, 1999). FCFS is also widely known as FIFO (First In First Out) in literature. In this study, we estimate the lead times under not only the FCFS but also two other scheduling rules. To the best of our knowledge, there is no study on lead time estimating related problems where hybrid flowshop systems with distinct dispatching rules are considered. This paper is organized as follows: in the next section, the considered problem is presented in detail and the simulation model
G.-C. Lee / Computers & Industrial Engineering 56 (2009) 1668–1674
used in this study is introduced in Section 3. In Section 4, we present the proposed estimation methods for three dispatching rules. The results of the computational experiments are presented in Section 5. The last section concludes this paper with summary and some extensions. 2. Problem description In this study, typical hybrid flowshops are considered, which can have up to 90 serial stages with identical parallel machines. In the considered shop, we assume that orders arrive dynamically. The specification of each order, i.e., its product type, size, and due date, is known on its arrival. The product type of the order is determined from one of the pre-defined product types. The order size, i.e., the number of products to be produced for the order, and the due date of the order are also given in a certain manner. Some assumptions are made to simplify the complexity of real production environments in this study: a newly arrived order is split into multiple lots at the first stage; the number of products in each lot is assumed to be one; the routing information of all product types is the same. That is, all lots must visit every stage in the same serial sequence for processing; a lot is loaded and processed on one of the parallel machines at a stage; each stage has infinite size of buffer so that there is no blocking; machine breakdown is not considered. The considered shop can have three scheduling environments depending on which dispatching rule is used. In this study, one of following dispatching rules is used for the scheduling, i.e., FCFS, EDD (Earliest Due Date), or SPT (Shortest Processing Time). We don’t consider the case where the rules are not applied in mixed ways. Rather, the same one dispatching rule is used at all stages in the shop. Please see Blackstone, Phillip, and Hogg (1982) for more information about the dispatching rules. Given the situation in which the hybrid flowshops with dynamic orders are considered, what we need to decide is the estimated lead times of orders on their arrival. For this decision, three estimation methods are proposed and their performances are compared with several existing methods. When evaluating the performance, two measures are used in this paper. One introduced by Moses et al. (2004) is a relative error ratio (RER) which Li j=Li , where Li and ^ Li are the actual and the estiis defined as jLi ^ mated lead times of order i, respectively. We can investigate the performance of an estimation method by calculating mean RER of orders which arrive and finish during the pre-specified time period in the simulation run. In this paper, the set of orders which arrive after the warm-up period and finish before the end time of the simulation run is called valid orders. Along with the mean RER, we also investigate the correlation coefficient (CC) between the actual and the estimated lead times of valid orders, which is used in Sha and Hsu (2004) for the performance comparison. In summary, in this study, we estimate the lead times of valid orders, which arrive at a hybrid flowshop, as accurate as possible, so that the mean RER and the CC of the valid orders are to be minimized and maximized, respectively, under three scheduling disciplines, i.e., FCFS, EDD, and SPT.
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In the model, some aspects can be pre-specified by the user and others have to be generated randomly in a certain fashion. The pre-specified items are the number of product types used (C), the number of stages in the considered shop (K), the maximum size of orders (Q), and the due date tightness of orders (T). To control the duration of each simulation run, warm-up period and simulation end time are also pre-specified. These are the parameters to be set by the user before the simulation run. There are some randomly generated configurations which are also determined before the simulation run. The first one is the number of machines at stage k (mk), which is randomly generated from DU(1, 10), where DU(LB, UB) is the discrete uniform distributions with range [LB, UB]. Also, the processing time of one lot in order i at stage k (pik) is randomly generated from DU(1 mk, 10 mk). Once the shop configuration is determined, now we can simulate arrivals and productions of orders in the shop. To simulate the arrivals of orders, the interarrival times of orders are exponentially generated according to a certain arrival rate, which is determined considering the shop capability. Also the product type, the size and the due date of an order is randomly determined on its arrival in the model. The product type of the order is selected arbitrarily as one of C product types and the size of order i (Qi) is generated from DU(1, Q). To generate the due date of order i, we Li , where Ai is the arrival use the following function, i.e., Ai þ T ~ time of order i at the shop and ~ Li is the predicted lead time of order i without considering inventory status. The due date tightness (T) is generated from DU(6, 10) in this study. The developed model is a typical discrete-event simulation model so that the simulation clock progresses according to the time of the next earliest event. There are three types of events in the model and the first one is the order arrival. When an order arrives, the arrival time of the next order is determined. This event always triggers the second type of event, i.e., the lot arrival. Once a lot arrives at a stage, the lot becomes inventory at the stage if there is no available machine. If there is an available machine, the lot is loaded on that machine and the lot completion time is determined, which is the third types of event, i.e., the completion of processing a lot on a machine. Once a lot is completed, the finished lot is sent to the next stage and it also triggers the lot arrival event. Then, the machine which becomes available finds the next lot to be processed using the dispatching rules. In this manner, all the lots generated by an order arrival continue to flow from the first stage to the last stage until the simulation clock reaches up to the simulation end time. In the model, not only the shop configurations but also the dispatching rules and the estimation methods introduced in the next section are implemented. Thus, the model knows which estimation method and which dispatching rule will be used before each simulation run. Once the simulation starts, the procedure of the prespecified estimation method is executed whenever a new order arrives at the shop. Newly arrived order which needs to be quoted is called as target order and any lot in the target order is called as target lot in this paper.
4. Estimation methods 3. Simulation model To see how the proposed estimation methods work in the considered shops, we develop a simulation model which can represent several configurations of hybrid flowshops. The model simulates order arrivals and lot flows, such as, waiting, processing, transferring and so on, in the pre-configured hybrid flowshops. The simulation model used in this study was developed with C language.
In this study, we develop three estimation methods and each method is devised for distinct dispatching rule used in the shop. To estimate order lead times, generally, information about the specification of the order and the shop status, i.e., inventory level, is considered. Additionally, in this study, we consider which dispatching rule is used in the shop for more accurate estimation. Fig. 1 shows the input and output of the suggested estimation methods.
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preceding the bottleneck as the inventory of the bottleneck stage. Using the above notation, we could estimate the lead time of order a under FCFS, with the following equation:
Order specification (Process plan, due date, etc.)
Shop information 1. Inventory status (Number of waiting and processed lot at each stage)
Estimation Method
2. Scheduling scheme (Dispatching rule used at the shop)
Estimated lead time of the arrived order
^La ¼ F b þ a
K X
k þ pak Þ þ paK Q a =mK ðw
ð1Þ
k¼bþ1
The equation is composed of three terms. The first term, i.e., F ba , represents the estimated flow time of the target order from stage 1 to stage b. The second term represents the estimated flow time of k can be obone lot from stage b + 1 to stage K. In the parenthesis, w tained from the periodic statistics gathering during the simulation run. The third term estimates the flow time of the target order at the last stage in the shop.
Fig. 1. Input/output of estimation method.
4.2. Estimation method for EDD (EM-E) Actually, the lead time of a target order is mainly affected by the current inventory level rather than the time to process the order. Such a phenomenon is more vivid if the shop is highly utilized and has long serial processes like the hybrid flowshops. Thus, we need to measure the amount of the current inventory and the time to complete that inventory. However, among the current inventory throughout the shop, not every waiting lot would cause the waiting of the target order. The impact of the inventory on the target order would change depending on where the inventory is located and which dispatching rule is used in the shop. Even orders which arrive in the future could affect the flow of the target order. In the following, we present how each proposed method considers the inventory for the lead time estimation.
