1. INTRODUCTION

1.1 OPERATIONS RESEARCH

‘Operations Research’ was coined during the World War II, but the

scientific origin of the subject dates much further back. Economist Quesnay

in 1759 and Walras in 1874 have developed primitive mathematical

programming models. More sophisticated economic models of a similar

genre were proposed by Von Newmann in 1937 and Kantrovich in 1939. The

mathematical foundations of linear models were established near the turn of

the 19th century by Jordan in 1873, Minkowski in 1896 and Farkas in 1903.

Many definitions of Operations Research are available. The following are

a few of them. In the words of T.L Saaty, “operations research is the art of

giving bad answers to problem which otherwise have worse answers”.

According to Fabrycky and Torgersen, “operations research is the application

of scientific methods to problems arising from the operations involving

integrated system by man, machine and materials. It normally utilizes the

knowledge and skill of an interdisciplinary research team to provide the

managers of such systems with optimum operating solutions”. Churchman,

Ackoff and Arnoff observe, “operations research in the most general sense

can be characterized as the application of scientific methods, techniques and

tools to problems involving the operations of a system so as to provide those

in control of the operations with optimum solutions to the problems”.

In a nutshell, operations research is the discipline of applying advanced

analytical methods to help make better decisions. The rapid growth of

operations research during and after World War II stemmed from the same

root with the application of mathematics to build and understand models that

only approximate the reality being studied. During World War II, the military

depots had the problems of maintaining their inventory such as their

materials, arms, ammunition and fuel etc., and hence the optimal utilization

of the same was needed with a view to minimize their costs. So, the military

management called-on Scientists from various disciplines and organized

them into teams to assist in solving strategic and tactic problems.

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Operations research as a field has always tried to maintain its

multidisciplinary character and its uniqueness. Operations research

comprises of various branches which includes Inventory control, Queuing

theory, Mathematical Programming, Game theory and Reliability methods. In

all these branches many real life problems are conceptualized as

mathematical and stochastic models. In operations research, a model is

almost always a mathematical and necessarily an approximate

representation of reality. Operations research gives the executive’s power to

make more effective decisions and build more productive systems based on

More complete data, Consideration of all available options, Careful

predictions of outcomes and estimates of risk and finally on the latest

decision tools and techniques.

During model building in operations research, the researcher draws upon

the latest analytical technologies, such as i) Probability and Statistics

for helping measure risk, mine data to find valuable connections, insights,

test conclusions and make reliable forecasts. ii) Simulation for giving the

ability to try out approaches and test ideas for improvement. iii) Optimization

for narrowing choices to the best when there are virtually innumerable

feasible options.

Operations researcher and computer scientists have been implementing

inventory systems, while the economists have been focusing on the effect of

inventories in the business cycle rather than inventory policies. Mainly,

operations research provides tools to (i) analyze the activity (ii) assist in

decision making, (iii) enhancement of organisations and experiences all

around us. Application of operations research involves better scheduling of

airline crews, the design of waiting lines at Disney theme parks, two-person

start-ups to Fortune 500® leaders and global resource planning decisions to

optimizing hundreds of local delivery routes. All benefit directly from

operations research decision.

Inventory control is one of the most developed fields of operations

research. Many sophisticated methods of practical utility were developed in

inventory management by using tools of mathematics, stochastic process

and probability theory. The primary motivation of this thesis is to analyse the

few inventory model from Hanssman F [33] using the stochastic concept with

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varying demand distribution. Hence this study is followed in the succeeding

chapters.

1.2 INVENTORY THEORY

Inventory has been defined by Monks, as idle resources that have certain

economic value. Usually, it is an important component of the investment

portfolio of any production system. Keeping an inventory for future sales and

utilizing it whenever necessary is common in business. For example, Retail

firms, wholesalers, manufacturing companies and blood banks generally

have a stock on hand. Quite often, the demand rate is decided by the

amount of the stock level. The motivational effect on the people is caused by

the presence of stock at times. Large quantities of goods displayed in

markets according to seasons, motivate the customers to buy more. Either

insufficient stock or stock in excess, both situations fetch loss to the

manufacturer.

1.3 DEFINITION

This section lists the factors that are important in making decisions

related to inventories and establishes some of the notation that is used in this

thesis. Additional model dependent notations are introduced in the

subsequent Chapters.

1. Holding cost (φ1): This is the cost of holding an item in inventory for some

given unit of time. It usually includes the loss investment income caused by

having the asset tied up in inventory. For example, if c is the unit cost of the

product, this component of the cost is cα , α is the discount or interest rate.

