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Order release and sequence-dependent setup times
intemationaljournalof
production economics ELSEVIER
ht.
J. Production Economics 49 (1997) 131-143
Order release and sequence-dependent
setup times
H. Missbauer” Institute of Industry and Production Management, University qf Linz, A-4040 Linz, Austria
Received: 26 June 1995 accepted: I1 November 1996
Abstract Sequence-dependent setup times lead to a potential for savings in setups if an appropriate sequencing rule is used at the detailed scheduling level. The benefits of savings in setups have to be taken into account by the higher planning levels of a production planning and control system. If we assume a complex production planning situation (complex products, multi-stage production, job-shop production), a workload control concept is often a good way to keep flow times and capacity utilization under control. In such a planning concept, order release has the task to keep work-in-process (WIP) at an appropriate level. In the case of sequence-dependent setup times, the total amount of setup time (for given lot sizes) depends on the level of WIP in the shop, and this relationship must be known to the order release function to determine the appropriate level of WIP in the shop. In this paper, the functional relationship between WIP and total setup time is examined. An approximate queueing model is developed to show the function for a certain setup-saving sequencing rule, and the results of the model are compared to simulation results. Keywords:
Setup times; Sequencing; Order release; Workload control; Queues
1. Problem description Sequence-dependent setup times are an import issue in production planning and control. They are a consequence of varying degrees of technical similarity between products, which often reflects an obvious grouping of the products into product groups or product families. Due to the technical similarities of the products within a product group, when a setup from one product to another occurs
on a machine, the machine parameters which have to be changed during a setup differ according to the production sequence, which leads to sequence-
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dependent setup times and consequently to sequence-dependent setup costs. To reap the potential benefits of the sequence dependence (saving setup time by choosing an optimized production sequence), the sequence dependence has to be taken into account on two levels of the planning process in production planning and control: on the sequencing level and on the higher levels of a production planning and control decision hierarchy. The planning problem on the detailed scheduling level can be stated as follows: The production orders have to be sequenced on one or more facilities so as to minimize an objective function which contains - at least in one term - the setup times which can be influenced by sequencing decisions.
1997 Elsevier Science B.V. All rights reserved
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This problem is well-known (for an overview of the literature see Cl]) and is not our topic here. The topic of this paper is the role of sequencedependent setup times in the decision making of the higher planning levels in a production planning and control (PPC) system, which place the restrictions on the detailed scheduling decisions mentioned above. The consideration of sequencedependent setup times in the higher levels of a PPC system depends on the structure of the PPC system as a whole, and therefore we have to describe the production planning and control structure in some more detail. We consider a complex production environment which is characterized by a complex product structure, multi-stage production and a large number of manufacturing facilities. For such a situation, the MRP (or MRP II) planning logic is applied in most cases (see [2]; for a short description [3]). If the detailed scheduling decisions for the manufacturing departments (e.g., component production, assembly) are decentralized and performed on the shop-floor level, which is often the case, then the order release decisions are the interface between the control of the material flow throughout the entire plant (performed by the MRP logic) and the detailed scheduling decisions performed on a local basis in the manufacturing departments. The relevant parts of such a system can be sketched as shown in Fig. 1. We consider the lot sizes (production orders) and their planned due dates as given by the MRP system. We assume an order release system which is consistent with the workload control concept [4,33: The sequencing decisions are decentralized (performed on the shop-floor level, seperately for each manufacturing department, e.g. component manufacturing). The order release decisions have to maintain an appropriate state in the manufacturing departments (more precisely, in the production units as described in [S, 41). The state in the manufacturing system can be controlled by controlling the level of work-in-process (WIP) in the system, because once the average WIP level is determined, the mean flow times and the average utilization are determined as well by the properties of the manufacturing system. This functional relationship can
49 (1997) 131-143
Aggregate Production Planning Master Production Scheduling Material Requirements Planning
I
Order release dates
Sequencing (shop-floor-level)
I
Manufacturing System
1
Fig 1. The role of order release in a workload control system.
be derived from queueing models (see, e.g., [6]) or from empirical studies [7], but often simulation is used. Fig. 2 shows a well-known example. Models of this kind (frequently termed metamodeling; for an overview see [S]) consider the manufacturing system as a black box; only the operational characteristics of the manufacturing system are used for order release decisions [S, p. 91. Consequently, all properties of the manufacturing system, including the sequence-dependent setup times and the sequencing procedure used by the dispatcher, have to be included in these functions in order to select the appropriate WIP level. Once the desired level of WIP has been determined, the order release decisions can be supported by methods like input/output control [9], load-oriented order release [lo] or CONWIP [ll, 123. (For an overview of order release algorithms, see [13].) The determination of the WIP level is equivalent to finding the best balance between the conflicting
H.Mw&werjlnt.
J Production
Economics 49 (1997) 131- I43
goals of output maximization, on the one hand, and minimization of flow times and WIP, on the other hand. If these conflicting goals are represented by different deparlments of the company (e.g. logistics department with goals refated to WIP and Aow times, manufacturing department with goals related to output), then the WIP level is a norm which coordinates the conflicting goals (for this aspect, see
133
[S]). This explains the importance of the functions shown in Fig. 2. So we have to explore the consequences of sequence-dependent setup times on the manufacturing goals 4s RXBZand more WIP is released to the shop. The following consideration is essential. An Increase in the level of WIP in the shop (longer queues) leads to more possibilities for
150:
average throughput time aclual value l >/p load limit in % of capacity
100
300 / 0
250, /I
1000
waifing
f~
time
/
100 ,507 zS=*-
actual value
I
.-.-.-
I p&essing
1_. time
I 1 2000
.
8 4000
6000
transportation t!me I! I , . z 8000 10000 std 12000 average wtP
x3
(hours)
Basis: 3000 orckrs, simulationperrod 16 w4&s
50 machinegroups, 90 machines Fig 2. Functional relationship between average WIP, mean flow time and output (see [22, p. 2471).
134
H. Mssbauerllnt.
J. Production
the dispatcher to optimize the job sequence with respect to setups. If he is able to take advantage of this potential, this will have two consequences: Increasing WIP level (more work released to the shop floor) will lead to lower setup times for a given output (and to lower setup costs, if the setup time is associated with money flow). Increasing WIP leads to lower utilization of the facilities for a given output (because of the setup time savings), so the maximum output for a given WIP level will be higher than without sequence optimization. If the functional relationship between WIP level and output (called clearing function by Karmarkar [14]) correctly reflects the properties of the manufacturing system, then the effects of sequence optimization on the output of the shop are included in this function. But the first effect mentioned above, the savings of setup time and of the associated costs with increasing WIP level, has not been treated in the literature up to now. Bertrand/Wijngaard [S] and ZHpfel/Missbauer [3] briefly mention this effect, and a detailed description of this effect is the main topic of this paper. So our focus is different from that of Kim/ Bobrowski [ 151, who compare different order release mechanisms in a sequence-dependent setup job shop. We concentrate on the impact of different levels of WIP in the shop on the total setup time. The amout of WIP in the shop is controlled by the paramenter setting of the order release method which performs the order release task, but we shall not focus on the question which order release algorithm is used. In the next section a queueing model is used to calculate the setup time as a function of work-inprocess and to show the effects of sequence-dependent setup times and of sequence optimization. The model contains some approximations and is an extension of a model originally develaped by Kekre [16,17] for a different purpose (exploring the effects of an increased product mix on the performance of a manufacturing cell). Note that the assumptions which are necessary for the application of the model are not relevant to the planning concept (workload control when sequence-dependent setup times occur), but they l
Economics 49 (1997) 131-143
are only effects of means of tions are functions model are simulations.