Unlike in FCFS, some of target lots can be passed by other lots whose orders arrived later than the target order at the shop in EDD environment. Thus, we need to consider not only the current inventory status but also orders with earlier due dates which arrive in the near future. However, in this study, we do not analyze the exact arrival pattern of future orders for the estimation, rather we consider the relative urgency of the target order. For the equation of the EM-E, we use the equation of the EM-F and add/deduct a certain amount of time according to the relative urgency of the target order. Before introducing the equation of the EM-E, notation is described as follows:
4.1. Estimation method for FCFS (EM-F)
Ai Di I
Under FCFS, lots are dispatched in order of their arrival times and no lot can be processed earlier than another lot which arrived earlier. Therefore, the amount of the current inventory, i.e., number of waiting lots, in the shop is a major factor when estimating the lead time of a target order. Before introducing the equation for estimating lead time, notation used in the equation is described as follows: A pik mk Qi qkn
Nk k p b k w F bi
index of newly arrived order which needs to be quoted, i.e., index of a target order. processing time of a lot in order i at stage k. number of machines at stage k. number of lots in order i, i.e., order size. order index of nth waiting lot at stage k, (e.g., the lot with the second earliest arrival time among the waiting lots at stage 1 is an element of order q12). number of waiting lots at stage k. mean processing time of a lot at stage k. stage index of the bottleneck stage, which can be obtained as k =mk g. arg maxk fp average waiting time of a lot at stage k. estimated flow time of order i from the first stage to the bottleneck stage, which can be obtained as Nk b P P pqkn b þ pib Q i Þ=mb . ð k¼1 n¼1
In the notation, information about the inventory status, such as, qkn and Nk, is real-time data at the time when the target order arrives. That is, we know the number of waiting lots and the specification of each waiting lot for every stage, before the lead time estimation. The bottleneck stage plays a big role in the estimation, because much of inventory in the shop is expected to be located in the bottleneck. Therefore, we consider the inventory of all stages
ui
arrival time of order i at the shop. due date of order i. average interval between arrival times and due dates of orders which have arrived so far. relative urgency of order i, which is set as ðDi Ai Þ I.
In the notation, ui represents the relative urgency of order i. If the value is negative, it means the order has relatively high urgency so that we can expect that the lead time would be shorter. Otherwise, we can expect that the order would stay longer in the shop due to its low urgency. Under EDD, the lead time of order a can be estimated by the following equation:
( ^La ¼
F ba
þ
K X
) k þ pak Þ þ paK Q a =mK ðw
þ ua
ð2Þ
k¼bþ1
Note that, Eqs. (1) and (2) are the same except the relative urgency term so that the performance of EM-E is largely dependent on that of EM-F. 4.3. Estimation method for SPT (EM-S) SPT environment is much different from FCFS/EDD environments. First, the dispatching priority value of a lot can fluctuate throughout the shop, because processing times of a lot are different depending on stages. Also, under SPT, priorities are absolute values while priorities by EDD are somewhat proportion to the arrival times. That means, in SPT environment, there is a possibility that target lots can be passed by other lots which arrive way far later than the target lot, which is rarely happened in EDD environment. In an extreme case, a lot with long processing time can be deadlocked at a certain stage, if other lots with shorter processing times continue to arrive. Under SPT environment, two sources of workload can affect the lead time of a target order. One is the current inventory and the other is the future orders. First, we propose the equation where
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only the current inventory is considered. Before introducing the equation, notation is described as follows:
Uik W cik
set of the waiting lots at stage k, whose processing times are shorter than or equal to that of a lot in order i. estimated waiting time of order i at stage k due to some of P pqkn k =mk . the current inventory and can be obtained as n2Uik
ð3Þ
k¼1
The above equation shows the lead time of a target order under SPT can be estimated by the sum of the estimated waiting time of the target order and the processing time of a lot in the target order for all stages. However, for more accurate estimation, we need to devise an equation which considers the future orders, because they can also affect the lead time of the target order. Before introducing the equation which considering both the current inventory and the future orders, notation is introduced as follows.
K Q cik
C lik
Sik zik = 1, W fik
^La ¼
K X ðW cak þ W fak þ pak Þ
ð4Þ
k¼1
Because, at a certain stage, lots in the target order have to wait until all inventory lots with the shorter and equal processing times are at least finished, we first find the set of such waiting lots at the stage, and then compute the sum of the processing times of all lots in the set. Considering only the current inventory, we can introduce the equation for estimating the lead time of target order a under SPT as follows: K X ^La ¼ ðW cak þ pak Þ
k =mk Þ. In summary, the waitW cik Sik þ W cik S2ik , where Sik ¼ lik ðp ing time of order i at stage k, which affected by some of the future orders is estimated as W cik Sik ð1 þ Sik zik Þ. Adding this portion to Eq. (3) for all stage, the final equation for estimating the lead time of target order a under SPT is proposed as follows:
mean arrival rate of orders. maximum order size. rank of the processing time of a lot in order i at stage k in ascending order. (e.g., if a lot in order i has the second shortest processing time at stage k, then cik = 2). number of product types. expected number of lots per unit time, which will arrive and probably causethe waiting of order i at stage k, and can be obtained as k Q2 cikC1 . estimated time to process workload lik, which can be k p obtained as lik m . k if the processing time of a lot in order i is the longest at stage k; 0, o/w. estimated waiting time of order i at stage k due to some of the future orders, and can be obtained as W cik Sik þ W cik S2ik zik .
To find out the waiting time of a target order by the future orders at a stage, we first compute the amount of lots which inflow into at the stage averagely, that can be obtained by multiplying the mean arrival rate of orders and the average order size, i.e., k(Q/2). Among them, the portion of lots with processing times shorter than that of a lot in target order i at stage k would be (cik 1)/C, thus fk ðQ =2Þg fðcik 1Þ=C g is the amount of future lots actually causing the waiting of the target order per unit time, i.e., lik. Thus, the number of lots that will arrive at stage k during W cik , i.e., the waiting time of order i by the current inventory, would be lik W cik . k =mk , the Because the average tact time of a lot at stage k is p k =mk Þ and this time to consume lik W cik lots would be lik W cik ðp can be considered as the estimated waiting time of order i at stage k due to some of the future orders, i.e., W fik . However, additional fuk =mk Þ at some ture orders could arrive again during lik W cik ðp k =mk Þþ could be extended as lik W cik ðp stages, W fik 2 k =mk Þ2 . In this study, we use lik W cik ðp k =mk Þ for W fik lik W cik ðp for all stages except one stage where the target lot has the longest processing time. For that stage, W fik is obtained from
Using the Eqs. (1), (2), and (4), we can estimate the lead time of the target order under three different dispatching rules. Because the estimated lead time is computed from the closed form equations, the results will be obtained instantly.