The holding cost may also include the cost of storage, insurance and other

factors that are proportional to the amount stored in inventory.

2. Shortage cost (φ2): When a customer seeks the product and finds the

inventory empty, the demand can either go unfulfilled or be satisfied later

when the product becomes available. The former case is called a lost sale,

and the latter is called a backorder. Although lost sales are often important in

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inventory analysis. The total backorder cost is assumed to be proportional to

the number of units backordered and time the customer must wait.

3. Ordering cost (C (Z )): This is the cost of placing an order to an outside

supplier or releasing a production order to a manufacturing shop. The

amount ordered is Z and its function is given as C (Z ).

4. Setup cost (K ): A common assumption is that the ordering cost consists

of a fixed cost that is independent of the amount ordered, and a variable cost

is dependent on the amount ordered.

5. Product cost (c): This is the unit cost of purchasing the product as part of

an order. If the cost is independent of the amount ordered, the total cost is

cz , c is the unit cost and z is the amount ordered.

6. Demand rate (Q): This is the constant rate at which the product is

withdrawn from inventory.

7. Order level (Z): The maximum level reached by the inventory is the order

level. When backorders are not allowed, this quantity is the same as Q.

When backorders are allowed, it is less than Q.

8. Cycle time (τ ): The time between consecutive inventory replenishments is

the cycle time.

9. Cost per time (T ): This is the total of all costs related to the inventory

system that are affected by the decision under consideration.

10. Optimal Quantities (Q , Z , τ ,T ): The quantities defined above that

maximize profit or minimize cost for a given model are the optimal solution.

11. Shortages Backordered: The stochastic model considered in this thesis

allows shortages to be backordered. This situation is illustrated in figure 1.1.

In this model, when the inventory level decreases below the 0 level, then it

implies that a portion of the demand is backlogged. The maximum inventory

level is considered as S and occurs when the order arrives. The maximum

backorder level is Q – S and backorder is represented in the figure 1.1 by a

negative inventory level.

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Figure 1.1 Lot-size model with shortages allowed

12. Random Variable: A random variable, usually written as X , is a variable

whose possible values are numerical outcome of a random phenomenon.

There are two types of random variables, discrete and continuous.

13. Discrete random variable: A discrete random variable is one which may

take on only a countable number of distinct values such as 0, 1, 2, 3, 4,… If a

random variable can take only a finite number of distinct values, then it said

to be discrete. Examples for discrete random variables include the number of

children in a family, the number of patients in a doctor's surgery and the

number of defective light bulbs in a box of ten.

14. Continuous random variable: A continuous random variable is one,

which takes an infinite number of possible values. Continuous random

variables are usually measurements. Examples include height, weight, the

amount of sugar in an orange and the time required to run a mile. A

continuous random variable is not defined at specific values. Instead, it is

defined over an interval of values, and is represented by the area under a

curve. The probability of observing any single value is equal to 0, since the

number of values which may be assumed by the random variable is infinite.

15. Random Variable for Demand (Q): This is a random variable that is the

demand for a given period of time. The random variable defined for a

particular period may differ with the models considered.

16. Discrete Demand Probability Distribution Function (P(Q)): When

demand is assumed to be a discrete random variable, P ¿) gives the

probability that the demand equals Q.

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17. Discrete Cumulative Distribution Function (F (b )): The probability that

demand is less than or equal to b is F (b ) when demand is discrete then

F (b )=∑Q=0

b

P (¿Q)¿ (1.1)

18. Continuous Demand Probability Density Function (f (Q )): When

demand is assumed to be continuous, f (Q ) is its density function. The

probability that the demand is between a and b is

P (a≤Q≤b )=∫a

b

f (Q ) dQ (1.2)

When the demand is assumed to be nonnegative, then f (Q ) is zero for

negative values.

19. Continuous Cumulative Distribution Function (H (b )): The probability

that demand is less than or equal to b when demand is continuous then

H (b )=∫a

b

f (Q )dQ (1.3)

20. Standard Normal Distribution Function ϕ (Q) andΦ (Q): These are the

density function and cumulative distribution function for the standard normal

distribution.

The study of inventory control requires a practical example for better

understanding. Hence, in figure 1.2 two figures on sample path are shown

one in environment process and other in inventory process.

Figure 1.2: A sample path of the (environment–inventory) process

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A sample path of the environment-inventory level process of K. Yan et.al

[79] is illustrated in figure 1.2, where Z( t) is the production rate and X ( t )is

demand rate are associated with each state of the inventory system. ‘Si’ is

taken as the supply in the interval (0 , t). The inventory increases when the

production rate exceeds the demand rate, and decreases when the demand

rate exceeds the production rate. For example, the inventory level under

continuous review is viewed as a fluid process that fluctuates according to

the evolution of the underlying background environment.