a technical detail to demonstrate the sequence-dependent setup times by an analytical model. So the assumpnot relevant for situations where the which we calculate from the queueing derived from empirical studies or from
l
2. The Model 2.1. The model&g
approach
Order release algorithms which are based on the workload control concept aim at keeping work-inprocess in the manufacturing system (production unit) at a desired level. Usually, the WIP level is expressed as the remaining work (measured in hours) in the system, and the desired level of WIP is constant for a certain time horizon. We will assume this situation. So a manufacturing system in a workload control environment can be modelled as a queueing network in a stationary state with an order arrival process which depends on the order release mechanism. If the number of orders in the system is kept constant, then closed queueing network models can be used in order to calculate the performance measures [18]. If the order release function only controls the arrival rate of jobs into the system (not the release of individual orders, based on state information of the system), then an open queueing network model is appropriate. Since the possibilities for analytical modelling of sequence-dependent setup times and of sequence optimization are limited, we only consider one work center within the manufacturing system (production unit) and assume a stochastic arrival process of orders to this work center whose parameters are controlled by the order release function. This assumption means that we think of, e.g., a bottleneck facility which is no gateway work center; that is, the orders have to pass a number of work centers before they arrive at the facility under consideration, which means that the order release function has no direct control over the arrivals of individual jobs. The sensitivity of the results to this assumption is tested by simulation experiments.
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H.Missbauer/Int. J. Production Economics 49 (1997) 131-143
2.2. Description
of the model
We consider a production facility (single-server system) which produces different products. These products divide into I product groups, each product group consisting of Ji individual products, SO the number of different products is If= iJi. We assume that no setups are necessary within the product groups. Substantial setup times are caused by a change of the product group on the facility. We define r as the setup time when changing to product group i (independent of the individual product within product group i and of the product group k # i which was produced before). For simplicity, we assume the same setup time for each product group. A complete list of symbols is given in Appendix B. We assume a stationary Poisson arrival process for each item with arrival rate /zij for item j of product group i. So the arrival rate of product group i to the facility is pi = x:= 1lij and the overall arrival rate is I
J.
(1) This arrival rate i is not autonomous, but is under control of the order release function of the PPC system. More specifically, order release controls 3, by changing all Rij by the same factor, because on the order release level the product mix is given. Note that in this model order release does not directly control the arrival of individual orders, but the parameters of the arrival processes (see Section 2.1). The service rate and the service time distribution cannot be calculated in a straightforward way from the product data and the relations of the ibij because they depend on the sequencing decisions on the detailed scheduling level. Since sequence optimization is a complicated task, the sequencing decisions depend on the skill and experience of the dispatcher, but first of all they depend on the objectives on the scheduling level: Saving setup time, on the one hand, and due date performance, on the other hand, are conflict-
ing goals, because the necessity to produce in cycles which are efficient with respect to setup times decreases the flexibility to give high priorities to urgent jobs. We consider two sequencing rules (see also [ 171): (a) FCFS, that is, no importance is given to saving setup time. (b) Maximum importance of saving setup time. This means that if on completion of a job there is another job of the same product group in the queue, then this job is processed first (or, equivalentely, if an arriving order encounters an order of the same product group already in the system, then it is scheduled immediately after this order (Fig. 3)). The product group is only changed if this is necessary to continue production. (See also the SIMSET rule by Wilbrecht/Prescott [ 191 and the “Repetitive Lots” procedure by Jacobs/Bragg [24], which use the same logic). The real decision rule at the sequencing level will be somewhere in between these two extremes, but possibilities for an analytical treatment of more sophisticated rules are not known yet. The consequences of the two rules on the total amount of setup time, on the utilization of the facility and on the work waiting to be processed are described now.
2.2.1. FCFS
If we assume that a setup (from one product group to another) is necessary for each job, then the mean service time b of a job can be calculated
rl
QUEUE
OF
BATCHES
ARRIVING BATCHES
MACHINE
A
AmB
B-C ‘L.9
SERVING A
Fig 3. The assumed sequencing setups (117, p. 3331).
t
t
rule for maximum
savings
in
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H. Missbauerfht. .J. Production Economics 49 (1997) 131-143
from the processing time aij and the setup time r by II J / \
(2) and the mean service rate p=;.
This calculation ignores the savings of setup time if two jobs of the same product group arrive consecutively. The probability that a job of product group i has a job of the same product group directly in front, is the probability that a job is of product group i: p(job i has job i as predecessor) = 2.
(4)
This is the conditional probability that no setup is necessary for a job of product group i (probability that no setup is necessary for the arriving job given the condition that the arriving job is of product group i). The probability that an arriving job is of product group i is also ;1,/5 so the probability for setup time savings is (conditional probability times the probability of the condition, summed for all possible products i): 2
probability for setup time savings = i i=l
So the probability FCFS (SPrc-s) is SPFCFS
=
1-i; i=l
$
.
0
that a setup is necessary for
0
2
The actual utilization for FCFS processing @rcFs) can be calculated from the arrival rate L (Eq. (1)) and the mean service time b = l/p (Eq. (2)). The average setup time which can be saved ( = Y.(l - SPrcrs)) has to be subtracted from the service time b (Eq. (2)):
PFCFS = l*Cb- r-U - SPFCFSII.
(6)
In this case the probability of setup time savings and the utilization do not depend on the number of jobs in the queue.
2.2.2. Maximum savings of setup time Given the sequencing rule described above, a setup for a job of product group i is necessary if the job encounters no job of the same product group in the queue or in process at the time of arrival. This means that the probability that a setup is necessary depends on the state of the system (more precisely: on the number of jobs in the system). Since we assume a Poisson arrival of jobs for each individual product, the probability that the system is in a certain state at the arrival time of a job of any product is the same as the probability that the system is in a certain state at an arbitrary time (PASTA (Poisson arrivals see time averages) property; see [20, p. 543). The probability that a setup is necessary for a job of product group i can approximately be derived as follows (see also [17, p. 3381): If there is no job in the system at the arrival time of a job of product group i, then a setup is necessary if the last job was not of product group i. The probability of this is the same as for FCFS processing and is 1 - lil1. If there are n > 0 jobs in the system at the arrival time of a job of product group i, then a setup is necessary for that job if there is no job of product group i in the system at that time. We calculate the probability as follows: The probability of a job of product group i at the server or at a certain position in the queue = li/‘a. The probability that no job of product group i is in a certain position at the server or in the queue = 1 - Ai/1. So the probability that no job of product group is in the system given n jobs in the system is (1 - Li/n)“. This is the probability that a setup is necessary for a job of product group i if n jobs are in the system on arrival of the job. Summation over all possible states of the system n gives the probability that a setup is necessary for a job of product group i. If we denote P,, as the probability of n jobs in the system; the probability that a setup is necessary for SPi,OPT a job of product group i if the described sequencing rule is applied, we can write:
H.Missbauer/Int.
