A STOCHASTIC MODELLING WITH VARYING
DEMAND DISTRIBUTIONS IN INVENTORY CONTROL
A THESIS REPORT
Submitted by
DOWLATH FATHIMA
Under the guidance of
Dr. P.S SEHIK UDUMAN
in partial fulfillment for the award of the degree of
DOCTOR OF PHILOSOPHY in
DEPARTMENT OF MATHEMATICS
B.S.ABDUR RAHMAN UNIVERSITY (B.S. ABDUR RAHMAN INSTITUTE OF SCIENCE & TECHNOLOGY)
(Estd. u/s 3 of the UGC Act. 1956) www.bsauniv.ac.in
SEPTEMBER 2013
B.S.ABDUR RAHMAN UNIVERSITY (B.S. ABDUR RAHMAN INSTITUTE OF SCIENCE & TECHNOLOGY)
(Estd. u/s 3 of the UGC Act. 1956)
www.bsauniv.ac.in
Date: 16.04.2014
SUBMITTED TO THE DEAN (AR)
Sub: Thesis submission of Dowlath Fathima (RRN: 0989205)-Reg.
Ref.: Lr No. 291 / DEAN (AR) /2014 dated 03.03.2014.
This is to certify that all the corrections and suggestions pointed out by the external
examiners are incorporated in the thesis entitled A Stochastic modelling with varying
demand distributions in inventory control submitted by Mrs. Dowlath Fathima (RRN
0989205).
SIGNATURE
Dr. P.S SEHIK UDUMAN
RESEARCH SUPERVISOR
Department of Mathematics
B.S Abdur Rahman University
Vandalur, Chennai-600048
PROCEEDINGS OF THE Ph.D VIVA-VOCE EXAMINATION OF Mrs. DOWLATH FATHIMA HELD AT 11.00 A.M
ON 16.04.2014 IN SEMINAR HALL, EEE DEPARTMENT
_____________________________________________________________________________________
The Ph.D. Viva-Voce Examination of Mrs Dowlath Fathima (RRN.0989205) on her Ph.D. Thesis Entitled “A
Stochastic Modelling with Varying Demand Distributions in Inventory Control” was conducted on 16.04.2014 at
11.00 A.M in the Department of EEE, Seminar Hall.
The following Members of the Oral Examination Board were present:
1. Indian Examiner 2. Subject Expert
Dr.D.Arivudainambi Dr.P.Vijayaraju
Associate Professor Professor Department of Mathematics Department of Mathematics Anna University, Chennai-600025 Anna University, Chennai-600025
3. Supervisor & Convener
Dr. P.S Sehik Uduman
Professor Department of Mathematics B.S Abdur Rahman University
The research scholar, Mrs. Dowlath Fathima presented the salient features of her Ph.D. work. This was
followed by questions from the board members. The questions raised by the Foreign and Indian Examiners were
also put to the scholar. The scholar answered the questions to the full satisfaction of the board members.
The corrections suggested by the Indian/Foreign examiner have been carried out and incorporated in the
Thesis before the Oral examination.
Based on the scholar’s research work, her presentation and also the clarifications and answers by the
scholar to the questions, the board recommends that Mrs. Dowlath Fathima be awarded Ph.D. degree in the Faculty
of Mathematics
1. Indian Examiner
2. Subject Expert
3. Supervisor & Convener
B.S.ABDUR RAHMAN UNIVERSITY (B.S. ABDUR RAHMAN INSTITUTE OF SCIENCE & TECHNOLOGY)
(Estd. u/s 3 of the UGC Act. 1956) www.bsauniv.ac.in
BONAFIDE CERTIFICATE
Certified that this thesis report A STOCHASTIC MODELLING WITH
VARYING DEMAND DISTRIBUTIONS IN INVENTORY CONTROL is the bonafide
work of DOWLATH FATHIMA (RRN: 0989205) who carried out the thesis work
under my supervision. Certified further, that to the best of my knowledge the work
reported herein does not form part of any other thesis report or dissertation on the
basis of which a degree or award was conferred on an earlier occasion on this or any
other candidate.
SIGNATURE SIGNATURE
Dr. P.S.SEHIK UDUMAN Dr. S.SRINIVASAN
RESEARCH SUPERVISOR HEAD OF THE DEPARTMENT
Professor Professor & Head
Department of Mathematics Department of Mathematics
B.S. Abdur Rahman University B.S. Abdur Rahman University
Vandalur, Chennai – 600 048 Vandalur, Chennai – 600 048
v
ACKNOWLEDGEMENT
Firstly, I am fortunate to have Dr. P. S Sehik Uduman, Professor,
Department of Mathematics, B.S Abdur Rahman University, Chennai as my
supervisor. His flexibility in scheduling, gentle encouragement made a good
working environment and the impetus for me to finish my research. He has been
a strong and supportive advisor throughout my research.
I am grateful to the Management, Vice-chancellor, Pro Vice-Chancellor,
Registrar, Deans and Directors of B.S Abdur Rahman University for their
encouragement throughout the course of my research.