The subject of inventory control is a major consideration in many

situations, because of its practical and economic importance. Questions

must be constantly answered as to when and how much raw material should

be ordered, when a production order should be released to the plant, what

level of safety stock should be maintained at a retail outlet, or how in-process

inventory is to be maintained in a production process. These questions are

amenable to quantitative analysis with the help of inventory theory.

The modern inventory theory offers a variety of economical and

mathematical models of inventory systems together with a number of

methods and approaches aimed at achieving an optimal inventory policy.

The main steps in applying a systematic inventory control are outlined as

follows.

a) Formulating a mathematical model by describing the behavior of the

inventory system.

b) Seeking an optimal inventory policy with respect to the model.

c) Using a computerized information processing system to maintain a

record of the current inventory levels.

d) Using this record of current inventory levels, applying the optimal

inventory policy to indicate when and how much to replenish

inventory.

In the conceptualization of inventory control, various costs and different

variables such as control variables and non-control variables are

incorporated. It is quite interesting to observe that the inventory model can

be either deterministic or probabilistic. If the model is probabilistic in nature

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then, the probability theory and stochastic processes plays a vital role in the

formulation of the model and also in the determination of optimal solution.

Optimization techniques such as dynamic programming and calculus

based methods to find optimal inventory policies have been studied by Arrow

K.J et.al [6]. Using linear programming principles and competitive bidding

methods many models have been developed by Hanssmann F et.al [32].

Arrow K.J et.al [6, 7] has studied a generalized model of inventory control

encompassing many inventory situations. A model for the optimal discharge

of water from a reservoir has been developed in Little J.D.C [40]. A

systematic review of such models is seen in Whitin T. M [77]. After a period

of dormancy in the 1960’s and 1970’s, empirical work on inventories has

enjoyed resurgence in the 1980’s and 1990’s. Inventory control model in the

literature is classified according to its deterministic and continuous nature.

1.4 CLASSIFICATION OF INVENTORY CONTROL MODEL

The study on inventory control deals with two types of problems such as

single-item and multi-item problems. Concerning the process of demand for

single-items, the mathematical inventory models are divided into two large

categories deterministic and stochastic models which is shown in figure 1.3

Figure 1.3: Classification of inventory control model

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In single-item stochastic models, the rate of demand for products

stocked by the system is considered to be known with uncertainty and it is

called stochastic demand and when the demand is known with certainty it is

considered to be deterministic. Also in single-item, the deterministic demand

is either a constant quantity i.e., deterministic static model or a known

function of time i.e., deterministic dynamic model. Multi-period is further

subdivided into periodic review and continuous review.

Many of the available stochastic models and their solutions are used here

to conceptualize some interesting new problems and solve them. The

problems which are conceptualized on certain hypothetical assumptions are

in Inventory Control, Reliability Theory and Queuing theory. All these

disciplines depend more and more for their development and sophistication,

the use of advanced probability theory for which stochastic process is a basic

structure. Many of the real life problems which are governed by chance

mechanism are deeply involved with the concept of stochastic process. An

important aspect in the theory of stochastic process is the renewal theory

which is from the mathematical view point and at the same time is a handy

tool to solve many problems of stochastic process.

One of the inventory models that have recently received renewed

attention is the Newsboy problem and Base stock system problem. Hadley G

et.al [29] and Hanssman F [33] have been credited for the seminal work on

the classical version of these problems. Their models have been the

foundation for many subsequent works by extending the original models to

other diverse scenarios and applications. Nevertheless, despite its

importance and the numerous publications related to the Newsboy problem

or the multi-product Newsboy model and its variations remain limited.

The basic problem of inventory control or inventory management is to

determine the optimal stock size and optimal reorder size. Determination of

the time to reorder is also a question. A very detailed and application

oriented treatment of this subject is seen in Hanssman F [33].

1.5 CLASSIFICATION OF CLASS OF INVENTORIES

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The classification of the class I, II, III, IV and V of inventories are

discussed in form of Table 1.1.

Table 1.1 Classification of inventories

Class Inventory I 0 Supply process a (Q) Demand b (Q)

I Raw Material Supplier Production

II Work in process Production Production

III Finished goods Production Wholesaler

IV Wholesale Manufacturer Retailer

V Retailer Wholesaler Consumer

The inventory on hand at any time ‘t’ is given by

I ( t )=I 0+∫0

t

[a (Q )−b (Q ) ]dQ (1.4)

Where

a (Q ) = supply rate / unit time

b (Q ) = demand rate / unit time

I 0 = initial or starting inventory level.

