- Email: [email protected]
A production ordering system for two-item, two-stage, capacity-constraint production and inventory model
prod&ion economics
ELSEVIER
Int. J. Production
Economics
44 (1996) 119-128
A production ordering system for two-item, two-stage, capacity-constraint production and inventory model Kazuyoshi IshiP *, Shigeki Imorib aDepartment of Industrial Engineering, Kanazawa Institute of Technology, Ohgigaoka 7-l. Nonoichi, Ishikawa 921, Japan b Planning and Coordinating
Center, Ishihmva Seisakusyo Ltd., Owari-cho I-2-40, Kanazawa. Ishikawa 920, Japan
Abstract This paper proposes an effective production ordering system for two-item, two-stage, capacity-constraint production and inventory systems, which reduces the fluctuations in total work load and inventory levels. The model system developed manufactures two final products whose component parts are a standard one and one with optional specifications chosen by the customer. As a result, simulation shows the influences of the safety stock of the optional component parts, average total work load ratio, and coefficient of variation of demand upon standard deviations of total work load and inventory levels in each stage. Multi-item multi-stage production system; Capacity-constraint Standard/optional components, Total workload; Structural formulation.
Keywords:
1. Introduction In recent years, the diversity of users’ needs has caused the market to become more segmented. This
has led to an increase in the number of items produced by each manufacturer as well as to increased fluctuations in product demand. Rapid technological improvement has resulted in extensive automation and specialization of production equipment, and in more complex parts’ configurations. In such an environment, many industries have additional stages for production and inventory. Companies must continue to reduce the lead
* Corresponding
author.
09255273/96/$15.00 Copyright SSDI 0925-5273(95)00097-6
0
1996 Elsevier
model;
Push-type
ordering
system;
time and the inventory levels of the product and component parts and increase the return on investment in order to succeed in the intensely competitive market. One of the effective solutions for the problem mentioned above is the production system which manufactures products by assembling two kinds of component parts. One is a component part which is standard for all products. The other is a component part with optional specifications chosen by the customer from specifications listed in a catalogue. However, the production system is problematic in the sense that the possibility of accumulating dead stock of the optional component part increases. Many production ordering systems for multistage production and inventory systems [l-3] have
Science B.V. All rights reserved
120
K. Ishii, S. Imori/Int.
J. Production
been developed and reported in the literature. They are classified into two types. One is the ‘push’-type [46] production ordering system, which may be called a ‘centralized’ production ordering system, such as the Material Requirements Planning (MRP) system. The other is the ‘pull’-type [7, S] production ordering system, which may be called a ‘decentralized’ production ordering system, such as the Kanban system. It has been pointed out by Muramatsu et al. [2] that the pull-type system has no amplification of the production and inventory level, whereas the push-type system has more diffuse amplification in the stage closer to the final production stage depending on the degree of factors such as forecasting error and difference between forecasted demands estimated at different periods. However, the pull-type system requires a long lead time and many management operations to cover the case of rapidly changing products in the market [9]. Furthermore, few studies [S, 10, 111 dealt with the multi-item product, multi-stage capacity-constraint production and inventory system. This paper focuses on decreasing the variation of total work load, taking into account forecasting error, the difference between forecasted values and the backlog quantity for which allocation of production capacity among items causes a discrepancy between production ordering and actual production levels. In this paper, an effective push-type ordering system for a two-stage, two-item production and inventory system with capacity constraint which reduces the variances of total work load at the production stage and the inventory level, is proposed and the following steps are taken in the analysis. (1) The structural formulations of production ordering and inventory levels are presented and the influences of forecasting error, the difference between forecasted demands, safety stock which prevents a shortage in supply on demand and the backlog upon the production ordering and inventory levels, are clarified. (2) The influences of the average total work load ratio, coefficient of variation of demand and safety stock of optional component parts upon the standard deviations of total work load at each
Economics
44 (1996) 119-128
production stage and inventory level of standard component parts/products, are clarified by a simulation.
2. Model descriptions 2.1. Assumptions
(1) The production system under consideration manufactures two kinds of products Ak (k = 1,2). The product has the family tree shown in Fig. 1. (2) The production system consists of two stages. The first stage produces products AI and A, in a processing time independent of the finished item. The second stage produces standard component “a” and optional components bk (k = 1,2) in a processing time dependent on each finished item. The flow of materials and information in the system is shown in Fig. 2. (3) The production capacity of each stage is assumed to be constant for any period and the set-up
Product Ak
(k-1,2)
,--+
Camp. a
Camp, bk
Fig. 1. The family tree of the product.
Ordering system Dk
0
Production stage
+-
lnforlat ion flow
V
Inventory state
f--
Feedback flow
Fig. 2. Schematic system.
_ _.