J. Production
The probability that an arriving job is of product group i is Li/l, so the probability that a setup is necessary for an arbitrary job is:
Economics
49 (1997) 131-143
for pn. After some algebra (evaluation of the infinite sum) we get SPOPT=
SPOPT = ii12
’ SPi,OF’T.
Theorem 1. SPOPT
d %CFS,
that is, the setup-saving sequencing rule can never increase the probability that a setup is necessary for an order. Theorem 2. Zf I > 1 and li > Ofor all i, then SPopT < SPFCFs if pn > 0 for at least one n > 1. That is, setup time can be saved if there is a certain probability that a queue occurs at the facility.
Both theorems can be derived from the definition of the two sequencing rules: The rule for sequence optimization only changes the sequence of the jobs if this saves a setup. This can only be done if at least at some times there exists a queue, because otherwise there is no possibility for sequence optimization. The proof of Theorems 1 and 2 is given in Appendix A. If we approximate the probability of n jobs in the system by the M/M/l - formula: Pn =
(1 - PIP”
(9)
(which assumes FCFS discipline and is an approximation in our case), then the probability of a setup can be calculated by substituting (7) into (8) and (9)
’ li(3, - /ii)(/? + p/Ii) t1 - P)1 i= 2’ ’ (R - i,p + ,Oii) 1
(8)
Note that this is an approximation, because it assumes that the average proportion of jobs of product group i in the jobs in the queue is the same as in the arriving jobs, which is inconsistent with the assumed sequencing decision rule. Comparing the effects of the two sequencing rules (FCFS and maximum savings of setup time), we can state the following theorems.
137
In this case, the probability that a setup is necessary SPopT depends on the utilization p, which depends on the total amount of setup time and therefore on SPOPT.
2.3. Calculation of setup time and production volume as functions of WIP
As described in Section 1, we assume an order release algorithm which is based on the workload control concept and tries to keep work-in-process (WIP) at an appropriate level. We define WIP as the hours of work which the facility has to perform in order to complete all orders currently in the system. This is a useful definition in our case because most order release algorithms, like input/output control [9], load-oriented order release [21,10,22], the method of Bertrand/Wortmann [23] and CONWIP [ll, 121 also measure WIP in hours (see also Fig. 2). So we have to analyze The ratio of setup and processing time (total setup time/total processing time) as a function of WIP and shall also show the total processing time (finished work, measured in hours) as a function of WIP. These functions can be calculated as follows: The control variable of the system is the arrival rate (order release rate) h, which determines the probability that a setup is necessary (Eq. (10)) and the utilization of the server p: l
l
POPT = l[b - r(l- SpoP~)l.
(11)
So, PopT and SPopT can be calculated for a given L by solving the simultaneous Eqs. (10) and (11). In the general case described here (no restrictions on the Aij),an analytical solution of the equations is not possible, so a numerical solution must be used, which gives the values for PopT and SPopT
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J. Production Economics
sequencing, where
as functions of /2: (12)
SPCIPT=S1(&
SP FCFS
see
=
above)
(13)
POPT =f#
According to the workload control concept, the controlled variable, which reflects the state of the system, is the remaining hours of work (WR). Without the effect of setup savings, WR would be defined as WR = E(L,)b,
(14)
since b is the average service time (including the setup time) for a job in the queue and, because of the exponential distribution of the service times, also the average time until the job currently being processed is completed. The average service time which is really necessary for a job depends on E(L,) due to the setup savings, so the problem is more complicated. Since hardly an order release algorithm will stabilize WIP level which depends on itself, we simply define as the remaining work the processing time of the orders in the system: WR = E&)(6
- r).
(15)
(Equivalently, we could add an average setup time to the processing time b - r).Using the M/M/lmodel for E(L,), (15) becomes WR OPT
49 (1997) 131-143
=
1 IIqP';pT@
-
d.
(16)
Now we can calculate the setup time to processing time ratio for sequence optimization: r SPOPT SPRopT = ___ b-r
(17)
and the output, measured in hours of processing time: u = A.(b - I)
(18)
as a function of WRopT (Eq. (16) and Eq. (13) for PopT) by plotting parametriC fUnCtiOnS with 2 as control variable, WRopT at the x-axis, and SPRopT and u on the y-axis. To visualize the effects of the setup time saving sequencing rule, we compare it to FCFS
instead of SPopT =fi(A)
(see above)
(12)
So we get the setup time to processing time ratio for FCFS: r SPFCFS SPRrcFs = ___
(19)
b-r
For FCFS the remaining work is also measured in hours of processing time, but the number of jobs in the system depends on the utilization resulting from FCFS processing (Eq. (6)). So WRFCFS
=
1 ~~~~Fs(b
- 4
(20)
The diagrams are calculated for the following product data: (2 product groups) I = 2 J1 = 2 (2 products in product group 1) J2 = 3 (3 products in product group 2) I aI,1 = 10 ~i,i = 0.022863.f lzi,2 = 30 & = 0.0122377.f u2,i = 50 &i = 0.00655036.f a2,2 = 70 &2 = 0.00350615.f a2,3 = 90 AZ.3= 0.00187671.f r = 10 if the product group changes, otherwise r = 0. (f is the common factor which increases the iij as
the order release function increases the job input 2). The product data are determined in such a way that the (discrete) distribution of the processing times is similar to an exponential distribution (the /zij are proportional to the values of the probabiltiy density function of an exponential distribution with mean 32 at the processing times aij). This example is the basis for the following calculations. The setup time to processing time ratio for sequence optimization (Eq. (17)) and for FCFS (Eq. (19)) as a function of average WIP (Eq. (16) for optimization, Eq. (20) for FCFS) is depicted in Fig. 4. Furthermore the simulation result for the case of sequence optimization is given.
H.Missbauer/lnt. J. Production Economics 49 (I 997) I3 I - I43
SPR
FCFS, analytical model
0.08
WRoPT
0
50
Fig 4. Ratio of setup and processing
150
100 time as a function
of average
The simulation model represents exactly the same situation as the analytical model: singleserver system, product data as described above. For each simulation experiment (with a certain value for the control variable i) we get the total setup time and the total processing time. total setup time/total processing time, is the value on the y-axis (empirical value for SPRo&. the average remaining work, expressed as the average remaining processing time of the jobs in the system (analogous to Eq. (15)), is the value on the x-axis (empirical value for WRoPT). For the job currently in process, the remaining work is set to its average value (half the processing time). Each simulation experiment results in one point for the diagram. A series of simulation experiments (with increasing values for A) was performed, and the resulting points are joined with lines for the function in Fig. 4. The figure shows a decrease of the setup time as the WIP increases for the case of sequence optimization. The analytical model describes very accurately the relation between setup time and WIP. l
l
remaining
200 work for FCFS
250 and sequence
mFCFS
optimization
The model assumes Poisson arrivals of the orders, which signifies a poor performance of the order release algorithm - at least for this work center. We use the simulation model to examine the consequences of sequence optimization for two order release methods which are based on state information of the work center: Lower bound on remaining work: An order is released if the remaining work at the work center is below a certain minimum value. Constant number of jobs at the work center: An order is released if the number ofjobs at the work center is below a certain minimum value. (For this case the ratio of setup and processing time SPRopT can be calculated approximately from the Eqs. (7) (8) and (17) by setting the corresponding pn = 1). The orders are released in the same sequence as they arrive. In the simulations it is assumed that if a new order is released this takes place immediately after an order has been completed and the next order in the queue has been moved to the server. Fig. 5 compares the results of these methods to the results of Poisson arrivals.
