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2010第 13屆科際整合管理研討會
June 26, pp462-475
462
易腐性商品易腐性商品易腐性商品易腐性商品及原料及原料及原料及原料之生產之生產之生產之生產策略策略策略策略分析分析分析分析
Production Policies of Perishable Products and Perishable Raw
Material 林宇軒林宇軒林宇軒林宇軒 Yu Hsuan Lin1 吳吉政吳吉政吳吉政吳吉政 Jei-Zheng Wu
2 阮金祥阮金祥阮金祥阮金祥 Jinshyang Roan
3
Abstract
In this paper, we consider the products and raw materials have limit life time. We assume
the products and raw materials are following the constant deterioration rate and raw materials
are supplied by constant supplement. We also consider the raw materials can be resale. A
two-echelon cost model is developed for a single product production process, and the process of
find optimal solution is used the numerical searching method for finding optimal solution.
Keywords: deterioration, constant supply, disposal cost
摘要摘要摘要摘要 在本篇論文中,我們考慮產品與原物料兩者都會隨時間產生腐壞的情形並且假設產品與原物料都有固定的腐壞率,我們也假設腐壞的原物料是可以變賣的。原物料的供給己設式為固定供給模式。我們主要探討在產品與原物料都會腐壞的情況之下,對於最小成本化的生產將會造成何種影響。我們針對單期的生產模式發展一個二階層的成本模型並運用數值分析方法來幫助求出最佳的解答。 關鍵字:腐壞率、固定供給模式、處置成本 1. Introduction
The economic order quantity (EOQ) model is one of the conventional and stander
inventory-control techniques, which could help us to determine the optimal order quantity or the
time between orders. The EOQ model gives a concept to let the producer to determine the order
quantity and seek for the minimal total cost. The classical EOQ model is based on few
1 東吳大學企業管理學系企業管理研究所碩士生。 2 東吳大學企業管理學系助理教授(聯絡地址:100台北市貴陽街一段 56號,聯絡電話:02-23111531轉 3403,E-mail: jzwu@scu.edu.tw)。 3 東吳大學企業管理學系教授兼國際商管學程主任(聯絡地址:100台北市貴陽街一段 56號,聯絡電話:02-23111531轉 3423,E-mail:baroan@scu.edu.tw)。
易腐性商品及原料生產策略分析公司
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assumptions as follow:
The demand rate for the item is constant and known with certainty;
1) There are no constraints on the size of each lot;
2) The only two relevant costs are the inventory holding cost and the fixed cost per lot for
ordering or setup;
3) Decisions for one item can be made independently of decisions for other items;
4) There is no uncertainty in lead time or supply. The lead time is constant and known with
certainty.
Many production models have been developed in recent years. Most existing models were
based on EOQ models and added some decisional factors such as pricing and lot-sizing problem
and extended to more realistic situation such as imperfect production and reproduction. The
following are some studies: Goyal (1977) was the first one to develop the classical EOQ model
by considering some realistic situation. He developed an EOQ model to minimize the total
variable cost with constant demand rate for a single product system. Brill and Chaouch and Brill
(1995) presented a model that incorporates variations in the demand rate at random time points
with the inventory planning decision. They presented that supplier should adjust their inventory
to satisfied the demand change such as economic downturn, strikes and other disruptions.
The economic production quantity (EPQ) model is an extension of the EOQ model.
The difference being that the EPQ model assumes orders are received incrementally
during the production process. The function of this model is to balance the
inventory holding cost and the average fixed ordering cost
Most inventory models are assumed that stock items can be stored indefinitely to meet
future demands. However, some inventories may deteriorate in storage so that they may
partially or entirely unsuitable for consumption at that time. Perishable products or raw material
could be found anywhere in our life. Not only food but also medicine or cosmetics are all
perishable items. In this paper, we want to find out that how producers make the optimal
production decision when they produce a perishable product and even the perishable raw
material. Perishable items add extra cost to the producer. Since they cannot produce or sale
when product begin to deteriorate. Even these items could be sold; the price must be lower than
that of normal items. Since it is a cost for the producer, planning a best production policy to
avoid extra cost is necessary. Following give an example to show the affection of the
deterioration.
In this paper, we assume both inventory and raw material side will be deteriorated. We want
to find out,
1) A search method to find the minimal total cycle cost and optimal ordering quantity under
deterioration conditions.
