- Email: [email protected]
Replanning timing in hierarchical production planning
prod&ion economics
ELSEVIER
Int. J. Production
Economics
44 (1996) 53-61
Replanning timing in hierarchical production Hitoshi Department
of Management
Engineering,
Tsubone*,
Tokyo Metropolitan
Hirohisa
planning
Furuta
Institute of Technologv.
6-6, Asahigaoka,
Hino-city,
Tokyo 191. Japan
Abstract In this paper we discuss the relationship between the timing for replanning and the buffer inventory in terms of system performance in a multistage production system, where there are multiple opportunities for replanning within the accumulated production lead time. Two performance measures are used: the unfilled rate for market demand (c()and the
rate of change in production quantity due to replanning (8). The results obtained through simulation study yield a basis for choosing a replanning rule. Keywords:
Replanning;
Multistage
production
system; Buffer inventory
1. Introduction In the make-to-stock environment, there is a trade-off relationship between achieving stable production planning and being responsive to changes in market requirements due to demand uncertainty. Frequent adjustment to the production plan, caused by changes in customer requirements and sales forecasts, may induce instability and uncertainty in the production planning and control system, resulting in increased managerial costs associated with changes in production rate, emergency setups, and expedition due to disruption in production. Adversely, fixing the production plan over a long term might increase the required buffer inventory level of products or components/parts to enable maintenance of a desired customer service level, although this approach enables stable production planning. The hierarchical approach can ease
* Corresponding
author.
Tel.: 0425 83 5111. Fax: 0425 83 5119.
0925-5273/96/$15.00 Copyright SSDI 0925-5273(95)00092-5
0
such a conflicting problem. Aggregate production planning determines the aggregate quantity of products or components/parts to be produced over a relatively long term, in order to stabilize the production. On the other hand, detailed planning involves individual products or components/parts over a short planning period, in order to respond to the market demand, although the aggregate quantity to be produced is fixed [l-3]. Thus the manufacturing performance depends on the planning system for such design decisions as the replanning timing, replanning interval, and fixed period in production planning. There are several schemes regarding how to deal with demand uncertainty [4,5]. Lin and Krajewski [6], Sridharan and Berry [7], and Cambell [8] examined how the replanning interval, fixed interval and forecast window would affect the production costs relative to setup, inventory, change in production, etc., with an infinite loading capacity under a rolling schedule. Carlson and Yano [9, lo] clarified the interaction among safety stock, emergency setup and rescheduling without capacity
1996 Elsevier Science B.V. All rights reserved
54
H. Tsubone, H. Furuta/Int.
J. Production
Economics
44 (1996) 53-61
constraints. Tsubone et al. [ 1 l] analyzed how fixed and flexible production planning rules would affect the manufacturing performance with a finite loading capacity. Fundamental questions to be solved still remain in determining the timing involved in replanning under capacity constraints in a multistage environment, with respect to the following concerns. (1) How will the replanning timing impact the manufacturing performance, such as unfilled rate, setup frequency and inventory level required? (2) What inventory levels for products or components/parts which can control the unfilled rate for market demand within its acceptable limit are required under several replanning schemes? To answer the above questions, we examine the impact on the manufacturing performance for the replanning timing in a hierarchical production planning system, where higher-level decisions impose constraints on lower-level decisions. The production system is assumed to be such that there are multiple opportunities for replanning within the accumulated production lead time in a multistage environment. Two performance measures were used: the unfilled rate or market demand (IX)and the ratio of the changed production quantity to the planned quantity (/I).
period, and that all unfilled demands for products are backordered when there is no stock available. It is also assumed that one complete unit of each component/part is required for a finished product.
2. Model
2.2. Two-level production
2. I. Production
system
2. I. 1. Market demand It is assumed that the demands for finished products are stationary and independent during each
‘fabrication process
r_m
part fabrication process
2. I .2. Production process The production process can be modeled as a three-stage production process, as shown in Fig. 1. The materials are processed into different components/parts categories for respective finished products in the fabrication processes. Components/ parts are fed to the assembly lines to be assembled into finished products. There is a one-week time lag between the first-stage and second-stage fabrication processes, and also a one-week time lag between the second-stage fabrication process and the assembly line. Accordingly, parts which would be required on the assembly line at week (w + 2) should be processed during week (w + 1) in the second fabrication process, and should be processed during week w in the first fabrication process. Lost time for setup on the assembly line can be regarded as being negligible when switching from one kind of finished product to another, but in the two fabrication processes, lost time for setup cannot be considered negligible. One complete unit of each component/part is required to produce one unit of each finished product.
planning
system
A two-level production planning model is used to determine the number of units of finished products, components and part to be produced in the forthcoming planning period: monthly planning and weekly planning.
component
Fig. 1. Production process.
Assembly line
Stock of finished oroduct
H. Tsubone, H. Furuta/Int.
J. Production
2.2.1. Monthly planning Monthly planning determines the aggregate numbers of finished products, components and parts to be produced for the next month on the planning horizon. These numbers are determined by using the aggregate forecast demand for finished products. Monthly planning takes place once a month. The aggregate numbers of finished products, components and parts are determined as X,=&+SA-at_,,
(2.1)
where X, is the aggregate number of finished products to be produced during month t, fit is the forecast demand in month t, SA is the aggregate buffer inventory level for finished productszhich might serve to prevent any shortages and AI,_ 1 is the aggregate inventory level for finished products at the end of month (t - 1). The aggregate numbers of components and parts also correspond to the number of finished products. These aggregate numbers are fixed throughout the month in order to stabilize production. The aggregate number is divided into weekly portions, so that the total number of finished products, components or parts to be produced during the planning month will be equal to the corresponding aggregate numbers which have been determined during the monthly planning at the higher planning level. Y, = X,IL
(2.2)
where Y, is the aggregate number of finished products, components or parts to be produced in week w in planning month t and L is the number of weeks in month t.