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H. Missbauer/Int.
J. Production Economics 49 (1997) 131-143
SPR
lower bound on WR
0.08-
jobs in system
0.06-
Poisson arrivals
0.04-
-
0.02-
t 0
.
.
.
.
I
.
.
.
.
*
.
.
.
.
.
.
.
1.
.
.
.
I
50
in hours of processing time (Eq. (18)), as a function of average WIP (Eq. (16) for sequence optimization, Eq. (20) for FCFS) is shown in Fig. 6, together with simulation results for sequence optimization. For a given WIP level, sequence optimization increases the output over that of FCFS. The simulation results show a discrepancy to the analytical model. At 85% utilization level the analytical model underestimates the utilization by 4.7% in the case of sequence optimization. In the FCFS-case the discrepancy is only 2.2% due to the discrete processing time distribution.
3. Conclusions
1 +
SPR
(from (ll), (17), (18) or (6), (19), (18), respectively). In the case of Poisson arrivals, the utilization p depends on the average remaining work. For this case the total output of the work center, measured
A production manager must have a realistic (cognitive or formal) model of the manufacturing system. The determination of planned flow times, which means the determination of a planned level of WIP to be maintained, has grave consequences for the performance of a manufacturing system. The functional relationships between average WIP,
141
H.Missbauer~lnt. J. Production Economics 49 (1997) 131-143 U
lr OPT, analytical OPT, simulation 0.8-
FCFS, analytical
mPT
50
100
200
150
250
mFCFS
Fig 6. Output of the work center in the case of Poisson arrivals.
mean flow time, capacity utilization, etc. are important means to quantify and to visualize the conflicting goals the production manager must deal with. Sequence-dependent setup times have important consequences for these functional relationships which are not easy to estimate. In the case of sequence-dependent setup times, short average flow times, which can be obtained by a low level of WIP, can increase the setup times and costs. This effect, which is well-known in some firms, can be quantified by functions which are described in this paper. The production manager should know the quantitative shape of these functions for the manufacturing system, so we need methods to estimate these functions for real-world situations. Future research is needed to develop models covering more complex manufacturing systems and production control rules.
4. Acknowledgements The author thanks an anonymous referee for his valuable comments and suggestions.
Appendix A. Proof of Theorems 1 and 2 We will prove Theorems 1 and 2 by proving that (a) SPOPT takes its maximum value if p. = 0 for n > 1. For I > 1 and Ai > 0 for all i, SPopT gets smaller than this maximum value if P,, > 0 for at least one n > 1. (b) This maximum value of SPopT is SPFCFs. (1) SPi,oPr (Eq. (7)) is the sum of
( I 1 - !j#
times (p. + pr),
4 n
1-G
(
1”>
times pn for n > 1
and can be interpreted as a weighted average where the pn are the weights. If we assume more than one product group (I > 1) and a non-zero production volume for each product group (Ai> 0 for all i), then the following inequality holds for all i: (1-$~>(1-+~+1
forn=1,2,...
.
H. Missbauerllnt. J. Production Economics 49 (1997) 131-143
142
because (1 - 3&i/A) < 1. SO SPi,oPT is a maximum if p0 + p1 = 1. Otherwise the weight for (1 - AL/A)gets smaller and the weights for (1 - Ai/‘A)“,n > 1, get larger, which decreases and SPOPT. SPi,OPT (2) We get the maximum value of SPi,oPT from (Eq. (7)) by substituting p. + p1 = 1 and pn = 0 for n > 1: SPi,oPT = (
SPOPT
SPFCFS
Pll
WR
E&)
1- 3 ;1)
For this case, SPopT is (Eq. (8))
U
i, I
j, Ji
cf=1ii/A
= 1, SO we get: I
q2
SPopT = 1 - c i=l
r
)
References
0
[l]
which 1s the formula for SPFCFs(Eq. (5)). [2]
6. Appendix B. List of symbols [3] Y
aij
setup time when changing the product group processing time of an order of product j of product group i arrival rate of product j of product group i arrival rate of product group i ij total arrival rate
P
PFcFS POPT
SPi,OPT
probability that a setup is necessary for a job for sequence optimization probability that a setup is necessary for a job for FCFS processing probability of n jobs in the system average WIP (in hours of work) waiting to be processed (remaining work) mean number of jobs in the system utilization caused by processing time (without setup time) index for product groups and number of product groups, respectively index for products and number of products within product group i, respectively
mean service time if a setup is necessary for each job mean service rate actual utilization for FCFS processing actual utilization for sequence optimization probability that a setup is necessary for a job of product group i if the described sequencing rule is applied
[4]
[S]
[6]
[7]
[S] [9] [lo]
Kim, S.C. and Bobrowski, P.M., 1994. Impact of sequencedependent setup time on job shop scheduling performance Int. J. Prod. Res., 32(7): 1503-1520. Vollman, T.E., Berry, W.L. and Whybark, D.C., 1988. Manufacturing Planning and Control Systems. Dow Jones Irwin, Homewood, IL. Zlpfel, G. and Missbauer, H., 1993. New concepts for production planning and control, European J. Oper. Res., 67: 297-320. Bertrand, J.W.M., Wortmann, J.C. and Wijngaard, J., 1990. Production Control: A Structural and Design Oriented Approach. Elsevier, Amsterdam-Oxford-New York. Bertrand, J.W.M. and Wijngaard, J., 1986. The structuring of production control systems, Int. J. Oper. Prod. Mgmt., 6(2): 5-20. Missbauer, H., 1994, Das Konzept der Bestandsregelung als Ausgangspunkt fiir eine Neugestaltung von PPS-Systemen. Habilitationsschrift University of Linz, Austria. Fine, C.H. and Graves, S.C., 1989. A tactical planning model for manufacturing subcomponents of mainframe computers, J. Manuf. Oper. Mgmt., 2: 4-34. Yu, B. and Popplewell, K., 1994. Metamodels in manufacuring: A review, Int. J. Prod. Res., 32(4): 787-796. Wight, O.W., 1974. Production and Inventory Management in the Computer Age. CBI, Boston. Bechte, W., 1988. Theory and practice of loadoriented manufacturing control, Int. J. Prod. Res., 26(3): 375-395.