I’i
model
of the
production
/ L
FWaterial
and
flow
inventory
121
K. Ishii, S. Imori/Int. J. Production Economics 44 (1996) 119-128
time at each stage is negligible compared to the processing time. (4) Machine breakdown does not occur at any production stage and the handling time between processes is assumed to be zero. (5) The raw material inventory and the safety stocks of a product and standard component are always sufficient. (6) The production lead time at each production stage is one planning period. The production ordering level of each stage determined at the end of the tth period is produced in the stage during the (t + 1)th period, and is supplied to the inventory stage at the beginning of the (t + 2)th period. (7) Both standard and optional component parts are produced under the periodical scheduling system where the required quantity of both component parts is based on a forecast. The production of the optional component parts in the second stage is given priority. (8) The demand for a product at each period is a stationary time series whose average and variance are known and constant.
2.2. Notation
0: ^k
D f,t+i
S n,k Hn
CA tkm
B,
n-k
0 n,k f,t+Z
the actual demand for product Ak during the tth period which has an average pk and a standard deviation 0,‘ the forecasted demand for product Ak for the (t + i)th period determined at the end of the tth period the safety stock of product/component part k at the nth stock point the production capacity of the nth stage which is measured as a function of hours in a planning period the assembly time of a product the processing time of a component part (k = 0: standard part, k = 1,2: optional part) the end-of-term inventory level of product/component part k at the end of the tth period the production ordering level of a product/component part k for the (t + 2)th
period determined at the end of the tth period at the nth production stage the actual production level of a prodP n,k f,f+Z uct/component part k during the (t + 1)th period; this quantity becomes the inventory at the beginning of the (t + 2)th period n,k the quantity out of the production orR, dering level which cannot be produced because of the restriction of the allocation of production capacity during the (t + 1)th period at the nth stage; this is called the backlog the total work load for the tth period at L: the nth production stage the standard deviation of * SD(*) Cov(*, **) the covariance between * and ** the largest integer less than the value in Cl the brackets
2.3.
Model formulations
(1) Production ordering level The production ordering level is based on a periodical model and the push-type ordering model. 1.k 0 t,t+2
-
@,+2
-
0
2.k f,f+z
-
+
@,:,t+,
-
B;,k + S”pk
R,t+,
+
@,:,t+z
:A2
=
k&
Rt+,
(k = -
-B;9k+S19k 0
p:~:,,+l)
1,2)
(1)
p:~kl,,+l)
(k=1,2) +
(kil
%+2
(2) -
PX,*+,>
_ B;.O + S’s0
(2) Actual production I,1 P t,t+z
-
min(0:;i2,
Big’
(3) level +
p$‘1,,+1,
CHIItAI)
(4) pl.2 t,t+z
--
[(HI 2,l P t,t+z
-
Bf,’
min(O:,;:z,
min
- P:%+I (0:;:
2,
+ P:~‘l,,+z, x
t&J)
CHh!J)
(5) (6)
122
K. Ishii, S. Imori/Int. J. Production Economics 44 (1996) 119-128
p2.2
-
1,1+2
min(O?Z2,
= min
P:;“,,
CW2
O:i”,,,
P?h2
Hz-
(
tom
-
i
K
I)
x
h!NJ)
(7)
L: = i (o:~kl,,+,x LA),
P:,;“+,xtk,
k=l
63)
)i
L:
B/-k = Bilk, + P2,k f 2.1
-
Bi.0 = B;:0, + P;$,
- ,iI
P::kl,r+l
R1.2
= max(O, O:,i$, -(B:*’
f
-
(k
(9) =
192)
(11)
ordering level + P:L\,,+,),
CHlItLl)
= max(O, Oj,i:, - [WI -
(121 Ok;:,
- (BjV2 + Pf~:,,+1),
p:‘t:2
x
(13)
tA)/tAl)
B2.1 = maxto,O:iL - CHdt,!,l) f R2.2
f
= max(0,0?;‘,2
-
[(H2
-
J’Ei:2
(14) x
t,Q/til) (15)
tom
SD(B:.k)
i
(20)
x t:,.
(o:lkl,,+l
I)
(16)
n = 1,2, (n=O;k=1,2)
3. Analysis and results 3.1 Structural formulations of the inventory and production ordering levels
(1) Analysis In order to analyze the standard deviations of the total work load and inventory level, the influences of forecasting error, the difference between forecasted demands, safety stock and the backlog of production ordering level upon the inventory and production ordering levels in each stage are clarified through the derivation of structural formulations. Forecasting error and the difference between forecasted demands are defined by Eqs. (21) and (22). k Et-i,* = @_i,t - D:y
(5) Evaluation functions To evaluate the performance of systems, the standard deviations defined in Eqs. (17) and (18) are used. If the deviation of the inventory level of product can be decreased, it is possible to reduce the safety stock while maintaining the level which meets the actual product demand. If the deviation of the inventory level of component parts can be decreased, it is possible to reduce safety stock of component parts while maintaining the level which prevents assembling stage from stopping due to the shortage of the component parts. If the deviation of total work load can be decreased, it is possible to improve the return on investment. SD@:)
=
(10)
P::“lJ+ 1
(4) Backlog of production
0 ti:2
(19)
k=l
k=O
(3) End-of-term inventory level B0.k = B;:kI + P::k2,t - 0: (k = 1,2) t
R:.’
where
(17) and (n=l;k=O), (18)
5:‘ki,l_j
=
.6:-i,*
-
(21)
b:_j,f,
i > j.