2) The effect of different deterioration rates of perishable product and raw material to total cost.
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3) The error of total solution using different approach method.
4) The relative relation between deterioration rates of product and raw material.
2. Literature Review
Perishable product in inventory research has studied for many researchers. These researches
not only considered perishable in product inventory affection but also considered some factors
to close the practice.
2.1 Perishable product
Perishable products and raw material issues are an important part of production process that
affected inventory ordering and operation strategy. There were two classifications of
perishability:Fixed -lifetime and Random- lifetime. According to Nahmias (1982), the former
category includes those cases where the lifetime is known as a specified number previously
independent of all other parameters of the system. The latter category includes exponential
decay as a special case and will also include those cases where the product lifetime is a random
variable with a specified probability distribution. He gave the definitions for two kinds of
perishability as following:
1) The fixed lifetime perishable:Units may be retained in stock to satisfy demand for some
specified fixed time after which they must be discarded. Existing models assume that all
units which have not expired are of equal utility (Nahmias 1982).
2) The random lifetime perishable:For many inventories the exact lifetime of stock items
cannot be determined in advance. Items are discarded when they spoil and the time to
spoilage may be uncertain (Nahmias 1982).
Shah (1977) developed EPQ model with both constant and vary decay rate by allowing for
backordering. He also assumed immediate replacement when stock became available. Chung
and Hou (2003) developed a model to determine an optimal run time for deteriorating
production system under allowable shortage. They extended EPQ model with imperfect
processes to describe shortage. They show that there exists a unique optimal production run
time to minimize the total relevant cost function. They also pointed the bounds for optimal
production run time could be easily found by used the bisection method.
2.1.1 Exponential decay
Exponential decay is the basic assumption of perishability. Ghare and Schrader (1963)
pointed out that production deterioration could affect inventory cost. He developed an EOQ
model with production deterioration rate of negative exponential distribution under constant
易腐性商品及原料生產策略分析公司
465
demand and the condition of no backordering. They assumed that demand rate at time t is D(t).
The on hand inventory level, P(t), declines due to the simultaneous effects on both demand and
decay according to the ordinary differential equation
� P�t��t
� � P�t� D�t�
When D(t) = D, independent of t, the solution obtained is
P�t� D�� C2 · e���
This relationship may then be used to develop an expression for the cost incurred per unit
time. The authors' procedure for determining the order quantity is based on approximating the
exponential function by the first three terms of the Taylor series expansion. Hollier and Mak
(1983) developed two mathematical models that deterioration rate was constant and demand rate
followed negative exponential distribution. Heng et al. (1991) also assumed a constant
deterioration rate and used Taylor’s series approximation of expression in model. Raafat et al.
(1991) assumed a constant deterioration rate but obtained a closed form expression without
using Taylor series approximating. Exponential decay gave us an easy way to describe the
deterioration and also because it’s formula is simple than other type, so it easily to combine
more conditions such as quantity discount or vary demand, etc. Abad (1996) developed an
inventory model for perishable product that the price changed with partial backordering. He
assumed that a reseller could change the prices during the inventory cycle by partially backorder.
Chu and Chen (2002) study inventory replenishment policies for deteriorating items with fixed
partial backordering in a declining market. They suggest an approximate solution for
determining the optimal policy in these inventory systems. Abad (2003) developed an inventory
model for perishable product with partial backordering and found the optimal solution. Teng et
al. (2002) develop an inventory model for deteriorating items with time-varying demand in
which unsatisfied demand is partially backordered. They impose an additional condition on this
function to guarantee the existence of an optimal solution. Goyal and Giri (2003) assumed that
the demand, production and deterioration rate were vary with time and allowed backlogged.
They used some bivariate search method to solve the numerical equations.