5. Performance evaluation 5.1. Benchmark methods In this study, we use four existing lead time estimation methods to investigate the performances of the proposed methods. Among the benchmark methods, three of them are parametric methods so that the parameters, k1 and k2, are determined by the linear regression method. These methods were used in many due date assignment related works including Sha and Hsu (2004). The fourth one introduced by Lawrence (1995) is based on the exponential smoothing method where the lead time is estimated with the latest and the second latest completed orders’ lead times. Each method is described briefly with the equation in the following subsections. 5.1.1. Total Workload (TWK) TWK is one of the most basic and simple methods in estimating the lead time of an order. Including TWK, some other basic methods are presented in the survey paper by Cheng and Gupta (1989). It just considers the total processing time of the target order and find out the relationship between the lead time and the total processing time with the linear regression method. The equation of TWK is as follows:
^La ¼ k1
K X
pak þ k2
ð5Þ
k¼1
5.1.2. Job In Queue (JIQ) Unlike TWK, inventory status, i.e., the number of waiting lots throughout the shop, is considered in JIQ. The equation of JIQ is as follows:
^La ¼ k1
Nk K X X
pqkn k þ
k¼1 n¼1
K X
pak þ k2
ð6Þ
k¼1
5.1.3. Job In Bottleneck Queue (JIBQ) Instead all the inventory workload is considered in JIQ, in this method, the inventory only at the bottleneck stage is considered. The equation of JIBQ is as follows:
^La ¼ k1
Nb X n¼1
pqbn b þ
K X
pak þ k2
ð7Þ
k¼1
5.1.4. Exponentially Smoothed Flow time (ESF) Unlike the above three parametric methods, this method uses two of the past history data of the order lead times. In ESF, at the time of estimating, the actual lead times of two orders that completed latest and completed second latest are used. Let L1 and L2 be the lead time of the latest and the second latest completed or-
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ders, respectively. The equation of ESF is as follows (in the equation, the value of smoothing constant a is set to 0.5 after the pre-test with several alternative values):
^La ¼ a L þ ð1 aÞ L 1
2
ð8Þ
Among the benchmark methods, ESF is expected to show good performance, if the considered production system has long serial processes like hybrid flowshops. However such reasoning would be valid, only if the scheduling scheme is FCFS. 5.2. Test results For each dispatching rule, we investigated the performance of the corresponding proposed estimation method and those of the benchmark methods. Also, the other two proposed methods were tested for that rule environment to see the results of inappropriate estimations. For example, under EDD, we tested not only EM-E and four benchmarks, but also EM-F and EM-S. Therefore, for one simulation instance, combinations of three dispatching rules and seven estimation methods, i.e., 21 combinations were tested. We increased the number of the simulation instances by varying the number of stages, the number of product types, and the maximum order size as follows: three levels of stage size, i.e., 30, 60, 90; three levels of product type size, i.e., 10, 50, 100; two levels of maximum order size, i.e., 50, 100. Additional ten replications for each combination led the total number of instances to 180 (3 3 2 10). For each simulation run, mean RER and CC were measured for the set of valid orders. In the following tables, the method which shows the mean RER close to zero and CC close to one is the better performing method. Summary of the overall test results is presented in Table 1. Under each dispatching rule, the corresponding proposed estimation method showed the best performances in terms of both measures (the best result in each case is written in bold). Among the benchmarks, ESF showed the fairly good performance in FCFS and EDD environments comparing to the proposed methods. Under SPT, most methods showed poor results except the EM-S in both mean RER and CC. Although the mean RER result of EM-S (i.e., about 33%) is still away from the best, that of other methods are much larger. Moreover, in terms of CC, EM-S made the correlation coefficient of the actual and estimated lead times close up to 90%, which is quite an achievement considering other numbers just hang around 0%. To see any performance variations according to the problem conditions, we performed the ANOVA on mean RER values for each dispatching rule environment and the results are summarized in Table 2. As you can see from the table, we notice that the performance of the methods varies across the problem conditions in most cases. To see the pattern of the performance change, the detail results of the test are summarized depending on the problem condition. Table 3 shows the performance results as the number of stages varies. Under FCFS and EDD conditions, most methods
Table 2 ANOVA results. Source of variation (a) Under FIFO Method (m) Number of stages (s) Number of product types (p) Max order size (o) ms mp mo Residuals (b) Under EDD Method (m) Number of stages (s) Number of product types (p) Max order size (o) ms mp mo Residuals (c) Under SPT Method (m) Number of stages (s) Number of product types (p) Max order size (o) ms mp mo Residuals
EM-F
EM-E
EM-S
TWK
FCFS
0.11a,c (0.99)b
0.14 (0.94)
0.36 (0.58)
0.63 (0.00)
0.32 (0.79)
0.56 (0.22)
0.16 (0.95)
EDD
0.13 (0.93)
0.10 (0.98)
0.36 (0.57)
0.65 (0.09)
1.36 (0.84)
0.84 (0.06)
0.18 (0.87)
SPT
2.66 (0.02)
2.62 (0.04)
0.33 (0.87)
0.66 (0.15)
6.17 (0.01)
6.49 (0.03)
1.75 (0.02)
a b c
JIQ
JIBQ
Mean square
F value
6 2 2
46.65 0.06 0.02
7.77 0.03 0.01
500.13 2.02 0.56
***
1 12 12 6 1218
0.23 0.50 1.22 0.22 18.93
0.23 0.04 0.10 0.04 0.02
14.88 2.70 6.56 2.34 NA
***
6 2 2
231.20 1.88 3.76
38.53 0.94 1.88
491.11 11.96 23.98
***
1 12 12 6 1218
0.26 8.71 8.43 0.35 95.57
0.26 0.73 0.70 0.06 0.08
3.31 9.25 8.96 0.75 NA
6 2 2
6586.53 1137.22 2649.13
1097.75 568.61 1324.57
99.03 51.29 119.49
***
1 12 12 6 1218
483.01 801.31 2754.56 433.14 13501.70
483.01 66.78 229.55 72.19 11.09
43.57 6.02 20.71 6.51 NA
***
** *** *
*** ***
*** ***
*** ***
*** *** ***
Statistically different at the significance level of 0.05. Statistically different at the significance level of 0.01. *** Statistically different at the significance level of 0.001. **
Table 3 The performance results for the different number of stages. Rules
Number of stages
FCFS
30 60 90
EDD
30 60 90
SPT
30
90
Rules
Sum of square
*
60 Table 1 Overall test results.
Degree of freedom
EM-F
EM-E
EM-S
TWK
JIQ
JIBQ
ESF
0.08 (0.99) 0.10 (0.99) 0.14 (0.99)
0.11 (0.95) 0.13 (0.94) 0.17 (0.93)
0.41 (0.51) 0.34 (0.60) 0.32 (0.64)
0.61 (0.01) 0.64 (0.01) 0.64 (0.00)
0.34 (0.80) 0.30 (0.80) 0.32 (0.76)
0.56 (0.23) 0.56 (0.26) 0.58 (0.16)
0.14 (0.94) 0.16 (0.94) 0.18 (0.97)
0.11 (0.94) 0.12 (0.93) 0.16 (0.92)
0.07 (0.98) 0.09 (0.98) 0.13 (0.99)
0.41 (0.51) 0.34 (0.58) 0.33 (0.61)
0.63 (0.07) 0.65 (0.09) 0.66 (0.11)
1.06 (0.76) 1.38 (0.86) 1.63 (0.90)
0.87 (0.13) 0.80 (0.04) 0.85 (0.08)
0.17 (0.88) 0.18 (0.87) 0.20 (0.87)
3.95 (0.01) 2.49 (0.01) 1.53 (0.07)
3.87 (0.02) 2.47 (0.03) 1.52 (0.09)
0.34 (0.88) 0.34 (0.85) 0.31 (0.88)
0.83 (0.16) 0.62 (0.13) 0.55 (0.15)
8.12 (0.01) 6.12 (0.01) 4.25 (0.03)
9.89 (0.05) 5.33 (0.05) 4.25 (0.00)
2.54 (0.01) 1.55 (0.01) 1.16 (0.04)
ESF
Average of mean relative error ratios. Average of correlation coefficients between actual and estimated lead times. Best result in each case is shown in bold.
including the best performed methods (i.e., EM-F and EM-E in each case) showed the mean RER increase, which means the performance worsening, as the number of stages increases. On the contrary, under SPT, the performances of most methods are getting better, as the number of stages increases. Comparing to the mean RER, CC values (in parenthesis) do not change much even though the problem condition varies. Note that the above ANOVA tests were done on RER mean values.