Economics
44 (1996) 53-61
55
parts, under conditions which satisfy the components/ parts requirements from the assembly line. The assumption tends to minimize the total setup time throughout the year. Tables 1 and 2 show the model formulation for production planning rules on the assembly line, and indicate both component and part fabrication processes, respectively.
2.3. Replanning policy Replanning is carried out to correct/adjust cast errors. Table 1 Formulation
for finished
products
assembly
(2.3)
(2.4)
fb+ z is the expected inventory level for finished product k at the end of week (w + 2), R&,+3 is the prearranged, at the end of week W, number of units of finished product k to be produced in week (w + 3), JL+, is the forecast market demand for finished product k in week (w + 3), Y,+ 3 is the aggregate number of units of finished products or components/parts to be produced in week (w + 3) and K” is the set of finished product k to be produced in the planning week.
Table 2 Formulation
&{
for part/component
production
SET,.d;+s
2.2.2. Weekly production planning Since the setup time is negligible on the assembly line, the number of finished products to be assembled in week (w + 1) is determined so that the expected inventory levels for the expected requirement ratios at the end of planning week (w - 2) are equal. This rule aims at minimizing the unfilled rate of finished products necessary to meet the market demands. The number of each component/part to be produced each week is determined to be proportional to the setup time and the expected weekly requirements for individual components/
fore-
subject 1
B$‘!k.
(2.3
}+min
to
B!%
=
Yw+3
(2.6)
LtKo
Rk,,w+3
inthecaseofn=2
1 + a@” ,_+, in thecaseofn=
(2.7) 1
(2.8)
SETk is the set up time for part/component k, Bt:“z is the number of parts to be produced in week (w + 2) at the fabricais the expected inventory level for tion number n, @!‘:, parts/components k at the end of week (w + 2) at fabrication stage n and n is the fabrication stage (n = 1, 2 in this paper).
56
H. Tsubone, H. Furuta/Int.
J. Production
Fig. 2 provides an illustration of the timing for replanning in the multistage process. Consider weekly planning carried out at the end of week (w - 2) as shown in Fig. 2(a). At the first stage in weekly planning, the weekly planning model determines the numbers of each finished product to be assembled in week (w + 1). At the second stage, the planning model determines the numbers of individual components to be produced at the secondstage process in week w, on the basis of finished product requirements which have been planned at the first planning stage. At the third stage, the planning model determines the numbers of each part to be produced at the first process in week (w - l), on the basis of component requirements which have been determined at the second stage of planning. At the end of week (w - l), as shown in Fig. 2(b), after 1 week has elapsed, the weekly planning model similarly determines the number of individual finished products, components and parts to be produced, respectively, in week (w + 2) week (w -I- 1) and week w. There are four scenarios or planning rules concerning weekly planning policy. Rule 1 is referred to
(b)
Planning cycle 2
R p
R
(cl
Rule
1
Rule
2
&”
B’*’
B’l’
Cd)
Fig. 2. Replanning rules.
Economics
44 (1996) 53-61
Under Rule 1, the respective as “fixing planning”. numbers of each finished product, component and part are fixed over the accumulated planning periods, as shown in Fig. 2(b). That is, the individual number of units of finished products, components and parts, actually to be produced in week (w + l), week w and week (w - l), respectively, are equal to the respective units which have been planned at the end of week (w - 2). Since Rule 1 does not allow variation in the respective number of units to be produced, such planning might ease the shop-floor control by avoiding “nervous” replanning. However, there is room to adjust or correct the number of units of finished products and components to actually be produced after 1 week has elapsed, although the individual numbers of parts cannot be varied. Rules 2,3 and 4 are referred to as “replanning”, since an updated inventory level and a more accurate forecast demand for finished products are reflected in the planning. Rule 2 may change only the numbers of each finished product to be produced in week w on the assembly line, which has already been determined at the end of week (w - 3), and regards the production plan determined at the end of week (w - 2) as “tentative”, as shown in Fig. 2(c). However, the number of finished products can be varied up to the number for which respective components can be supplied. Rule 3 may change the respective number of finished products and components to be produced, respectively, in week (w + 1) and week w, which have been planned at the end of week (w - 2), as shown in Fig. 2(d). However, this planning cannot change the number of finished products to be produced in week w, which was already determined at the end of week (w - l), since that number is “fixed”. Rule 4 may change the number of all the finished products and components that can be replanned, as shown in Fig. 2(e). That is, the following three categories can be replanned: the number of components to be produced in week w which was determined at the end of week (w - 2), and the numbers of finished products to be produced in week w and in week (w + 1) which had been determined in week (w - 2). The algorithms for
H. Tsubone. H. Furutallnt.
individual planning pendixes A-E.
2.4. Manufacturing
rules
are
performance
expressed
J. Production
in Ap-
E f { - min(ZL, 0)} 2 2 dk,. w=l k=l I w=l k=l
(2) Rate (8) of change in production to replanning,
w=l
quantity
(2.9) due
k=l
/ w=l
44 (1996) 53-61
the effects of timing for replanning turing performance.
3.2. Experimental
criteria
Although Lin and Krajewski [6] use setup costs and expedition costs due to replanning as the measure to evaluate manufacturing performance, such a cost presentation sometimes fails to completely express the production planning effect on manufacturing performance. Additionally, there are difficulties in estimating such tangible costs. Thus, we use the unfilled rate for meeting market demands (IX) and the change rate in production quantity due to replanning (/?). They are expressed as follows. (1) Unfilled rate (a) for meeting market demands, CI=
Economics
k=l
57
on the manufac-
conditions
The series of experiments was conducted under the following experimental conditions. The fluctuations in demand are a normally distributed random variable with zero mean and known variance (a:): od = p . ii (where p is the coefficient of variance and is fixed at 0.2 in this experiment). The forecasted demand was set equal to the expected demand in both monthly and weekly planning. Thus, the standard deviation in the demand od is equivalent to the standard deviation in the forecasted demand error. The buffer inventory levels for finished products and both components and parts at fabrication processes 1 and 2, and the assembly line, respectively, as design parameters, were set proportional to the weekly expected market demand. Five hundred planning periods are used as available working weeks. It was assumed that 1 month has 4 weeks, i.e., the L value was set to 4. The first 100 periods were discarded in computing the average value of the unfilled rate and rate of change in production quantity.