(22)
The production ordering level of product A is defined by Eq. (1). Equations (4) and (5) are transformed into Eq. (23) by the use of Eqs. (12) and (13): 1.k P f,f+2
-
Q’,” t,1+2
_
R;;kl,
(23)
Equation (1) is transformed into Eq. (24) by the use of Eqs. (l), (21) and (22): 0
1.k t,t+2
_ -
bk
-
t,t+2
(&*
-
(&,+I
-
-
K+1,
0:) f R:::
= b,:,*+, - <;‘:;; - E;_~,~ + R;lkI.
(24)
Furthermore, in a manner similar to that shown above, the inventory and production ordering levels of the product and component parts are transformed into the equations given in Table 1.
K. Ishii, S. Imorillnt.
J. Production
Economics
44 (1996) 119-128
123
Table 1 The structural formulations of the inventory and production ordering levels
RI.0
t
(2) Results and discussion From Table 1, the following information is obtained. (a) The structure of the equation for end-of-term inventory level for a product and that for a component part are different. That is, the inventory level of a component part is affected by forecasting error, the difference between forecasted demands, and the backlogs of both the first and the second production stages. On the other hand, that of a product is affected only by forecasting error and the backlog of the first stage. The difference between standard and optional component parts is the sum of the forecasting error, the difference between forecasted demands and the backlog of the first stage. The number represents the component commonality. (b) The structure of the equation for the production level for a product and that for a component part are different. That is, the production ordering
level of a product is affected by forecasted demand, forecasting error, the difference between forecasted demands and the backlog of the first stage. Such influence causes production ordering level to differ for products and component parts by the difference between forecasted demands and the backlog at the first stage. The difference between the production ordering levels of standard and optional component parts is the same as that of the inventory levels. (c) The backlog at the first stage is affected by forecasted demand, forecasting error, the difference between forecasted demands, safety stock of optional component parts, production capacity and production priority. On the other hand, the backlog at the second stage is not affected by safety stock of optional component parts. Therefore, safety stock of optional components affects the inventory and production ordering levels of a product and both component parts through the backlog at the first stage.
K. Ishii, S. Imorillnt.
124
3.2. Standard deviations work load
J. Production Economics 44 (1996) 119-128
of inventory level and total
(1) Theoretical analysis The above analysis shows that when R:*k = 0 for all k, n and t, the inventory and production ordering levels are determined by the forecasting error, the difference between forecasted demands and forecasted demand. The standard deviations of forecasting error and the difference between forecasted demands of the evaluation functions, can be derived as Eqs. (28) and (29) under the conditions that the demand is a stationary time series with given average and variance, and is forecasted using the average value of product demand shown in Eqns. (25)-(27): 0: N
N(pk,
Cov(D:,
(k
0:)
=
Cov(Df,
SD(C
=
i,,) =
Jk$ld.
(32)
(27)
ck,
(28)
= 0.
(29)
SD(t:?i,,-j)
tA
(26)
(i = 1, 2, . . . ),
p(k,
=
Df+J = 0,
(i = 1,2, . . . ), bf,,+i
tA/v(&f-l)
(25)
1,2),
and
0:) = 0
=
By the use of Eqs. (28) and (29), the standard deviations of total work load and inventory level can be derived as the following equations.
+ i
bk
-
‘&,t-2bk,
k=l
SD(BPPk) = SD(E:_~,~_~ + E:-~,~ + SoVk) = ,,I/&
- D:ml + & - D: + S”‘k)
= ,&(D:-
I) + V(D:)>
= J(M)
SD@+‘)
(33)
= ,/2
* 0,‘
= SD
i
(k = 1, 2).
(30)
(E:_~,~_~ + E:_~,~_~
[k=l
=
Ji J’
2
1
@k-D:-2
k=l
-
D;el + S”vk)
+Pk
When a backlog occurs due to the restrictions on the safety stock of optional component parts, production capacity and production priority, it is difficult to analyze the standard deviations theoretically. Because the backlog includes stochastic and dynamic characteristics and is nonlinear, the backlog of the evaluation functions cannot be derived exactly as functions with parameters such as S”,k, p. and ok. Therefore, the simulation focuses on the terms of the backlog under the following additional conditions.
K. Ishii, S. ImorilInt.
J. Production
Economics
44 (1996) 119-128
125
7
Start
Reading;
t:. tA.,H., S”’ k, ,Uk, u
k,
T
t-o
Reading;
1
/t-t+1
Reading:
D:
Calculating B:.*,B:*" and B:” by Table 1 I
I
Calculating O:::+, ,Of::+, and 0:: :+a by Table 1
I
I I
Calculating R:”
and R:”
by Table 1
1
I
II
I
Calculating R:”
and R:,*
by Table 1
I I
I
1
Calculating R:” by Table 1
1
Fig. 3. The flow chart
Calculatinz”)
for calculation
of the standard
and SD(L:)
deviations
]
of total work load and inventory.
126
K. Ishii, S. Imorillnt.
(2) Simulations and conditions (a) The coefficient of variation mand is defined as follows: pk
=
ak//ik.