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G.-C. Lee / Computers & Industrial Engineering 56 (2009) 1668–1674 Table 4 The performances results for the different number of product types. Rules
Number of product types
FCFS
EM-F
10 50 100
EDD
10 50 100
SPT
10 50 100
EM-E
FCFS
Max order size 50 100
EDD
50 100
SPT
50 100
TWK
JIQ
JIBQ
ESF
0.05 (1.00) 0.14 (0.98) 0.14 (0.99)
0.09 (0.97) 0.16 (0.92) 0.16 (0.93)
0.41 (0.55) 0.35 (0.56) 0.31 (0.64)
0.66 (0.00) 0.62 (0.01) 0.61 (0.00)
0.28 (0.94) 0.36 (0.67) 0.32 (0.75)
0.59 (0.22) 0.55 (0.29) 0.56 (0.15)
0.19 (0.97) 0.15 (0.93) 0.15 (0.94)
0.10 (0.96) 0.15 (0.91) 0.14 (0.92)
0.05 (0.99) 0.12 (0.98) 0.12 (0.98)
0.43 (0.54) 0.34 (0.54) 0.30 (0.62)
0.68 (0.06) 0.64 (0.10) 0.63 (0.09)
1.52 (0.85) 1.31 (0.86) 1.24 (0.81)
1.12 (0.23) 0.81 (0.04) 0.58 (0.09)
0.21 (0.92) 0.17 (0.84) 0.17 (0.85)
4.30 (0.04) 2.33 (0.02) 1.35 (0.10)
4.26 (0.04) 2.28 (0.04) 1.31 (0.13)
0.28 (0.88) 0.34 (0.88) 0.37 (0.85)
0.77 (0.21) 0.62 (0.12) 0.60 (0.11)
10.40 (0.03) 4.33 (0.00) 3.77 (0.06)
12.26 (0.05) 4.87 (0.05) 2.34 (0.00)
2.41 (0.01) 1.59 (0.02) 1.25 (0.05)
Table 5 The performances results for the different max order sizes. Rules
EM-S
EM-F
EM-E
EM-S
TWK
JIQ
JIBQ
ESF
0.09 (0.99) 0.13 (0.98)
0.12 (0.95) 0.16 (0.92)
0.36 (0.58) 0.35 (0.59)
0.63 (0.00) 0.63 (0.01)
0.31 (0.86) 0.32 (0.72)
0.52 (0.28) 0.60 (0.16)
0.15 (0.97) 0.17 (0.93)
0.11 (0.94) 0.15 (0.91)
0.08 (0.99) 0.12 (0.98)
0.36 (0.57) 0.36 (0.57)
0.64 (0.10) 0.66 (0.08)
1.31 (0.92) 1.41 (0.77)
0.85 (0.10) 0.83 (0.02)
0.17 (0.90) 0.20 (0.84)
3.22 (0.01) 2.10 (0.04)
3.18 (0.02) 2.06 (0.06)
0.32 (0.89) 0.35 (0.85)
0.69 (0.17) 0.64 (0.13)
7.83 (0.01) 4.50 (0.02)
7.76 (0.05) 5.22 (0.02)
2.01 (0.01) 1.49 (0.03)
Tables 4 and 5 show the performance results for the different number of product types and maximum order size used in the tests. Similar with the results of Table 3, under FCFS and EDD, the performance slightly getting worse in most methods, as the product types and the maximum order size are getting bigger. In SPT case, conversely, the mean RER of all the methods except the best performing method (EM-S) are getting smaller. Still, the CC values do not show any clear up- and down-tendency according to the problem conditions. From the computational tests, we can verify the outperformance of the proposed methods. In various cases, the proposed methods always show the most superior results in terms of both accuracy measures. Particularly, the CCs of EM-F/E show almost perfect scores (i.e., one) under FCFS/EDD and the performance of EM-S is especially outstanding compared to others under SPT. 6. Conclusions In this paper, we consider estimation of the order lead time in hybrid flowshops with different dispatching rules. Estimating the accurate lead times is essential to assign appropriate due date of the orders. One of the major factors affecting the lead time is scheduling policy used in the production systems. The complexity of the estimation problem have forced most previous studies assume the FCFS policy as their scheduling schemes. In this study, we proposed three lead time estimation methods which can cope
with three scheduling conditions, i.e., FCFS, EDD, and SPT. Each proposed method is devised considering the workload which might have effects on the flow of the target order. The computational results using the simulation model showed the superiorities of the proposed methods over the benchmarks under any of three dispatching rules. The proposed methods can be directly applied to order promising (or due date assignment) problems. A naive approach of the due date assignment could be just adding the estimated lead time of an order to the arrival time, i.e., Di ¼ Ai þ ^Li . In addition, the ideas of the suggested methods can be used in general scheduling methods. Many scheduling algorithms, even simple dispatching rules, use the remaining flow time of a job as key information in their logic. More accurate information about the remaining lead time would enhance the performance of the scheduling methods. Even though the suggested lead time estimation methods consider dispatching rules other than FCFS, they are still simple dispatching discipline. Thus, further research about the lead time estimating under general scheduling schemes is needed. Because decisions of the scheduling and the lead time estimating affect each other, some kinds of integrated approaches would be expected eventually, which can show good performance in terms of both scheduling and estimation measures. Acknowledgement This work was supported by the faculty research fund of Konkuk University in 2006. References Alisantoso, D., Khoo, L. P., & Jiang, P. Y. (2003). An immune algorithm approach to the scheduling of a flexible PCB flow shop. International Journal of Advanced Manufacturing Technology, 22, 819–827. Blackstone, J. H., Phillip, D. T., & Hogg, D. L. Jr., (1982). A state-of-the-art survey of dispatching rules for manufacturing jobshop operations. International Journal of Production Research, 25, 1143–1156. Cheng, T. C. E., & Gupta, M. C. (1989). Survey of scheduling research involving due date determination decisions. European Journal of Operational Research, 38, 156–166. Gordon, V., Proth, J. M., & Chu, C. (2002). A survey of the state-of-the-art of common due date assignment and scheduling research. European Journal of Operational Research, 139, 1–25. Hopp, W. J., & Sturgis, M. R. (2001). A simple, robust leadtime-quoting policy. Manufacturing & Service Operations Management, 3(4), 321–336. Lawrence, S. R. (1995). Estimating flowtimes and setting due-dates in complex production systems. IIE Transactions, 27, 657–668.
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Moses, S., Grant, H., Gruenwald, L., & Pulat, P. (2004). Real-time due-date promising by build-to-order environments. International Journal of Production Research, 42, 4353–4375. Quadt, D. (2004). Lot-sizing and scheduling for flexible flow lines. Berlin Germany: Springer. Sha, D. T., & Hsu, S. Y. (2004). Due-date assignment in wafer fabrication using artificial neural networks. International Journal of Advanced Manufacturing Technology, 23, 768–775.
Spearman, M. L., & Zhang, R. Q. (1999). Optimal lead time policies. Management Science, 45, 290–295. Wang, H. (2005). Flexible flow shop scheduling: optimum, heuristics and artificial intelligence solutions. Expert Systems, 22(2), 78–85. Zhao, Z. M., Ball, O., & Kotake, M. (2005). Optimization based available-to-promise with multi-stage resource availability. Annals of Operational Research, 135, 65–85.