3.3. Experimental
results
(2.10) where 1”, is the actual inventory level for finished product k at the end of week w, dk, is the actual demand for finished product k in week w and the symbol I ) means absolute value.
3. Numerical
experiments
3. I. Experimental
and results
purpose
The pupose of experiments is to clarify how the manufacturing performance will be affected by four planning scenarios under the conditions whereby the buffer inventories for finished products and both components and parts are maintained at specific levels. Simulations were conducted to study
The experimental results are represented as the mean observed values of the performance measures for each level of the experimental factors. These mean values are plotted in Figs. 3-8. Fig. 3 represents the relationship between the buffer inventory level for components/parts and the unfilled rate for market demand under Rules 1 and 2. Under Rule 1, the unfilled rate (a) is not affected by buffer inventory levels for either components or parts (BI”) and BI”‘), because Rule 1 is fixed planning. The unfilled rate can be decreased under Rule 2, as the buffer inventory level for components increases independent of the buffer inventory for parts in the first process. Fig. 4 compares Rule 2 and Rule 3. The unfilled rate is affected by both the buffer inventory levels for components and parts under Rule 3, although it is only affected by the
H. Tsubone, H. Furutallnt.
58 0.
J. Production Economics 44 (1996) 53-61
18
0. 18 ,Aule
2
1
$0.17
2 0. 16 2 p 0.15 2 E 5
0.14 0. 13 0. 12 “.
Buffer inventory for components
level
Buffer inventory for co.ponent*
level
1.0
1.0
J
Buffer inventory for part*
level
Fig. 3. Relationship between the buffer inventory level for components/parts and unfilled rate (under Rules 1 and 2).
Fig. 5. Relationship between the buffer inventory level for components/parts and unfilled rate (under Rules 2 and 4).
Fig. 4. Relationship between the buffer inventory level for component/parts and unfilled rate (under Rules 2 and 3).
Fig. 6. Relationship between the buffer inventory level for components/parts and unfilled rate (under Rules 3 and 4).
buffer inventory level for components under Rule 2. However, Rule 3 is inferior to Rule 2, in spite of it involving two kinds of buffer inventory, except for the condition that the buffer inventory of components is controlled at a low level. Fig. 5 compares Rule 2 and Rule 4. Fig. 6 compares Rule 3 and Rule 4. The following points were clarified in Figs. 3-6. (1) There is no difference in the unfilled rate among the four rules when there are no buffer inventories for either components or parts. (2) The unfilled rate is not affected by the buffer inventory for parts in the first process under Rule 1 and Rule 2, and it can be decreased to a low level at the same degree under either Rule 3 or
Rule 4 by maintaining a buffer inventory of parts only when there is no buffer inventory for components. (3) There is little difference in the unfilled rates between Rules 2 and 4 when the buffer inventory for only components is maintained where there is no buffer inventory for parts. (4) The unfilled rate can be controlled at a low level in the order of Rule 4, Rule 2, Rule 3, and Rule 1 under the conditions that both buffer inventories for parts and components are maintained. However, the unfilled rates under Rule 2 and Rule4 become equal when the buffer inventory for components is maintained at a very high level.
H. Tsubone, H. Furuta/Int.
J. Production
Economics
44 (1996) 53-61
59
0. m
-
s
0.
c :: 0. g
2 * ”
0.6
0.7
0. B
0.9
1.0
Buffer inventory level for finished product
1.1
Fig. 7. Relationship between the buffer inventory level for both the finished product and components/parts, which can control the unfilled rate within a specified value (tl = 0.01).
Fig. 7 represents the relationship between the buffer inventory levels for finished products and the combination of those of parts and components, which can control the unfilled rate at a specified value (U = 0.01). Rule 4 can decrease the required buffer inventory to the lowest level among these three rules, since it has the largest extent in replanning. In the comparison between Rule 2 and Rule 3, Rule 2 can control the required buffer inventory at a lower level than Rule 3, even though Rule 2 has a smaller extent in replanning. Fig. 8 represents the relationship between the buffer inventory level for components/parts and the rate of change in production quantity. As the inventory level increases under any replanning rule, the rate of change in production quantity increases to respective constant values. Concerning Rules 1, 3 and 4, the greater extent the planning rule has in replanning, the larger the rate of change in production quantity. The rate of change in production quantity is the highest under Rule 2, although Rule 2 has a smaller extent in replanning than Rules 3 and 4. The reason is that the production quantity is replanned at a point just before assembling products under Rule 2. On the other hand, under Rules 3 and 4, the production quantity has already been replanned at an earlier time than Rule 2. Through a series of experiments, the following points were clarified. (1) When the buffer for components/parts is maintained at a certain inventory level, Rule 4 can
0.
0
0.4 0.2 0 Buffer inventory
0.6
level
0.8
for
1.0
1.2
1.4
components/Parts
Fig. 8. Relationship between the buffer inventory level for components/parts and the changed rate in production quantity.
control the unfilled rate at the lowest level among the four rules. The next is Rule 2 or Rule 3 depending on the inventory level, followed by Rule 1. (2) The rate of change in production quantity depends on the timing and extent of replanning. Rule 3 can control the rate of change at the lowest level among the three rules, except for Rule 1. The next is Rule 4 and the last is Rule 2.
