易腐性商品及原料之生產策略分析2010 13 June 26, pp462-475 462...

2010第 13屆科際整合管理研討會

June 26, pp462-475

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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: [email protected])。 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:[email protected])。

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


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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


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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,


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�� 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


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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


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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 eλ��"��, 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


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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,


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

�eε��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


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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.


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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��



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.


r unit time. The setup cost per production run is 150. Suppose the raw





materials need to pay for disposal. So the deteriorated raw materials disposal cost is $0/mg.


=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.


subject to �B�=��

!

�3��



in Roan (2001) to illustrate the solution procedures given last


r unit time. The setup cost per production run is 150. Suppose the raw





disposal cost is $0/mg.


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.

References

Adab, P. L., Optimal pricing and lot-sizing under conditions of perishability and partial

Management Science, 42(8), 1996, 1093-1104.

Abad, P. L., Optimal pricing and lot-sizing under conditions of perishability, finite production

and partial backordering and lost sale. European Journal of Operational Research

ch, B. A., An EOQ model with random variations in demand.

, 1995, 41, 927- 936.


Naval Research LogisTPCs, 1977, 24, 257-268.

. The TC is show a convex graphic and

deteriorating product and

he numerical example has

is effected by deterioration

f production and raw materials and the graphic had show the cost function is convexity.

the same method as 3rd

limit. This method

of product deterioration rate and production time can be ignored. But

So we use the original

easy way to simplify

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.

sizing under conditions of perishability and partial

sizing under conditions of perishability, finite production

European Journal of Operational Research, 144,

ch, B. A., An EOQ model with random variations in demand.



475

Covert, R. P. and Philip, G. C., AEOQ model for items with Weibull distribution deterioration.

AIIE Transactions, 5(4), 1973, 323-326.

Chu, P. and Chen, P.S., A note of inventory replenishment policies for deteriorating items in an

exponentially declining market. Computers and Operations Research, 2002, 29,

1827–1842.

Eilon, S. and Mallaya, R.V., Issuing and pricing policy of semi-perishables. Proceedings of 4th

International conference on operational research, 1966.

Goyal, S. K., An integrated inventory model for a single production system. Opl Res. Q., 28,

1977, 539-545.

Goyal, .K and Gunasekaran, A., An integrated production-inventory-marketing model for

deteriorating items. Computers and Industrial Engineering, 28(4), 1995, 755-762.

Ghare, P. M. and Schrader, G. F., A model for exponentially decaying inventory. The Journal of

Industrial Engineering, 14(5), 1963, 238-243.

Huang, J. Y. and Yao, M. J., A New Algorithm for Optimally Determining Lot-Sizing Policies

for a Deteriorating Item in Integrated Production-Inventory System. Computers &

Mathematics with applications, 15, 2006, 83-104.

Kotler, P., in Marketing Decision Making: A Model Building Approach. Holt, Rinehart, Winston,

New York. 1971.

Kang, S. and Skim, I. T., A study on the price and production level of the deteriorating inventory

system. International Journal of Production Research, 1983, 21, 899-908.

Luo, W., An integrated inventory system for perishable goods with backordering. Computers

and Industrial Engineering, 34, 1998, 685-693.

Misra, R. B., Optimum production lot size for a system with deteriorating inventory.

International Journal of Production Research, 13(5), 1975, 495-505.

Mehra, S., Agrawal, P. and Rajagopalan, M., Some comments on the validity of EOQ formula

under inflationary conditions. Decision Sciences, 1991, 22, 206.

Nahmias, S., Perishable inventory theory：A Review. Operations Research, 1982, 30, 680-708.

Park, K. S., Integrated production inventory model for decaying raw materials. International

Journal of Systems Science, 1983, 14, 801-806.

Raafat, K., Survey of Literature on Continuously Deteriorating Inventory Models. The Journal

of the Operational Research Society, 1991, 42, 27-37.

Roan, J., Process mean and production policies determination under constant supply of

perishable raw material. 2001

Skouri, K. and Papachristos, S., A continuous review inventory model, with deteriorating items,

time-varying demand, linear replenishment cost, partially time-varying backlogging.

Applied MathemaTPCal Modelling, 2002, 26, 603–617.

Teng, J.T., Chang, H.J., Dye, C.Y. and Hung, C. H., An optimal replenishment policy for

deteriorating items with time-varying demand and partial backlogging. Operations

Research Letters, 2002, 30, 387–393.

Wee, H. M., Deteriorating inventory model with quantity discount, pricing and partial

backordering. International Journal of Production Economics, 59, 1999, 511-518.

Yao, M. C. and Wang, Y. C, THEORETICAL ANALYSIS AND A SEARCH PROCEDURE

FOR THE JOINT REPLENISHMENT PROBLEM WITH DETERIORATING

PRODUCTS. Journal of Industrial and Management Optimization, 2005, 1, 359-375

Yang, P. C. and Wee, H. M., A single-vendor and multiple-buyers production-inventory policy

for a deteriorating item. European Journal of Operational Research, 143, 2002, 570-581

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