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Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie
Estimating order lead times in hybrid flowshops with different scheduling rules Geun-Cheol Lee * College of Business Administration, Konkuk University, 1 Hwayang-dong Gwangjin-gu, Seoul 143-701, South Korea
a r t i c l e
i n f o
Article history: Received 24 June 2008 Received in revised form 30 October 2008 Accepted 30 October 2008 Available online 6 November 2008 Keywords: Lead time estimating Dispatching rules Hybrid flowshop Dynamic orders
a b s t r a c t In this study, we consider a problem of estimating order lead time in hybrid flowshops, where orders arrive dynamically. When an order arrives at the shop, the time duration between the arrival and completion (i.e., lead time) of the order is to be estimated. For good estimation, not only the order specification and the inventory status but also the scheduling method used in the shop should be taken into account. In this study, we consider three most common dispatching rules, i.e., FCFS, EDD, and SPT, for the shop scheduling, and develop three distinct estimation methods for the corresponding dispatching rules. A series of computational experiments were carried out and the results show that the proposed methods outperform the existing benchmarks in terms of two accuracy measures. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Today, as the customer satisfaction policy is strengthened in many companies, providing products and services on time becomes more important. In most practice environments, once a new order arrives, the due date of the order is determined after negotiation between the customer and a sales representative. When deciding the due date of the order, it is important that the promised date should be long enough for the capability of the production line as well as short enough for the customer’s expectation. This trade-off feature is already stated in several previous studies like Hopp and Sturgis (2001) and Spearman and Zhang (1999). In order to determine an appropriate due date considering the above feature, it is essential to use an accurately estimated lead time. In this study, we propose methods of estimating lead times of orders which arrive at the considered shop dynamically. A lot of studies have been done on order promising (or due date assignment) problems. Recently, the decision is also called Available-To-Promise (ATP) and many of production planning/scheduling solutions have modules that play the role of ATP. One point of view that distinguishes ATP from previous order promising is its short response time (Zhao, Ball, & Kotake 2005). Even though some manufacturing systems have products with complex process plans, order promising operations still need to be done in very short period of time. Because the order promising always requires the order lead time estimation (Moses, Grant, Gruenwald, & Pulat 2004), we need an estimation method which is not only accurate but also efficient. The scope of this study includes lead time estimating in com-
* Tel.: +82 2 450 4100; fax: +82 2 450 4141. E-mail address: [email protected]. 0360-8352/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2008.10.016
plex production environments not order promising which is left for the future research. In this study, we consider a lead time estimating problem in hybrid flowshops which are same with the traditional flowshops except there can be one more machines at each stage. Many researchers have taken into consideration in hybrid flowshop-type production systems due to their many applications. Especially the production lines of high-tech electronic products, such as semiconductor, TFT-LCD, PCB, can be considered as hybrid flowshops (Alisantoso, Khoo, & Jiang 2003; Quadt 2004). Actually, motivation of this study arises from the recent demand for good order promising methods from these industries. More studies on hybrid flowshoprelated scheduling can be found in the recent survey by Wang (2005). Although many studies related with the lead time estimating were surveyed by Gordon, Proth, and Chu (2002), the review mainly focused on single and parallel machines shops, because majority of the research dealt with those production shops. There is not much research of the lead time estimating (or due date assignment) on the complex processing systems like hybrid flowshops. Even though some researchers consider multi-resource shops for the due date assignments, most of them use FCFS (First Come First Served) as their scheduling schemes (Lawrence, 1995; Moses et al., 2004; Sha & Hsu, 2004; Spearman & Zhang, 1999). FCFS is also widely known as FIFO (First In First Out) in literature. In this study, we estimate the lead times under not only the FCFS but also two other scheduling rules. To the best of our knowledge, there is no study on lead time estimating related problems where hybrid flowshop systems with distinct dispatching rules are considered. This paper is organized as follows: in the next section, the considered problem is presented in detail and the simulation model
G.-C. Lee / Computers & Industrial Engineering 56 (2009) 1668–1674
used in this study is introduced in Section 3. In Section 4, we present the proposed estimation methods for three dispatching rules. The results of the computational experiments are presented in Section 5. The last section concludes this paper with summary and some extensions. 2. Problem description In this study, typical hybrid flowshops are considered, which can have up to 90 serial stages with identical parallel machines. In the considered shop, we assume that orders arrive dynamically. The specification of each order, i.e., its product type, size, and due date, is known on its arrival. The product type of the order is determined from one of the pre-defined product types. The order size, i.e., the number of products to be produced for the order, and the due date of the order are also given in a certain manner. Some assumptions are made to simplify the complexity of real production environments in this study: a newly arrived order is split into multiple lots at the first stage; the number of products in each lot is assumed to be one; the routing information of all product types is the same. That is, all lots must visit every stage in the same serial sequence for processing; a lot is loaded and processed on one of the parallel machines at a stage; each stage has infinite size of buffer so that there is no blocking; machine breakdown is not considered. The considered shop can have three scheduling environments depending on which dispatching rule is used. In this study, one of following dispatching rules is used for the scheduling, i.e., FCFS, EDD (Earliest Due Date), or SPT (Shortest Processing Time). We don’t consider the case where the rules are not applied in mixed ways. Rather, the same one dispatching rule is used at all stages in the shop. Please see Blackstone, Phillip, and Hogg (1982) for more information about the dispatching rules. Given the situation in which the hybrid flowshops with dynamic orders are considered, what we need to decide is the estimated lead times of orders on their arrival. For this decision, three estimation methods are proposed and their performances are compared with several existing methods. When evaluating the performance, two measures are used in this paper. One introduced by Moses et al. (2004) is a relative error ratio (RER) which Li j=Li , where Li and ^ Li are the actual and the estiis defined as jLi ^ mated lead times of order i, respectively. We can investigate the performance of an estimation method by calculating mean RER of orders which arrive and finish during the pre-specified time period in the simulation run. In this paper, the set of orders which arrive after the warm-up period and finish before the end time of the simulation run is called valid orders. Along with the mean RER, we also investigate the correlation coefficient (CC) between the actual and the estimated lead times of valid orders, which is used in Sha and Hsu (2004) for the performance comparison. In summary, in this study, we estimate the lead times of valid orders, which arrive at a hybrid flowshop, as accurate as possible, so that the mean RER and the CC of the valid orders are to be minimized and maximized, respectively, under three scheduling disciplines, i.e., FCFS, EDD, and SPT.
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In the model, some aspects can be pre-specified by the user and others have to be generated randomly in a certain fashion. The pre-specified items are the number of product types used (C), the number of stages in the considered shop (K), the maximum size of orders (Q), and the due date tightness of orders (T). To control the duration of each simulation run, warm-up period and simulation end time are also pre-specified. These are the parameters to be set by the user before the simulation run. There are some randomly generated configurations which are also determined before the simulation run. The first one is the number of machines at stage k (mk), which is randomly generated from DU(1, 10), where DU(LB, UB) is the discrete uniform distributions with range [LB, UB]. Also, the processing time of one lot in order i at stage k (pik) is randomly generated from DU(1 mk, 10 mk). Once the shop configuration is determined, now we can simulate arrivals and productions of orders in the shop. To simulate the arrivals of orders, the interarrival times of orders are exponentially generated according to a certain arrival rate, which is determined considering the shop capability. Also the product type, the size and the due date of an order is randomly determined on its arrival in the model. The product type of the order is selected arbitrarily as one of C product types and the size of order i (Qi) is generated from DU(1, Q). To generate the due date of order i, we Li , where Ai is the arrival use the following function, i.e., Ai þ T ~ time of order i at the shop and ~ Li is the predicted lead time of order i without considering inventory status. The due date tightness (T) is generated from DU(6, 10) in this study. The developed model is a typical discrete-event simulation model so that the simulation clock progresses according to the time of the next earliest event. There are three types of events in the model and the first one is the order arrival. When an order arrives, the arrival time of the next order is determined. This event always triggers the second type of event, i.e., the lot arrival. Once a lot arrives at a stage, the lot becomes inventory at the stage if there is no available machine. If there is an available machine, the lot is loaded on that machine and the lot completion time is determined, which is the third types of event, i.e., the completion of processing a lot on a machine. Once a lot is completed, the finished lot is sent to the next stage and it also triggers the lot arrival event. Then, the machine which becomes available finds the next lot to be processed using the dispatching rules. In this manner, all the lots generated by an order arrival continue to flow from the first stage to the last stage until the simulation clock reaches up to the simulation end time. In the model, not only the shop configurations but also the dispatching rules and the estimation methods introduced in the next section are implemented. Thus, the model knows which estimation method and which dispatching rule will be used before each simulation run. Once the simulation starts, the procedure of the prespecified estimation method is executed whenever a new order arrives at the shop. Newly arrived order which needs to be quoted is called as target order and any lot in the target order is called as target lot in this paper.
4. Estimation methods 3. Simulation model To see how the proposed estimation methods work in the considered shops, we develop a simulation model which can represent several configurations of hybrid flowshops. The model simulates order arrivals and lot flows, such as, waiting, processing, transferring and so on, in the pre-configured hybrid flowshops. The simulation model used in this study was developed with C language.
In this study, we develop three estimation methods and each method is devised for distinct dispatching rule used in the shop. To estimate order lead times, generally, information about the specification of the order and the shop status, i.e., inventory level, is considered. Additionally, in this study, we consider which dispatching rule is used in the shop for more accurate estimation. Fig. 1 shows the input and output of the suggested estimation methods.