2.1.2 Weibull decay
This class of models extends the basic model by considering that the deterioration rate is
non- constant. The two-parameter Weibull distribution is used because the items are assumed to
have a varying rate of deterioration. Covert and Philip (1973) generalized Ghare and Schrader's
model to describe deterioration rate by two-parameter Weibull distribution, and also
considerable constant demand and no backordering model. This is equivalent to saying that item
2010第 13屆科際整合管理研討會
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lifetimes have a Weibull distribution. Their approach also involves solving an appropriate
differential equation. That is,
dP(t)/dt +αβtβ-1P(t) =-A,
Where α and β are the shape parameters of the Weibull distribution. The optimal order
quantity is given by an infinite series expansion which can be approximated by Newton's
method. Misra (1975) developed an EPQ model in both a constant and varying decay rate by
using Weibull distribution with no backordering. By the way, the exponential distribution which
has a constant rate of deterioration is a particular case of the two-parameter Weibull distribution
(β=1). Elsayed and Teresi (1983) developed an inventory model with Weibull deterioration rate
and allowed backlogged of demand. Luo (1998) developed an integrated inventory model with
Weibull deterioration rate and allowed backordering. He uses an enumeration procedure for
finding the optimal lot size and the maximum backordering level. He showed that if production
is perishable and unique, backordering may be a way to reduce inventory cost. Wee (1999)
developed a model with Weibull deterioration rate and considerable pricing problem that the
supplier gives quantity discount.
2.1.3 Exponential approach method
In traditional EPQ model, the relation between two production time and total cycle time
was clear. The traditional EPQ model used the boundary condition between two time stages.
The time of production stage can be verified to be the multiplication between the total cycle
time and the ratio of demand rate and production rate. But when the product is deteriorating, the
relation between two stages cannot be verified by a simple relation. The time variable and the
cost function usually have some exponential parameters. Luo (1998) used the boundary
condition of production stage and production rest stage to verify the time variable without using
any approach method. In some studies, the exponential parameters and time variable were
expressed by using Taylor series. Misra (1975) used the boundary condition of production stage
and production rest stage. He used the Taylor series expression to 3rd and ignored the higher
order terms. And he also assumed the multiplication between deterioration rate and time is very
small. That is,
��� � 1 � �� ������
2, 0 � �� � 1
He verified the relation between production stage and production rest stage,
T� ��r D�
DT��1 �
λ T�2�
Wee and Yang (2002) used Misra (1975)’s method to verify the time variable and the
exponential parameters in multiple-buyer model. Yao and Wang (2005) proved that using the
Taylor series express to 4th and ignoring higher order terms was more precise. That is,
易腐性商品及原料生產策略分析公司
467
��� � 1 � �� ������
2������
6, 0 � �� � 1
Huang and Yao (2006) also used the Taylor series expression to 4th and ignored higher orders
terms to verify the time variable and exponential parameters in multiple-buyers problem.
2.2 Raw Material
Above articles all considered that deterioration only happened on finished product, but in
practice deterioration also happened on raw material. When deterioration happened on raw
material, the quantity of raw material purchased and raw material stock level will be affected,
producers may need to order more raw material than they actually need to use or increase the
stock level to make sure the production cycle could work. However, the perishable raw material
will increase the production cost and the management cost. Park (1983) developed a production
inventory model for decayed raw material which was based on Goyal’s integrated inventory
model for a single product system. This model considered the inventory problem of decayed
raw materials and a single finished product. It was assumed that finished product does not decay
and produced in batches, only the raw materials decay at a constant rate; the effect of in-process
deterioration was not considered. Raw material is ordered from outside suppliers and the arrival
of raw material coincides with the start of the production run. Roan (2001) determined process
mean and production policies with constant supply of perishable raw material for a
container-filling process. He assumed that the deterioration rate of raw material followed
exponential distribution. He found that the optimal process mean and optimal supply rate of raw
material are less sensitive to the perishability. And optimal production run size increases when
raw material is perishable.
3. Model Formulation
The production system considered in this paper is a single stage production system in which
a perishable product is being produced and sold. In this system, backordering is not allowed.
The raw material ordering lead time is zero and the producer could resale the deteriorated raw
material. The integrated model is developed in the following section. The objective of this
model is to find out the optimal production quantity in several situations.
3.1 Notations
The notations are summarized as follow.
T1 is production process time
T2 is production rest time
t1 is time of production process
2010第 13屆科際整合管理研討會
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t2 is time of production end
r is the production rate per unit time
D is the demand of products rate per unit time
H is the holding cost per unit inventory
S is the setup cost per production cycle
α is the added value of raw materials to product per unit inventory
β is the supply rate of raw materials per unit time
ε is the deterioration rate of raw materials per unit time
λ is the deterioration rate of inventory per unit time
c is the cost of raw materials per unit raw material per unit time
CdP is the decay cost of product per unit inventory
CdR is the decay cost of raw materials per unit raw material
a is the raw materials required for producing a product
h1 is the holding cost each unit of the raw material for unit time
h is the holding cost of raw materials per unit time
PHC is average holding cost of product
PDC is average decay cost of product
TPC is average total cost of product system
RHC is average holding cost of raw materials
RDC is average disposal cost of raw materials
RTC is average total cost of raw materials
TC is total cost of system
3.2 Product production system
3.2.1 Assumption
1) Production rate and demand rate is constant and given.