In an inventory system, if the supply and demand is from a single

source, then it is called a single station model. If there are many supply

sources and similarly several sources of demand and a number of stations

operate simultaneously then it is called a system of parallel stations model.

A system of stations is called a series of station model, if the output of one

station is the input for the next, which are in series. The solution to any

model depends upon these three characteristics. If the supply and demand

namely a (Q ) and b (Q ) are constant over time, then it is called a static system,

otherwise it is called a dynamic one. The inventory problems in real life

situation, is conceptualized as a stochastic model and involves the

optimization of inventory problem.

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1.6 OPTIMIZATION OF AN INVENTORY PROBLEM

In the case of stochastic models, the periodic approach of expressing

demand is preferred to be a continuous (demand rate) approach. In doing so,

the two costs namely the cost of excess inventory which is also known as the

salvage cost and shortage cost is incorporated into the model. If the demand

is more than the supply the shortage may arise and hence the stock-out cost

is incorporated. The solution is derived by using the standard mathematical

tools and techniques. If the derived solution is optimal, then process of

solution is complete. The objective of obtaining the optimal solution is to

determine the solution which minimizes the overall cost. It is known as the

optimal policy. In addition, the cost of reordering, the optimal reorder size as

well the time at which the reordering is to be made has been incorporated by

many authors for the optimization of inventory problem.

It may be observed that the demand depends upon many factors like

market conditions, availability of substitutes etc., and hence it is not under

the control of the decision maker. On the other hand the supply is under the

control of the decision maker and hence called the control variable. The

demand and supply are two different variables associated with the inventory

model. If the demand is assumed to be a random variable then the demand

is called the probabilistic demand. Another aspect is the static or dynamic

aspect of demand and also the supply. If the demand and supply do not

change with the passage of time, it is called static demand and static supply,

respectively otherwise it is called dynamic.

In many problems of inventory control, obtaining the optimal size of

the supply is a prime interest. Hence the optimal solution is often the

determination of the supply size. A similar approach is to determine the time

of reorder and quantity of reorder. If the demands as well as the supply are

probabilistic in nature then the probability distributions are taken into account

and the expected cost is obtained. The solution which minimizes the

expected cost is the optimal solution.

It may be noted that the recent approach to find the optimal solution

takes into consideration another fact. The demand distribution may undergo

a parametric change, after a particular value of the random variable involved

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in the model and the point at which the change occurs is called the truncation

point. Sometimes after the truncation point, the distribution of demand which

is a random variable can undergo a change of distribution itself. Such facts

are also incorporated in the model and the optimal solution is derived.

Another interesting area of research in inventory control has come up

recently. It is the so called perishable inventory theory. There are many

products such as vegetables, food products, fruits and pharmaceutical

products in which deterioration occurs. After a certain period the entire lot

unsold will deteriorate completely and hence cannot be sold. In such models,

the rate of deterioration is an important aspect of consideration and these

models were studied using exponential and Weibull distribution.

In this thesis, the contribution follows the following tools for analysis

of inventory systems subject to supply disruptions such as i) exact and

approximate expected cost functions when supply is disrupted and demand

is stochastic. ii) A closed-form approximation for the optimal base-stock level

when supply is disrupted and demand is stochastic. iii) A closed-form

approximation for the optimal base-stock level when demand is disrupted

and supply is stochastic.

Hence, this thesis involves the concept of closed form in chapter 4

and chapter 5 with the application of stochastic process.

1.7 STOCHASTIC PROCESS

Stochastic process is concerned with the sequence of events

governed by probabilistic laws. Many applications of stochastic process are

available in Physics, Engineering, Mathematical Analysis and other

disciplines. In some cases, arising in certain industries or military installations

not only the demand for a particular commodity is a stochastic variable but its

supply as well. In these cases it is convenient to consider the inventory level

resulting from the interaction of supply and demand as a stochastic variable.

The variation of the inventory level in time can be considered as a stochastic

process.

If the process is ergodic, the total inventory cost over a certain time T

may be represented as a function of the mean inventory level. This mean

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level can then be manipulated in such a way as to minimize the total

inventory cost. In case of a stochastic process, if a specific ordering policy is

introduced then the resultant fluctuating inventory level is a stochastic

phenomenon. Also it becomes a problem to investigate the transient and

stationary characteristics of the underlying stochastic process.

