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A STUDY ON
INVENTORY
MANAGEMENT
SYSTEM

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ABSTRACT
Centuries ago, inventories were viewed as measures of wealth and power of a
country or an individual. In recent past it was also viewed as a measure of business
failure. Businessmen therefore, have started to put larger emphasis on the liquidation for
fast turnover. Now a day’s due to fast advancement of technologies, inventories are
viewed as a large potential risk rather than a measure of wealth. Thus it is required to use
scientific techniques in management of inventories known as ‘Inventory Control’. It is the
technique of maintaining stock – items at desired level.
Thus inventory management system is the means by which material of correct
quality and quantity is made available as and when required with regard to economy in
the shortage costs, set up costs, manufacturing costs, purchase prices and market capital.
The basic idea of the present thesis work is to study certain aspects of inventory
management system. Particularly, the objective of the thesis is to discuss the inventory
models of deteriorating items. Further consideration of exponential and weibull
distribution under time varying and price dependent demand of actual degree of
deterioration under the influence of inflation is one of the significant works in the present
study. Also amelioration being a natural phenomenon in life stock inventory, so inventory
model of ameliorating items is incorporated in the present study. To avoid inadequacy in
demand, inventory models under fuzzy environment has also studied at the end.

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The present thesis is organized in eight chapters. The first chapter presents the
introduction and literature survey on the related work where the background literature has
been described briefly in the light of which the investigation of the author has been
presented in the subsequent chapters. Chapter 2 to chapter 7 deals with the contribution of
the author in the field of Inventory management system. The last chapter is a brief
conclusion and outlines of some proposed models for further investigation. The
bibliography section and the research publications of the authors are presented at the end.
The chapter wise summary of the present work is given below.
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In chapter – 2, we have established a deteriorating inventory model of perishable
items under time varying demand condition. The objective of the model is to discuss an
inventory model for deteriorating items and obtain the total optimal cost of the model
with time varying type of demand rate having time proportional deterioration without
shortage. The time varying holding cost is assumed to be a linear function of time. The
result is illustrated with numerical example and the sensitivity analysis of various
parameters is carried out.
In chapter – 3, inventory model of deteriorating items for nonlinear holding cost
with time dependent demand has been discussed. The objective of the model is to
investigate the inventory system for perishable items where time proportional
deterioration rate is considered. The Economic order quantity is determined for
minimizing the average total cost per unit time. Time dependent demand rate is used with
finite time horizon. Nonlinear holding cost with shortage is considered. The result is
illustrated with numerical example.
In chapter – 4, an inventory model for weibull ameliorating, deteriorating items under
the influence of inflation has been analyzed. The objective of this model is to discuss the
development of an inventory model for ameliorating items. Generally fast growing animals
like duck, pigs, broiler etc. in poultry farm, high-bred fishes are these types of items. This
paper investigates an instantaneous replenishment model for the above type of items under
cost minimization in the influence of inflation and time value of money. A time varying type
of demand rate with infinite time horizon, constant deterioration and without shortage is
considered. The result is illustrated with numerical example.
In chapter – 5, we have studied a deteriorating inventory model with shortages
under price dependent demand and inflation. The objective of this model is to discuss the
inventory system for deteriorating items with price dependent demand pattern where
weibull rate of deterioration is considered. Here shortages are assumed and are partially

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backlogged with inflation. The Economic order quantity is determined for maximizing the
profit per unit time. The result is illustrated with numerical example.
In chapter – 6, we have discussed optimal inventory model with single item under
various demand conditions. The objective of this model is to discuss the inventory model
for time varying demand and constant demand; and time dependent holding cost and
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constant holding cost for case 1 and case 2 respectively. Mathematical model has been
developed for determining the optimal order quantity, the optimal cycle time and optimal
total inventory cost for both cases. Numerical examples are given for both the cases to
validate the proposed model. Sensitivity analysis is carried out to analyze the effect of
changes in the optimal solution with respect to change in various parameters.
In chapter–7, an Inventory control model of deteriorating items in Fuzzy
environment has been discussed. The objective of this model is to investigate the
inventory system for perishable items where fixed deterioration rate and stock dependent
demand rate is used with finite time horizon. The analytical development is provided to
obtain the fuzzy optimal solution to minimize the total cost per time unit and for
defuzzification, graded unit-preference integration method is used.
In chapter – 8, the concluding remarks of the investigations have been presented
and then future scope of searching out new results in this area has been addressed.
Keywords: Inventory system, Fuzzy Inventory system, deterioration, Amelioration,
Weibull distribution, optimal control, time varying demand, Shortage, Inflation, holding
cost.

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Chapter 1

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INTRODUCTION AND
LITERATURE SURVEY

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Chapter-1
INTRODUCTION AND LITERATURE SURVEY
1.1 Introduction
The study of science emerges out of the practical need as well as of a convergence
interest in some class of problems of human beings. Operations Research (OR) is of no
exception. It is the gift of the Armed Forces to the modern society. This is a humble effort of
an army officer Prof. P.L.S. Blackett, who has spent years in teaching. In 1940, during Second
World War, the scientists from different disciplines were called by the military management of
United Kingdom to assist in solving various problems associated with military context. The
point was to discover the most suitable assignment utilizing the rare assets to the different
military operations. The result incorporated the best possible utilization of recently concocted
radar, designation of British Air Force artworks to missions and further determination of best
examples for seeking submarines. The empowering aftereffects of such endeavors lead to the
advancement of more groups in British Armed administrations and the outline of such
experimental gatherings was a huge commitment to western Allies.
Soon after the war many of the scientists who were engaged in the military groups
diversified their attention for applying similar approaches to civilian-problems like business,

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industry, agriculture, aeronautical engineering, management science, economics, genetic
engineering etc. to investigate complicated real world systems with an aim of improving ideal
solution to optimum level. As an application of operations research the production system was
evolved in real world problem. The most important thing is the stocking of items from a
smaller retailer to a larger production firm or shop to meet up the customer’s demand as far as
possible. Again, stocking of items depends on different factors such as deterioration,
amelioration, demand, replenishment of order etc. The handling of such type of problems is
known as inventory management.

