2.1 literature review and survey ofshodhganga.inflibnet.ac.in/bitstream/10603/18501/4/chapter...

2.1 LITERATURE REVIEW AND SURVEY OF

INVENTORY MODELS

Review of literature and survey of the developed inventory models

are being treated for consideration in this chapter. Although there is an

abundance of the literature of general inventory models, however an

attempt has been made to cover some of them which are related to our

problems. The aim of this chapter is to present a complete and update

picture of inventory theory related to the problems of the thesis. The

review of the related literature is an essential aspect of a research work.

2.1.1 Survey of Inventory Models with Inflation

In the classical inventory it is assumed that all the costs associated

with the inventory system remain constant over time. Since most decision

makers think that the inflation does not have a significant influence on the

inventory policy and most of the inventory models developed so far does

not include inflation and time value of money as parameters of the

system. But, due to large scale of inflation the monetary situation in

almost of the countries has changed to an extent during the last thirty

years. Nowadays inflation has become a permanent feature in the

inventory system. Inflation enters in the picture of inventory only because

it may have an impact on the present value of the future inventory cost.

Chapter 2 Literature Review

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Thus the inflation plays a vital role in the inventory system and

production management though the decision makers may face difficulties

in arriving at answers related to decision making. At present, it is

impossible to ignore the effects of inflation and it is necessary to consider

the effects of inflation on the inventory system.

Buzacott (1975) developed the first EOQ (Economic Order

Quantity) model taking inflationary effects into account. In this model, a

uniform inflation was assumed for all the associated costs and an

expression for the EOQ was derived by minimizing the average annual

cost. Misra (1975-a, 1979) investigated inventory systems under the

effects of inflation. Bierman and Thomas (1977) suggested the inventory

decision policy under inflationary conditions.

Economic analysis of dynamic inventory models with non-

stationary costs and demand was presented by Hariga (1994). The effect

of inflation was also considered in this analysis. An economic order

quantity inventory model for deteriorating items was developed by Bose

et al. (1995). Authors developed inventory model with linear trend in

demand allowing inventory shortages and backlogging. The effects of

inflation and time-value of money were incorporated into the model. The

inventory policy was discussed over a finite time-horizon with several

reorder points. It was assumed that the goods in the inventory deteriorate


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over time at a constant rate. The results were discussed with a numerical

example and sensitivity analysis of the optimal solution with respect to

the parameters of the system was carried out. Several particular cases of

the model were discussed in brief.

Effects of inflation and time-value of money on an inventory model

was discussed by Hariga (1995) with linearly increasing demand rate and

shortages. Hariga and Ben-Daya (1996) then discussed the inventory

replenishment problem over a fixed planning horizon for items with

linearly time-varying demand under inflationary conditions. Ray and

Chaudhuri (1997) developed a finite time-horizon deterministic

economic order quantity inventory model with shortages, where the

demand rate at any instant depends on the on-hand inventory at that

instant. The effects of inflation and time value of money were taken into

account. A generalized dynamic programming model for inventory items

with Weibull distributed deterioration was proposed by Chen (1998). The

demand was assumed to be time-proportional, and the effects of inflation

and time-value of money were taken into consideration. Shortages were

allowed and partially backordered. The effects of inflation and time-value

of money on an economic order quantity model have been discussed by

Moon and Lee (2000). Authors have considered the normal distribution

as a product life cycle in addition to the exponential distribution. The


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two-warehouse inventory models for deteriorating items with constant

demand rate under inflation were developed by Yang (2004). The

shortages were allowed and fully backlogged in the models. Some

numerical examples for illustration were provided.