A special class of problems arises, if a situation where the system is

already in a stationary state is assumed, and where the acquisition policy

has no apparent relation to the inventory level. In the case discussed above

the mean inventory level becomes a decision variable. As an example liquid

flowing in random fashion in and out of storage tank is considered. The

fluctuation of the inventory level is then a stochastic process.

Recently, the problem of how to determine optimum mean inventory

levels has arisen frequently in large industrial concerns, where it appears to

be a consequence of the institutional framework of the modern firm. In many

of the integrated companies of today, the principle of decentralized

management has become a well established fact. This has led with necessity

in many cases to the practice of sub-optimization, because if a large

industrial enterprise is subdivided for administrative purpose into several

rather independent acting departments, such as production, transportation,

manufacturing, distribution, sales-department etc.

It will often happen that the different decision parameters, which are

necessary to decide upon in order to achieve an overall optimization, are

controlled by different departments. For example, in an integrated oil

company, the size and composition of the crude oil inventories held by the

manufacturing department at the refineries are the result of the interaction of

the crude oil supply from overseas areas. It is managed and controlled by the

production and transportation departments on the one hand and the demand

for the finished goods coming from the distribution and sales departments on

the other hand. Thus, the manufacturing department is left with just one

decision variable under its direct control which is the mean inventory level.

This is in general manipulated by exchange with oil companies. This concept

of decentralisation is discussed in chapter 3.

Uncertainty plays an important role in most inventory management

situations. The retail merchant needs enough supply to satisfy customer

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demands, but ordering too much increases holding costs and the risk of

losses through obsolescence or spoilage. A fewer order increases the risk of

lost sales and unsatisfied customers. For example, the water resources

manager must set the amount of water stored in a reservoir at a level that

balances the risk of flooding and the risk of shortages. Hence, this concept of

shortage and holding is analyzed throughout the thesis.

The company manager sets a master production schedule

considering the imprecise nature of forecasts of future demands and the

uncertain lead time of the manufacturing process. These situations are

common and the answer one gets from a deterministic analysis varies often

when uncertainty prevails. The decision maker faced with uncertainty may

not act in the same way as the one who operates with perfect knowledge of

the future.

The inventory model in which the stochastic nature of demand is

explicitly recognized is dealt. In inventory theory, demand for the product is

considered to be one of the features of uncertainty. In this thesis, the

demand is assumed to be unknown and the probability distribution of

demand is known. Mathematical derivation determines the optimal policies in

terms of the distribution and selecting an appropriate distribution for the

study is very important.

1.8SELECTING A DISTRIBUTION

In this thesis, the prime motivation is to study which distribution may

be suitable for the representation of demand. A common assumption is that

individual demand occurs independently. This assumption leads to the

Poisson distribution when the expected demand in a time interval is small

and the normal distribution when the expected demand is large. Later the

uniform distribution and the exponential distribution were used for their

analytical simplicity. Erlang distribution was prime interest for the solution of

inventory problem in 2000’s. Hence, the literature suggests that other

distributions can be assumed for demand.

Hence, motivated from the view of usage of other distribution, this

thesis involves the study of single-period model and multi-period model using

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SCBZ property, renewal reward theory, truncated exponential distribution,

exponential order statistics and generalized gamma distribution with bessel’s

function. Once decided on the demand distribution to be applied, the next

aim is to find the total expected cost of the inventory problems under study of

this thesis.

1.9STOCHASTIC INVENTORY MODEL

Often, there is some concern about the relation of demand during

some time period which is relative to the inventory level at the beginning of

the time period. If the demand is less than the initial inventory level and there

is an inventory remaining at the end of the interval then the condition of

excess incurs. If the demand is greater than the initial inventory level then

the condition of shortage incurs. At some point, the inventory level is

assumed to be a positive value Z. During some interval of time, the demand

is a random variable Q with PDF f (Q) and CDF F (Q). The mean and

standard deviation of this distribution are μ and σ respectively. With the given

distribution, the probability of a shortage PS and the probability of excess PE

are computed. For a continuous distribution, PE and PS is given as

PE=P {Q≤ Z }=∫0

Z

f (Q ) dQ=F (Z) (1.5)

PS=P {Q>Z }=∫Z

∞

f (Q )dQ=1−F (Z) (1.6)

In some cases it may be interesting to obtain expected shortageψ1 (Q ).