4. Conclusion We discussed the relationship between the timing for replanning and the buffer inventory in terms of system performance in a multistage production system, where there are multiple opportunities for replanning within the accumulated production lead time. A simulation model was developed in order to clarify the timing for replanning, which would affect manufacturing performance. Experimental results indicate the following: (1) The total required buffer inventory for finished products and components and parts can be controlled at a comparatively low level under such a replanning rule (Rule 2) that limits a portion of the production process, the assembly line, although the production quantity to be changed becomes maximal. (2) On the other hand, the production quantity to be changed can be decreased under such a replanning rule (Rule 3) that manages the overall production processes to change the production quantity, although the buffer inventory is required to be maintained at a higher level than the former replanning
H. Tsubone, H. Furutallnt.
60
J. Production
rule. (3) The total required buffer inventory for finished products and components/parts can be controlled at the lowest level under such a replanning rule (Rule 4) which combines the two rules mentioned above. That is, this rule replans two times, just before and one week before the assembly line. The production quantity to be changed can be controlled at a low level in comparison with rules that replan once on the basis of information on immediate market demand. However, the management may have to withstand instability and uncertainty in production control due to the high frequency of replanning. The obtained results also offer a basis for choosing a replanning rule through the trade-off between the costs regarding the unfilled rate, production quantity to be changed due to replanning, and the required buffer inventory of components/parts and finished products.
Appendix A: Planning Algorithm product production
for finished
ewv+3 =
(
1
f:+z
ksK"
Step 6. Compute,
stop.
Appendix B: Planning Algorithm parts production Step 1. Q+ Step 2. Let
K” = {kl@$
< Rk,,w+3}
If K” = f#~go to step 8. Otherwise go to step 3. Step 3. Compute b$y2 for all k E K”
Step 4. Compute for all k E K”.
if Ak > 0
If yes, B’,“!“2= ,$!“2
for all keK”
stop. Otherwise,
) -fl,+2
if ykW,,,,+3 3 0.
If yes,
K’ = {kJAk < 0}
Go to step 6. Step 6. Let B’,“y2 = b$‘y2 - Ak
for all keK’
Update for all kEKnK’ k E K” is optimal; Stop.
Otherwise go to step 5. Step 5. If K” = 8 go to step 6. Otherwise
K” = {kl&,w+3> 0)
Go to step 3.
Q+Q+
c
kcKnF
Ak
Let K” = (klAk >O}
and go to step 3. Step 7. Compute for all keK”
Let
Rk,,w+3 =0
for components/
Yw+3
Step 5. Check, for all keK”,
.a;+,
kp+3
Step 4. Check, for all kEK”,
44 (1996) 53-61
Ak = b’,“y2 +3’,“!“2 - R:,w+3
Step 1. Q+ Yw+, Step 2. k E K” for all k. Step 3. Compute r”,,,,,+ 3 for all k E K”. Q-t
Economics
for kEKn??. stop.
H. Tsubone. H. Furuta/Int.
Appendix C: Planning Algorithm
J. Production
for Rule 2
Economics
44 (1996) 53-61
Appendix E: Planning Algorithm
61
for Rule 4
Rule 3 is carried out after Rule 2.
Step 1. QC Y,+l Step 2. Compute Rk,, ,,,+1 using steps l-6 of App-
endix A. References
Step 3. For all k E K”, let
K’ = {k 1Rk,, ,,,+1 3 %ii;,” 1 + B$jk} K” = {kIRk,,w+l
< Ii?:?,
+ B’,“‘k}
[l] [Z]
IfK’=qJ [3]
R:,,N
= Rk,,w+l
for all k E K” is optimal; Stop. Otherwise, Rk,:w+l = @!‘_“, + B$!jk for all keK’
[4]
[S]
Step 4. If K” = 0 stop.
Otherwise, let
Q+Q-_;,R:w,,
[6]
and carry out steps 3-6 of Appendix A, and go to step 2.
[7]
[8]
Appendix D: Planning Algorithm
for Rule 3
Rk,++ 2 using the algorithm of Appendix A. Step 2. Calculate &!k, using the algorithm of Appendix B. Step 3. Decide Bcy, using the same algorithm of steps 3-5 of Appendix B. Step 4. Replanning Rk,,w+2 according to Rule 2.
[9]
Step 1. Calculate
[lo]
[ll]
Bitran, B. and Hax, A., 1977, On the design of hierarchical production planning system. Dec. Sci., 8: 28-55. Birtan, G., Hass, E. and Hax, A., 1981. Hierarchical production planning system: A single-stage system. Oper. Res., 29: 717-741. Birtan, G., Hass, E. and Hax, A., 1982. Hierarchical production planning system: A two-stage system. Oper. Res., 30: 232-251. Murthy, D.N.P. and Ma, L., 1991. MRP with uncertainty: A review and some extensions. Int. J. Prod. Econom., 25: 5 l-64. Narasimhan, R., Melnyk, S.A. and Carter, P.L., 1992. Research framework for investigating the effectiveness of dampening procedures to cope with MRP system nervousness. Int. J. Oper. Prod. Mgmt., 12: 3G43. Lin, N. and Lee, K., 1991. A model for master production, scheduling in uncertain environments. Dec. Sci., 23: 839-861. Sridharan, V. and William, L.B., 1989. Freezing the master production schedule under demand uncertainty. Dec. Sci., 21: 97-120. Campbell, G.M., 1991. Master production scheduling under rolling planning horizons with fixed order intervals. Dec. Sci., 23: 312-331. Carlson, R.C. and Yano, C.A., 1986. Safety stock in MRPsystems with emergency setups for components. Mgmt. Sci., 32: 403-412. Yano, C.A. and Carlson, R.C., 1987. Interaction between frequency of rescheduling and the role of safety stock in materials requirements planning systems. Int. J. Prod. Res., 25: 221-232. Tsubone, H., Matsuura, H. and Tsutsu, T., 1991. Hierarchical production planning system for a two-stage process. Int. J. Prod. Res., 29: 769-785.