Chang (2004) proposed an inventory model under a situation in

which the supplier has provided a permissible delay in payments to the

purchaser if the ordering quantity is greater than or equal to a

predetermined quantity. Shortage was not allowed and the effect of the

inflation rate, deterioration rate and delay in payments were discussed as

well. Models for ameliorating / deteriorating items with time-varying

demand pattern over a finite planning horizon were proposed by Moon et

al. (2005). The effects of inflation and time value of money were also

taken into account. An inventory model for deteriorating items with

stock-dependent consumption rate with shortages was produced by Hou

(2006). Model was developed under the effects of inflation and time

discounting over a finite planning horizon. The results were discussed

with a numerical example and particular cases of the model were

discussed in brief. Sensitivity analysis of the optimal solution with

respect to the parameters of the system was carried out.

Jolai et al. (2006) presented an optimization framework to derive

optimal production over a fixed planning horizon for items with a stock-


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dependent demand rate under inflationary conditions. Deterioration rate

was taken as two parameter Weibull distribution function of time.

Shortages in inventory were allowed with a constant backlogging rate.

Two-warehouse partial backlogging inventory models for deteriorating

items were discussed by Yang (2006). The inflationary effect was

considered in the models. Deterioration rates in both the warehouses were

taken as constant. Some numerical examples for illustration were

provided and sensitivity analysis on some parameters was made.

Jaggi et al. (2007) presented the optimal inventory replenishment

policy for deteriorating items under inflationary conditions using a

discounted cash flow (DCF) approach over a finite time horizon.

Shortages in inventory were allowed and completely backlogged and

demand rate was assumed to be a function of inflation. Optimal solution

for the proposed model was derived and the comprehensive sensitivity

analysis has also been performed to observe the effects of deterioration

and inflation on the optimal inventory replenishment policies. Two stage

inventory problem over finite time horizon under inflation and time value

of money was discussed by Dey et al. (2008). Chern, Yang, Teng, and

Papachristos, (2008) developed an inventory lot-size model for

deteriorating items with partial backlogging and time value of money.

Roy, Pal and Maiti, (2009) developed a production inventory model


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with inflation and time value of money. Demand of the item was stock

dependent and lifetime of the product was taken as random in nature with

exponential distribution. Learning effect on production and setup cost

was incorporated. In their study, model was formulated to maximize the

expected profit during the whole planning horizon. Yang, Teng, and

Chern, (2010) developed an economic order quantity model, in which

shortages were allowed with partial backlogging. The effects of inflation

and time value of money were taken into consideration.

2.1.2 Survey of Inventory Models for Deteriorating Items

In recent years, mathematical ideas have been used in different

areas in real life problems, particularly for controlling inventory. One of

the important concerns of the management is to decide when and how

much to order or to manufacture so that the total cost associated with the

inventory system should be minimum. This is somewhat more important,

when the inventory under go decay or deterioration. Most of the

researchers in inventory system were directed towards non-deteriorating

products. However, there are certain substances, whose utility does not

remain same with the passage of time. Deterioration of these items plays

an important role and items cannot be stored for a long time.

Deterioration of an item may be defined as decay, evaporation,

obsolescence, loss of utility or marginal value of an item that results in


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the decreasing usefulness of an inventory from the original condition.

When the items of the commodity are kept in stock as an inventory for

fulfilling the future demand, there may be the deterioration of items in the

inventory system, which may occur due to one or many factors, i.e.,

storage conditions, weather conditions or due to humidity.

Commodities such as fruits, vegetables and foodstuffs suffer from

depletion by direct spoilage while kept in store. Highly volatile liquids

such as alcohol, gasoline etc. undergo physical depletion over time

through the process of evaporation. Electronic goods, photographic film,

grains, chemicals, pharmaceuticals etc. deteriorate through a gradual loss

of potential or utility with the passage of time. Thus, deterioration of

physical goods in stock is very realistic feature. As a result, while

determining the optimal inventory policy of that type of products, the loss

due to deterioration cannot be ignored. In the literature of inventory

theory, the deteriorating inventory models have been continually

modified so as to accumulate more practical features of the real inventory

systems.