This depend on whether the demand is greater or less than Z

Items short ={ 0 , if Q≤ ZQ−Z , if Q>Z (1.7)

Then ψ1 (Q ) is the expected shortage and is

ψ1 (Q )=∫Z

∞

(Q−Z ) f (Q )dQ (1.8)

Similarly for excess, the expected excess is ψ2 (Q )

ψ2 (Q )=∫0

Z

(Z−Q ) f (Q )dQ (1.9)

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Also the expected excess can be represented in terms of ψ1 (Q )

ψ2 (Q )=∫0

∞

(Z−Q ) f (Q )dQ-∫z

∞

(Z−Q ) f (Q ) dQ

¿ Z−μ+ψ1 (Q ) (1.10)

Hence, this concept of stochastic process has similarity with the model

discussed in Hanssman F [33] which is the prime motivation behind this

research work.

1.10 PRELIMINARY CONCEPTS AND RESULTS

The following are some of the basic, existing and recently developed

concepts in Mathematics and Statistics that are used to analyse some

inventory models in this thesis.

1. SETTING THE CLOCK BACK TO ZERO (SCBZ) PROPERTY: In

stochastic process when considering sequence of random variables each

random variable has an associated probability distribution. So, the probability

distribution function of random variableQ is denoted as f (Q ). For every

probability distribution there are corresponding one or more parameters. The

corresponding distribution function is denoted as F (Z ), and S (Z )=1−F (Z ) is

called the survivor function and it gives the probability that a random variable

Q.

For example, if a random variable Q is distributed as exponential with

parameter θ then Q f (Q ,θ )=θe−θQ. Hence, exponential distribution satisfy

the lack of memory property and there is slight modification of this property

known as Setting the Clock Back to Zero (SCBZ) property which was

introduced by Raja Rao et.al [53].This property is given as, a family of life

distribution { f (Q,θ ) ,Q≥0 , θ∈Ω }, (where Ω is the space parameter) is said to

have the ‘Setting the Clock Back to Zero’ (SCBZ) property if f (Q ,θ ) remains

unchanged except for the value of the parameters under the following three

cases,

(i) Truncating the original distribution at some point Q0≥0

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(ii) Considering the observable distribution for inventory control Q≤Q0

andQ>Q0

(iii) Let Q0be a truncation point and Q be fixed. If f (Q ,θ )→f (Q ,θ¿ ) ,θ¿∈Ω,

then

Q f (Q ,θ1) When Q≤Q0

Q f (Q ,θ2) When Q>Q0 (1.11)

Setting the clock back to zero property is the prime interest of study

throughout the thesis and it is discussed in chapter 3 and 5.

2. CHANGE OF DISTRIBUTION AT A CHANGE POINT: The concept of

SCBZ property indicates that a random variable Q with density function f (Q )

undergoes a parametric change after a certain value of Q say Q0 which is

called the truncation point. This is a slight modification of the lack of memory

property. An extension of this concept is change of distribution after a

change point.

For example, if Q is a random variable denoting the life time of the

component and f (Q,θ ) is the probability density function then the random

variable undergoes a change of distribution after a change point, when the

following condition is satisfied.

The random variable Q has a PDF f (Q) with CDFF (Q), whenever

Q≤Q0 and it has PDF h(Q) with CDF H (Q) if Q>Q0. Here Q0 is called the

change point. It can be noted that

∫0

Q0

f (Q)dQ+∫Q0

∞

h(Q)dQ=1 (1.12)

The concept of change of distribution is discussed in Stagnl D.K [71].

An application of this property in shock model cumulative damage process

has been introduced by Suresh Kumar R [72]. The detailed study on this

concept is given in chapter 7.

3. TRUNCATED EXPONENTIAL DISTRIBUTION: Suppose that Q is a

random variable with exponential Probability Density Function (PDF) of mean

(1θ ) then the PDF of the random variable Q is truncated on the right at Q0 is

given by Deemer W.L et.al [18] and the maximum likelihood estimator of the

parameter θ is derived in the form of truncated exponential distribution as

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f (Q ,θ )={θ exp (−θQ ) (1−e−θQ 0 ) ,0<Q<Q0

0 , otherwise(1.13)

4. RENEWAL REWARD THEORY: Chang H.C et.al [12] revisited the work

of Wee H.M et al [76] and adopted the suggestion of Maddah B et.al [41] to

use renewal reward theorem to derive the expected profit per unit time for

their model. Exact closed-form solutions were derived for the optimal lot size,

backordering quantity and maximum expected profit. Given the attention

received by the Salameh M.K et.al [61], it was important to enhance it and

correct any flaws in the problems. Renewal theory to obtain the exact

expression for the expected profit is applied. This approach leads to a

simpler expression for the optimal order quantity than that in Salameh et.al

[61]. The annual profit function in the simplified way is given by

f (Z ,Q )=(φ¿¿1+φ2)ψ

2Q [Z−Qφ1

(φ1φ2 )ψ ]+ φ1+Q2ψ [ψ+ 2ψ

Q0−

φ1

φ1+φ2 ]¿

(1.14)

Truncation exponential distribution and renewal reward concepts are

discussed in chapter 4, 5 and 7.