H.Missbauer/Int. _I Production Economics 49 (I 997) 131-143
[ll]
Spearman, M.L., Hopp, W.J. and Woodruff, D.L., 1989. A hierarchical control architecture for constant work-inprocess (CONWIP) production systems, J. Manuf. Oper. Mgmt., 2(3): 1477171. [I21 Spearman, M.L., Woodruff, D.L. and Hopp, W.J., 1990. CONWIP: A pull alternative to kanban, Int. J. Prod. Res., 2X(5): 8799894. [ 133 Philipoom, R.R. and Fry, T.D., 1992. Capacity based order review/release strategies to improve manufacturing performance, Int. J. Prod. Res., 30: 255992572. [14] Karmarkar, U.S., 1989. Capacity loading and release planning with work-in-progress (WIP) and leadtimes. J. Manuf. Oper. Mgmt.. 2: 1055123. [IS] Kim, SC. and Bobrowski. P.M., 1995. Evaluating order release mechanisms in ajob shop with sequence-dependent setup times. Prod. Oper. Mgmt., 4(2): 1633180. [16] Kekre, Sunder, 1984. The Effect of Number of Items processed at a Facility on Manufacturing Lead Time. Working Paper Series No. QM 8418, University of Rochester. [ 171 Kekre, Sunder, 1987. Performance of a manufacturing cell with increased product mix, IIE-Trans., 19(3): 3299339.
[lS]
143
Suri, R., Sanders, J.L. and Kamath, M., 1993. Performance evaluation of production networks. In: S.C. Graves, A.H.G. Rinnooy Kan and P.H. Zipkin (Eds.): Logistics of Production and Inventory. North-Holland, Amsterdam, pp. 1999286. [19] Wilbrecht, J.K. and Prescott, W.B., 1969. The influence of set-up time on job shop performance. Mgmt. Sci.. 16(4): B274--B280. [20] Buzacott, J.A. and Shantikumar, J.G., 1993. Stochastic Models of Manufacturing Systems. Prentice-Hall, Englewood Cliffs. N.J. [21] Bechte. W., 1980. Steuerung der Durchlaufzeit durch belastungsorientierte Auftragsfreigabe bei Werkstattfertigung. Diss. TU Hannover. [22] Wiendahl. H.-P, 1987. Belastungsorientierte Fertigungssteuerung. Hanser, Miinchen-Wien. [23] Bertrand. J.W.M. and Wortmann, J.C., 1981. Production Control and Information Systems for Component-Manufacturing Shops. Elsevier, Amsterdam-Oxford-New York. 1241 Jacobs, F.R. and Bragg, D.J., 1988. The repetitive lots: Flow-time reductions through sequencing and dynamic batch sizing. Decision Sci.. 19: 2X1-294.
production economics ELSEVIER
ht.
J. Production Economics 49 (1997) 131-143
Order release and sequence-dependent
setup times
H. Missbauer” Institute of Industry and Production Management, University qf Linz, A-4040 Linz, Austria
Received: 26 June 1995 accepted: I1 November 1996
Abstract Sequence-dependent setup times lead to a potential for savings in setups if an appropriate sequencing rule is used at the detailed scheduling level. The benefits of savings in setups have to be taken into account by the higher planning levels of a production planning and control system. If we assume a complex production planning situation (complex products, multi-stage production, job-shop production), a workload control concept is often a good way to keep flow times and capacity utilization under control. In such a planning concept, order release has the task to keep work-in-process (WIP) at an appropriate level. In the case of sequence-dependent setup times, the total amount of setup time (for given lot sizes) depends on the level of WIP in the shop, and this relationship must be known to the order release function to determine the appropriate level of WIP in the shop. In this paper, the functional relationship between WIP and total setup time is examined. An approximate queueing model is developed to show the function for a certain setup-saving sequencing rule, and the results of the model are compared to simulation results. Keywords:
Setup times; Sequencing; Order release; Workload control; Queues
1. Problem description Sequence-dependent setup times are an import issue in production planning and control. They are a consequence of varying degrees of technical similarity between products, which often reflects an obvious grouping of the products into product groups or product families. Due to the technical similarities of the products within a product group, when a setup from one product to another occurs
on a machine, the machine parameters which have to be changed during a setup differ according to the production sequence, which leads to sequence-
* Tel.: + 43173212468-9463;fax: + 43173212468-9422. 0925-5273/97/$17.00 Copyright 0 PII
SO925-5273(97)00008-X
dependent setup times and consequently to sequence-dependent setup costs. To reap the potential benefits of the sequence dependence (saving setup time by choosing an optimized production sequence), the sequence dependence has to be taken into account on two levels of the planning process in production planning and control: on the sequencing level and on the higher levels of a production planning and control decision hierarchy. The planning problem on the detailed scheduling level can be stated as follows: The production orders have to be sequenced on one or more facilities so as to minimize an objective function which contains - at least in one term - the setup times which can be influenced by sequencing decisions.
1997 Elsevier Science B.V. All rights reserved
132
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This problem is well-known (for an overview of the literature see Cl]) and is not our topic here. The topic of this paper is the role of sequencedependent setup times in the decision making of the higher planning levels in a production planning and control (PPC) system, which place the restrictions on the detailed scheduling decisions mentioned above. The consideration of sequencedependent setup times in the higher levels of a PPC system depends on the structure of the PPC system as a whole, and therefore we have to describe the production planning and control structure in some more detail. We consider a complex production environment which is characterized by a complex product structure, multi-stage production and a large number of manufacturing facilities. For such a situation, the MRP (or MRP II) planning logic is applied in most cases (see [2]; for a short description [3]). If the detailed scheduling decisions for the manufacturing departments (e.g., component production, assembly) are decentralized and performed on the shop-floor level, which is often the case, then the order release decisions are the interface between the control of the material flow throughout the entire plant (performed by the MRP logic) and the detailed scheduling decisions performed on a local basis in the manufacturing departments. The relevant parts of such a system can be sketched as shown in Fig. 1. We consider the lot sizes (production orders) and their planned due dates as given by the MRP system. We assume an order release system which is consistent with the workload control concept [4,33: The sequencing decisions are decentralized (performed on the shop-floor level, seperately for each manufacturing department, e.g. component manufacturing). The order release decisions have to maintain an appropriate state in the manufacturing departments (more precisely, in the production units as described in [S, 41). The state in the manufacturing system can be controlled by controlling the level of work-in-process (WIP) in the system, because once the average WIP level is determined, the mean flow times and the average utilization are determined as well by the properties of the manufacturing system. This functional relationship can
49 (1997) 131-143
Aggregate Production Planning Master Production Scheduling Material Requirements Planning
I
Order release dates
Sequencing (shop-floor-level)
I
Manufacturing System
1
Fig 1. The role of order release in a workload control system.
be derived from queueing models (see, e.g., [6]) or from empirical studies [7], but often simulation is used. Fig. 2 shows a well-known example. Models of this kind (frequently termed metamodeling; for an overview see [S]) consider the manufacturing system as a black box; only the operational characteristics of the manufacturing system are used for order release decisions [S, p. 91. Consequently, all properties of the manufacturing system, including the sequence-dependent setup times and the sequencing procedure used by the dispatcher, have to be included in these functions in order to select the appropriate WIP level. Once the desired level of WIP has been determined, the order release decisions can be supported by methods like input/output control [9], load-oriented order release [lo] or CONWIP [ll, 123. (For an overview of order release algorithms, see [13].) The determination of the WIP level is equivalent to finding the best balance between the conflicting
H.Mw&werjlnt.