J. Production Economics
44 (1996) 119-128
of product de-
(34)
(b) The average total work load ratio of the nth production stage is defined by Eqs. (35) and (36): PI =
k$PkXt*/Hl,
p2
ki,
=
(/Ah X t:
+
(35) /Lk X &J/Hz.
(36)
(c) The normalized factor of the safety stock of optional component parts bk is defined by Eq. (37): vlVk= S”k//&. (k = 1, 2)
(37)
(d) Simulation follows the procedure shown in (3) Results and discussion Fig. 4 shows the relationships between the factor of the safety stock of optional component parts and the standard deviations of inventory levels of product AI and standard component part under the conditions of average total work load ratio and coefficient of variation of demand. Those of product A2 show results similar to those of product AI. Fig. 5 shows the relationships between the factor of safety stock of optional component parts and standard deviations of total work loads of the product assembly stage and of the production stage of component parts. From Figs. 4 and 5, the following information is obtained. (a) The higher the average total work load ratio and coefficient of variation of demand are, the greater the effect of the safety stock of the optional component parts upon the reduction of standard deviations of product inventory level and total work load of the first production stage is. (b) The increase of the safety stock of optional component parts causes increases of the standard deviations of inventory of the standard component part and total work load at the second production stage.
A
0 dC
a V
n cn
(v) Fig. 4. The effects of the safety stock level of optional component part on the standard deviations of the inventory levels.
(c) Based on the above results, it must be considered when determining an optimal safety stock of the optional component parts, that there are trade-offs in the safety stock levels of product versus standard component part, and in the standard deviations of total work loads at the first production stage versus the second production stage. (d) The production priority affects the backlogs at both the stages. For the model analyzed in
K. Ishii, S. Imorillnt.
J. Production Economics
this paper, the production of optional component parts in the second stage is given priority. The backlogs upon the production ordering of optional parts do not occur in the simulation. It may be one of the important problems for multi-item, multistage, capacity-constraint production and inventory systems that an effective control method on the backlogs at the second stage should be discussed for future research.
44 (1996) 119-128
121
4. Conclusions In order to propose an effective production ordering method for two-item, two-stage production and inventory systems with capacity constraint, the standard deviations of total work load and inventory level were analyzed and the following conclusions were reached. (1) The influence of safety stock of optional components upon the production ordering and inventory levels is different for products and component parts. The safety stock might reduce the standard deviations of total work load and inventory level by controlling the backlog created by forecasting error, the difference between forecasted demands, production capacity and production priority. (2) The effect of the safety stock of optional component parts on reducing standard deviations of total work load and inventory level differs according to the average total work load ratio and variation of product demand. Furthermore, there is a trade-off in the standard deviations of total work load and inventory level of a product versus those of a component part. (3) To determine the optimum safety stock of optional component parts so that actual product demand will be met, it is necessary to obtain more information concerning inventory costs of products and component parts, and investment cost with respect to production capacity. Acknowledgement
This study was supported by the Scientific Research Fund of the Japanese Ministry of Education, Science and Culture. References
Cl1 Axslter,
Fig. 5. The effects of the safety stock level of optional component part on the standard deviations of the total work loads.
S., Schneeweiss, C.A. and Silver, E.A. (eds.), 1986. Multi-stage production planning and inventory control. Springer, Berlin, p. 264. PI Muramatsu, R., Ishii, K. and Takahashi, K., 1985. Some ways to increase flexibility in production systems. Int. J. Prod. Res., 23(4): 691. c31Krajewski, L., King, J., Ritzman, B.E. and Wong, D.S., 1987. Kanban, MRP and shaping the manufacturing environment. Management Science, 33(l): 39.
128
K. Ishii, S. Imorillnt.
J. Production
[4] Takahashi, K., Muramatsu, R. and Ishii, K., 1987. Feedback method of production ordering system in multistage production and inventory systems. Int. J. Prod. Res., 25(6): 925. [S] Billington, P.J., McClain, J.O. and Thomas, L.T., 1983. Mathematical programming approaches to capacity-constrained MRP systems - Review, formulation and problem reduction. Management Science., 29(10): 1126. [6] Ishii, K., Minaki, T. and Hiraki, S., 1993. Method of production smoothing for two-stage production and inventory system. Int. J. Prod. Econom., 29: 139. [7] Hiraki, S., Ishii, K., Takahashi, K. and Muramatsu, R., 1992. Designing a pull-type parts procurement system for international co-operative knockdown production system. Int. J. Prod. Res., 30(2): 337.
Economics
44 (1996) 119-128
[S] Kimura, 0. and Terada, H., 1981. Design and analysis of pull system: a method of multistage production control. Int. J. Prod. Res., 19(3): 241. [9] Ishii, K. Takahashi, K. and Muramatsu, R., 1988, Integrated production, inventory and distribution system. Int. J. Prod. Res., 26(3): 473. [lo] Ishii, K. and Muramatsu, R., 1985. A study on designing mixed buffer production systems for manufacturing the components of prefabricated houses via precast concrete panels. Int. J. Prod. Res., 23(3): 457. [11] Tabe, T. and Muramatsu, R., 1981. The economical assessment and design method of some production ordering system in a two-stage production inventory model, in D.M. Zelenovic, (Ed.), Proc. Int. Conf. on Production Research, Novi Sad, Yugoslavia, 1, p. 329.