2) Units from production are immediately available.
3) Units start deteriorating only when they are received in to inventory.
4) Deteriorated products can resale, and deterioration raw materials can resale.
5) Deterioration rate of products and raw materials are positive.
6) Products deterioration rate is constant. Raw material deterioration rate is constant.
3.2.2 Product level
This product system extended by Luo (1998), but we consider the deterioration rate is
constant λ not original Weibull distribution. The shape at 0~t1 is concave, because at this
易腐性商品及原料生產策略分析公司
469
interval produce quantity were over than the demand when the product increased the
deteriorated quantity was also increase. The shape at t1~t2 is convex, because the production is
end and demand still exist so the quantity decrease then the deteriorated quantity decrease. We
set the total cycle time is T, the production time is T1 and the rest time is T2 or T-T1. The basic
formulation was as follow. The product level change,
d P(t) = r – I (t)λ- D, t > 0,
The product level can be written by,
P��t� �r D�λ
! �1 e�"����, t0 # t # t1,
And
P��t� Dλ! �eλ��$��� 1�, t1 # t # t2
3.2.3 The product system cost function
We consider the total product system cost with following conditions:1) production setup
cost; 2) product decay cost; 3) product holding cost.
1) Production setup cost. We conditioned each production cycle consider only one setup cost,
the production setup cost in one production cycle is S. The total set up cost is
% S
�t� t'�
2) Product holding cost. The product holding cost per unit time was expressed by the product
level per unit time and product holding cost per unit time. Let H be the product cost per unit
item. And let c be the cost of raw materials per unit raw material per unit time, a is the raw
materials required for producing a product. Then ca was the raw material cost required for
an item of finished product. So the direct cost of producing an item was ()*. Let h1 is the
holding cost each unit of the raw material for unit time and h is the holding cost of raw
materials per unit time. The cost of holding raw material is h h�c. In other words, the cost
of holding a monetary unit of raw material is h� h c- . Assume the costs of holding a
monetary unit of raw material and finished product are the same. Then, the cost of holding a
conforming item for a unit time is
H hc�αca� hαa
The product holding cost is
2010第 13屆科際整合管理研討會
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PHC H01 P��t� dt
�3�"
� 1 P��t� dt�$�3
4
�T� � T��
3) Product decay cost. In our conditions, the deteriorated products have to be abandoned. The
number of deteriorate products could be determined easily by compared the product level at
t1 with decay and without decay. The total number of deteriorated products in one product
cycle is
�r D��t� t'� D�t� t��
we assumed C56 be the product decay cost per unit time and then the decay product cost
per unit time were given by
C56�T� � T��
0�r D��t� t'� D�t� t��4
But not only the product decay cost have considered we also need to consider the raw
materials. Here we use the product decay number to determine the decay cost of raw
materials. αca is the direct cost of producing an item. So the decay cost of raw materials is
αca�T� � T��
0�r D��t� t'� D�t� t��4
The total decay cost is
PDC αca � C56�T� � T��
0�r D��t� t'� D�t� t��4
Consequently, the total product system cost.
TPC�t� S � PHC � PDC
3.2.4 Time variables
In this paper, we use time variable and raw materials supply rate be our key decision
variables. In the past, many studies used production run size, production rate and demand rate to
express the time. But in this paper, we cannot only use production run size and production rate
to express the time of producing process stage. In the fact, Luo (1998) had used the boundary
condition of production model that the time of rest stage and production stage also can be
expressed by above variable, but that let the function became hard to read and analysis. Some
studies used Taylor series to solve above problem: Misra (1975) showed the clear procedure to
find the relation between two stages by using Taylor series but the relation of time variable only
can work when the multiplication of product deterioration rate and production time is very small.