The analysis of deteriorating inventory began with Ghare and

Schrader (1963), who established the classical no-shortage inventory

model with a constant rate of decay. However, it has been empirically

observed that failure and life expectancy of many items can be expressed


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in terms of Weibull distribution. This empirical observation has prompted

researchers to represent the time to deterioration of a product by a

Weibull distribution. Covert and Philip (1973) extended Ghare and

Schraders (1963) model and obtain an economic order quantity model

for a variable rate of deterioration by assuming a two-parameter Weibull

distribution. Philip (1974) presented an EOQ model for items with

Weibull distribution deterioration rate.

Pierskalla and Roach (1972) suggested optimal issuing policies

for perishable inventory. Misra (1975-b) presented a production lot size

model for an inventory system with deteriorating items with variable rate

of deterioration while rate of production was finite. An order level

inventory model for a system with constant rate of deterioration was

presented by Shah and Jaiswal (1977). Aggarwal (1978) and Aggarwal

and Jaggi (1989) presented inventory ordering policies for deteriorating

items. Goyal (1986) and Gupta and Vrat (1986) suggested inventory

models with variable rates of demand. Dave (1986) presented an order

level inventory model for deteriorating items. Model was continuous in

units but allowed discrete opportunities for replenishment. In this model,

demand kept on changing from time unit to time unit and occurs

instantaneously at the beginning of each time unit. Deterioration rate was

assumed to be constant and lead-time was zero.


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A deterministic inventory system with stock dependent demand

rate was formulated by Baker and Urban (1988). An inventory model

concerning a single item was suggested by Mandal and Phaujdar (1989)

for deteriorating items with a variable rate of deterioration. Model was

developed with stock-dependent consumption rate and uniform

production rate. Shortage was allowed and the excess demand was

backlogged as well. The rate of deterioration was assumed first as

constant and then variable. Dutta and Pal (1990) presented a note on an

inventory level dependent demand rate. Inventory model for deteriorating

items with stock dependent demand rate was proposed by Pal et al.

(1993). Inventory models for perishable items with stock dependent

selling rate were suggested by Padmanabhan and Vrat (1995). The

selling rate was assumed to be a function of current inventory level and

rate of deterioration was taken to be constant with complete, partial

backlogging and without backlogging. Balkhi and Benkherouf (1996)

presented a production lot size model for deteriorating items and arbitrary

production and demand rates. In this model demand and production were

allowed to vary with time in an arbitrary way and shortages were

allowed. The rate of deterioration was taken as constant and the method

was illustrated with the help of numerical example.


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Mandal and Maiti (1997) presented an inventory model for

damageable items with fully backlogged shortages for both linear and

non-linear damage and demand functions. The authors considered

production rate as finite and solved the model with profit maximization

criteria together with sensitive analysis. An economic order quantity

model for perishable products with nonlinear holding cost was proposed

by Giri and Chaudhuri (1998). The demand rate was taken as function

of on-hand inventory.

An EOQ inventory model with ramp-type demand rate for items

with Weibull deterioration was formulated by Wu et al. (1999).

An order level inventory model for deteriorating items was

proposed by Gupta and Agarwal (2000). The demand was taken as a

linear function of time and production rate was taken as demand

dependent. A production inventory model for deteriorating items with

exponential declining demand was discussed by Kumar and Sharma

(2000). Time horizon was fixed and the production rate at any instant was

taken as the linear combination of on-hand inventory and demand.

Aggarwal and Jain (2001) presented an inventory model for

exponentially increasing demand rate with time. The items were

deteriorating at a constant rate and shortages were allowed. The authors

solved the model by two methods. The first method was efficient for


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getting accurate results, although, lengthy and complicated while the

second method was faster than the first method and thus easy to apply.

Goyal and Giri (2001-a) presented a review of the advances of

deteriorating inventory literature since early 1990s. The models available

in the relevant literature have been suitably classified by shelf-life

characteristic of the inventoried goods. A single-vendor and multiple-

buyers production-inventory policy for a deteriorating item was

formulated by Yang and Wee (2002). Production and demand rates were

taken to be constant. A mathematical model incorporating the costs of

both the vender and the buyers was developed.