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Chapter- 1 Introduction and Literature Survey
Inventory is the physical stock of products that a business continues with keeping in
mind the end goal to make advance the smooth and adequate running of its undertakings. It
might be held before the generation cycle, as crude material stock; at a middle of the road stage
in the creation cycle as in procedure stock; or toward the end of creation cycle, as completed
products stock. The hypothesis on stock was created in 1920s. At in the first place, it had
exceptionally straightforward models that utilized just a couple of parameters to catch the key
elements. Later these models were decorated to incorporate more points of interest by
including more parameters yet overlooked variability and vulnerability. Step by step
probabilistic models were produced in 1950s to catch the effects of eccentric request and lead
times.
All of these models suffered from one limitation they dealt with only one product at a
time. The genuine stock confronting numerous individuals was to oversee on huge assortment
of things, with enough interrelations among them to represent an administration issue.
Consequently, a separate data processing oriented subject called inventory control or inventory
management evolved. Here the major concern was for organizing and maintaining records.
Gradually the optimizing performance has been enhanced.
1.2 Operations Researchin Production Systems
Organizations and associations much of the time face testing operational issues whose
effective arrangement requires certain aptitude in connected measurements, advancement,
stochastic displaying, or a blend of these ranges. To outline, an organization might need to plan
an inspecting arrangement so as to meet particular quality control goals. In an assembling
situation, operations that seek the same assets must be planned in a way that due dates are not

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abused. The director of a general store must decide what number of checkout lines to keep
open at different times amid the day and night so customers are not pointlessly deferred or as a
last illustration, the extent of the zones held for putting away work in procedure must be
resolved so that a smooth stream of work results, even at the busiest (top) generation times.
The range of operations research that focuses on certifiable operational issues is called
generation frameworks. Creation frameworks issues might emerge in settings that incorporate,
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however are not restricted to, assembling, information transfers, human services conveyance,
office area and format, and staffing. In an assembling situation, operations that go after the
same assets must be planned in a way those due dates are not abused. The chief of a market
must decide what number of checkout lines to keep open at different times amid the day and
night with the goal that customers are not pointlessly postponed or as a last sample, the extent
of the regions held for putting away work in procedure must be resolved so that a smooth
stream of work results, even at the busiest (top) creation times. The range of operations
research that focuses on true operational issues is called creation frameworks. Creation
frameworks issues might emerge in settings that incorporate, yet are not constrained to,
assembling, information transfers, medicinal services conveyance, office area and format, and
staffing. At the present time, the field of OR is greatly powerful and always developing. To
give some examples of the contemporary (primary)research ventures, momentum work in OR
tries to create programming for material stream examination and outline of adaptable
assembling offices utilizing design acknowledgment and chart hypothesis calculations.
Further, approaches for the outline of re-configurable assembling frameworks and progressive
automation of discrete assembling frameworks are being worked on. Extra OR activities
concentrate on the mechanical organization of computer based strategies for sequential
construction system adjusting, scope quantification, pull booking, and setup diminishment,
principally through the coordination of the methods of insight of the Theory of Constraints and
Lean Manufacturing.
Before the industrial revolution, most business and industry consisted of small
enterprises, each directed by a single boss who did the purchasing, planned and supervised
production, sold the product, hired and fired personal etc. Such mechanization of production
led to rapid growth of industrial enterprises that it became impossible for one man to perform
all these managerial function took place. Inventory control is one such consequence.

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The management of every economic sector gained interest after World War II to study
inventory management system due to much risk factor and uncertainty. Consequently, the
study of inventory literature has been enlarged day by day and various inventory models have
been published in different journals. In 1951, Arrow et. al. [9] established a result on
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mathematical analysis of inventory models on “Optimal Inventory Policy”. Whitin [142]
published an EOQ model on stochastic version in 1954. Further, Whitin [143] analyzed a
review of the inventory system on “The Theory of Inventory Management”. The collection and
processing of large quantity of data are needed in every inventory problem. After 1950, several
researchers like Star, M.and Miller, D.W, [125], Hadley and Whitin [55], Fabrycky and Banks
[13], Silver et. al. [123] etc. have published different articles in this direction.
1.3 Inventory ManagementSystems
Inventory is simply stock of physical asserts having some economic value. It is an idle
resource as long as it is not used. It may be used as those goods that are stored and used for day
to day working of an organization.
Centuries ago, inventories were viewed as measures of wealth and power of a country
or an individual. In recent past it was also viewed as a measure of business failure.
Businessmen therefore, have started to put larger emphasis on the liquidation for fast turnover.
Now a day’s due to fast advancement of technologies, inventories are viewed as a large
potential risk rather than a measure of wealth. In this manner it is required to utilize
experimental strategies in administration of inventories known as 'Inventory Control'. It is the
system of looking after stock – things at craved level.

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Thus inventory management system is the means by which material of correct quality
and quantity is made available as and when required with regard to economy in the shortage
costs, set up costs, manufacturing costs, purchase prices and market capital.
In general Inventory management takes the responsibility of the availability of right
quantity at right time at minimum cost.
In addition to the above, the control and maintenance of any inventory is a common
problem to all organizations and economic sectors. For instance, inventories are maintained in
Industry, Business, Military, and Agriculture etc. Some of the benefits of keeping of the
inventories are as follows:
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i. It helps in proper and efficient running of organization.
ii. It gives the immediate service to the customer.
iii. It minimizes the chance of loss due to change in price of items.
iv. It helps in minimizing the loss because of the deterioration, obsolescence and
damages.
v. It uses as a buffer stock when raw materials are delay to receive or due to delay in
supply to the market.
vi. It minimizes the product cost due to an advantage of batching.
vii. It keeps up the economy by enduring a portion of the deviations when the interest of
a thing varies.
viii. It directs how to use of available money in a most efficient way and avoid an
unnecessary expenditure on high inventories.
In broad sense term stock might be partitioned into two sorts
Direct inventories
These inventories include those things which assume an imperative part in the
production and turned into an essential piece of completed merchandise. For example, crude
material stock, work–in-process stock and completed products stock.

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Indirect inventories
Such inventories involve various items which are essential for manufacturing the
component of finished production such as oil, grease, lubricants etc.
Inventories are used to meet the seasonal fluctuation in demand economically.
Production of specialized items like crackers well before Diwali, fans and coolers before on set
of summers, fruits and vegetables are harvested a few months in each year to meet demand for
the entire year. Hence inventory control is the processes to maintain the stock within desired
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limits. It also standardizes and centralized information on stock levels. It has timely records of
inventories of all items.
In Inventory management system, Inventory models contain the different decision
variables such as quantity ordering and stock. Then there arises three basic questions such as
i. How much quantities of an item need to be ordered?
ii. What is the right time that the ordering to be placed?
iii. How do you maintain the completion stage of stocked items?
But in real business, it is quite difficult to determine an appropriate procuring policy.
An inventory management problem is a decision making problem which answers the above
questions. On the other hand the aim of an Inventory model is to obtain an optimal order
request which minimizes the total inventory expense. i.e., an inventory problem deals with
decisions that maximizes the total profit achieved while meeting the customer’s demand.
1.4 MathematicalTreatment
The inventory system involves the following steps:
i. Development of a mathematical model of the system
ii. Obtaining a solution of the system

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iii. Determination of the properties of the system
iv. Derive relationships between different related parameters of inventory
v. Study of optimality
The above steps have been taken into account during the analysis of the problems
presented in the thesis. It is a starting work to build up a Mathematical model of the stock. This
model depends on different suppositions and approximations. It is a difficult device to operate
accurately. In practice, it is not easy to construct a truly realistic model with completely
accuracy and very often it may not be amenable to a mathematical treatment. Hence it is
necessary to take some approximation and simplification while constructing a feasible model.
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In addition to, the inventory problem contains a set of parameters which optimize the
aggregate expense of the framework. Sometimes it is difficult to obtain the values of these
parameters due to general nature of the problem undertaken or due to some other reasons. So
in these cases, simple differential calculus, numerical techniques, scientific calculator, search
techniques, different algorithms from various software’s may be used to optimize the objective
function.
1.5 Economic Parameters
In Inventory management system there are different economical parameters. The
various economical parameters that are associated with the present work are given below.
1.5.1 Ordering or Setup or Replenishment Cost
The expense connected with placing so as to acquire the materials of a request or
getting or assembling or setting up types of gear before beginning the generation is typically
called requesting or setup or renewal cost. It is signified by c (z), where the sum requested is z
and the capacity c (z) might be straight or nonlinear. The expenses connected with order,
follow up getting the products, quality control and so on are autonomous of amount requested.
In some cases, it may depend on the quantity of goods procured with price discounts or
quantity discounts or transportation cost etc.
1.5.2. Holding Cost