Huang and Yao (2006) used the boundary condition and the Taylor series to find the relation
between the time variable. We will discuss above two Taylor series approach methods later. In
this paper, we use the method same as Luo (1998) even that method is hard to read and analysis,
易腐性商品及原料生產策略分析公司
471
but this method is more precise. Following show the procedures to find the relation between
time variables.
The boundary condition (12) can be use to verify the relation between the time variable,
�7 8��
! 91 �����3�: 8�! �������3� 1�
Then,
�� �
�;<�1 � �=>=?@A
B�
3.3Raw material system
3.3.1 Raw material level
In this paper, we extend Roan (2001)’s model with a constant perishable rate ε. And we
assumed the raw material arrived by a constant rate β. We consider the constant supply model in
this paper, because in practice, many producers will make a contract with the raw materials
supplier to make sure the raw materials will no shortage.
We assumed the raw materials were supplied by a fixed raw materials supplement. That
means there had a constant raw materials supply rate during any time to satisfy the production
need and increase the raw materials. The raw materials level was decay by a constant
deterioration rate. The raw materials level was consider by supply rate, production demand rate
and deterioration rate. And when the production process is idle, the raw material will decrease to
zero.
The raw materials level functions could be verified to,
R��t� �ar β�ε
�e�3��� 1�, t0 # t # t�
And
R��t� β
ε�1 eε��3����, t1 # t # t�
Let h be the raw material holding cost per unit time. Since the raw materials level functions
had showed before. We could easily verify the average raw materials level was,
RHC h ! 01 R��t� �t
�3�"
� 1 R��t� �t�$�3
4
�T� � T��
Let C5D be the disposal cost per raw material unit. In product system cost we already
2010第 13屆科際整合管理研討會
472
consider the decay cost of raw materials, but in fact, the deteriorated raw materials usually
resoled for other use, like deteriorated vegetables could be reproduce to fertilizer. So the
disposal cost we denoted was considered a negative cost factor. To determine the number of raw
material decay was similar with production product decay number. We could compare the raw
materials level at time t0 and t2 with decay and without decay. The decay number of raw
materials was,
0β ! �t� t�� � �ar β��t� t'�4
So the disposal cost of raw materials is,
RDC C5D ! 0β ! �t� t�� � �ar β��t� t'�4
�T� � T��
Since the supplement of raw materials in this case was constant supplement and the raw
materials was arrived every time without ordering. So the setup cost was zero in this case. The
raw materials purchasing cost could be easily to express by raw materials supply rate and the
cost of raw materials per unit raw material per unit time, that is βc.
Consequently, the total raw material cost per unit time is
TRC RHC � RDC � RPC
The total cost function,
TC % � PHC � PDC � RHC � RDC � RPC
3.3.2 Raw material supply rate
In this section, we set the raw material supply rate is a constant supply rate. And the supply
could be determined by some conditions in our model. Roan (2001) used the boundary
condition and Taylor series to determine the raw material supply rate. Here we only use
boundary condition to determine the supply rate. In the raw materials we have two raw materials
level functions at two stages and the both level functions are equal at time 0 and T, thus
E��0� E���� Then,
�*7 F�G
9�H��3� 1: FG�1 ��H����3��
Then,
F *7�1 ��H�3�1 ��H�
F *7�1 �1 � 8 � 8���
7 ��H��
1 ��H�
The optimal solution is given by,
Minimize �N��� % � OPN
91 �����3�: =
�! �������
Since a closed form analytical solution cannot be obtained for objective function, a computer
program is developed to find the optimal T and Total cost using an exhaustive search method.
An example here is same as
section. The example will also be used in the sensitivity analysis next section.
Assume the unit time is month. The product demand rate and the production rate are 5000
items and 7500 items per unit time. The setup cost per production run is 150. Suppose the raw
materials are purchase from vendor and the raw material cost is $1/mg. The add value of a
product is 1.2 and a product required 4 mgs raw material to produce. The product deterioration
rate and raw materials deterioration rate are 0.01 and 0.01. Assume the decay cost of product is
$1.5 per item. Consider deteriorated raw materials can be resale but the deteriorated raw
materials need to pay for disposal. So the deteriorated raw materials
Furthermore, the holding cost of product is $1.44 and the cost of holding raw material is 0.03
per unit time.
The result is T*=8.72, q = 44,227 and
The optimal raw material supply rate is 20,573 mgs and the production
The total cost is $4,720.34.