An order level inventory model for deteriorating items with an

exponential increasing demand was proposed by Sharma and Singh

(2003). An order-level inventory problem for a deteriorating item with

time dependent quadratic demand was presented by Khanra and

Chaudhuri (2003). The inventory was assumed to deteriorate at a

constant rate and shortages were not allowed. A production-inventory

model for a deteriorating item over a finite planning horizon was

presented by Sana et al. (2004). Deterioration rate was taken as constant

fraction of the on-hand inventory. The demand was taken to be linear

time-varying function. An economic production quantity model for

deteriorating items was discussed by Teng and Chang (2005). Demand


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rate was taken as dependent on the display stock level and the selling

price per unit. A deteriorating multi-item inventory model with limited

storage capacity was suggested by Mandal et al. (2006). The demand

rate for the items was finite and deterioration rate was taken to be

constant. The replenishment rate was taken as constant. The problem was

supported by numerical examples. A note on the inventory models for

deteriorating items with ramp type demand was developed by Deng et al.

(2007). They have proposed an extended inventory model with ramp type

demand rate and its optimal feasible solution. Liao (2008) considered an

EOQ model with non-instantaneous receipt and exponentially

deteriorating items under two-level trade credit. Chung (2009) explored a

complete proof on the solution procedure for non-instantaneous

deteriorating items with permissible delay in payment.

2.1.3 Survey of Inventory Models with Permissible Delay in

Payments

There have been a lot of discussions in the existing literature for

the possible extension of the basic economic order quantity model to a

number of situations where its assumptions are not valid up to now. A

business either manufactures the products it sells or it purchases the

products to sell from other businesses. In either case, an increase in

inventory usually involves a corresponding increase in accounts payable.


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Raw materials used in the production process are purchased on credit, and

many other manufacturing costs are not paid for immediately. Products

from other businesses are bought on credit. Instead of making immediate

cash payment when inventory is increased, a business delays payment,

perhaps by a month or so.

that the customers are allowed some grace period before they settle the

account with the supplier. This gives a very advantage to the customers

due to the fact that they do not have to pay the supplier immediately after

receiving the product, but instead, they can delay their payment until the

end of the allowed period. The customer pays no interest during the fixed

period they have to settle the account, but if the payment is delayed

beyond that period, interest will be charged. The permissible delay in

payments brings some advantages to the buyer, as he would try to earn

some interest for the payment received during this period. When a

supplier allows a fixed time for settling the account, he is actually giving

a loan to the buyer without interest during this period. Therefore, it is

economical to delay in the settlement of accounts to the last moment of

the permissible delay in payments.

Inventory policy with trade credit financing was formulated by

Haley and Higgins (1973). The effect of payment rules on ordering and


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stock holding in purchasing was suggested by Kingsman (1983).

Different inventory policies with trade credit was developed by

Chapman et al. (1984) and Chapman and Ward (1988). Goyal (1985)

was the first to develop the economic order quantity under conditions of

permissible delay in payments. Author has assumed that the unit selling

price and the purchase price are equal. In practice, the unit selling price

should be greater than the unit purchasing price. Aggarwal and Jaggi

(1995) developed ordering policies of deteriorating items under

permissible delay in payments. The demand and deterioration were

exponentially deteriorating products under the condition of permissible

delay in payments was discussed by Hwang and Shinn (1997). Jamal et

al. (2000) presented optimal payment time for a retailer under permitted

delay of payment by the wholesaler. The wholesaler allowed a

permissible credit period to pay the dues without paying any interest for

the retailer. In the study, a retailer model was considered with a constant

rate of deterioration. An inventory model for initial-stock-dependent

consumption rate was suggested by Liao et al. (2000). Shortages were

not allowed. The effects of inflation rate, deterioration rate, initial stock-

dependent consumption rate and delay in payments were discussed.