5. PHASE TYPE DISTRIBUTIONS: Poisson process and exponential

distribution have mathematical properties that make the inventory models as

demand process or service time or replenishment time distribution. However,

in applications these assumptions are highly restrictive. Neuts M.F [48]

developed the theory of PH-distributions and related point process as an

alternative of the above distributions. In stochastic modelling, PH-

distributions lend themselves naturally to algorithmic implementations and

have closure properties along with a related matrix formulation to utilize in

practice. In this thesis, concept of PH-distributions is discussed in Chapter 6.

6. GENERALIZED GAMMA DISTRIBUTION WITH BESSEL FUNCTION: In Nicy Sebastian [50], a new probability density function associated with

a Bessel function is introduced, which is the generalization of a gamma-type

distribution. Some of its special cases are also mentioned in this thesis. The

author also introduced Multivariate analogue, conditional density, best

predictor function, Bayesian analysis, etc., connected with this new density.

From Nicy Sebastian [50], the probability density function is given as

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f (Q )={ α β

eδα Γ (β)

e−αQQβ−1∑k=0

∞ (δQ)k

(β )kk !;α>0 , β , δ>0 ,Q≥0

0 ; elsewhere

(1.15)

This concept is discussed in chapter 7.

7. EXPONENTIAL ORDER STATISTICS: The ordering decision in each

period is affected by a single setup cost k, a linear variable ordering cost

d1 , d2 ,…,dn=d. In stock level is given as a1 ,…,an=a at the beginning of a

period. Let an inventory system whose time to shortage and holding of the

items is considered which is the prime interest. If the experiment with a

single new component at time zero be started and it is replaced upon loss by

a new component and so on which is represented by Exponential Order

statistics ∑i=1

n

Qi is independent and the key to model when there is joint PDF

is

f (Q )= α β

eδα Γ (β)

e−αQQβ−1∑k=0

∞ (δQ)k

( β)k k !

−αβQ

,0<Q<∞ (1.16)

Suppose Q1 :n ,Q2 :n ,Q3 :n ,…Qn :n are the order statistics of a random variable of

size n arising from f (Q ) along with the distribution of the form

Q1 :n,Q2 :n ,Q3 :n ,…Qn :n=dr ;r=1,2 ,…,n ;Q 0:n=0 (1.17)

Then dr will constitute the renewal process. Considering the joint probability

density function of all order to be given by

f 1: 2 ,… ,n :n(Q1 ,…,Qn)=n! Z α β

eδα Γ ( β)

e−αQQβ−1∑k=0

∞ (δQ)k

(β)kk !

−αβ∑i=1

n

Q i

(1.18)

Chapter 7 involves the use of these concepts in obtaining the optimal

expected cost.

1.11 ARRANGEMENT OF THE CHAPTERS

In Chapter 1, a brief introduction about operations research, inventory

control and its practical applications to real life problems is studied. The

results of stochastic process using varying demand distribution are applied in

this thesis.

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In Chapter 2, a brief summary on research papers published by

various authors is given as the review of literature.

In Chapter 3, Single period Newsboy problem using SCBZ Property is

discussed. The Newsboy problem is discussed assuming that the demand

distribution satisfies SCBZ property. The Newsboy problem is one under the

finite inventory process. In this problem it is assumed that there is a one-time

supply of items and demand is probabilistic. Each unit of items produced but

not sold is called salvage cost and if the supply is less than demand, it

results in stock-out cost. This model has been discussed by Hanssman F

[33]. At the beginning of each period of time the stock level of each item is

reviewed and a decision to order or not to order is made. The cost elements

that affect the ordering decision in each period are salvage cost and stock-

out cost. The costs are charged on the basis of the stock levels at the end of

the period. Demand for the item in each period of time is described by a

continuous random variable with a joint density function which is

independently distributed from period to period.

An approximate closed-form solution is developed using a single

stochastic period of demand which is discussed. A Stationary Multi-

commodity inventory problem has been formulated from a single period

inventory model. Also a generalization of Newsboy problem for several

individual source of demand is discussed. It is assumed that the demand has

a probability distribution which satisfies the so called SCBZ property. Such

an assumption is justified since the demand distribution undergoes a change

with the size of the demand. Under this assumption the optimal supply size

is determined and the change in the optimal size consequent to the change

in the parameter involved in the distribution is illustrated numerically.