ELSEVIER
Int. J. Production
Economics
44 (1996) 53-61
Replanning timing in hierarchical production Hitoshi Department
of Management
Engineering,
Tsubone*,
Tokyo Metropolitan
Hirohisa
planning
Furuta
Institute of Technologv.
6-6, Asahigaoka,
Hino-city,
Tokyo 191. Japan
Abstract In this paper we discuss the relationship between the timing for replanning and the buffer inventory in terms of system performance in a multistage production system, where there are multiple opportunities for replanning within the accumulated production lead time. Two performance measures are used: the unfilled rate for market demand (c()and the
rate of change in production quantity due to replanning (8). The results obtained through simulation study yield a basis for choosing a replanning rule. Keywords:
Replanning;
Multistage
production
system; Buffer inventory
1. Introduction In the make-to-stock environment, there is a trade-off relationship between achieving stable production planning and being responsive to changes in market requirements due to demand uncertainty. Frequent adjustment to the production plan, caused by changes in customer requirements and sales forecasts, may induce instability and uncertainty in the production planning and control system, resulting in increased managerial costs associated with changes in production rate, emergency setups, and expedition due to disruption in production. Adversely, fixing the production plan over a long term might increase the required buffer inventory level of products or components/parts to enable maintenance of a desired customer service level, although this approach enables stable production planning. The hierarchical approach can ease
* Corresponding
author.
Tel.: 0425 83 5111. Fax: 0425 83 5119.
0925-5273/96/$15.00 Copyright SSDI 0925-5273(95)00092-5
0
such a conflicting problem. Aggregate production planning determines the aggregate quantity of products or components/parts to be produced over a relatively long term, in order to stabilize the production. On the other hand, detailed planning involves individual products or components/parts over a short planning period, in order to respond to the market demand, although the aggregate quantity to be produced is fixed [l-3]. Thus the manufacturing performance depends on the planning system for such design decisions as the replanning timing, replanning interval, and fixed period in production planning. There are several schemes regarding how to deal with demand uncertainty [4,5]. Lin and Krajewski [6], Sridharan and Berry [7], and Cambell [8] examined how the replanning interval, fixed interval and forecast window would affect the production costs relative to setup, inventory, change in production, etc., with an infinite loading capacity under a rolling schedule. Carlson and Yano [9, lo] clarified the interaction among safety stock, emergency setup and rescheduling without capacity
1996 Elsevier Science B.V. All rights reserved
54
H. Tsubone, H. Furuta/Int.
J. Production
Economics
44 (1996) 53-61
constraints. Tsubone et al. [ 1 l] analyzed how fixed and flexible production planning rules would affect the manufacturing performance with a finite loading capacity. Fundamental questions to be solved still remain in determining the timing involved in replanning under capacity constraints in a multistage environment, with respect to the following concerns. (1) How will the replanning timing impact the manufacturing performance, such as unfilled rate, setup frequency and inventory level required? (2) What inventory levels for products or components/parts which can control the unfilled rate for market demand within its acceptable limit are required under several replanning schemes? To answer the above questions, we examine the impact on the manufacturing performance for the replanning timing in a hierarchical production planning system, where higher-level decisions impose constraints on lower-level decisions. The production system is assumed to be such that there are multiple opportunities for replanning within the accumulated production lead time in a multistage environment. Two performance measures were used: the unfilled rate or market demand (IX)and the ratio of the changed production quantity to the planned quantity (/I).
period, and that all unfilled demands for products are backordered when there is no stock available. It is also assumed that one complete unit of each component/part is required for a finished product.
2. Model
2.2. Two-level production
2. I. Production
system
2. I. 1. Market demand It is assumed that the demands for finished products are stationary and independent during each
‘fabrication process
r_m
part fabrication process
2. I .2. Production process The production process can be modeled as a three-stage production process, as shown in Fig. 1. The materials are processed into different components/parts categories for respective finished products in the fabrication processes. Components/ parts are fed to the assembly lines to be assembled into finished products. There is a one-week time lag between the first-stage and second-stage fabrication processes, and also a one-week time lag between the second-stage fabrication process and the assembly line. Accordingly, parts which would be required on the assembly line at week (w + 2) should be processed during week (w + 1) in the second fabrication process, and should be processed during week w in the first fabrication process. Lost time for setup on the assembly line can be regarded as being negligible when switching from one kind of finished product to another, but in the two fabrication processes, lost time for setup cannot be considered negligible. One complete unit of each component/part is required to produce one unit of each finished product.
planning
system
A two-level production planning model is used to determine the number of units of finished products, components and part to be produced in the forthcoming planning period: monthly planning and weekly planning.
component
Fig. 1. Production process.
Assembly line
Stock of finished oroduct
H. Tsubone, H. Furuta/Int.
J. Production
2.2.1. Monthly planning Monthly planning determines the aggregate numbers of finished products, components and parts to be produced for the next month on the planning horizon. These numbers are determined by using the aggregate forecast demand for finished products. Monthly planning takes place once a month. The aggregate numbers of finished products, components and parts are determined as X,=&+SA-at_,,
(2.1)
where X, is the aggregate number of finished products to be produced during month t, fit is the forecast demand in month t, SA is the aggregate buffer inventory level for finished productszhich might serve to prevent any shortages and AI,_ 1 is the aggregate inventory level for finished products at the end of month (t - 1). The aggregate numbers of components and parts also correspond to the number of finished products. These aggregate numbers are fixed throughout the month in order to stabilize production. The aggregate number is divided into weekly portions, so that the total number of finished products, components or parts to be produced during the planning month will be equal to the corresponding aggregate numbers which have been determined during the monthly planning at the higher planning level. Y, = X,IL
(2.2)
where Y, is the aggregate number of finished products, components or parts to be produced in week w in planning month t and L is the number of weeks in month t.