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preceding the bottleneck as the inventory of the bottleneck stage. Using the above notation, we could estimate the lead time of order a under FCFS, with the following equation:
Order specification (Process plan, due date, etc.)
Shop information 1. Inventory status (Number of waiting and processed lot at each stage)
Estimation Method
2. Scheduling scheme (Dispatching rule used at the shop)
Estimated lead time of the arrived order
^La ¼ F b þ a
K X
k þ pak Þ þ paK Q a =mK ðw
ð1Þ
k¼bþ1
The equation is composed of three terms. The first term, i.e., F ba , represents the estimated flow time of the target order from stage 1 to stage b. The second term represents the estimated flow time of k can be obone lot from stage b + 1 to stage K. In the parenthesis, w tained from the periodic statistics gathering during the simulation run. The third term estimates the flow time of the target order at the last stage in the shop.
Fig. 1. Input/output of estimation method.
4.2. Estimation method for EDD (EM-E) Actually, the lead time of a target order is mainly affected by the current inventory level rather than the time to process the order. Such a phenomenon is more vivid if the shop is highly utilized and has long serial processes like the hybrid flowshops. Thus, we need to measure the amount of the current inventory and the time to complete that inventory. However, among the current inventory throughout the shop, not every waiting lot would cause the waiting of the target order. The impact of the inventory on the target order would change depending on where the inventory is located and which dispatching rule is used in the shop. Even orders which arrive in the future could affect the flow of the target order. In the following, we present how each proposed method considers the inventory for the lead time estimation.
Unlike in FCFS, some of target lots can be passed by other lots whose orders arrived later than the target order at the shop in EDD environment. Thus, we need to consider not only the current inventory status but also orders with earlier due dates which arrive in the near future. However, in this study, we do not analyze the exact arrival pattern of future orders for the estimation, rather we consider the relative urgency of the target order. For the equation of the EM-E, we use the equation of the EM-F and add/deduct a certain amount of time according to the relative urgency of the target order. Before introducing the equation of the EM-E, notation is described as follows:
4.1. Estimation method for FCFS (EM-F)
Ai Di I
Under FCFS, lots are dispatched in order of their arrival times and no lot can be processed earlier than another lot which arrived earlier. Therefore, the amount of the current inventory, i.e., number of waiting lots, in the shop is a major factor when estimating the lead time of a target order. Before introducing the equation for estimating lead time, notation used in the equation is described as follows: A pik mk Qi qkn
Nk k p b k w F bi
index of newly arrived order which needs to be quoted, i.e., index of a target order. processing time of a lot in order i at stage k. number of machines at stage k. number of lots in order i, i.e., order size. order index of nth waiting lot at stage k, (e.g., the lot with the second earliest arrival time among the waiting lots at stage 1 is an element of order q12). number of waiting lots at stage k. mean processing time of a lot at stage k. stage index of the bottleneck stage, which can be obtained as k =mk g. arg maxk fp average waiting time of a lot at stage k. estimated flow time of order i from the first stage to the bottleneck stage, which can be obtained as Nk b P P pqkn b þ pib Q i Þ=mb . ð k¼1 n¼1
In the notation, information about the inventory status, such as, qkn and Nk, is real-time data at the time when the target order arrives. That is, we know the number of waiting lots and the specification of each waiting lot for every stage, before the lead time estimation. The bottleneck stage plays a big role in the estimation, because much of inventory in the shop is expected to be located in the bottleneck. Therefore, we consider the inventory of all stages
ui
arrival time of order i at the shop. due date of order i. average interval between arrival times and due dates of orders which have arrived so far. relative urgency of order i, which is set as ðDi Ai Þ I.
In the notation, ui represents the relative urgency of order i. If the value is negative, it means the order has relatively high urgency so that we can expect that the lead time would be shorter. Otherwise, we can expect that the order would stay longer in the shop due to its low urgency. Under EDD, the lead time of order a can be estimated by the following equation:
( ^La ¼
F ba
þ
K X
) k þ pak Þ þ paK Q a =mK ðw
þ ua
ð2Þ
k¼bþ1
Note that, Eqs. (1) and (2) are the same except the relative urgency term so that the performance of EM-E is largely dependent on that of EM-F. 4.3. Estimation method for SPT (EM-S) SPT environment is much different from FCFS/EDD environments. First, the dispatching priority value of a lot can fluctuate throughout the shop, because processing times of a lot are different depending on stages. Also, under SPT, priorities are absolute values while priorities by EDD are somewhat proportion to the arrival times. That means, in SPT environment, there is a possibility that target lots can be passed by other lots which arrive way far later than the target lot, which is rarely happened in EDD environment. In an extreme case, a lot with long processing time can be deadlocked at a certain stage, if other lots with shorter processing times continue to arrive. Under SPT environment, two sources of workload can affect the lead time of a target order. One is the current inventory and the other is the future orders. First, we propose the equation where
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only the current inventory is considered. Before introducing the equation, notation is described as follows:
Uik W cik
set of the waiting lots at stage k, whose processing times are shorter than or equal to that of a lot in order i. estimated waiting time of order i at stage k due to some of P pqkn k =mk . the current inventory and can be obtained as n2Uik
ð3Þ
k¼1
The above equation shows the lead time of a target order under SPT can be estimated by the sum of the estimated waiting time of the target order and the processing time of a lot in the target order for all stages. However, for more accurate estimation, we need to devise an equation which considers the future orders, because they can also affect the lead time of the target order. Before introducing the equation which considering both the current inventory and the future orders, notation is introduced as follows.
K Q cik
C lik
Sik zik = 1, W fik
^La ¼
K X ðW cak þ W fak þ pak Þ
ð4Þ
k¼1
Because, at a certain stage, lots in the target order have to wait until all inventory lots with the shorter and equal processing times are at least finished, we first find the set of such waiting lots at the stage, and then compute the sum of the processing times of all lots in the set. Considering only the current inventory, we can introduce the equation for estimating the lead time of target order a under SPT as follows: K X ^La ¼ ðW cak þ pak Þ
k =mk Þ. In summary, the waitW cik Sik þ W cik S2ik , where Sik ¼ lik ðp ing time of order i at stage k, which affected by some of the future orders is estimated as W cik Sik ð1 þ Sik zik Þ. Adding this portion to Eq. (3) for all stage, the final equation for estimating the lead time of target order a under SPT is proposed as follows:
mean arrival rate of orders. maximum order size. rank of the processing time of a lot in order i at stage k in ascending order. (e.g., if a lot in order i has the second shortest processing time at stage k, then cik = 2). number of product types. expected number of lots per unit time, which will arrive and probably causethe waiting of order i at stage k, and can be obtained as k Q2 cikC1 . estimated time to process workload lik, which can be k p obtained as lik m . k if the processing time of a lot in order i is the longest at stage k; 0, o/w. estimated waiting time of order i at stage k due to some of the future orders, and can be obtained as W cik Sik þ W cik S2ik zik .
To find out the waiting time of a target order by the future orders at a stage, we first compute the amount of lots which inflow into at the stage averagely, that can be obtained by multiplying the mean arrival rate of orders and the average order size, i.e., k(Q/2). Among them, the portion of lots with processing times shorter than that of a lot in target order i at stage k would be (cik 1)/C, thus fk ðQ =2Þg fðcik 1Þ=C g is the amount of future lots actually causing the waiting of the target order per unit time, i.e., lik. Thus, the number of lots that will arrive at stage k during W cik , i.e., the waiting time of order i by the current inventory, would be lik W cik . k =mk , the Because the average tact time of a lot at stage k is p k =mk Þ and this time to consume lik W cik lots would be lik W cik ðp can be considered as the estimated waiting time of order i at stage k due to some of the future orders, i.e., W fik . However, additional fuk =mk Þ at some ture orders could arrive again during lik W cik ðp k =mk Þþ could be extended as lik W cik ðp stages, W fik 2 k =mk Þ2 . In this study, we use lik W cik ðp k =mk Þ for W fik lik W cik ðp for all stages except one stage where the target lot has the longest processing time. For that stage, W fik is obtained from
Using the Eqs. (1), (2), and (4), we can estimate the lead time of the target order under three different dispatching rules. Because the estimated lead time is computed from the closed form equations, the results will be obtained instantly.