In Chapter 4, the single period Newsboy problem discussed in

chapter 3 is extended using Truncated Exponential Distribution and Renewal

Reward Theory. In this chapter, a study on the salvage cost undergoing a

change using the Truncated Exponential Distribution and the use of Renewal

Reward Theory for obtaining the solution involving the occurrence of partial

backlogging due to stock-out is carried out. The objective is to derive the

optimal stock level and numerical illustration with corresponding figure is

provided.

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In Chapter 5, the Truncated Exponential Distribution discussed in

chapter 4 is used to study the base-stock for patient customer. In the base

stock system the total inventory on hand is to be taken as the sum of the

actual inventory on ground and inventory due to orders for replenishment.

The customers do not cancel the orders if shortage occurs but waits till the

supply is received. The patient customer case is studied, where all unfilled

demand is backlogged. Immediate delivery of orders and complete

backlogging of all unfilled demands is assumed. The optimal expected cost

of base-stock system for patient customer is obtained when the demand

distributions are distributed exponentially before the truncation point and

Erlang2 after the truncation point. The objective is to derive the optimal stock

level and also numerical illustration is provided.

So far in chapter 5, the base-stock system for patient customer is

discussed, but in real life there are also customers who are impatient. Hence

in chapter 6, a study on the base-stock system for impatient customer is

carried out.

In Chapter 6, the Base-stock impatient customer using finite-horizon

models is studied. So far the Base-stock for impatient customer leaded to a

discrete case but in this work is extended for a continuous case. Also a way

of optimizing the average cost per day by balancing cost of empty beds

against cost of delay patients is analysed which is discussed. The upper and

lower echelon case of the impatient customer in base-stock policy is

discussed. In this chapter, the base-stock is viewed as the number of initial

inventory facility in stock. Here the demand is considered as the Poisson

fashion i.e., one demand at a time. The probability lead time for a reordered

item corresponds to the service time and its distribution is assumed to be

Erlang type. At the upper echelon is a supplier with a single production

facility which manufactures to order with a fixed production time on a first-

come first-served basis and the numbers of non-identical and independent

retailer is considered at the lower echelon. The objective is to derive the

optimal stock level and numerical illustration is provided.

So far in the above chapter continuous single-period models are

discussed and in chapter 7, the multi-period or the multi-item problems is

studied.

21

In Chapter 7, the multi-period stochastic model is discussed with two

varying demand models. The m-dimensional convolution method which was

introduced by Hanssman F [33] is used for study of generalisation concept of

the ordering convolution operation. Now in this chapter, the multi-period or

the multi demand case is discussed when ψ (C ) has the form ψ (C )=∑i=1

N

ψ i (C i )

where each of it is continuous and differentiable. The function ψ (C )is the cost

charged over a given period of time excluding the ordering cost and in

general it is the holding and shortage costs. Considering the case when the

salvage and stock-out cost for each item is linear. Let for item (

i=1 ,2 ,3 ,…, N ¿ an inventory model is discussed under the following

assumptions regarding the model.

(i) There is a onetime supply at the start of the period (0 ,T ).

(ii) The demands occur at N th random epochs in (0 , T ) and the

magnitudes of the demands are random variables denoted as Qi=1 ,2 ,3 ,…N

(iii) If the cumulative demand ∑Qi≤Z, then salvage occurs and if

∑Qi>Z ,then stock-out occur during (0 , T ). The random variable

representing demand namely Qi has PDF f (.)and CDF F ( . ) . Q1 ,Q2 ,Q3,…,QN

is identically independently distributed random variables. This chapter the

demand and lead time is considered a constant and a random variable. By

assuming exactly N th demand epochs in (0 , T ), and using renewal theory the

optimal value of Z is obtained. Another extension discussed in this chapter

is by the assuming that the random variable Q has a distribution initially but

there a change of distribution after a truncation. The optimal one time supply

during the interval (0 , T ) using the generalized gamma distribution with

Bessel’s function and a multi-commodity inventory system with periodic

review operating under a stationary policy using the exponential order

statistics is discussed. The optimal inventory level is determined for the multi-

period demands. Also adequate numerical analysis shows its effectiveness.

The result of this study, especially the properties are hoped to be of

great use in determining the transient and stationery distribution of the stock

level prior to making ordering decision.

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In Chapter 8, a brief summary of the results and conclusions drawn

hereby are furnished.

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