Economics
44 (1996) 53-61
55
parts, under conditions which satisfy the components/ parts requirements from the assembly line. The assumption tends to minimize the total setup time throughout the year. Tables 1 and 2 show the model formulation for production planning rules on the assembly line, and indicate both component and part fabrication processes, respectively.
2.3. Replanning policy Replanning is carried out to correct/adjust cast errors. Table 1 Formulation
for finished
products
assembly
(2.3)
(2.4)
fb+ z is the expected inventory level for finished product k at the end of week (w + 2), R&,+3 is the prearranged, at the end of week W, number of units of finished product k to be produced in week (w + 3), JL+, is the forecast market demand for finished product k in week (w + 3), Y,+ 3 is the aggregate number of units of finished products or components/parts to be produced in week (w + 3) and K” is the set of finished product k to be produced in the planning week.
Table 2 Formulation
&{
for part/component
production
SET,.d;+s
2.2.2. Weekly production planning Since the setup time is negligible on the assembly line, the number of finished products to be assembled in week (w + 1) is determined so that the expected inventory levels for the expected requirement ratios at the end of planning week (w - 2) are equal. This rule aims at minimizing the unfilled rate of finished products necessary to meet the market demands. The number of each component/part to be produced each week is determined to be proportional to the setup time and the expected weekly requirements for individual components/
fore-
subject 1
B$‘!k.
(2.3
}+min
to
B!%
=
Yw+3
(2.6)
LtKo
Rk,,w+3
inthecaseofn=2
1 + a@” ,_+, in thecaseofn=
(2.7) 1
(2.8)
SETk is the set up time for part/component k, Bt:“z is the number of parts to be produced in week (w + 2) at the fabricais the expected inventory level for tion number n, @!‘:, parts/components k at the end of week (w + 2) at fabrication stage n and n is the fabrication stage (n = 1, 2 in this paper).
56
H. Tsubone, H. Furuta/Int.
J. Production
Fig. 2 provides an illustration of the timing for replanning in the multistage process. Consider weekly planning carried out at the end of week (w - 2) as shown in Fig. 2(a). At the first stage in weekly planning, the weekly planning model determines the numbers of each finished product to be assembled in week (w + 1). At the second stage, the planning model determines the numbers of individual components to be produced at the secondstage process in week w, on the basis of finished product requirements which have been planned at the first planning stage. At the third stage, the planning model determines the numbers of each part to be produced at the first process in week (w - l), on the basis of component requirements which have been determined at the second stage of planning. At the end of week (w - l), as shown in Fig. 2(b), after 1 week has elapsed, the weekly planning model similarly determines the number of individual finished products, components and parts to be produced, respectively, in week (w + 2) week (w -I- 1) and week w. There are four scenarios or planning rules concerning weekly planning policy. Rule 1 is referred to
(b)
Planning cycle 2
R p
R
(cl
Rule
1
Rule
2
&”
B’*’
B’l’
Cd)
Fig. 2. Replanning rules.
Economics
44 (1996) 53-61
Under Rule 1, the respective as “fixing planning”. numbers of each finished product, component and part are fixed over the accumulated planning periods, as shown in Fig. 2(b). That is, the individual number of units of finished products, components and parts, actually to be produced in week (w + l), week w and week (w - l), respectively, are equal to the respective units which have been planned at the end of week (w - 2). Since Rule 1 does not allow variation in the respective number of units to be produced, such planning might ease the shop-floor control by avoiding “nervous” replanning. However, there is room to adjust or correct the number of units of finished products and components to actually be produced after 1 week has elapsed, although the individual numbers of parts cannot be varied. Rules 2,3 and 4 are referred to as “replanning”, since an updated inventory level and a more accurate forecast demand for finished products are reflected in the planning. Rule 2 may change only the numbers of each finished product to be produced in week w on the assembly line, which has already been determined at the end of week (w - 3), and regards the production plan determined at the end of week (w - 2) as “tentative”, as shown in Fig. 2(c). However, the number of finished products can be varied up to the number for which respective components can be supplied. Rule 3 may change the respective number of finished products and components to be produced, respectively, in week (w + 1) and week w, which have been planned at the end of week (w - 2), as shown in Fig. 2(d). However, this planning cannot change the number of finished products to be produced in week w, which was already determined at the end of week (w - l), since that number is “fixed”. Rule 4 may change the number of all the finished products and components that can be replanned, as shown in Fig. 2(e). That is, the following three categories can be replanned: the number of components to be produced in week w which was determined at the end of week (w - 2), and the numbers of finished products to be produced in week w and in week (w + 1) which had been determined in week (w - 2). The algorithms for
H. Tsubone. H. Furutallnt.
individual planning pendixes A-E.
2.4. Manufacturing
rules
are
performance
expressed
J. Production
in Ap-
E f { - min(ZL, 0)} 2 2 dk,. w=l k=l I w=l k=l
(2) Rate (8) of change in production to replanning,
w=l
quantity
(2.9) due
k=l
/ w=l
44 (1996) 53-61
the effects of timing for replanning turing performance.
3.2. Experimental
criteria
Although Lin and Krajewski [6] use setup costs and expedition costs due to replanning as the measure to evaluate manufacturing performance, such a cost presentation sometimes fails to completely express the production planning effect on manufacturing performance. Additionally, there are difficulties in estimating such tangible costs. Thus, we use the unfilled rate for meeting market demands (IX) and the change rate in production quantity due to replanning (/?). They are expressed as follows. (1) Unfilled rate (a) for meeting market demands, CI=
Economics
k=l
57
on the manufac-
conditions
The series of experiments was conducted under the following experimental conditions. The fluctuations in demand are a normally distributed random variable with zero mean and known variance (a:): od = p . ii (where p is the coefficient of variance and is fixed at 0.2 in this experiment). The forecasted demand was set equal to the expected demand in both monthly and weekly planning. Thus, the standard deviation in the demand od is equivalent to the standard deviation in the forecasted demand error. The buffer inventory levels for finished products and both components and parts at fabrication processes 1 and 2, and the assembly line, respectively, as design parameters, were set proportional to the weekly expected market demand. Five hundred planning periods are used as available working weeks. It was assumed that 1 month has 4 weeks, i.e., the L value was set to 4. The first 100 periods were discarded in computing the average value of the unfilled rate and rate of change in production quantity.