J Production
Economics 49 (1997) 131- I43
goals of output maximization, on the one hand, and minimization of flow times and WIP, on the other hand. If these conflicting goals are represented by different deparlments of the company (e.g. logistics department with goals refated to WIP and Aow times, manufacturing department with goals related to output), then the WIP level is a norm which coordinates the conflicting goals (for this aspect, see
133
[S]). This explains the importance of the functions shown in Fig. 2. So we have to explore the consequences of sequence-dependent setup times on the manufacturing goals 4s RXBZand more WIP is released to the shop. The following consideration is essential. An Increase in the level of WIP in the shop (longer queues) leads to more possibilities for
150:
average throughput time aclual value l >/p load limit in % of capacity
100
300 / 0
250, /I
1000
waifing
f~
time
/
100 ,507 zS=*-
actual value
I
.-.-.-
I p&essing
1_. time
I 1 2000
.
8 4000
6000
transportation t!me I! I , . z 8000 10000 std 12000 average wtP
x3
(hours)
Basis: 3000 orckrs, simulationperrod 16 w4&s
50 machinegroups, 90 machines Fig 2. Functional relationship between average WIP, mean flow time and output (see [22, p. 2471).
134
H. Mssbauerllnt.
J. Production
the dispatcher to optimize the job sequence with respect to setups. If he is able to take advantage of this potential, this will have two consequences: Increasing WIP level (more work released to the shop floor) will lead to lower setup times for a given output (and to lower setup costs, if the setup time is associated with money flow). Increasing WIP leads to lower utilization of the facilities for a given output (because of the setup time savings), so the maximum output for a given WIP level will be higher than without sequence optimization. If the functional relationship between WIP level and output (called clearing function by Karmarkar [14]) correctly reflects the properties of the manufacturing system, then the effects of sequence optimization on the output of the shop are included in this function. But the first effect mentioned above, the savings of setup time and of the associated costs with increasing WIP level, has not been treated in the literature up to now. Bertrand/Wijngaard [S] and ZHpfel/Missbauer [3] briefly mention this effect, and a detailed description of this effect is the main topic of this paper. So our focus is different from that of Kim/ Bobrowski [ 151, who compare different order release mechanisms in a sequence-dependent setup job shop. We concentrate on the impact of different levels of WIP in the shop on the total setup time. The amout of WIP in the shop is controlled by the paramenter setting of the order release method which performs the order release task, but we shall not focus on the question which order release algorithm is used. In the next section a queueing model is used to calculate the setup time as a function of work-inprocess and to show the effects of sequence-dependent setup times and of sequence optimization. The model contains some approximations and is an extension of a model originally develaped by Kekre [16,17] for a different purpose (exploring the effects of an increased product mix on the performance of a manufacturing cell). Note that the assumptions which are necessary for the application of the model are not relevant to the planning concept (workload control when sequence-dependent setup times occur), but they l
Economics 49 (1997) 131-143
are only effects of means of tions are functions model are simulations.
a technical detail to demonstrate the sequence-dependent setup times by an analytical model. So the assumpnot relevant for situations where the which we calculate from the queueing derived from empirical studies or from
l
2. The Model 2.1. The model&g
approach
Order release algorithms which are based on the workload control concept aim at keeping work-inprocess in the manufacturing system (production unit) at a desired level. Usually, the WIP level is expressed as the remaining work (measured in hours) in the system, and the desired level of WIP is constant for a certain time horizon. We will assume this situation. So a manufacturing system in a workload control environment can be modelled as a queueing network in a stationary state with an order arrival process which depends on the order release mechanism. If the number of orders in the system is kept constant, then closed queueing network models can be used in order to calculate the performance measures [18]. If the order release function only controls the arrival rate of jobs into the system (not the release of individual orders, based on state information of the system), then an open queueing network model is appropriate. Since the possibilities for analytical modelling of sequence-dependent setup times and of sequence optimization are limited, we only consider one work center within the manufacturing system (production unit) and assume a stochastic arrival process of orders to this work center whose parameters are controlled by the order release function. This assumption means that we think of, e.g., a bottleneck facility which is no gateway work center; that is, the orders have to pass a number of work centers before they arrive at the facility under consideration, which means that the order release function has no direct control over the arrivals of individual jobs. The sensitivity of the results to this assumption is tested by simulation experiments.
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2.2. Description
of the model
We consider a production facility (single-server system) which produces different products. These products divide into I product groups, each product group consisting of Ji individual products, SO the number of different products is If= iJi. We assume that no setups are necessary within the product groups. Substantial setup times are caused by a change of the product group on the facility. We define r as the setup time when changing to product group i (independent of the individual product within product group i and of the product group k # i which was produced before). For simplicity, we assume the same setup time for each product group. A complete list of symbols is given in Appendix B. We assume a stationary Poisson arrival process for each item with arrival rate /zij for item j of product group i. So the arrival rate of product group i to the facility is pi = x:= 1lij and the overall arrival rate is I
J.
(1) This arrival rate i is not autonomous, but is under control of the order release function of the PPC system. More specifically, order release controls 3, by changing all Rij by the same factor, because on the order release level the product mix is given. Note that in this model order release does not directly control the arrival of individual orders, but the parameters of the arrival processes (see Section 2.1). The service rate and the service time distribution cannot be calculated in a straightforward way from the product data and the relations of the ibij because they depend on the sequencing decisions on the detailed scheduling level. Since sequence optimization is a complicated task, the sequencing decisions depend on the skill and experience of the dispatcher, but first of all they depend on the objectives on the scheduling level: Saving setup time, on the one hand, and due date performance, on the other hand, are conflict-
ing goals, because the necessity to produce in cycles which are efficient with respect to setup times decreases the flexibility to give high priorities to urgent jobs. We consider two sequencing rules (see also [ 171): (a) FCFS, that is, no importance is given to saving setup time. (b) Maximum importance of saving setup time. This means that if on completion of a job there is another job of the same product group in the queue, then this job is processed first (or, equivalentely, if an arriving order encounters an order of the same product group already in the system, then it is scheduled immediately after this order (Fig. 3)). The product group is only changed if this is necessary to continue production. (See also the SIMSET rule by Wilbrecht/Prescott [ 191 and the “Repetitive Lots” procedure by Jacobs/Bragg [24], which use the same logic). The real decision rule at the sequencing level will be somewhere in between these two extremes, but possibilities for an analytical treatment of more sophisticated rules are not known yet. The consequences of the two rules on the total amount of setup time, on the utilization of the facility and on the work waiting to be processed are described now.
2.2.1. FCFS
If we assume that a setup (from one product group to another) is necessary for each job, then the mean service time b of a job can be calculated
rl
QUEUE
OF
BATCHES
ARRIVING BATCHES
MACHINE
A
AmB
B-C ‘L.9
SERVING A
Fig 3. The assumed sequencing setups (117, p. 3331).
t
t
rule for maximum
savings
in
136
H. Missbauerfht. .J. Production Economics 49 (1997) 131-143
from the processing time aij and the setup time r by II J / \
(2) and the mean service rate p=;.
This calculation ignores the savings of setup time if two jobs of the same product group arrive consecutively. The probability that a job of product group i has a job of the same product group directly in front, is the probability that a job is of product group i: p(job i has job i as predecessor) = 2.