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Chang and Dye (2001) developed an inventory model for

deteriorating items with partial backlogging and permissible delay in

payments. Chung et al. (2002) have discussed the inventory decision for

EOQ inventory model under permissible delay in payments. An EOQ

model for deteriorating items with credit policy was discussed by Chang

et al. (2003). Lot sizing decision policy was presented by Chung and

Liao (2004). In this policy trade credit was depending on the ordering

quantity. Teng et al. (2005) presented an optimal pricing and ordering

policy under permissible delay in payments. Deterioration rate was taken

as constant and shortages were not allowed in the study. The optimal

ordering policy in a DCF analysis for deteriorating items was suggested

by Chung and Liao (2006). In this policy trade credit was also

depending on the ordering quantity.

Huang (2007)

levels of trade credit policy within the economic production quantity

(EPQ) framework. Author has assumed that the supplier would offer the

retailer a delay period and the retailer also adopted the trade credit policy

replenishment model. Replenishment rate was taken as finite. Sana and

Chaudhuri (2008)

Increasing deterministic demands were discussed in the environment of


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permissible delay in payment and discount offer to the retailer. An

inventory model under two levels of trade credit policy was proposed by

Jaggi et al. (2008) with credit-linked demand.

2.1.4 Survey of Supply Chain Inventory Models

One of the most important topics in the study of the management

of contemporary manufacturing and distribution is supply chain

management (SCM). Inventory management is an essential process for all

parties engaged in supply chain activities, from the procurement of raw

materials through to the delivery of finished goods. The effective

execution of this process has a major influence on both the financial and

operational performance of an organization.

While many may argue that inventories are undesirable and should

be eliminated from the supply chain, the fact remains that organizations

require inventory in order to operate effective and to ensure the smooth

operation of their day-to-day business. Notwithstanding the critical

importance of inventories, companies have to keep in mind that

inventories can also cause adverse effects and often disguise other

underlying issues in the supply chain. Managing inventory can be both a

complicated and critical process given the conflicting business objectives

that can occur with in an organization and across multiple enterprises.


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Supply cha

management is the coordination of production, inventory, location and

transportation among the participants in a supply chain to achieve the best

mix of responsiveness and efficiency for the market being served

into widespread use in the 1990s. Prior to that time, business used terms

There is a difference between the concept of supply chain

management and traditional concept of logistics. Logistics typically refers

to activities that occur with in the boundaries of a single organization and

supply chains refer to networks of companies that work together and

coordinate their actions to deliver a product to market. Also, traditional

logistics focuses its attention on activities such as procurement,

distribution, maintenance and inventory management. Supply chain

management acknowledges all the traditional logistics and also includes

activities such as marketing, new product development, finance and

customer service.

In the wider view of supply chain thinking, these additional

activities are now seen as part of the work needed to fulfill customer

requests. Supply chain management views the supply chain and the


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organizations in it as a single entity. It brings a systems approach to

understanding and managing the different activities needed to coordinate

the flow of products and services to best serve the ultimate customer.

This systems approach provides the framework in which to best respond

in business requirements that otherwise would seem to be in conflict with

each other.

Taken individually, different supply chain requirements often have

conflicting needs. For instance, the requirement of maintaining high

levels of customer service calls for maintaining high levels of inventory,

but then the requirement to operate efficiently calls for reducing

inventory levels. It is only when these requirements are seen together as

part of a large picture that ways can be found to effectively balance their

different demands.

Effective supply chain management requires simultaneous

improvements in both customer service levels and the internal operating

efficiencies of the companies in the supply chain. Customer service at its

most basic level means consistently high order fill rates, high on-time

delivery rates and a very low rate of products returned by customers for

whatever reason. Internal efficiency for organizations in a supply chain

means that these organizations get an attractive rate of return on their

investments in inventory and other assets.


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The effective management of supply channel inventories is perhaps

the most fundamental objective of supply chain management.

Manufacturers procure raw material and process them into finished

goods, and sell the finished goods to distributors, then to retailer and/or

customer. When an item moves through more than one stage before

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A large amount of researches on multi-echelon inventory control has

appeared in the literature during the last decades.