ELSEVIER
Int. J. Production
Economics
44 (1996) 119-128
A production ordering system for two-item, two-stage, capacity-constraint production and inventory model Kazuyoshi IshiP *, Shigeki Imorib aDepartment of Industrial Engineering, Kanazawa Institute of Technology, Ohgigaoka 7-l. Nonoichi, Ishikawa 921, Japan b Planning and Coordinating
Center, Ishihmva Seisakusyo Ltd., Owari-cho I-2-40, Kanazawa. Ishikawa 920, Japan
Abstract This paper proposes an effective production ordering system for two-item, two-stage, capacity-constraint production and inventory systems, which reduces the fluctuations in total work load and inventory levels. The model system developed manufactures two final products whose component parts are a standard one and one with optional specifications chosen by the customer. As a result, simulation shows the influences of the safety stock of the optional component parts, average total work load ratio, and coefficient of variation of demand upon standard deviations of total work load and inventory levels in each stage. Multi-item multi-stage production system; Capacity-constraint Standard/optional components, Total workload; Structural formulation.
Keywords:
1. Introduction In recent years, the diversity of users’ needs has caused the market to become more segmented. This
has led to an increase in the number of items produced by each manufacturer as well as to increased fluctuations in product demand. Rapid technological improvement has resulted in extensive automation and specialization of production equipment, and in more complex parts’ configurations. In such an environment, many industries have additional stages for production and inventory. Companies must continue to reduce the lead
* Corresponding
author.
09255273/96/$15.00 Copyright SSDI 0925-5273(95)00097-6
0
1996 Elsevier
model;
Push-type
ordering
system;
time and the inventory levels of the product and component parts and increase the return on investment in order to succeed in the intensely competitive market. One of the effective solutions for the problem mentioned above is the production system which manufactures products by assembling two kinds of component parts. One is a component part which is standard for all products. The other is a component part with optional specifications chosen by the customer from specifications listed in a catalogue. However, the production system is problematic in the sense that the possibility of accumulating dead stock of the optional component part increases. Many production ordering systems for multistage production and inventory systems [l-3] have
Science B.V. All rights reserved
120
K. Ishii, S. Imori/Int.
J. Production
been developed and reported in the literature. They are classified into two types. One is the ‘push’-type [46] production ordering system, which may be called a ‘centralized’ production ordering system, such as the Material Requirements Planning (MRP) system. The other is the ‘pull’-type [7, S] production ordering system, which may be called a ‘decentralized’ production ordering system, such as the Kanban system. It has been pointed out by Muramatsu et al. [2] that the pull-type system has no amplification of the production and inventory level, whereas the push-type system has more diffuse amplification in the stage closer to the final production stage depending on the degree of factors such as forecasting error and difference between forecasted demands estimated at different periods. However, the pull-type system requires a long lead time and many management operations to cover the case of rapidly changing products in the market [9]. Furthermore, few studies [S, 10, 111 dealt with the multi-item product, multi-stage capacity-constraint production and inventory system. This paper focuses on decreasing the variation of total work load, taking into account forecasting error, the difference between forecasted values and the backlog quantity for which allocation of production capacity among items causes a discrepancy between production ordering and actual production levels. In this paper, an effective push-type ordering system for a two-stage, two-item production and inventory system with capacity constraint which reduces the variances of total work load at the production stage and the inventory level, is proposed and the following steps are taken in the analysis. (1) The structural formulations of production ordering and inventory levels are presented and the influences of forecasting error, the difference between forecasted demands, safety stock which prevents a shortage in supply on demand and the backlog upon the production ordering and inventory levels, are clarified. (2) The influences of the average total work load ratio, coefficient of variation of demand and safety stock of optional component parts upon the standard deviations of total work load at each
Economics
44 (1996) 119-128
production stage and inventory level of standard component parts/products, are clarified by a simulation.
2. Model descriptions 2.1. Assumptions
(1) The production system under consideration manufactures two kinds of products Ak (k = 1,2). The product has the family tree shown in Fig. 1. (2) The production system consists of two stages. The first stage produces products AI and A, in a processing time independent of the finished item. The second stage produces standard component “a” and optional components bk (k = 1,2) in a processing time dependent on each finished item. The flow of materials and information in the system is shown in Fig. 2. (3) The production capacity of each stage is assumed to be constant for any period and the set-up
Product Ak
(k-1,2)
,--+
Camp. a
Camp, bk
Fig. 1. The family tree of the product.
Ordering system Dk
0
Production stage
+-
lnforlat ion flow
V
Inventory state
f--
Feedback flow
Fig. 2. Schematic system.
_ _.
I’i
model
of the
production
/ L
FWaterial
and
flow
inventory
121
K. Ishii, S. Imori/Int. J. Production Economics 44 (1996) 119-128
time at each stage is negligible compared to the processing time. (4) Machine breakdown does not occur at any production stage and the handling time between processes is assumed to be zero. (5) The raw material inventory and the safety stocks of a product and standard component are always sufficient. (6) The production lead time at each production stage is one planning period. The production ordering level of each stage determined at the end of the tth period is produced in the stage during the (t + 1)th period, and is supplied to the inventory stage at the beginning of the (t + 2)th period. (7) Both standard and optional component parts are produced under the periodical scheduling system where the required quantity of both component parts is based on a forecast. The production of the optional component parts in the second stage is given priority. (8) The demand for a product at each period is a stationary time series whose average and variance are known and constant.