(4)
This is the conditional probability that no setup is necessary for a job of product group i (probability that no setup is necessary for the arriving job given the condition that the arriving job is of product group i). The probability that an arriving job is of product group i is also ;1,/5 so the probability for setup time savings is (conditional probability times the probability of the condition, summed for all possible products i): 2
probability for setup time savings = i i=l
So the probability FCFS (SPrc-s) is SPFCFS
=
1-i; i=l
$
.
0
that a setup is necessary for
0
2
The actual utilization for FCFS processing @rcFs) can be calculated from the arrival rate L (Eq. (1)) and the mean service time b = l/p (Eq. (2)). The average setup time which can be saved ( = Y.(l - SPrcrs)) has to be subtracted from the service time b (Eq. (2)):
PFCFS = l*Cb- r-U - SPFCFSII.
(6)
In this case the probability of setup time savings and the utilization do not depend on the number of jobs in the queue.
2.2.2. Maximum savings of setup time Given the sequencing rule described above, a setup for a job of product group i is necessary if the job encounters no job of the same product group in the queue or in process at the time of arrival. This means that the probability that a setup is necessary depends on the state of the system (more precisely: on the number of jobs in the system). Since we assume a Poisson arrival of jobs for each individual product, the probability that the system is in a certain state at the arrival time of a job of any product is the same as the probability that the system is in a certain state at an arbitrary time (PASTA (Poisson arrivals see time averages) property; see [20, p. 543). The probability that a setup is necessary for a job of product group i can approximately be derived as follows (see also [17, p. 3381): If there is no job in the system at the arrival time of a job of product group i, then a setup is necessary if the last job was not of product group i. The probability of this is the same as for FCFS processing and is 1 - lil1. If there are n > 0 jobs in the system at the arrival time of a job of product group i, then a setup is necessary for that job if there is no job of product group i in the system at that time. We calculate the probability as follows: The probability of a job of product group i at the server or at a certain position in the queue = li/‘a. The probability that no job of product group i is in a certain position at the server or in the queue = 1 - Ai/1. So the probability that no job of product group is in the system given n jobs in the system is (1 - Li/n)“. This is the probability that a setup is necessary for a job of product group i if n jobs are in the system on arrival of the job. Summation over all possible states of the system n gives the probability that a setup is necessary for a job of product group i. If we denote P,, as the probability of n jobs in the system; the probability that a setup is necessary for SPi,OPT a job of product group i if the described sequencing rule is applied, we can write:
H.Missbauer/Int.
J. Production
The probability that an arriving job is of product group i is Li/l, so the probability that a setup is necessary for an arbitrary job is:
Economics
49 (1997) 131-143
for pn. After some algebra (evaluation of the infinite sum) we get SPOPT=
SPOPT = ii12
’ SPi,OF’T.
Theorem 1. SPOPT
d %CFS,
that is, the setup-saving sequencing rule can never increase the probability that a setup is necessary for an order. Theorem 2. Zf I > 1 and li > Ofor all i, then SPopT < SPFCFs if pn > 0 for at least one n > 1. That is, setup time can be saved if there is a certain probability that a queue occurs at the facility.
Both theorems can be derived from the definition of the two sequencing rules: The rule for sequence optimization only changes the sequence of the jobs if this saves a setup. This can only be done if at least at some times there exists a queue, because otherwise there is no possibility for sequence optimization. The proof of Theorems 1 and 2 is given in Appendix A. If we approximate the probability of n jobs in the system by the M/M/l - formula: Pn =
(1 - PIP”
(9)
(which assumes FCFS discipline and is an approximation in our case), then the probability of a setup can be calculated by substituting (7) into (8) and (9)
’ li(3, - /ii)(/? + p/Ii) t1 - P)1 i= 2’ ’ (R - i,p + ,Oii) 1
(8)
Note that this is an approximation, because it assumes that the average proportion of jobs of product group i in the jobs in the queue is the same as in the arriving jobs, which is inconsistent with the assumed sequencing decision rule. Comparing the effects of the two sequencing rules (FCFS and maximum savings of setup time), we can state the following theorems.
137
In this case, the probability that a setup is necessary SPopT depends on the utilization p, which depends on the total amount of setup time and therefore on SPOPT.
2.3. Calculation of setup time and production volume as functions of WIP
As described in Section 1, we assume an order release algorithm which is based on the workload control concept and tries to keep work-in-process (WIP) at an appropriate level. We define WIP as the hours of work which the facility has to perform in order to complete all orders currently in the system. This is a useful definition in our case because most order release algorithms, like input/output control [9], load-oriented order release [21,10,22], the method of Bertrand/Wortmann [23] and CONWIP [ll, 121 also measure WIP in hours (see also Fig. 2). So we have to analyze The ratio of setup and processing time (total setup time/total processing time) as a function of WIP and shall also show the total processing time (finished work, measured in hours) as a function of WIP. These functions can be calculated as follows: The control variable of the system is the arrival rate (order release rate) h, which determines the probability that a setup is necessary (Eq. (10)) and the utilization of the server p: l
l
POPT = l[b - r(l- SpoP~)l.
(11)
So, PopT and SPopT can be calculated for a given L by solving the simultaneous Eqs. (10) and (11). In the general case described here (no restrictions on the Aij),an analytical solution of the equations is not possible, so a numerical solution must be used, which gives the values for PopT and SPopT
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J. Production Economics
sequencing, where
as functions of /2: (12)
SPCIPT=S1(&
SP FCFS
see
=
above)
(13)
POPT =f#
According to the workload control concept, the controlled variable, which reflects the state of the system, is the remaining hours of work (WR). Without the effect of setup savings, WR would be defined as WR = E(L,)b,
(14)
since b is the average service time (including the setup time) for a job in the queue and, because of the exponential distribution of the service times, also the average time until the job currently being processed is completed. The average service time which is really necessary for a job depends on E(L,) due to the setup savings, so the problem is more complicated. Since hardly an order release algorithm will stabilize WIP level which depends on itself, we simply define as the remaining work the processing time of the orders in the system: WR = E&)(6
- r).
(15)
(Equivalently, we could add an average setup time to the processing time b - r).Using the M/M/lmodel for E(L,), (15) becomes WR OPT
49 (1997) 131-143
=
1 IIqP';pT@
-
d.
(16)
Now we can calculate the setup time to processing time ratio for sequence optimization: r SPOPT SPRopT = ___ b-r
(17)
and the output, measured in hours of processing time: u = A.(b - I)
(18)
as a function of WRopT (Eq. (16) and Eq. (13) for PopT) by plotting parametriC fUnCtiOnS with 2 as control variable, WRopT at the x-axis, and SPRopT and u on the y-axis. To visualize the effects of the setup time saving sequencing rule, we compare it to FCFS
instead of SPopT =fi(A)
(see above)
(12)
So we get the setup time to processing time ratio for FCFS: r SPFCFS SPRrcFs = ___
(19)
b-r
For FCFS the remaining work is also measured in hours of processing time, but the number of jobs in the system depends on the utilization resulting from FCFS processing (Eq. (6)). So WRFCFS
=
1 ~~~~Fs(b
- 4
(20)
The diagrams are calculated for the following product data: (2 product groups) I = 2 J1 = 2 (2 products in product group 1) J2 = 3 (3 products in product group 2) I aI,1 = 10 ~i,i = 0.022863.f lzi,2 = 30 & = 0.0122377.f u2,i = 50 &i = 0.00655036.f a2,2 = 70 &2 = 0.00350615.f a2,3 = 90 AZ.3= 0.00187671.f r = 10 if the product group changes, otherwise r = 0. (f is the common factor which increases the iij as
the order release function increases the job input 2). The product data are determined in such a way that the (discrete) distribution of the processing times is similar to an exponential distribution (the /zij are proportional to the values of the probabiltiy density function of an exponential distribution with mean 32 at the processing times aij). This example is the basis for the following calculations. The setup time to processing time ratio for sequence optimization (Eq. (17)) and for FCFS (Eq. (19)) as a function of average WIP (Eq. (16) for optimization, Eq. (20) for FCFS) is depicted in Fig. 4. Furthermore the simulation result for the case of sequence optimization is given.