易腐性商品及原料生產策略分析公司
473
Fig. 1 Total system cost
The optimal solution is given by,
OPN � O8N � EPN � E8N � EON subject
�3� 1�,�UB�V�
H9�H��3� 1: V
H�1 ��H����
Since a closed form analytical solution cannot be obtained for objective function, a computer
program is developed to find the optimal T and Total cost using an exhaustive search method.
4. Numerical example
An example here is same as in Roan (2001) to illustrate the solution procedures given last
section. The example will also be used in the sensitivity analysis next section.
Assume the unit time is month. The product demand rate and the production rate are 5000
r unit time. The setup cost per production run is 150. Suppose the raw
materials are purchase from vendor and the raw material cost is $1/mg. The add value of a
product is 1.2 and a product required 4 mgs raw material to produce. The product deterioration
rate and raw materials deterioration rate are 0.01 and 0.01. Assume the decay cost of product is
$1.5 per item. Consider deteriorated raw materials can be resale but the deteriorated raw
materials need to pay for disposal. So the deteriorated raw materials disposal cost is $0/mg.
Furthermore, the holding cost of product is $1.44 and the cost of holding raw material is 0.03
=8.72, q = 44,227 and β = 20,573.
he optimal raw material supply rate is 20,573 mgs and the production run size is 44,227 items.
易腐性商品及原料生產策略分析公司
subject to �B�=��
!
�3��
Since a closed form analytical solution cannot be obtained for objective function, a computer
program is developed to find the optimal T and Total cost using an exhaustive search method.
in Roan (2001) to illustrate the solution procedures given last
Assume the unit time is month. The product demand rate and the production rate are 5000
r unit time. The setup cost per production run is 150. Suppose the raw
materials are purchase from vendor and the raw material cost is $1/mg. The add value of a
product is 1.2 and a product required 4 mgs raw material to produce. The product deterioration
rate and raw materials deterioration rate are 0.01 and 0.01. Assume the decay cost of product is
$1.5 per item. Consider deteriorated raw materials can be resale but the deteriorated raw
disposal cost is $0/mg.
Furthermore, the holding cost of product is $1.44 and the cost of holding raw material is 0.03
run size is 44,227 items.
2010第 13屆科際整合管理研討會We use above example to draft a graphic in figure
the example solution is the minimal of the TC.
In this paper, we develop a
deteriorating raw materials with constant raw materials
show the optimal solution for minimizing the total production cost
rates of production and raw materials and the graphic had show the cost function is con
The sensitive analysis is developing.
Taylor or 4th Taylor series to verify the time variable.
assumes the multiplication of product deterioration rate and production time can be ignored.
if the multiplication is large, the optimal solution is not
model to avoid the errors in this paper.
the model. In this paper, we consider the production system with deteriorating products and raw
materials. Therefore, the backordering of products,
quantity discount raw materials are the interest
Adab, P. L., Optimal pricing and lot
backordering. Management
Abad, P. L., Optimal pricing and lot
and partial backordering and lost sale.
2003, 677-685.
Brill, P. H., and Chaouch, B. A., An EOQ model with random variations in demand.
Management Science, 1995,
Cohen, M. A., Joint pricing and ordering policy for exponentially decaying inventories with
known demand. Naval Research Logis
屆科際整合管理研討會
474
We use above example to draft a graphic in figure 5. The TC is show a convex graphic and
the example solution is the minimal of the TC.
Fig. 2 TC with T
5. Conclusion
e develop a two-echelon production model for deterioratin
with constant raw materials supplement. The numerical example has
optimal solution for minimizing the total production cost is effected by deterioration
f production and raw materials and the graphic had show the cost function is con
is developing. In some studies’ model which used the same method as
to verify the time variable. But this method has a limit.
of product deterioration rate and production time can be ignored.
if the multiplication is large, the optimal solution is not fit our condition. So we use the original
the errors in this paper. However, those methods give an easy
n this paper, we consider the production system with deteriorating products and raw
herefore, the backordering of products, multiple orders of raw materials
quantity discount raw materials are the interesting research conditions in the future research.
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. The TC is show a convex graphic and
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he numerical example has
is effected by deterioration
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So we use the original
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n this paper, we consider the production system with deteriorating products and raw
orders of raw materials or the
ing research conditions in the future research.
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