5. Performance evaluation 5.1. Benchmark methods In this study, we use four existing lead time estimation methods to investigate the performances of the proposed methods. Among the benchmark methods, three of them are parametric methods so that the parameters, k1 and k2, are determined by the linear regression method. These methods were used in many due date assignment related works including Sha and Hsu (2004). The fourth one introduced by Lawrence (1995) is based on the exponential smoothing method where the lead time is estimated with the latest and the second latest completed orders’ lead times. Each method is described briefly with the equation in the following subsections. 5.1.1. Total Workload (TWK) TWK is one of the most basic and simple methods in estimating the lead time of an order. Including TWK, some other basic methods are presented in the survey paper by Cheng and Gupta (1989). It just considers the total processing time of the target order and find out the relationship between the lead time and the total processing time with the linear regression method. The equation of TWK is as follows:
^La ¼ k1
K X
pak þ k2
ð5Þ
k¼1
5.1.2. Job In Queue (JIQ) Unlike TWK, inventory status, i.e., the number of waiting lots throughout the shop, is considered in JIQ. The equation of JIQ is as follows:
^La ¼ k1
Nk K X X
pqkn k þ
k¼1 n¼1
K X
pak þ k2
ð6Þ
k¼1
5.1.3. Job In Bottleneck Queue (JIBQ) Instead all the inventory workload is considered in JIQ, in this method, the inventory only at the bottleneck stage is considered. The equation of JIBQ is as follows:
^La ¼ k1
Nb X n¼1
pqbn b þ
K X
pak þ k2
ð7Þ
k¼1
5.1.4. Exponentially Smoothed Flow time (ESF) Unlike the above three parametric methods, this method uses two of the past history data of the order lead times. In ESF, at the time of estimating, the actual lead times of two orders that completed latest and completed second latest are used. Let L1 and L2 be the lead time of the latest and the second latest completed or-
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ders, respectively. The equation of ESF is as follows (in the equation, the value of smoothing constant a is set to 0.5 after the pre-test with several alternative values):
^La ¼ a L þ ð1 aÞ L 1
2
ð8Þ
Among the benchmark methods, ESF is expected to show good performance, if the considered production system has long serial processes like hybrid flowshops. However such reasoning would be valid, only if the scheduling scheme is FCFS. 5.2. Test results For each dispatching rule, we investigated the performance of the corresponding proposed estimation method and those of the benchmark methods. Also, the other two proposed methods were tested for that rule environment to see the results of inappropriate estimations. For example, under EDD, we tested not only EM-E and four benchmarks, but also EM-F and EM-S. Therefore, for one simulation instance, combinations of three dispatching rules and seven estimation methods, i.e., 21 combinations were tested. We increased the number of the simulation instances by varying the number of stages, the number of product types, and the maximum order size as follows: three levels of stage size, i.e., 30, 60, 90; three levels of product type size, i.e., 10, 50, 100; two levels of maximum order size, i.e., 50, 100. Additional ten replications for each combination led the total number of instances to 180 (3 3 2 10). For each simulation run, mean RER and CC were measured for the set of valid orders. In the following tables, the method which shows the mean RER close to zero and CC close to one is the better performing method. Summary of the overall test results is presented in Table 1. Under each dispatching rule, the corresponding proposed estimation method showed the best performances in terms of both measures (the best result in each case is written in bold). Among the benchmarks, ESF showed the fairly good performance in FCFS and EDD environments comparing to the proposed methods. Under SPT, most methods showed poor results except the EM-S in both mean RER and CC. Although the mean RER result of EM-S (i.e., about 33%) is still away from the best, that of other methods are much larger. Moreover, in terms of CC, EM-S made the correlation coefficient of the actual and estimated lead times close up to 90%, which is quite an achievement considering other numbers just hang around 0%. To see any performance variations according to the problem conditions, we performed the ANOVA on mean RER values for each dispatching rule environment and the results are summarized in Table 2. As you can see from the table, we notice that the performance of the methods varies across the problem conditions in most cases. To see the pattern of the performance change, the detail results of the test are summarized depending on the problem condition. Table 3 shows the performance results as the number of stages varies. Under FCFS and EDD conditions, most methods
Table 2 ANOVA results. Source of variation (a) Under FIFO Method (m) Number of stages (s) Number of product types (p) Max order size (o) ms mp mo Residuals (b) Under EDD Method (m) Number of stages (s) Number of product types (p) Max order size (o) ms mp mo Residuals (c) Under SPT Method (m) Number of stages (s) Number of product types (p) Max order size (o) ms mp mo Residuals
EM-F
EM-E
EM-S
TWK
FCFS
0.11a,c (0.99)b
0.14 (0.94)
0.36 (0.58)
0.63 (0.00)
0.32 (0.79)
0.56 (0.22)
0.16 (0.95)
EDD
0.13 (0.93)
0.10 (0.98)
0.36 (0.57)
0.65 (0.09)
1.36 (0.84)
0.84 (0.06)
0.18 (0.87)
SPT
2.66 (0.02)
2.62 (0.04)
0.33 (0.87)
0.66 (0.15)
6.17 (0.01)
6.49 (0.03)
1.75 (0.02)
a b c
JIQ
JIBQ
Mean square
F value
6 2 2
46.65 0.06 0.02
7.77 0.03 0.01
500.13 2.02 0.56
***
1 12 12 6 1218
0.23 0.50 1.22 0.22 18.93
0.23 0.04 0.10 0.04 0.02
14.88 2.70 6.56 2.34 NA
***
6 2 2
231.20 1.88 3.76
38.53 0.94 1.88
491.11 11.96 23.98
***
1 12 12 6 1218
0.26 8.71 8.43 0.35 95.57
0.26 0.73 0.70 0.06 0.08
3.31 9.25 8.96 0.75 NA
6 2 2
6586.53 1137.22 2649.13
1097.75 568.61 1324.57
99.03 51.29 119.49
***
1 12 12 6 1218
483.01 801.31 2754.56 433.14 13501.70
483.01 66.78 229.55 72.19 11.09
43.57 6.02 20.71 6.51 NA
***
** *** *
*** ***
*** ***
*** ***
*** *** ***
Statistically different at the significance level of 0.05. Statistically different at the significance level of 0.01. *** Statistically different at the significance level of 0.001. **
Table 3 The performance results for the different number of stages. Rules
Number of stages
FCFS
30 60 90
EDD
30 60 90
SPT
30
90
Rules
Sum of square
*
60 Table 1 Overall test results.
Degree of freedom
EM-F
EM-E
EM-S
TWK
JIQ
JIBQ
ESF
0.08 (0.99) 0.10 (0.99) 0.14 (0.99)
0.11 (0.95) 0.13 (0.94) 0.17 (0.93)
0.41 (0.51) 0.34 (0.60) 0.32 (0.64)
0.61 (0.01) 0.64 (0.01) 0.64 (0.00)
0.34 (0.80) 0.30 (0.80) 0.32 (0.76)
0.56 (0.23) 0.56 (0.26) 0.58 (0.16)
0.14 (0.94) 0.16 (0.94) 0.18 (0.97)
0.11 (0.94) 0.12 (0.93) 0.16 (0.92)
0.07 (0.98) 0.09 (0.98) 0.13 (0.99)
0.41 (0.51) 0.34 (0.58) 0.33 (0.61)
0.63 (0.07) 0.65 (0.09) 0.66 (0.11)
1.06 (0.76) 1.38 (0.86) 1.63 (0.90)
0.87 (0.13) 0.80 (0.04) 0.85 (0.08)
0.17 (0.88) 0.18 (0.87) 0.20 (0.87)
3.95 (0.01) 2.49 (0.01) 1.53 (0.07)
3.87 (0.02) 2.47 (0.03) 1.52 (0.09)
0.34 (0.88) 0.34 (0.85) 0.31 (0.88)
0.83 (0.16) 0.62 (0.13) 0.55 (0.15)
8.12 (0.01) 6.12 (0.01) 4.25 (0.03)
9.89 (0.05) 5.33 (0.05) 4.25 (0.00)
2.54 (0.01) 1.55 (0.01) 1.16 (0.04)
ESF
Average of mean relative error ratios. Average of correlation coefficients between actual and estimated lead times. Best result in each case is shown in bold.
including the best performed methods (i.e., EM-F and EM-E in each case) showed the mean RER increase, which means the performance worsening, as the number of stages increases. On the contrary, under SPT, the performances of most methods are getting better, as the number of stages increases. Comparing to the mean RER, CC values (in parenthesis) do not change much even though the problem condition varies. Note that the above ANOVA tests were done on RER mean values.