H.Missbauer/lnt. J. Production Economics 49 (I 997) I3 I - I43
SPR
FCFS, analytical model
0.08
WRoPT
0
50
Fig 4. Ratio of setup and processing
150
100 time as a function
of average
The simulation model represents exactly the same situation as the analytical model: singleserver system, product data as described above. For each simulation experiment (with a certain value for the control variable i) we get the total setup time and the total processing time. total setup time/total processing time, is the value on the y-axis (empirical value for SPRo&. the average remaining work, expressed as the average remaining processing time of the jobs in the system (analogous to Eq. (15)), is the value on the x-axis (empirical value for WRoPT). For the job currently in process, the remaining work is set to its average value (half the processing time). Each simulation experiment results in one point for the diagram. A series of simulation experiments (with increasing values for A) was performed, and the resulting points are joined with lines for the function in Fig. 4. The figure shows a decrease of the setup time as the WIP increases for the case of sequence optimization. The analytical model describes very accurately the relation between setup time and WIP. l
l
remaining
200 work for FCFS
250 and sequence
mFCFS
optimization
The model assumes Poisson arrivals of the orders, which signifies a poor performance of the order release algorithm - at least for this work center. We use the simulation model to examine the consequences of sequence optimization for two order release methods which are based on state information of the work center: Lower bound on remaining work: An order is released if the remaining work at the work center is below a certain minimum value. Constant number of jobs at the work center: An order is released if the number ofjobs at the work center is below a certain minimum value. (For this case the ratio of setup and processing time SPRopT can be calculated approximately from the Eqs. (7) (8) and (17) by setting the corresponding pn = 1). The orders are released in the same sequence as they arrive. In the simulations it is assumed that if a new order is released this takes place immediately after an order has been completed and the next order in the queue has been moved to the server. Fig. 5 compares the results of these methods to the results of Poisson arrivals.
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J. Production Economics 49 (1997) 131-143
SPR
lower bound on WR
0.08-
jobs in system
0.06-
Poisson arrivals
0.04-
-
0.02-
t 0
.
.
.
.
I
.
.
.
.
*
.
.
.
.
.
.
.
1.
.
.
.
I
50
in hours of processing time (Eq. (18)), as a function of average WIP (Eq. (16) for sequence optimization, Eq. (20) for FCFS) is shown in Fig. 6, together with simulation results for sequence optimization. For a given WIP level, sequence optimization increases the output over that of FCFS. The simulation results show a discrepancy to the analytical model. At 85% utilization level the analytical model underestimates the utilization by 4.7% in the case of sequence optimization. In the FCFS-case the discrepancy is only 2.2% due to the discrete processing time distribution.
3. Conclusions
1 +
SPR
(from (ll), (17), (18) or (6), (19), (18), respectively). In the case of Poisson arrivals, the utilization p depends on the average remaining work. For this case the total output of the work center, measured
A production manager must have a realistic (cognitive or formal) model of the manufacturing system. The determination of planned flow times, which means the determination of a planned level of WIP to be maintained, has grave consequences for the performance of a manufacturing system. The functional relationships between average WIP,
141
H.Missbauer~lnt. J. Production Economics 49 (1997) 131-143 U
lr OPT, analytical OPT, simulation 0.8-
FCFS, analytical
mPT
50
100
200
150
250
mFCFS
Fig 6. Output of the work center in the case of Poisson arrivals.
mean flow time, capacity utilization, etc. are important means to quantify and to visualize the conflicting goals the production manager must deal with. Sequence-dependent setup times have important consequences for these functional relationships which are not easy to estimate. In the case of sequence-dependent setup times, short average flow times, which can be obtained by a low level of WIP, can increase the setup times and costs. This effect, which is well-known in some firms, can be quantified by functions which are described in this paper. The production manager should know the quantitative shape of these functions for the manufacturing system, so we need methods to estimate these functions for real-world situations. Future research is needed to develop models covering more complex manufacturing systems and production control rules.
4. Acknowledgements The author thanks an anonymous referee for his valuable comments and suggestions.
Appendix A. Proof of Theorems 1 and 2 We will prove Theorems 1 and 2 by proving that (a) SPOPT takes its maximum value if p. = 0 for n > 1. For I > 1 and Ai > 0 for all i, SPopT gets smaller than this maximum value if P,, > 0 for at least one n > 1. (b) This maximum value of SPopT is SPFCFs. (1) SPi,oPr (Eq. (7)) is the sum of
( I 1 - !j#
times (p. + pr),
4 n
1-G
(
1”>
times pn for n > 1
and can be interpreted as a weighted average where the pn are the weights. If we assume more than one product group (I > 1) and a non-zero production volume for each product group (Ai> 0 for all i), then the following inequality holds for all i: (1-$~>(1-+~+1
forn=1,2,...
.
H. Missbauerllnt. J. Production Economics 49 (1997) 131-143
142
because (1 - 3&i/A) < 1. SO SPi,oPT is a maximum if p0 + p1 = 1. Otherwise the weight for (1 - AL/A)gets smaller and the weights for (1 - Ai/‘A)“,n > 1, get larger, which decreases and SPOPT. SPi,OPT (2) We get the maximum value of SPi,oPT from (Eq. (7)) by substituting p. + p1 = 1 and pn = 0 for n > 1: SPi,oPT = (
SPOPT
SPFCFS
Pll
WR
E&)
1- 3 ;1)
For this case, SPopT is (Eq. (8))
U
i, I
j, Ji
cf=1ii/A
= 1, SO we get: I
q2
SPopT = 1 - c i=l
r
)
References
0
[l]
which 1s the formula for SPFCFs(Eq. (5)). [2]
6. Appendix B. List of symbols [3] Y
aij
setup time when changing the product group processing time of an order of product j of product group i arrival rate of product j of product group i arrival rate of product group i ij total arrival rate
P
PFcFS POPT
SPi,OPT
probability that a setup is necessary for a job for sequence optimization probability that a setup is necessary for a job for FCFS processing probability of n jobs in the system average WIP (in hours of work) waiting to be processed (remaining work) mean number of jobs in the system utilization caused by processing time (without setup time) index for product groups and number of product groups, respectively index for products and number of products within product group i, respectively
mean service time if a setup is necessary for each job mean service rate actual utilization for FCFS processing actual utilization for sequence optimization probability that a setup is necessary for a job of product group i if the described sequencing rule is applied
[4]
[S]
[6]
[7]
[S] [9] [lo]
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