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Holding expense is the upkeep expense of the stock. It depends upon both the size of
the inventory and time for which item is placed in stock. It is denoted by h per unit time. This
cost includes the cost of storage, salaries of staff engaged in (store, security, etc.) interest on
capital blocked on purchase of inventory, insurance, deterioration, obsolescence, amelioration,
etc. The holding expense might likewise incorporate the expense of capacity, protection, and
different elements that are relative to the sum put away in stock.
.
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1.5.3. Shortage or Stock-Out Cost
The shortage cost is the penalty incurred for being unable to fulfill a demand where it
happens. For this, the enterprise has to pay not only the penalty in terms of money but also lose
the good will of the customers. It is denoted by p per unit time. At the point when a client
looks for the item and finds the stock exhaust, the interest can either go unfulfilled or be
fulfilled later when the item gets to be accessible. The previous case is known as a lost deal,
and the last is known as a delay purchase.
1.5.4 Purchase Cost
The unit cost of an item obtained either from an external source or from the unit
replenishment cost of internal production is called the purchase cost. It is not necessarily
constant. It is also denoted by c (z), where the amount purchased is z. The unit purchasing price
depends on the quantity procured happening in many practical situations. For examples, in a
competitive market, there is a price discounting on purchasing of big lot size so also there is a
reduction of replenishment cost per unit when production happens in large scale.
1.5.5. Selling Price
The unit price of an item sold to a customer on demand is called the selling price. It
results revenue from selling the commodity. It is not necessarily constant but sometimes
depends on the decision maker.

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1.5.6. Salvage Value
During the storage period, some items spoiled partially and some items lose their utility
after their selling period. It is observed that some revenue is collected by the management
selling these items at a less cost than the purchasing cost to a certain group of customers. This
revenue is called salvage value.
Apart from the above cost components, some other important components are also used
while designing an inventory model.
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1.5.7. Demand
The main purpose of holding of inventory is to meet the demands of the market.
Demands are not within the control of the organization. A demand is said to be deterministic if
it is fixed and known. But this is not very common situation because the demand may be
changed due to large number of factors beyond the control of the organization such as
consumer’s tastes, technological developments, government policy and other factors related
with the business environment. The demand which is assumed to be variable is called
stochastic demand. The probabilistic interest happens when the interest over a specific period
is not typically known but rather it can be clarified by a likelihood conveyance. A probabilistic
interest might be either settled or not altered after some time.
1.5.8. Lead Time
It is the time crevice between the submitting of a request and accepting of the stock is
called lead-time. It may not be essentially a consistent.
1.5.9. Time Horizon
It is the time duration during which an optimal policy is to be determined to maximize
the benefit or minimize the aggregate stock expense. It may be finite or infinite depending
upon the nature of the inventory system.
1.5.10. Replenishment

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It is the rate of quantities added to the stock. The replenishment quantity may be
constant or variable depending upon the nature of the inventory system. Instantaneous
replenishment happens when the stock is added up from external resource point in no time.
1.5.11. Buffer or Safety Stock
As demand is uncertain and uncontrolled, some extra stock is advised to be maintained
to overcome any shortages occur during the lead time. This extra stock is called as Buffer
stock.
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1.6 Inventory ControlMethods
In everyday business undertakings the normal stock circumstance confronted by
producers, wholesalers and retailers is such, the span of stock levels are lessened incredibly
after some time and afterward renewed by bringing of a cluster of new units. A typical model
for overcoming such circumstance is the Economic Ordering Quantity model or economic lot-
size model, in short EOQ model. The fundamental EOQ model has been founded on the
suppositions that request rate is known and is steady per unit time, deficiencies are not
considered, lead-time is zero or constant, and the recharging of request amount arrives just
once when the stock level drops to zero. In any case, all things considered, applications these
suspicions are not for the most part confronted. The economic lot-size model does not mull
over of the interest example of the deciding item before deciding the stock levels of parts and
materials. This is one of the weaknesses of this model. The other mainstream model which has
been utilized by some Japanese organizations is the in the nick of time model, in short JIT
model. This model thinks about of different uncertainties, for example, request vacillations,
machine disappointment, and stochastic set-up times and so on. In a matter of seconds, it has
been watched that numerous organizations in various parts of the world are redoing the path in
which they deal with their inventories. The uses of Operations Research have been an intense
device around there for picking up a focused edge.
1.6.1 Economic Order Quantity (EOQ)
EOQ model is an extremely normal model for getting the amount to be requested and
when to be requested. By request amount it is implied that the amount delivered or obtained
per generation cycle. With, the expansion in size of request, the requesting costs, obtaining

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costs and so on will diminishes where as the stock conveying costs, stockpiling costs and so
forth increases. Along these lines during the time spent generation there are two inverse costs,
one empowers the expansion in size of request and different demoralizes. Along these lines
EOQ is that size of request which minimizes the aggregate yearly expenses of requesting and
conveying expenses of stock. The following figure 1.1 indicates that the minimum cost occurs
at the point where the ordering cost and carrying costs of inventory are equal. This
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mathematical approach of EOQ model makes a balance in between ordering and holding costs
and giving the minimum order level.
Let
S = purchasing cost
D = Annual usage
I = Holding cost as percentage of stock value
C = Unit purchasing cost
Q = Economic order quantity (EOQ)
Consequently, purchasing cost per unit =
S
,
Q
Yearly obtaining cost =
S D
,
Q
Average amount held every year =
Q
, Annual carrying cost =
I C Q
,
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Total inventory costs = annual carrying costs + annual purchasing costs.
Fig. 1.1: Economic Order Quantity

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The Economic Order Quantity model is a principal model for stock control. It is
extremely easy to execute, nonetheless, it depends on unrealistic suppositions.
1.6.2 Just-in-Time (JIT) System
Now a day, due to the advancement of new technologies it has been a potential risk to
keep large number of inventories in stock. Thus it is required to get the inventory in the nick of
time as and when required which leads to Just-in-time system. It is a well developed technique
for managing inventory system. This has been regarded as one of the most familiar systems,
adopted so far in new technologies. This framework assumes a crucial part to reduce stock
level to a base thus supplying the things without a moment to spare as and when required. This
strategy is here and there misconstrued as being contradictory with utilizing an EOQ model as
the later gives a huge request amount when the setup expense is vast, they really are reciprocal.
When all is said in done a JIT stock framework stresses on discovering intends to decrease the
set up costs so that the ideal request amount will be little. Accordingly this system concentrates
on maintaining a strategic distance from waste at whatever point it may happen in the creation
process. Waste is taken in its most broad sense and incorporates time and assets and
additionally materials. Most research endeavors have concentrated on depictions of JIT
rationality. There were theoretical and test concentrates, scientifically models and recreations
by and large. A few researchers exhibited a complete audit of JIT frameworks. These
exploration endeavors show that the JIT procedure is an exceedingly viable strategy for
controlling the stream of stock on shop floors as it lessens the stock level to an absolute
minimum and consequently decreases creation costs.
1.6.3 Material Requirement Planning (MRP)