2.2. Notation
0: ^k
D f,t+i
S n,k Hn
CA tkm
B,
n-k
0 n,k f,t+Z
the actual demand for product Ak during the tth period which has an average pk and a standard deviation 0,‘ the forecasted demand for product Ak for the (t + i)th period determined at the end of the tth period the safety stock of product/component part k at the nth stock point the production capacity of the nth stage which is measured as a function of hours in a planning period the assembly time of a product the processing time of a component part (k = 0: standard part, k = 1,2: optional part) the end-of-term inventory level of product/component part k at the end of the tth period the production ordering level of a product/component part k for the (t + 2)th
period determined at the end of the tth period at the nth production stage the actual production level of a prodP n,k f,f+Z uct/component part k during the (t + 1)th period; this quantity becomes the inventory at the beginning of the (t + 2)th period n,k the quantity out of the production orR, dering level which cannot be produced because of the restriction of the allocation of production capacity during the (t + 1)th period at the nth stage; this is called the backlog the total work load for the tth period at L: the nth production stage the standard deviation of * SD(*) Cov(*, **) the covariance between * and ** the largest integer less than the value in Cl the brackets
2.3.
Model formulations
(1) Production ordering level The production ordering level is based on a periodical model and the push-type ordering model. 1.k 0 t,t+2
-
@,+2
-
0
2.k f,f+z
-
+
@,:,t+,
-
B;,k + S”pk
R,t+,
+
@,:,t+z
:A2
=
k&
Rt+,
(k = -
-B;9k+S19k 0
p:~:,,+l)
1,2)
(1)
p:~kl,,+l)
(k=1,2) +
(kil
%+2
(2) -
PX,*+,>
_ B;.O + S’s0
(2) Actual production I,1 P t,t+z
-
min(0:;i2,
Big’
(3) level +
p$‘1,,+1,
CHIItAI)
(4) pl.2 t,t+z
--
[(HI 2,l P t,t+z
-
Bf,’
min(O:,;:z,
min
- P:%+I (0:;:
2,
+ P:~‘l,,+z, x
t&J)
CHh!J)
(5) (6)
122
K. Ishii, S. Imori/Int. J. Production Economics 44 (1996) 119-128
p2.2
-
1,1+2
min(O?Z2,
= min
P:;“,,
CW2
O:i”,,,
P?h2
Hz-
(
tom
-
i
K
I)
x
h!NJ)
(7)
L: = i (o:~kl,,+,x LA),
P:,;“+,xtk,
k=l
63)
)i
L:
B/-k = Bilk, + P2,k f 2.1
-
Bi.0 = B;:0, + P;$,
- ,iI
P::kl,r+l
R1.2
= max(O, O:,i$, -(B:*’
f
-
(k
(9) =
192)
(11)
ordering level + P:L\,,+,),
CHlItLl)
= max(O, Oj,i:, - [WI -
(121 Ok;:,
- (BjV2 + Pf~:,,+1),
p:‘t:2
x
(13)
tA)/tAl)
B2.1 = maxto,O:iL - CHdt,!,l) f R2.2
f
= max(0,0?;‘,2
-
[(H2
-
J’Ei:2
(14) x
t,Q/til) (15)
tom
SD(B:.k)
i
(20)
x t:,.
(o:lkl,,+l
I)
(16)
n = 1,2, (n=O;k=1,2)
3. Analysis and results 3.1 Structural formulations of the inventory and production ordering levels
(1) Analysis In order to analyze the standard deviations of the total work load and inventory level, the influences of forecasting error, the difference between forecasted demands, safety stock and the backlog of production ordering level upon the inventory and production ordering levels in each stage are clarified through the derivation of structural formulations. Forecasting error and the difference between forecasted demands are defined by Eqs. (21) and (22). k Et-i,* = @_i,t - D:y
(5) Evaluation functions To evaluate the performance of systems, the standard deviations defined in Eqs. (17) and (18) are used. If the deviation of the inventory level of product can be decreased, it is possible to reduce the safety stock while maintaining the level which meets the actual product demand. If the deviation of the inventory level of component parts can be decreased, it is possible to reduce safety stock of component parts while maintaining the level which prevents assembling stage from stopping due to the shortage of the component parts. If the deviation of total work load can be decreased, it is possible to improve the return on investment. SD@:)
=
(10)
P::“lJ+ 1
(4) Backlog of production
0 ti:2
(19)
k=l
k=O
(3) End-of-term inventory level B0.k = B;:kI + P::k2,t - 0: (k = 1,2) t
R:.’
where
(17) and (n=l;k=O), (18)
5:‘ki,l_j
=
.6:-i,*
-
(21)
b:_j,f,
i > j.