Clark and Scarf (1960) were the first to study the two-echelon

inventory model. They proved the optimality of a base stock policy for

the pure serial inventory system and developed an efficient decomposing

method to compute the optimal base stock ordering policy. Sherbrooke

(1968) considered an ordering policy of two-echelon model for

warehouse and retailer. It is assumed that stock-outs at the retailers are

completely backlogged. Van der Heijden et al. (1997) presented stock

allocation policies in general single-item and N-echelon distribution

systems, where it is allowed to hold stock at all levels in the network.

Diks and De Kok (1998) determined a cost optimal replenishment policy

for a divergent multi-echelon inventory system. A joint replenishment

policy for multi-echelon inventory control was proposed by Axsater and

Zhang (1999). A multi-echelon inventory model for a deteriorating item


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was developed by Rau et al. (2003). An optimal joint total cost has been

derived from an integrated perspective among the supplier, the producer

and the buyer. A deteriorating item inventory model in a supply chain

was proposed by Wu and Wee (2005). Shortages in inventory were

allowed and fully backlogged. Two-echelon inventory model with lost

sales was proposed by Hill et al. (2007). A two level supply chain in

which production interruptions for restoring of the quality of the

production process coordinated by Ahmed et al.(2008). Wong et al.

(2009) proposed a two-echelon supply chain with a single supplier

serving multiple retailers in vendor-managed inventory (VMI)

partnership.

2.1.5 Survey of Partial Backlogging Inventory Models

An important issue in the inventory theory is related to how to deal

with the unfulfilled demands which occur during shortages or stock outs.

In most of the developed models researchers assumed that the shortages

are either completely backlogged or completely lost. The first case,

known as backordered or backlogging case, represent a situation where

the unfulfilled demand is completely back ordered. In the second case,

also known as lost sale case, we assume that the unfulfilled demand is

completely lost.


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Furthermore, when the shortages occur, some customers are willing

to wait for backorder and others would turn to buy from other sellers. In

many cases customers are conditioned to a shipping delay and may be

willing to wait for a short time in order to get their first choice. For

instance, for fashionable commodities and high-tech products with short

product life cycle, the willingness of a customer to wait for backlogging

is diminishing with the length of the waiting time. Thus, the length of the

waiting time for the next replenishment would determine whether the

backlogging would be accepted or not. In many real life situations, during

a shortage period, the longer the waiting time is, the smaller is the

backlogging rate would be. Therefore, for realistic business situations the

backlogging rate should be variable and dependent on the waiting time

for the next replenishment. Many researchers have modified inventory

Some researchers have developed inventory models in which the

proportion of customers who would like to accept backlogging is the

reciprocal of a linear function of the waiting time up to the arrival of next

lot. Some researches have been done in which backlogging rate is taken

as exponential decreasing function of time. The inventory involving

backlogging and lost sales is called mixture inventory. Mixture inventory

means mixture of backorders and lost sales. Therefore, in the third case, it


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is assumed that a fixed fraction of the unfulfilled demand during the stock

out stage is back ordered while the remaining fraction is lost. Thus,

concept of partial backlogging should not be ignored in inventory models

of shortage.

Zangwill (1966) developed a production multi-period production

scheduling model with backlogging. Inventory models with a mixture of

backorders and lost sales were formulated by Montgomery et al. (1972).

Rosenberg (1979) presented the analysis of a lot-size model with partial

backlogging. Mak (1987) proposed optimal production-inventory control

policies for an inventory system. Shortages in inventory were allowed

and partially backlogged.

Economic production lot size model for deteriorating items with

partial backordering was suggested by Wee (1993). Abad (1996)

formulated a generalized model of dynamic pricing and lot-sizing for

perishable items. Shortage was allowed and the demand was partially

backlogged. The effects of lost sales on composite lot sizing were

discussed by Sharma and Sadiwala (1997).