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MRP is an arrangement of systems for changing over estimate interest for a
manufactured item into a schedule for getting segments, sub-assemblies and crude materials.
From a logistics perspective, the motivation behind MRP is to avoid, as much as possible in
carrying these items in inventory. The MRP approach underlines on a reasonable expert
generation plan (MPS) to facilitate the creation stages as far as volume of creation that keeps
the formation of uncommon instrastage slack. MRP additionally weights on figuring
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subordinate interest. It keeps the production of unnecessary interstage slack because of uneven
parcel estimating of coordinated arrangement of parts. Along these lines just wellbeing stocks,
which are permitted in such a methodology, are stocks at the MPS level. MRP chooses the
accompanying two elements: timing (when to request) and quantity (how much to order). With
respect to the timing decision, requests are made as late as could be expected under the
circumstances, but never order before one needs to, never permit to stock out. In this way MRP
adequately decreases the expense of generation by minimizing the stock levels.
1.7 The Deterministic Model
In This model, the demand is static. i.e., the demand is assumed to be fixed and
completely predetermined. Such models are referred to as economic lot size model. Thus the
inventory model without uncertainty has been addressed under this model and the behavior is
described in Fig. 1.2. Here it is assumed that at a fixed rate the items are withdrawn from the
inventory and lots are of a fixed size Q, where lead time is zero or constant. Because of the
simplicity of the model, the optimal solution to a deterministic model can be obtained under
several operating assumptions.

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Fig. 1.2: The inventory pattern without uncertainty
1.7.1 Economic Lot Size Model with no Shortages
The objective for study of this type of model is to find an optimal economic lot size that
minimizes the total inventory cost. In this inventory model, it is assumed that demand is
continuous and consumed at constant rate, orders are placed at periodical intervals of equal
size, replenishment rate is instantaneous, costs are invariant and shortages are not allowed. The
purchase price per unit is same regardless of the request size. In the accompanying Fig. 1.3, we
have portrayed the suppositions considered in the model by demonstrating a plot of stock level
as a function of time. Here the stock level lies in the range from 0 to Q. As shortages are not
allowed so the inventory level never goes below 0. For replenishment of the inventory, orders
are placed periodically in a continuous process. The request is accepted to arrive immediately,
that causes the stock level to reach from 0 to the sum Q. In the middle of the requests stock
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level declines at a consistent rate ‘a’. The required time in between the requests is known as the
cycle duration.
Fig. 1.3: Lot size model with no shortages
Let
D = Yearly demand in units
C0 = Ordering cost per request

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Cc = Carrying cost / unit per year
P = Purchase price / unit
Q = Order size
Then we have
The number of orders per year = D/Q
Average inventory = Q/2
Cost ordering per year = (D/Q) C0
Purchase cost per year = DP
Total inventory cost per year = (D/Q) C0 + (Q/2) C0 + DP
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By simple Differential Calculus we obtain
Optimal lot size
i. Q=
2c0 D
C 0
ii. Optimum no of orders placed per year n
DC c
2C0
iii. Optimum length of time between orders T
2C 0
DC c
iv. Minimum total inventory cost TC 2C0Cc D
1.7.2 Economic Lot Size Model with Shortages
The objective for study of this type of model is to find an optimal economic lot size that
minimizes the total inventory cost. In this inventory model, it is assumed that demand is
continuous and consumed at constant rate, orders are placed at periodical intervals of equal

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size, replenishment rate is instantaneous, and costs are invariant. The headache of the inventory
manager in this model is the occurrence of the shortages of inventory i.e., stock out of demand.
Here the present demand cannot fulfill the requirement currently as the inventory is depleted.
In this model the unhappiness of the customers occurs further it is required to keep extra record
for backorders i.e. to keep the record of arranging and filling the demand later after the
replenishment of the inventory. Additionally here the purchase price per unit is same regardless
of the request size. In the accompanying Fig. 1.4, we have depicted the suppositions considered
in the model by demonstrating a plot of stock level as a function of time.
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Fig. 1.4: Lot-size model with deficiencies permitted.
The aggregate expense per unit time is:
Cost/time = Product cost + Setup cost + Holding cost + Backorder cost

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T
aK
ac
hS 2 p(Q S)2
Q 2Q 2Q
The component multiplying h in this expression is average on-hand inventory level.
This is the positive part of the stock bend appeared in Fig. 1.4. Since all cycles are the same,
the normal close by stock registered for the principal cycle is the same with respect to record-
breaking. We see the primary cycle in Fig. 1.5.
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Fig. 1.5: The first cycle of EOQ model with backorders.
We have some more notations
C S = Shortage cost per unit per period
Q1= Most extreme stock
Q2= maximum stork – stock out
T 1 = Period of shortage
T 2 = Period of shortage
T = T1 +T 2 = Cycle time
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By simple calculation we obtain
i. Q=
2C0 Cs Cc
CS CC
ii. Q1
2C0 CS D
CC CS CC
iii. Q 2 Q Q 1
iv. T=
Q
D
v. T1
Q1
D
vi. T2
Q
2
D
D
vii. n= =no of optimal orders per period.

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Q
1.8 Stochastic Inventory Models
In our day to day life requirements of future demand is uncertain. In the business
world, it is a matter of challenges to meet the uncertain demand of the customers which leads
to study stochastic inventory management system. Here the stock is observed by continuous
monitor process and another request is set when the stock level drops to reorder point. This
inventory management system is also referred as modern inventory management system. In
this framework computers are widely utilized for constant survey of stock framework and it
highly prevalent the items that are adequately imperative to warrant a formal stock approach.
This model is similar to the EOQ model with arranged deficiencies with the significant
deviation that the interest for pulling back units from stock to offer them when lead time is
uncertain. However, the probability distribution is estimated or known.
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1.8.1 Probability Distribution for Demand
To overcome the uncertainty in demand probability distribution has been used. In
deterministic model demand is known yet here we accept that request is obscure, however the
likelihood dispersion (Probability Distribution) of demand is known.
Random Variable (x):
Random variable is a real valued function whose outcome depends upon the chance.
The random variable for demand under a given period of time is considered and is denoted by
x. Attention must be given to perceive the period for which the random variable is
characterized as it contrasts among the models considered.
Discrete Demand & Probability Mass Function:
When the interest is not immediate i.e., limited in nature it alludes to be a discrete
random variable and f(x) gives the probability that the demand equivalents x.
Discrete Demand & Probability Function (F (b)):
At the point when the demand is of
then the probability that demand ≤ b is F(b),
discrete sort with probability mass capacity f(x),
cumulatively given as:

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b
F (b) f (x)
x 0
Continuous Demand & Probability Density Function:
At the point when demand is happened continuously with f(x) as its probability density
function (pdf), then probability that the demand lies in between ‘a’ and ‘b’ is given by:
b
P(a X b) f (x)dx.
a
We expect that demand is nonnegative; subsequently f(x) is zero for negative values.
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Continuous Demand & Probability Distribution Function:
When the demand is of continuous type, then the cumulative probability distribution
(CDF) that demand ≤ b is given by:
b
F(b) f (x)dx
0
Standard Normal Distribution Function:
At the point when the uncertain demand is considered with given mean and standard
deviation then normal distribution is used. The density function and cumulative distribution
function for the standard normal distribution is denoted by (x) and (x) respectively.
1.8.2 Selecting a Distribution
Dealing with uncertain demand it is important to choose a distribution function for the
demand. It is assumed that the individual demand events occur independently. At the point
when the normal in a period of time interval is small, the Poisson distribution is used and the
normal distribution is used when the expected demand is substantial. Let ‘a’ be the average
demand rate, then for a given period of time t, the expected demand is at. The Poisson
distribution function is given by:

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P(x) (at)x e-(at)
.
x!
For at is large, the Poisson distribution can be approximated with a typical normal distribution
having mean and standard deviation
at, and at.
Further values of F (b) can be evaluated by using standard normal distribution tables.
The Weibull distribution has been utilized to describe fatigue failure, vacuum tube
failure, ball – bearing failure and so forth. At present it is one of the most widely used
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parametric family of failure distribution. It is because of the choice of the shape parameter
, the probability density function for Weibull distribution is
f (t) t 1 exp( t )
The prompt rate of enhancement of the stock in time t is A(t) and it complies with the Weibull
distribution, i.e.,
A(t)
f (t)
t 1 .
1 F(t)
Also it is known as the hazard rate of the distribution and is given by
h(t) t 1
.
Where ‘ ’ is the shape parameter and ‘ ’ is the scale parameter.
Further different distribution can likewise be expected for various requests. Most regular
suppositions are the normal distribution with known mean and standard deviation, the
exponential distribution, and the uniform distribution. The last two are helpful as a result of
their diagnostic effortlessness.

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1.8.3 Finding the Expected Shortage and the Expected Excess
We are for the most part agonized over the connection of interest amid some time
period regarding the stock level toward the beginning of the time period. On the off chance
that the interest is not exactly the underlying stock level, there is stock staying toward the end
of the interim. This is the condition of plenitude. On the off chance that the interest is more
noteworthy than the underlying stock level, we have the condition of insufficiency.
1.9 Fuzzy Set Theory
In real life situations, mostly exact data are inadequate for a mathematical model.
Ordinarily human choices including inclinations are typically ambiguous and which can't be
evaluated as far as inclinations with definite numerical information. The capacity of fuzzy sets
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to express slow moves from membership to nonmember boat and the other way around has an
expansive utility. It gives us an important estimation of instabilities, as well as with a
significant and intense representation of ambiguous ideas (vague concepts) communicated in
regular way. In fact, expression of the essential idea of fuzzy set, a concept that is both basic
and pleasing and which forms a generalization of classical or crisp set. A crisp set ascents or
falls unexpectedly, having its components thoroughly disjoint with other individual from the
universe. It is defined in such a way to have a binary classification of the individuals in some
given universal set decomposed into two groups: members (those belong to the set) and non-
members (those do not belong to the set. While watching a physical action, we attempt to
model it with various types of recognition. While displaying it, the model might be a straight
forward articulation, a figure; a piece chart and so forth and the parameters of the model can
never be freshly (crisply) characterized.
The aggregate issue space from the littlest to the biggest permissible estimation of
the variable and thought is known as the universe of discourse. Truly likelihood hypothesis has
been the essential test for speaking to uncertainty in mathematical models. In view of this all
instability was expected to take after the attributes of arbitrary random uncertainty. An
irregular procedure was one where the result of a specific acknowledgment of the procedure is
entirely a matter of chance; an expectation of a grouping of occasions is impractical. Fuzzy set
theory is an astounding device for displaying the sort of instability connected with vagueness
imprecision and absence of data in regards to a specific issue in hand.
A fuzzy set can be characterized mathematically by relegating to every conceivable
individual in the universal set a value representing its grade of membership. This evaluation
relates to the extent to which that individual is comparative or perfect with the idea spoke to by
the fuzzy set. Therefore, people might have a place in the fuzzy set to a more noteworthy or

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lesser degree as showed by a bigger or littler participation grade. These membership grades are
represented by real number values extending in the closed interval between zero to one.
Fuzzy Set Theory was presented by L.A. Zadeh [151] as an augmentation of traditional
set (crisp set or classical set). A fuzzy set is a class of items having a continuum of evaluations
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of grades of membership. Such a set is portrayed by membership (characteristics) function that
assigns to each object a grade of membership extending somewhere in between zero and one.
Every crisp set is a fuzzy set but the converse is not true. Like the idea of embedding
the real numbers into complex plane, the mathematical embedment of ordinary set hypothesis
into fuzzy sets is as natural as the thought. Thus the idea of fuzziness is one of gradual
enrichment but not a replacement.
The significance of fuzzy variable is that they encourage gradual transitions between
states and therefore, have a characteristic ability to express and manage perceptions and
estimation instabilities.
1.9.1 Basic terminologies associatedwith fuzzy sets
Fuzzy set:
A fuzzy set in the universal set X is described by its membership function denoted by
i.e.,
and is understood as the “membership grade” of element in fuzzy set , .
α - cut:

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The set of elements belonging to the fuzzy set whose membership grade is not less than
‘α’ is called the α-level set or α-cut i.e.,
Fuzzy Numbers:
A fuzzy subset in the real line having membership
function fuzzy number if it fulfills the accompanyingproperties as below:
i s cal leda
is normal, i.e., an element for which .
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a. is fuzzy convex, i.e.,
b. is upper continuous, and
c. Supp is bounded, where Supp .
Generalized Fuzzy Number:
A generalized fuzzy number is a fuzzy subset of a real line such that the membership
function fulfills the accompanying conditions:
is a continuous mapping from to the closed interval [0, 1].
i. , .
ii. is increasing strictly in [ ].
iii. , .

