Page 1 of 10
European Journal of Business &
Social Sciences
Available at https://ejbss.org/
ISSN: 2235-767X
Volume 07 Issue 03
March 2019
Available online: https://ejbss.org/ P a g e | 1156
Inventory Models for Deteriorating Items with Shortages
SAMRITI
M.Sc (Mathematics), CSIR – NET
ABSTRACT
Inventory modeling is an important part of Operation Research, which may be used in variety of
problems. To make it applicable in real life situations researchers are engaged in modifying the
existing models on different parameters under various circumstances. This paper reviewed the
inventory models with shortages of different types for deteriorating items with different demand
patterns and proposed future need of research in this direction.
Keywords: Inventory Models, Deterioration, Shortage, Operation Research.
INTRODUCTION
One of the most developed fields of Operations Research is inventory modeling. Inventory has
been defined as idle resources that posses economic value by Monks (1987). Usually, it is an
important component of the investment portfolio of any production system. Keeping an
inventory for future sales or use is very common in business. Retail firms, wholesalers,
manufacturing companies and even blood banks generally have a stock of goods on hand.
Usually the demand rate is decided by the amount of the stock level. The motivational effect on
the people may be caused by the presence of stock at times. Large quantities of goods displayed
in markets according to seasons motivate the customers to buy more. If the stock is insufficient
the customers may prefer some other brands, as shortages will fetch loss to the producers. The
shortage or stock out cost is the penalty incurred for being unable to meet the demand when
occurs. It has two wings internal and external shortage. Internal shortage occurs when an order of
a group or department within the organization is not filled. External shortages can incur
backorder cost, present profit loss and future profit loss. Internal shortages can result in lost
production and delay in a completion date. On the other hand, deterioration is an important
Page 2 of 10
European Journal of Business &
Social Sciences
Available at https://ejbss.org/
ISSN: 2235-767X
Volume 07 Issue 03
March 2019
Available online: https://ejbss.org/ P a g e | 1157
natural phenomenon and the consequent loss due to decay of items may be quite significant.
Mainly when, physical goods are stocked for future use, in some items such as medicines,
foodstuff, dairy items, volatile liquids, the process of deterioration is observed. Hence effect of
deterioration is very important in many inventory systems.
Research in this direction began with the work of Whitin (1957) who considered fashion goods
deteriorating at the end of a prescribed storage period. Ghar and Schrader (1963) developed an
inventory model with a constant rate of deterioration. An order level inventory model for items
deteriorating at a constant rate was discussed by Shah and Jaiswal (1977). In all these models,
the demand rate and the deterioration rate were constant and shortages were not allowed.
An interesting subset of inventory modeling is the mathematical modeling of deteriorating items
with shortages. The literature related to deteriorating items with shortages is scattered and no
comprehensive up to date discussion of these models is available. This paper presents a complete
survey of the published literature in mathematical modeling of deteriorating items with - 52 -
shortages and proposed future research directions needed in this field.
Numerous researches have been carried out incorporating time demand patterns into inventory
models. The time dependent demand patterns, used in existing models are, mainly,
linearly time dependent
exponentially time dependent
The time dependent demand patterns reported above are linear, that is, the demand increases
continuously with time or decreases continuously along with the time. Dave and Patel (1981),
Dutta and Pal (1992) considered time proportional demand. Hariga and Benkherouf (1994)
considered exponential time varying demand for deteriorating items.
Hill(1995) proposed a time dependent demand pattern by considering it as the combination of
linearly time dependent and exponentially time dependent of demand in two successive time
periods over the entire time horizon and termed as “ramp- type” time dependent demand pattern.
Then, inventory models with ramp type demand rate also studied by Mandal and Pal (1998), Wu
et al(1999), Wu and Ouyang(2000), Wu(2001), are worth mentioning. In these papers, the
determination of the optimal replenishment policy requires the determination of the time point,
when the inventory level falls to zero. So the following two cases should be examined:
Page 3 of 10
European Journal of Business &
Social Sciences
Available at https://ejbss.org/
ISSN: 2235-767X
Volume 07 Issue 03
March 2019
Available online: https://ejbss.org/ P a g e | 1158
This time point occurs before the point, where the demand is stabilized,
This time point occurs after the point, where the demand is stabilized.
Almost all of the researchers examine only the first case. Deng et al (2006) reconsidered the
inventory model of Mandal and Pal (1998) and the models of Wu and Ouyang(2001) and studied
it exploring these two cases. Skouri et al (2009) extended the work of Deng et al(2007) by
introducing a general ramp type demand and considering Weibull distribution deterioration rate.
The works done by Roy(2008), Sabahno(2009), Mirzazadeh(2010), Gayen and Pal(2009) are
some of the models for deteriorating items based on different realistic situations.
Deterioration is defined as decay, spoilage, loss of utility of the product as defined by Shah and
Shukla(2009) . Product such as vegetables, fish, medicine, blood, radio-active chemicals have
finite self life and start to deteriorate once they are produced.
Obsolescence refers to inventories that become obsolete at a certain time, for instance because of
rapid changes in technology, or the introduction of a new product by a competitor. If the rate of
obsolescence, deterioration or amelioration is not sufficiently low, its impact on modeling of
such an inventory system cannot be ignored. Moon et al(2005) considered ameliorating/
deteriorating items on an inventory model with time varying demand pattern. The model of
Balkhi(2004) and Lo et al.(2007) may also be mentioned.
In the literature survey by Raafat(1991), Shah and Shah(2000), Goyal and Giri(2001), the details
can be found for deteriorating items.
Dave (1985) modified the model of Dave and Pandya(1985) with special sales incorporating
shortages. Later Dutta and Pal (1991) designed the similar problem with shortages where the
deterioration rate of the items is constant and deterioration is assumed to start only after a fixed
period of time from the instant of their arrival in stock. Beckherouf (1995) developed an
algorithm for solving the problem of determining an optimal replenishment policy for inventories
of perishable goods with constant rate of deterioration and with decreasing demand rate over a
certain time period.
Manna and Chaudhuri (2003) noted that ramp type demand pattern is generally followed by new
brand of consumer goods coming in the market. But for fashionable products as well as for
seasonable products, the steady demand will never be continued indefinitely. Rather it would be