Deteriorating inventory model with quantity discount, pricing and

partial backordering was developed by Wee (1999). In this study, demand

was assumed to decrease with the increment in price for the product. The


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numerical example was also given to illustrate the model. Optimal price

and order size inventory policy for a reseller under partial backlogging

was proposed by Abad (2000-a). The problem of determining the lot size

for a perishable good under finite production was formulated by Abad

(2000-b). Shortage in inventory was allowed with partial backordering

and lost sale. Author has taken a problem of determining the production

lot size of a perishable product that decays at an exponential rate. Wu

(2000) developed inventory model for items with Weibull distribution

deterioration rate, time dependent demand and partial backlogging.

An EOQ inventory model for items with Weibull distribution

deterioration rate and ramp type demand was formulated by Wu (2001).

Shortages in inventory were allowed and assumed to be partially

backlogged. Ouyang and Chang (2002) presented stochastic continuous

review inventory model with variable lead time and partial backorders to

capture the reality of uncertain backorders. An EOQ model for

deteriorating items with time-varying demand and partial backlogging

was suggested by Teng et al. (2003). Papachristos and Skouri (2003)

considered a model where the demand rate was a convex decreasing

function of the selling price and backlogging rate was a time dependent

function. The time of deterioration of the item was distributed as two

parameter Weibull distribution function of time.


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A general time-varying demand inventory lot-sizing model with

waiting-time-dependent backlogging and a lot-size-dependent

replenishment cost was developed by Zhou et al. (2004).

An EOQ model with time varying deterioration and linear time

varying demand over finite time horizon was proposed by Ghosh and

Chaudhuri (2005). Shortages in inventory were allowed and partially

backlogged with waiting time dependent backlogging rate. Ouyang et al.

(2005) discussed an EOQ inventory mathematical model for deteriorating

items with exponentially decreasing demand. The shortages were allowed

and partially backordered. The backlogging rate was variable and

dependent on the waiting time for the next replenishment.

Pal et al. (2006) have developed an inventory model for single

deteriorating item by considering the impact of marketing strategies such

as pricing and advertising as well as the displayed stock level on the

demand rate of the system. Shortages were allowed and the backlogging

rate was dependent on the duration of waiting time up to the arrival of

next lot. An economic production lot size model for deteriorating items

with stock-dependent demand was produced by Jolai et al. (2006) under

the effects of inflation. Shortages were allowed and partially

backlogged.


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A deterministic inventory model for deteriorating items with price

dependent demand was developed by Dye et al. (2007-a). The demand

and deterioration rates were continuous and differentiable function of

price and time, respectively. Shortages for unsatisfied demand were

allowed and backlogging rate was taken as negative exponential function

of the waiting time. Numerical example was also used to study the

problem numerically. Dye et al. (2007-b) presented a deterministic

inventory model for deteriorating items with two warehouses. The

deterioration rate in both the warehouses was taken as different.

Shortages in OW were allowed and the backlogging demand rate was

dependent on the duration of stockout. Authors have used numerical

example to illustrate the model and conclude the paper with suggestions

for possible future research.

An inventory lot-size model for deteriorating items with partial

backlogging was formulated by Chern et al. (2008). Authors have taken

time value of money in to consideration. The demand was assumed to

fluctuating function of time and the backlogging rate of unsatisfied

demand was a decreasing function of the waiting time. The effects of

inflation and time value of money were also considered in the model.

Thangam and Uthayakumar (2008) presented a two-level supply chain

model with partial backordering and approximated Poisson demand.


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Skouri et al. (2009) developed an inventory model with general ramp

type demand rate, time dependent (Weibull) deterioration rate and partial

backlogging of unsatisfied demand.


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[3] Aggarwal S.P. and Jaggi C.K. (1995). Ordering policies of

deteriorating items under permissible delay in payments. Journal of

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[4] Aggarwal, S.P. and Jain V. (2001). Optimal inventory management for

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two-level supply chain with production interruptions to restore process

quantity. Computers and Industrial Engineering; 54 (1), 95-109.

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