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G.-C. Lee / Computers & Industrial Engineering 56 (2009) 1668–1674 Table 4 The performances results for the different number of product types. Rules
Number of product types
FCFS
EM-F
10 50 100
EDD
10 50 100
SPT
10 50 100
EM-E
FCFS
Max order size 50 100
EDD
50 100
SPT
50 100
TWK
JIQ
JIBQ
ESF
0.05 (1.00) 0.14 (0.98) 0.14 (0.99)
0.09 (0.97) 0.16 (0.92) 0.16 (0.93)
0.41 (0.55) 0.35 (0.56) 0.31 (0.64)
0.66 (0.00) 0.62 (0.01) 0.61 (0.00)
0.28 (0.94) 0.36 (0.67) 0.32 (0.75)
0.59 (0.22) 0.55 (0.29) 0.56 (0.15)
0.19 (0.97) 0.15 (0.93) 0.15 (0.94)
0.10 (0.96) 0.15 (0.91) 0.14 (0.92)
0.05 (0.99) 0.12 (0.98) 0.12 (0.98)
0.43 (0.54) 0.34 (0.54) 0.30 (0.62)
0.68 (0.06) 0.64 (0.10) 0.63 (0.09)
1.52 (0.85) 1.31 (0.86) 1.24 (0.81)
1.12 (0.23) 0.81 (0.04) 0.58 (0.09)
0.21 (0.92) 0.17 (0.84) 0.17 (0.85)
4.30 (0.04) 2.33 (0.02) 1.35 (0.10)
4.26 (0.04) 2.28 (0.04) 1.31 (0.13)
0.28 (0.88) 0.34 (0.88) 0.37 (0.85)
0.77 (0.21) 0.62 (0.12) 0.60 (0.11)
10.40 (0.03) 4.33 (0.00) 3.77 (0.06)
12.26 (0.05) 4.87 (0.05) 2.34 (0.00)
2.41 (0.01) 1.59 (0.02) 1.25 (0.05)
Table 5 The performances results for the different max order sizes. Rules
EM-S
EM-F
EM-E
EM-S
TWK
JIQ
JIBQ
ESF
0.09 (0.99) 0.13 (0.98)
0.12 (0.95) 0.16 (0.92)
0.36 (0.58) 0.35 (0.59)
0.63 (0.00) 0.63 (0.01)
0.31 (0.86) 0.32 (0.72)
0.52 (0.28) 0.60 (0.16)
0.15 (0.97) 0.17 (0.93)
0.11 (0.94) 0.15 (0.91)
0.08 (0.99) 0.12 (0.98)
0.36 (0.57) 0.36 (0.57)
0.64 (0.10) 0.66 (0.08)
1.31 (0.92) 1.41 (0.77)
0.85 (0.10) 0.83 (0.02)
0.17 (0.90) 0.20 (0.84)
3.22 (0.01) 2.10 (0.04)
3.18 (0.02) 2.06 (0.06)
0.32 (0.89) 0.35 (0.85)
0.69 (0.17) 0.64 (0.13)
7.83 (0.01) 4.50 (0.02)
7.76 (0.05) 5.22 (0.02)
2.01 (0.01) 1.49 (0.03)
Tables 4 and 5 show the performance results for the different number of product types and maximum order size used in the tests. Similar with the results of Table 3, under FCFS and EDD, the performance slightly getting worse in most methods, as the product types and the maximum order size are getting bigger. In SPT case, conversely, the mean RER of all the methods except the best performing method (EM-S) are getting smaller. Still, the CC values do not show any clear up- and down-tendency according to the problem conditions. From the computational tests, we can verify the outperformance of the proposed methods. In various cases, the proposed methods always show the most superior results in terms of both accuracy measures. Particularly, the CCs of EM-F/E show almost perfect scores (i.e., one) under FCFS/EDD and the performance of EM-S is especially outstanding compared to others under SPT. 6. Conclusions In this paper, we consider estimation of the order lead time in hybrid flowshops with different dispatching rules. Estimating the accurate lead times is essential to assign appropriate due date of the orders. One of the major factors affecting the lead time is scheduling policy used in the production systems. The complexity of the estimation problem have forced most previous studies assume the FCFS policy as their scheduling schemes. In this study, we proposed three lead time estimation methods which can cope
with three scheduling conditions, i.e., FCFS, EDD, and SPT. Each proposed method is devised considering the workload which might have effects on the flow of the target order. The computational results using the simulation model showed the superiorities of the proposed methods over the benchmarks under any of three dispatching rules. The proposed methods can be directly applied to order promising (or due date assignment) problems. A naive approach of the due date assignment could be just adding the estimated lead time of an order to the arrival time, i.e., Di ¼ Ai þ ^Li . In addition, the ideas of the suggested methods can be used in general scheduling methods. Many scheduling algorithms, even simple dispatching rules, use the remaining flow time of a job as key information in their logic. More accurate information about the remaining lead time would enhance the performance of the scheduling methods. Even though the suggested lead time estimation methods consider dispatching rules other than FCFS, they are still simple dispatching discipline. Thus, further research about the lead time estimating under general scheduling schemes is needed. Because decisions of the scheduling and the lead time estimating affect each other, some kinds of integrated approaches would be expected eventually, which can show good performance in terms of both scheduling and estimation measures. Acknowledgement This work was supported by the faculty research fund of Konkuk University in 2006. References Alisantoso, D., Khoo, L. P., & Jiang, P. Y. (2003). An immune algorithm approach to the scheduling of a flexible PCB flow shop. International Journal of Advanced Manufacturing Technology, 22, 819–827. Blackstone, J. H., Phillip, D. T., & Hogg, D. L. Jr., (1982). A state-of-the-art survey of dispatching rules for manufacturing jobshop operations. International Journal of Production Research, 25, 1143–1156. Cheng, T. C. E., & Gupta, M. C. (1989). Survey of scheduling research involving due date determination decisions. European Journal of Operational Research, 38, 156–166. Gordon, V., Proth, J. M., & Chu, C. (2002). A survey of the state-of-the-art of common due date assignment and scheduling research. European Journal of Operational Research, 139, 1–25. Hopp, W. J., & Sturgis, M. R. (2001). A simple, robust leadtime-quoting policy. Manufacturing & Service Operations Management, 3(4), 321–336. Lawrence, S. R. (1995). Estimating flowtimes and setting due-dates in complex production systems. IIE Transactions, 27, 657–668.
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Moses, S., Grant, H., Gruenwald, L., & Pulat, P. (2004). Real-time due-date promising by build-to-order environments. International Journal of Production Research, 42, 4353–4375. Quadt, D. (2004). Lot-sizing and scheduling for flexible flow lines. Berlin Germany: Springer. Sha, D. T., & Hsu, S. Y. (2004). Due-date assignment in wafer fabrication using artificial neural networks. International Journal of Advanced Manufacturing Technology, 23, 768–775.
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