3.3. Experimental
results
(2.10) where 1”, is the actual inventory level for finished product k at the end of week w, dk, is the actual demand for finished product k in week w and the symbol I ) means absolute value.
3. Numerical
experiments
3. I. Experimental
and results
purpose
The pupose of experiments is to clarify how the manufacturing performance will be affected by four planning scenarios under the conditions whereby the buffer inventories for finished products and both components and parts are maintained at specific levels. Simulations were conducted to study
The experimental results are represented as the mean observed values of the performance measures for each level of the experimental factors. These mean values are plotted in Figs. 3-8. Fig. 3 represents the relationship between the buffer inventory level for components/parts and the unfilled rate for market demand under Rules 1 and 2. Under Rule 1, the unfilled rate (a) is not affected by buffer inventory levels for either components or parts (BI”) and BI”‘), because Rule 1 is fixed planning. The unfilled rate can be decreased under Rule 2, as the buffer inventory level for components increases independent of the buffer inventory for parts in the first process. Fig. 4 compares Rule 2 and Rule 3. The unfilled rate is affected by both the buffer inventory levels for components and parts under Rule 3, although it is only affected by the
H. Tsubone, H. Furutallnt.
58 0.
J. Production Economics 44 (1996) 53-61
18
0. 18 ,Aule
2
1
$0.17
2 0. 16 2 p 0.15 2 E 5
0.14 0. 13 0. 12 “.
Buffer inventory for components
level
Buffer inventory for co.ponent*
level
1.0
1.0
J
Buffer inventory for part*
level
Fig. 3. Relationship between the buffer inventory level for components/parts and unfilled rate (under Rules 1 and 2).
Fig. 5. Relationship between the buffer inventory level for components/parts and unfilled rate (under Rules 2 and 4).
Fig. 4. Relationship between the buffer inventory level for component/parts and unfilled rate (under Rules 2 and 3).
Fig. 6. Relationship between the buffer inventory level for components/parts and unfilled rate (under Rules 3 and 4).
buffer inventory level for components under Rule 2. However, Rule 3 is inferior to Rule 2, in spite of it involving two kinds of buffer inventory, except for the condition that the buffer inventory of components is controlled at a low level. Fig. 5 compares Rule 2 and Rule 4. Fig. 6 compares Rule 3 and Rule 4. The following points were clarified in Figs. 3-6. (1) There is no difference in the unfilled rate among the four rules when there are no buffer inventories for either components or parts. (2) The unfilled rate is not affected by the buffer inventory for parts in the first process under Rule 1 and Rule 2, and it can be decreased to a low level at the same degree under either Rule 3 or
Rule 4 by maintaining a buffer inventory of parts only when there is no buffer inventory for components. (3) There is little difference in the unfilled rates between Rules 2 and 4 when the buffer inventory for only components is maintained where there is no buffer inventory for parts. (4) The unfilled rate can be controlled at a low level in the order of Rule 4, Rule 2, Rule 3, and Rule 1 under the conditions that both buffer inventories for parts and components are maintained. However, the unfilled rates under Rule 2 and Rule4 become equal when the buffer inventory for components is maintained at a very high level.
H. Tsubone, H. Furuta/Int.
J. Production
Economics
44 (1996) 53-61
59
0. m
-
s
0.
c :: 0. g
2 * ”
0.6
0.7
0. B
0.9
1.0
Buffer inventory level for finished product
1.1
Fig. 7. Relationship between the buffer inventory level for both the finished product and components/parts, which can control the unfilled rate within a specified value (tl = 0.01).
Fig. 7 represents the relationship between the buffer inventory levels for finished products and the combination of those of parts and components, which can control the unfilled rate at a specified value (U = 0.01). Rule 4 can decrease the required buffer inventory to the lowest level among these three rules, since it has the largest extent in replanning. In the comparison between Rule 2 and Rule 3, Rule 2 can control the required buffer inventory at a lower level than Rule 3, even though Rule 2 has a smaller extent in replanning. Fig. 8 represents the relationship between the buffer inventory level for components/parts and the rate of change in production quantity. As the inventory level increases under any replanning rule, the rate of change in production quantity increases to respective constant values. Concerning Rules 1, 3 and 4, the greater extent the planning rule has in replanning, the larger the rate of change in production quantity. The rate of change in production quantity is the highest under Rule 2, although Rule 2 has a smaller extent in replanning than Rules 3 and 4. The reason is that the production quantity is replanned at a point just before assembling products under Rule 2. On the other hand, under Rules 3 and 4, the production quantity has already been replanned at an earlier time than Rule 2. Through a series of experiments, the following points were clarified. (1) When the buffer for components/parts is maintained at a certain inventory level, Rule 4 can
0.
0
0.4 0.2 0 Buffer inventory
0.6
level
0.8
for
1.0
1.2
1.4
components/Parts
Fig. 8. Relationship between the buffer inventory level for components/parts and the changed rate in production quantity.
control the unfilled rate at the lowest level among the four rules. The next is Rule 2 or Rule 3 depending on the inventory level, followed by Rule 1. (2) The rate of change in production quantity depends on the timing and extent of replanning. Rule 3 can control the rate of change at the lowest level among the three rules, except for Rule 1. The next is Rule 4 and the last is Rule 2.