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iv. is decreasing strictly in .
v. , .
Where and are real numbers.
Triangular Fuzzy Number:
The fuzzyset
wi t h triangularf uzzynumber,i fthemembershipf unctionof def inedon i sgivenby:
is called a
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Fig. 1.6: Triangular Fuzzy Number

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Trapezoidal Fuzzy Number:
The fuzzy set with is defined on is called
a trapezoidal fuzzy number, if the membership function of is given by:

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Fig. 1.7: Trapezoidal Fuzzy Number
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Operations on Fuzzy Number:
There are different methods available for the operation of fuzzy numbers. The most suitable
idea for operation of fuzzy numbers is function principle.
Function Principle:
To manage operations like addition, multiplication, subtraction and division of fuzzy
numbers the function principle was presented. Assume
and are two triangular fuzzy numbers, then
i. The addition of and is denoted by A B and is given by:

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, where and arereal
numbers.
ii. The product of and is denoted by A B and is given by:
iii. where and
If and are all non-zero positive real numbers, then
iv.
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v. Then the subtraction of and is denoted by A B and is given by:
, where and are real
numbers.
vi. where and are all non-zero real numbers,
then
vii. Let , then
Defuzzification:
The procedure of changing fuzzy values to crisp values is known as Defuzzification. This
strategy has been generally concentrated on for a few years and was connected to fuzzy
frameworks. The significant thought behind this technique was that from a given set, a typical
value is obtained according to certain specified characters. In general, this technique gives a
correspondence from the set of all fuzzy sets into the set of all real numbers. Generally, we use
the following methods for defuzzification:

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Graded mean integration representation
method. Signed Distance Method.
Median rule.
Centroid method etc.
1.10 Literature Review
There are different inventory models those played a vital role in the field of productions
system which are assumed to have great importance because of their applicability and
usefulness in the business world. Some literatures of particular models which are relevant to
our work are surveyed and briefly described below.
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A large portion of the stock models in literature can be put away inconclusively to
satisfy the future necessities. However, particular type of items deteriorates or ameliorates or
become outdated after some time and hence unstable. The deterioration or amelioration of
ordinarily utilized products like vegetables, fruits, foodstuffs, meat, fishes, mushrooms, broiler,
duck, pig, medicine, perfume, alcohol, chemical, gasoline, radioactive substances, electronics
component, photographic films, etc. is typically seen amid their ordinary stockpiling period or
at their ranch houses. In this manner, if the deterioration rate is not adequately low, its impact
on displaying of such a stock framework can't be disregarded. Inventory items generally
classified into four main categories as:
i. Obsolescence
ii. Deterioration/amelioration
iii. Deterioration
iv. Neither obsolescence nor deterioration
Due to rapid change of technology alternately entering another item by a contender, the
items those loss their values in course is known as obsolescence, the price of goods is
substantially reduced after the season is over or otherwise the goods are disposed off, for
instance, when replacement model is introduced the style goods, military air craft become
obsolete.
In recent days businessmen are quite aware of the requirement for exactness in the field
of stock control of falling apart things (deteriorating items). Deterioration means damage, rot,
out of date quality, decay, dryness vaporization and so on. Accordingly there emerges a

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reduction of convenience of the original item. The item, for example, green vegetable,
sustenance stuffs, human blood, photographic film and so on having greatest usable lifetime are
known as perishable item and the item like gas, liquor, radioactive substance and etc. having no
time frame of realistic usability are known as rotting items. Then again the time span of
usability of a few items can't be clear and thus they fit in with neither obsolescence nor
deterioration.
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In the present work, we have not entertained the recharging arrangements
(replenishment policies) for stock of things under oldness or obsolescence. Because once the
items become obsolete, they are not required to record. Very few researchers have attached to
study the inventory of obsolescence items. Covert and Phillip. [34] has created stock models
for quick moving thing subjected to sudden death obsolescence. In their past model, diverse
instances of outdated nature hazard have been considered under the instances of both
deficiencies and without deficiencies.
Numerous analysts have created stock models for falling apart things from time to time.
Around there Whitin [142] began his research work considering stylish products which break
down toward the end of the endorsed stockpiling period. Shah and jaiswal [122], set up a
request level-stock model for perishable things where the rate of deterioration is constant. Also
different works in this direction have been carried out by Benkherouf [15], [16], [17], Kim
[81], Giri et.al. [50], Balkhi [12], Chakrabarty et.al. [48], Jalan and Chaudhuri [73]. Such
models are all created for a boundless renewal rate. Further many researchers like Dave [39],
Bahari-kashani [11] Wee [137], Hariga and Benkharoof [61], Yang and Wee [146] have
studied inventory model of items that fall apart at a steady rate.
In our continuous review of inventory models the random life time has been discussed
with exponential decay belonging to the category of models under exponential distribution
lifetimes. Ghare and Scharader [46] have studied an exponentially decaying inventory model.
EOQ models with tow-parameter Weibull distribution deterioration was discussed by Covert et.
al. [34], Philip [103], Giri et. al. [50], Ghosh and Chaudhury [47] and many others. Further
Giri et. al. [48] and other researchers utilized three-parameter Weibull distribution deterioration
in their stock models. Researchers like Giri, Chakrabarty et. al. [48], Sana et. al. [116], [117],
has created stock models in this field.

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In any inventory model demand plays a vital role. Inventory of items exist due to the
demand for it. Therefore, decisions of inventory are to be made because of the present and
future demands. Based on the nature of the demand inventory models can be classified as
follows:
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(i) Deterministic inventory model
(ii) Stochastic inventory model.
If the magnitude of the demand is known and it has the dimension of quantity, then the
demand is called deterministic. Various types of deterministic demands are considered in
inventory literatures. Demand might be consistent, time-indigent, stock-subordinate, value
needy, stock-price-dependent and time-price-dependent etc. The various researchers have
identified and studied the types of demands required to suit different situations of real life.
The constant demand is valid, only when the phase of the product life cycle is matured
and also for finite periods of time. Covert and Philip [34], Misra [90], Dave [39], Hadley and
Whitin [55] etc. established stock models with consistent demand rate. Generally, this type of
demand exists for some particular goods. Raafat [106] added to a stock model for falling apart
things with steady request rate and limited renewal rate. Misra [91] and Shah [122] presented
models considering a limited recharging rate, consistent interest rate and exponential decay in
stock. Wee and Shum [139] concentrated on ideal renewal arrangement for deteriorating stock.
With the advancement of time, researchers created stock models with time-subordinate
demand rates. In this area, the work had been done by various researchers like Silver [123],
Deb and Chaudhuri [42], Hariga and Benkherouf [61], Jalan and Chaudhury [73] etc.
Time-varying demands were first considered by Silver and Meal [124]. In this model,
they established the Heuristic solution procedure. Also Silver and Meal [124] developed a
model for deterministic time-varying demand, which also gives an approximate solution
procedure termed as Silver – Meal Heuristic. Many research articles by Donaldson [43], Silver