(22)
The production ordering level of product A is defined by Eq. (1). Equations (4) and (5) are transformed into Eq. (23) by the use of Eqs. (12) and (13): 1.k P f,f+2
-
Q’,” t,1+2
_
R;;kl,
(23)
Equation (1) is transformed into Eq. (24) by the use of Eqs. (l), (21) and (22): 0
1.k t,t+2
_ -
bk
-
t,t+2
(&*
-
(&,+I
-
-
K+1,
0:) f R:::
= b,:,*+, - <;‘:;; - E;_~,~ + R;lkI.
(24)
Furthermore, in a manner similar to that shown above, the inventory and production ordering levels of the product and component parts are transformed into the equations given in Table 1.
K. Ishii, S. Imorillnt.
J. Production
Economics
44 (1996) 119-128
123
Table 1 The structural formulations of the inventory and production ordering levels
RI.0
t
(2) Results and discussion From Table 1, the following information is obtained. (a) The structure of the equation for end-of-term inventory level for a product and that for a component part are different. That is, the inventory level of a component part is affected by forecasting error, the difference between forecasted demands, and the backlogs of both the first and the second production stages. On the other hand, that of a product is affected only by forecasting error and the backlog of the first stage. The difference between standard and optional component parts is the sum of the forecasting error, the difference between forecasted demands and the backlog of the first stage. The number represents the component commonality. (b) The structure of the equation for the production level for a product and that for a component part are different. That is, the production ordering
level of a product is affected by forecasted demand, forecasting error, the difference between forecasted demands and the backlog of the first stage. Such influence causes production ordering level to differ for products and component parts by the difference between forecasted demands and the backlog at the first stage. The difference between the production ordering levels of standard and optional component parts is the same as that of the inventory levels. (c) The backlog at the first stage is affected by forecasted demand, forecasting error, the difference between forecasted demands, safety stock of optional component parts, production capacity and production priority. On the other hand, the backlog at the second stage is not affected by safety stock of optional component parts. Therefore, safety stock of optional components affects the inventory and production ordering levels of a product and both component parts through the backlog at the first stage.
K. Ishii, S. Imorillnt.
124
3.2. Standard deviations work load
J. Production Economics 44 (1996) 119-128
of inventory level and total
(1) Theoretical analysis The above analysis shows that when R:*k = 0 for all k, n and t, the inventory and production ordering levels are determined by the forecasting error, the difference between forecasted demands and forecasted demand. The standard deviations of forecasting error and the difference between forecasted demands of the evaluation functions, can be derived as Eqs. (28) and (29) under the conditions that the demand is a stationary time series with given average and variance, and is forecasted using the average value of product demand shown in Eqns. (25)-(27): 0: N
N(pk,
Cov(D:,
(k
0:)
=
Cov(Df,
SD(C
=
i,,) =
Jk$ld.
(32)
(27)
ck,
(28)
= 0.
(29)
SD(t:?i,,-j)
tA
(26)
(i = 1, 2, . . . ),
p(k,
=
Df+J = 0,
(i = 1,2, . . . ), bf,,+i
tA/v(&f-l)
(25)
1,2),
and
0:) = 0
=
By the use of Eqs. (28) and (29), the standard deviations of total work load and inventory level can be derived as the following equations.
+ i
bk
-
‘&,t-2bk,
k=l
SD(BPPk) = SD(E:_~,~_~ + E:-~,~ + SoVk) = ,,I/&
- D:ml + & - D: + S”‘k)
= ,&(D:-
I) + V(D:)>
= J(M)
SD@+‘)
(33)
= ,/2
* 0,‘
= SD
i
(k = 1, 2).
(30)
(E:_~,~_~ + E:_~,~_~
[k=l
=
Ji J’
2
1
@k-D:-2
k=l
-
D;el + S”vk)
+Pk
When a backlog occurs due to the restrictions on the safety stock of optional component parts, production capacity and production priority, it is difficult to analyze the standard deviations theoretically. Because the backlog includes stochastic and dynamic characteristics and is nonlinear, the backlog of the evaluation functions cannot be derived exactly as functions with parameters such as S”,k, p. and ok. Therefore, the simulation focuses on the terms of the backlog under the following additional conditions.
K. Ishii, S. ImorilInt.
J. Production
Economics
44 (1996) 119-128
125
7
Start
Reading;
t:. tA.,H., S”’ k, ,Uk, u
k,
T
t-o
Reading;
1
/t-t+1
Reading:
D:
Calculating B:.*,B:*" and B:” by Table 1 I
I
Calculating O:::+, ,Of::+, and 0:: :+a by Table 1
I
I I
Calculating R:”
and R:”
by Table 1
1
I
II
I
Calculating R:”
and R:,*
by Table 1
I I
I
1
Calculating R:” by Table 1
1
Fig. 3. The flow chart
Calculatinz”)
for calculation
of the standard
and SD(L:)
deviations
]
of total work load and inventory.
126
K. Ishii, S. Imorillnt.
(2) Simulations and conditions (a) The coefficient of variation mand is defined as follows: pk
=
ak//ik.