4. Conclusion We discussed the relationship between the timing for replanning and the buffer inventory in terms of system performance in a multistage production system, where there are multiple opportunities for replanning within the accumulated production lead time. A simulation model was developed in order to clarify the timing for replanning, which would affect manufacturing performance. Experimental results indicate the following: (1) The total required buffer inventory for finished products and components and parts can be controlled at a comparatively low level under such a replanning rule (Rule 2) that limits a portion of the production process, the assembly line, although the production quantity to be changed becomes maximal. (2) On the other hand, the production quantity to be changed can be decreased under such a replanning rule (Rule 3) that manages the overall production processes to change the production quantity, although the buffer inventory is required to be maintained at a higher level than the former replanning
H. Tsubone, H. Furutallnt.
60
J. Production
rule. (3) The total required buffer inventory for finished products and components/parts can be controlled at the lowest level under such a replanning rule (Rule 4) which combines the two rules mentioned above. That is, this rule replans two times, just before and one week before the assembly line. The production quantity to be changed can be controlled at a low level in comparison with rules that replan once on the basis of information on immediate market demand. However, the management may have to withstand instability and uncertainty in production control due to the high frequency of replanning. The obtained results also offer a basis for choosing a replanning rule through the trade-off between the costs regarding the unfilled rate, production quantity to be changed due to replanning, and the required buffer inventory of components/parts and finished products.
Appendix A: Planning Algorithm product production
for finished
ewv+3 =
(
1
f:+z
ksK"
Step 6. Compute,
stop.
Appendix B: Planning Algorithm parts production Step 1. Q+ Step 2. Let
K” = {kl@$
< Rk,,w+3}
If K” = f#~go to step 8. Otherwise go to step 3. Step 3. Compute b$y2 for all k E K”
Step 4. Compute for all k E K”.
if Ak > 0
If yes, B’,“!“2= ,$!“2
for all keK”
stop. Otherwise,
) -fl,+2
if ykW,,,,+3 3 0.
If yes,
K’ = {kJAk < 0}
Go to step 6. Step 6. Let B’,“y2 = b$‘y2 - Ak
for all keK’
Update for all kEKnK’ k E K” is optimal; Stop.
Otherwise go to step 5. Step 5. If K” = 8 go to step 6. Otherwise
K” = {kl&,w+3> 0)
Go to step 3.
Q+Q+
c
kcKnF
Ak
Let K” = (klAk >O}
and go to step 3. Step 7. Compute for all keK”
Let
Rk,,w+3 =0
for components/
Yw+3
Step 5. Check, for all keK”,
.a;+,
kp+3
Step 4. Check, for all kEK”,
44 (1996) 53-61
Ak = b’,“y2 +3’,“!“2 - R:,w+3
Step 1. Q+ Yw+, Step 2. k E K” for all k. Step 3. Compute r”,,,,,+ 3 for all k E K”. Q-t
Economics
for kEKn??. stop.
H. Tsubone. H. Furuta/Int.
Appendix C: Planning Algorithm
J. Production
for Rule 2
Economics
44 (1996) 53-61
Appendix E: Planning Algorithm
61
for Rule 4
Rule 3 is carried out after Rule 2.
Step 1. QC Y,+l Step 2. Compute Rk,, ,,,+1 using steps l-6 of App-
endix A. References
Step 3. For all k E K”, let
K’ = {k 1Rk,, ,,,+1 3 %ii;,” 1 + B$jk} K” = {kIRk,,w+l
< Ii?:?,
+ B’,“‘k}
[l] [Z]
IfK’=qJ [3]
R:,,N
= Rk,,w+l
for all k E K” is optimal; Stop. Otherwise, Rk,:w+l = @!‘_“, + B$!jk for all keK’
[4]
[S]
Step 4. If K” = 0 stop.
Otherwise, let
Q+Q-_;,R:w,,
[6]
and carry out steps 3-6 of Appendix A, and go to step 2.
[7]
[8]
Appendix D: Planning Algorithm
for Rule 3
Rk,++ 2 using the algorithm of Appendix A. Step 2. Calculate &!k, using the algorithm of Appendix B. Step 3. Decide Bcy, using the same algorithm of steps 3-5 of Appendix B. Step 4. Replanning Rk,,w+2 according to Rule 2.
[9]
Step 1. Calculate
[lo]
[ll]
Bitran, B. and Hax, A., 1977, On the design of hierarchical production planning system. Dec. Sci., 8: 28-55. Birtan, G., Hass, E. and Hax, A., 1981. Hierarchical production planning system: A single-stage system. Oper. Res., 29: 717-741. Birtan, G., Hass, E. and Hax, A., 1982. Hierarchical production planning system: A two-stage system. Oper. Res., 30: 232-251. Murthy, D.N.P. and Ma, L., 1991. MRP with uncertainty: A review and some extensions. Int. J. Prod. Econom., 25: 5 l-64. Narasimhan, R., Melnyk, S.A. and Carter, P.L., 1992. Research framework for investigating the effectiveness of dampening procedures to cope with MRP system nervousness. Int. J. Oper. Prod. Mgmt., 12: 3G43. Lin, N. and Lee, K., 1991. A model for master production, scheduling in uncertain environments. Dec. Sci., 23: 839-861. Sridharan, V. and William, L.B., 1989. Freezing the master production schedule under demand uncertainty. Dec. Sci., 21: 97-120. Campbell, G.M., 1991. Master production scheduling under rolling planning horizons with fixed order intervals. Dec. Sci., 23: 312-331. Carlson, R.C. and Yano, C.A., 1986. Safety stock in MRPsystems with emergency setups for components. Mgmt. Sci., 32: 403-412. Yano, C.A. and Carlson, R.C., 1987. Interaction between frequency of rescheduling and the role of safety stock in materials requirements planning systems. Int. J. Prod. Res., 25: 221-232. Tsubone, H., Matsuura, H. and Tsutsu, T., 1991. Hierarchical production planning system for a two-stage process. Int. J. Prod. Res., 29: 769-785.