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[123], Dave and Patel [40], Sachan [113], Goyal and Gunasekaran [53], Deb and Chaudhuri
[42], Dave [39], Bahari-Kashani [11], Giri et.al.[50], Hariga [59], Chung and Ting [31], Kim
[81], Giri et.al. [50], Jalan and Chaudhuri [73], Lin et.al. [87] etc. analyzed liner time-varying
demand. In the early 1990’s, Haping and Wang [57] added to a financial strategy model for
falling apart things with time corresponding demand and acquired a closed structure
arrangement of ideal request amount. Xu and Wang [145] introduced stock models for
exponentially decaying things with linearly time-varying demand. Hollier and Mak [64],
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Aggrawal and Bahari-Kashani [4], Hariga and Benkherouf [61], Wee and Shum[137], Jalan
and Chaudhuri [73] etc. developed inventory models in which exponential time-varying
demand has been taken. Later, Khanra and Chaudhuri [79], Ghosh and Chaudhuri [47] etc.
established their models in which quadratic time-varying demand was considered. Roy and
Choudhuri [111] established a non-static single-choice acquisition model for weakening things
in which time-changing ramp-type demand, consistent deterioration and deficiency were
considered. Recently, Begum et al. [14] added to a stock model with time-shifting quadratic
demand.
The variability in the holding expense was set up in the stock models by Van der Veen
[135], Muhlemann and Valtis-Spanopoulous [94] Weiss [141], Goh [52], Giri et.al. [50], Giri
and Chaudhuri [49] and many more researchers. Rosenblatt and Kapsi [108] and Queranne
[105] approached DTP (dynamic programming technique) to obtain minimum inventory cost
with this joint ordering cost. Neves [98] and Pratsini et.al. [104] analyzed an optimal inventory
policy with ordering costs that decrease gradually as ordering respected.
Prior to 1970, according to the classical EOQ model, it was trusted that inflation would
not impact the stock to any noteworthy degree. Following from the 1970's vitality emergency,
expansion has turned into an installed highlight of the economy all through the world and
extensive scale inflation rates are not uncommon in many countries. Therefore, in the present
monetary circumstance, the financial state of various nations has changed in such a way, to the
point that it has not been conceivable to disregard the impacts of expansion and time-estimation
of cash any more. The total inventory cost in the system may be affected by inflation.
Consequently, to reformulate the ideal stock approaches considering inflation and time
estimation of cash, a few endeavors have been made by numerous analysts. Buzacott [20] was

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the pioneer researcher in this direction. Then many researchers have extended their approach to
different interesting situation. Misra [90], [91], Aggrawal [1], Datta and Pal [37], Bose et. al.
[19], Roy et. al. [110], Chen [27] and many more have worked in this line. Further, Roy and
Chaudhuri [111] broke down a stock model for breaking down things with stock level-
subordinate demand in which expansion, time-reducing and shortage have been taken.
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Lin et. al. [87] contemplated a buying model for falling apart things with time-differing request
under time reducing and inflation.
In the genuine business world, suppliers for the most part offer some credit periods to
the retailers so as to quicken the interest of the items they having. They don't charge any
enthusiasm on the remarkable sum if the retailer pays altogether inside of the reasonable
postponement period. The retailers get some monetary point of preference gaining enthusiasm
from the income gathered amid the time of permissible delay in payment. Haley and Higins
[56] is the pioneer researchers of the EOQ model under states of admissible postponement in
installments. The impact of exchange credit on the ideal stock renewal arrangement under
different down to earth circumstances, have been examined by numerous researchers
thereafter. Aggarwal and Jaggi [2] have obtained an ideal stock strategy for deteriorating items
under discounted cash flows approach. Kim et. al. [82] proposed a stock model under the states
of allowable postponement in installment where retailers getting rate and loaning rate are
indistinguishable which is immaterial from down to earth perspective. Hwang and Shinn. [70]
have introduced an inventory model to determine the retailer's ideal cost and part measure at
the same time such that supplier permits delay in installments for a request of an item whose
demand rate is
D k . p , where k 0 is the scaling element
p Unit price
Index of price elasticity

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They have also proposed that the retailers borrowing rate is greater than equal to his
loaning rate. Aggarwal and Jaggi [3] embraced an ideal stock approach under a reasonable
deferral in installment without deficiencies. Jalan and Chaudhury [73] broadened the model of
Aggarwal and Jaggi [3] permitting deficiencies. Again Chu et. al. [30] enhanced the model by
setting up the aggregate cost capacity, which is piecewise convex yet not in general.
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Liao et. al. [86] have addressed on stock model for starting stock dependent
consumption rate, permitting reasonable postponement in installment such that (a) the credit
period is not exactly or equivalent to the process duration, (b) the credit period is more
noteworthy than process duration. Shah and Shah [120] inferred both deterministic and
stochastic stock models for exponentially decaying, under admissible postponement in
installment.
Settled lifetime items have deterministic timeframe of realistic usability. i.e., on the off
chance that an item stays unused up to its lifetime, it is thought to be out-dated and should be
arranged off. Essential samples are human blood utilized for transfusion, pharmaceutical items,
photographic films and so forth. The settled life time items in stock are normally exhausted
after either First-In-First out (FIFO) or lat-In-Last-Out (LIFO) issuing arrangement. In this
manner, it is a critical issue of discovering suitable issuing arrangement for such a perishable
stock framework. Stock models for altered lifetime perishable items have been concentrated on
by Nandakumar and Morton [97].
Amelioration is a natural phenomenon observing in life stock money, a few researchers
have focused on ameliorating system. The vast majority of the traditional stock models depend
on the rule that the estimation of stock stays consistent after some time. This is an uncommon
sort of stock model where scientists have concentrated on for both ameliorating and
deteriorating things, where items ameliorate when stay at breeding yard and deteriorates when
in the distribution systems. Hwang [68] developed a stock model for improving things only.
Again Hwang [69] added to a stock model for both ameliorating and deteriorating things
independently under the strategies FIFO and LIFO. Bhunia and Maiti [18] have planned and
tackled a deterministic stock model for enhancing things considering incomplete offering and
linear price dependent demand. Tadj et. al. [128] has considered a creation stock model for

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both enhancing and breaking down things. Professionals did not give much consideration for
quickly developing creatures such as broiler, ducks, pigs and so on in the poultry ranch,
highbred angles in berry (lake) which are known as improving things. At the point when these
things are away, the stock increments (in weight) because of development of the things.
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Furthermore the stock diminishes because of death, different illnesses or due to some different
components.
Generally, in any economic or business sector, it is not possible to proceed with a
certainty of the pattern of demand. Consequently, demand has been taken to be random or
stochastic in nature. Arrow, Harris and Marschak [9], Dvoretzky, Kiefer and Wolfowitz [44],
Whitin [142], Arrow et.al. [10], Banks and Fabrycky [13] etc. described uncertainties of the
demand under different distribution functions. Stock models for breaking down things with
stochastic interest were additionally created by Pakkala and Achary [99], Aliyu and Boukas
[8], Shah [121], Shah and Shah [120] respectively. Recently, Benkherouf et al. [17] and
Aggoun et al. [5] tended to stochastic bounce stock models for deteriorating things and created
ideal control strategies for them. Sana and Chaudhuri [117] examined monetary creation part
measure (EPLS) model with stochastic request and machine reliability quality.
In the literature, the fuzzy set hypothesis has been connected to stock issues to handle
the vulnerabilities identified with the interest or cost coefficients. An amplified audit of the
utilization of the fuzzy set hypothesis in stock administration can be found in chapter-6. The
upside of utilizing the fuzzy set hypothesis as a part of demonstrating the stock issues is its
capacity to measure dubiousness and imprecision.
In specific circumstances, instabilities are because of fuzziness, basically presented by
Zadeh [151], and are relevant. In 1970, Zadeh and Bellman [152] proposed a few
methodologies for choice making in fuzzy environment. Jain [72] chipped away at choice
making in the vicinity of fuzzy variables. Kacpryzk and Staniewski [76] examined some long-
term stock approach settling on through fuzzy choice making models. Wide uses of fuzzy set