J. Production Economics
44 (1996) 119-128
of product de-
(34)
(b) The average total work load ratio of the nth production stage is defined by Eqs. (35) and (36): PI =
k$PkXt*/Hl,
p2
ki,
=
(/Ah X t:
+
(35) /Lk X &J/Hz.
(36)
(c) The normalized factor of the safety stock of optional component parts bk is defined by Eq. (37): vlVk= S”k//&. (k = 1, 2)
(37)
(d) Simulation follows the procedure shown in (3) Results and discussion Fig. 4 shows the relationships between the factor of the safety stock of optional component parts and the standard deviations of inventory levels of product AI and standard component part under the conditions of average total work load ratio and coefficient of variation of demand. Those of product A2 show results similar to those of product AI. Fig. 5 shows the relationships between the factor of safety stock of optional component parts and standard deviations of total work loads of the product assembly stage and of the production stage of component parts. From Figs. 4 and 5, the following information is obtained. (a) The higher the average total work load ratio and coefficient of variation of demand are, the greater the effect of the safety stock of the optional component parts upon the reduction of standard deviations of product inventory level and total work load of the first production stage is. (b) The increase of the safety stock of optional component parts causes increases of the standard deviations of inventory of the standard component part and total work load at the second production stage.
A
0 dC
a V
n cn
(v) Fig. 4. The effects of the safety stock level of optional component part on the standard deviations of the inventory levels.
(c) Based on the above results, it must be considered when determining an optimal safety stock of the optional component parts, that there are trade-offs in the safety stock levels of product versus standard component part, and in the standard deviations of total work loads at the first production stage versus the second production stage. (d) The production priority affects the backlogs at both the stages. For the model analyzed in
K. Ishii, S. Imorillnt.
J. Production Economics
this paper, the production of optional component parts in the second stage is given priority. The backlogs upon the production ordering of optional parts do not occur in the simulation. It may be one of the important problems for multi-item, multistage, capacity-constraint production and inventory systems that an effective control method on the backlogs at the second stage should be discussed for future research.
44 (1996) 119-128
121
4. Conclusions In order to propose an effective production ordering method for two-item, two-stage production and inventory systems with capacity constraint, the standard deviations of total work load and inventory level were analyzed and the following conclusions were reached. (1) The influence of safety stock of optional components upon the production ordering and inventory levels is different for products and component parts. The safety stock might reduce the standard deviations of total work load and inventory level by controlling the backlog created by forecasting error, the difference between forecasted demands, production capacity and production priority. (2) The effect of the safety stock of optional component parts on reducing standard deviations of total work load and inventory level differs according to the average total work load ratio and variation of product demand. Furthermore, there is a trade-off in the standard deviations of total work load and inventory level of a product versus those of a component part. (3) To determine the optimum safety stock of optional component parts so that actual product demand will be met, it is necessary to obtain more information concerning inventory costs of products and component parts, and investment cost with respect to production capacity. Acknowledgement
This study was supported by the Scientific Research Fund of the Japanese Ministry of Education, Science and Culture. References
Cl1 Axslter,
Fig. 5. The effects of the safety stock level of optional component part on the standard deviations of the total work loads.
S., Schneeweiss, C.A. and Silver, E.A. (eds.), 1986. Multi-stage production planning and inventory control. Springer, Berlin, p. 264. PI Muramatsu, R., Ishii, K. and Takahashi, K., 1985. Some ways to increase flexibility in production systems. Int. J. Prod. Res., 23(4): 691. c31Krajewski, L., King, J., Ritzman, B.E. and Wong, D.S., 1987. Kanban, MRP and shaping the manufacturing environment. Management Science, 33(l): 39.
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[4] Takahashi, K., Muramatsu, R. and Ishii, K., 1987. Feedback method of production ordering system in multistage production and inventory systems. Int. J. Prod. Res., 25(6): 925. [S] Billington, P.J., McClain, J.O. and Thomas, L.T., 1983. Mathematical programming approaches to capacity-constrained MRP systems - Review, formulation and problem reduction. Management Science., 29(10): 1126. [6] Ishii, K., Minaki, T. and Hiraki, S., 1993. Method of production smoothing for two-stage production and inventory system. Int. J. Prod. Econom., 29: 139. [7] Hiraki, S., Ishii, K., Takahashi, K. and Muramatsu, R., 1992. Designing a pull-type parts procurement system for international co-operative knockdown production system. Int. J. Prod. Res., 30(2): 337.
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[S] Kimura, 0. and Terada, H., 1981. Design and analysis of pull system: a method of multistage production control. Int. J. Prod. Res., 19(3): 241. [9] Ishii, K. Takahashi, K. and Muramatsu, R., 1988, Integrated production, inventory and distribution system. Int. J. Prod. Res., 26(3): 473. [lo] Ishii, K. and Muramatsu, R., 1985. A study on designing mixed buffer production systems for manufacturing the components of prefabricated houses via precast concrete panels. Int. J. Prod. Res., 23(3): 457. [11] Tabe, T. and Muramatsu, R., 1981. The economical assessment and design method of some production ordering system in a two-stage production inventory model, in D.M. Zelenovic, (Ed.), Proc. Int. Conf. on Production Research, Novi Sad, Yugoslavia, 1, p. 329.