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hypothesis can be found in Zimmerman [154] and Park [102]. In essential EOQ model we
distinguish the request measure that minimizes the aggregate of yearly expenses of stock
holding and settled setup to place orders.
35

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1.11 Objective and Scope ofthe Thesis
A standout amongst the most critical issues confronted in stock administration is the
means by which to control and keep up the inventories of crumbling (deteriorating) things. The
impact of decay of physical merchandise can't be dismissed in any stock framework as all the
physical products fall apart after some time i.e., the quality and quantity of goods decreases in
course of time due to deterioration. The main objective of the thesis is to discuss the inventory
models of deteriorating items. Further consideration of exponential and weibull distribution
under time varying and price dependent demand of actual degree of deterioration under the
influence of inflation is one of the significant works in the present study. Also amelioration
being a natural phenomenon in life stock inventory so inventory model of ameliorating items is
incorporated in the present study. To avoid inadequacy in demand, inventory models under
fuzzy environment has also studied at the end.
1.12 Organizationof the Thesis
The thesis is embodied with eight chapters. The first chapter is the introduction and
literature survey on the related work where the background literature has been described
briefly in the light of which the investigation of the author has been presented in the
subsequent chapters. Chapter 2 to Chapter 7 deals with the contribution of the author in the
field of Inventory Management System. The last chapter is a brief conclusion and outlines of
some proposed models for further investigation. The bibliography section and the information
about the research publications of the papers are presented at the end. The chapter wise
summary of the proposed work is given below.

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In chapter – 2, we have established a stock model of decaying things under time
shifting interest condition. The goal of the model is to talk about a stock model for
deteriorating things and acquire the aggregate ideal expense (optimal cost) of the model with
time differing sort of demand rate having time relative deterioration without lack. The time
changing holding expense is thought to be a linear function of time. The outcome is
represented with numerical illustration and the sensitivity analysis of different parameters has
been studied to strengthen the model.
36

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In chapter – 3, inventory model of deteriorating items for nonlinear holding cost with
time dependent demand has been discussed. The target of the model is to examine the stock
framework for perishable things where time relative weakening rate is considered. The
Economic request amount is resolved for minimizing the normal aggregate expense per unit
time. Time dependent demand rate is used with finite time horizon. Nonlinear holding cost
with lack is considered. The outcome is shown with numerical examples.
In chapter – 4, a stock model for weibull ameliorating, deteriorating things affected by
inflation has been studied. The thought behind this model is to examine the improvement of a
stock model for enhancing things. For the most part quickly developing creatures like pigs,
duck, grill and so on in poultry ranch, high-reproduced fishes are these sorts of things. In this
section an immediate renewal model for the above kind of things under cost minimization in
the impact of swelling time estimation of cash has been concentrated on. Further a period
fluctuating kind of interest rate with unending time horizon, steady decay and without lack is
considered. Additionally the outcome is shown with numerical illustration.
In section – 5, we have examined a decaying stock model with deficiencies under price
dependent demand and inflation. The target of this model is to talk about the stock framework
for decaying things with price dependent demand pattern where weibull rate of crumbling
(deterioration) is considered. Here deficiencies are accepted and are partially backlogged with
inflation. The Economic request quantity is resolved for boosting the benefit per unit time. The
outcome is delineated with numerical illustration.
In chapter – 6, we have discussed optimal inventory model with single item under
various demand conditions. The goal of this model is to talk about the stock model for time

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changing interest and consistent request, and time subordinate holding expense and steady
holding cost for case 1 and case 2 individually. Mathematical model has been produced for
deciding the ideal request amount, the ideal process duration and ideal aggregate stock expense
for both cases. i.e. optimize the order quantity, the optimal cycle time and optimal total
inventory cost for both cases. Numerical illustrations are given for both cases to approve the
37

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Chapter- 1 Introduction and Literature
Survey
proposed model. Sensitivity analysis is done to dissect the impact of changes in the ideal
arrangement concerning change in different parameters.
In chapter – 7, an Inventory control model of breaking down (deteriorating) things in
Fuzzy environment has been discussed. The goal of this model is to examine the stock
framework for perishable things where fixed deterioration rate and stock dependent demand
rate is used with finite time horizon. The analytical development is provided to obtain the
fuzzy optimal solution to minimize the aggregate expense per time unit and for
defuzzification; graded unit-preference integration method is used.
In chapter – 8, the concluding remarks of the investigations have been presented and
then future scope of searching out new results in this area has been addressed.

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CONCLUSION AND SCOPE
OF FUTURE WORK

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Chapter - 8
CONCLUSION AND SCOPE OF FUTURE WORK
8.1 Conclusion
Inventory management is one of the important aspects in production system as well as
in business affairs. At present, the applications of inventory management covers almost all the
branches of scientific studies like oil and gas plant, atomic power plant, chemical power
plant, pharmaceutical power plant, ocean and space research, study of anthropology,
architectures, ancient human dynasties etc. The early period of inventory management system
was mainly focused towards the study of simple inventory models. But with the progress of
time the model becomes more and more complicated due to various factors.
In the present study different inventory models under various factors like time varying
demand condition, non linear holding cost etc. have been studied under the influence of
inflation. The ameliorating life – stock items should be purchased when they are young and
sold when they become grow up. It has also been reflected in the present thesis. The
customer’s impatience has also been realized allowing shortages and their painfulness to
purchase the materials due to inflation has also been presented in the present work. In real life
situations, the exact data are inadequate in many inventory models. Human judgments
including inclinations are frequently obscure and in this way can't be assessed with precise
numerical information. To avoid such ambiguity the vital decision regarding inventory
models under fuzzy environment has been present in the thesis.
8.2 Scope of Future Work

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The measure work in the present thesis is to discuss different inventory models of
deteriorating items under various conditions.
In chapter – 2, we have established an inventory model of deteriorating items under time
varying demand condition. Further it can be enriched by allowing shortages. In chapter –
3, inventory model of deteriorating items for nonlinear holding cost with time dependent
demand has been discussed. It can further be enriched by incorporating inflation and
allowing shortages. In chapter – 4, an inventory model for Weibull ameliorating,

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Chapter-8 Conclusion and Scope of Future Work
deteriorating items under the influence of inflation has been analyzed. In future it can be
extended by considering time value of money. In chapter – 5, we have studied a
deteriorating inventory model with shortages under price dependent demand and
inflation. It can further be enriched considering nonlinear demand. In chapter – 6, we
have discussed optimal inventory model with single item under various demand
conditions. It can further be enriched by incorporating inflation, effect of advertisement
and allowing shortages with completely backlogged. In chapter – 7, an Inventory control
model of deteriorating items in Fuzzy environment has been discussed. It can further be
enriched by imposing different demand conditions.

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