My sincere thanks are to the Doctoral committee members Dr. G. P
Youvraj, Associate Professor, University of Madras and Dr. S. Srinivasan,
Professor and Head, Department of Mathematics, B.S Abdur Rahman University
for their suggestions.
I wish to thank Dr. R. Sathiyamoorthy, Professor and Head(Retd),
Department of Statistics, Annamalai University, Chidambaram and Dr. I. Raja
Mohamed, Professor, Department of Physics, B.S Abdur Rahman University for
giving very useful suggestions during my course of research.
I am grateful to the University Grant Commission (UGC), for providing
me the research fellowship in the scheme of Maulana Azad National Fellowship
for minority community under award number MANF/TAM/MUS/4867.
I offer my thanks to Mr. Nazeer Ahmed, for the support during my Ph.D
admission. Also, I offer my thanks to the staff members and Research Scholar in
the Department of Mathematics.
Let me thank my family members especially my mom Mrs. Sultana
Begum for her blessings and encouragement. Also, I wish to thank my sister
Mrs. Dilara Fathima, better half Mr. Ameen Sherief and my son Mohammed
Sarfaraz Sherief for their support.
DOWLATH FATHIMA
vi
ABSTRACT
The thesis presents the studies on single-period and multi-period
demand models. Here keeping a stock of goods, manpower etc., is
necessary to meet the fluctuating demand and both the factors like salvage
and stock-out situations are equally important. Hence, depending upon the
problems that arise, suitable stochastic inventory models are analysed.
In Single-period demand model, finite inventory process models such
as Newsboy and Base-Stock models are studied. The contribution of this
thesis involves a study on the single-period demand model such as the
Newsboy model using the SCBZ property, truncated exponential distribution
and renewal reward theory. Truncated exponential distribution being the
appropriate distribution for the study of change point, hence this distribution
is analysed in case of base stock for patient customer. After analysing the
patient customer model, the discussion on impatient customer is carried out
and hence the model of impatience customer is discussed in form of queuing
model which is then extended to the continuous case. Also, the optimal stock
size is obtained along with the appropriate numerical illustrations.
Multi-period demand models are studied using truncated exponential
distribution and using the renewal theory with Nth epoch demand approach.
Also, the generalised gamma distribution with Bessel function and
exponential order statistics is analysed for its stochastic behaviour. The
overall objective of this study is to derive the optimal stock level or the
optimal reorder level. Obtaining the optimal expected cost is very important,
since it is cost effective. Hence, the optimal expected cost is derived along
with the appropriate numerical illustrations.
vii
TABLE OF CONTENTS
CHAPTER NO. TITLE PAGE NO.
ACKNOWLEDGEMENT v
ABSTRACT vi
LIST OF TABLES xiv
LIST OF FIGURES xv
LIST OF SYMBOLS xvii
LIST OF ABBREVATIONS xviii
1 INTRODUCTION 1
1.1 OPERATIONS RESEARCH 1
1.2 THE INVENTORY THEORY 3
1.3 DEFINITION 3
1.4 CLASSIFICATION OF INVENTORY CONTROL
MODEL 8
1.5 CLASSIFICATION OF CLASS OF INVENTORIES 9
1.6 OPTIMIZATION OF AN INVENTORY PROBLEM 10
1.7 STOCHASTIC PROCESS 12
1.8 SELECTING A DISTRIBUTION 14
1.9 STOCHASTIC INVENTORY MODEL 15
1.10 PRELIMINARY CONCEPTS AND RESULTS 16
viii
CHAPTER NO. TITLE PAGE NO.
1.11 ARRANGEMENT OF THE CHAPTERS 19
2 LITERATURE OVERVIEW 23
2.1 INTRODUCTION 23
2.2 EOQ MODELS 23
2.3 HIDDEN MARKOV MODELS(HMMS) 25
2.4 ORDER STATISTICS 26
2.5 SINGLE-PERIOD MODELS 27
2.5.1 Newsboy problem 28
2.5.2 Base-stock systems 31
2.6 MULTI-PERIOD DEMAND MODELS 34
2.7 GENERAL OVERVIEW 39
3 SINGLE PERIOD NEWSBOY PROBLEM WITH
STOCHASTIC DEMAND AND PARTIAL
BACKLOGGING
43
3.1 INTRODUCTION 43
3.2 ASSUMPTIONS AND NOTATIONS 44
3.3 BASIC MODEL 46
3.4 FINITE PROCESS INVENTORY MODEL USING
SCBZ PROPERTY 47
3.5 OPTIMALITY OF TOTAL EXPECTED COST USING
ix
CHAPTER NO. TITLE PAGE NO.
SCBZ PROPERTY 51
3.5.1 Numerical illustration 56
3.5.2 Inference 57
3.5.3 Numerical illustration 58
3.5.4 Inference 59
3.6 OPTIMALITY FOR HOLDING COST USING SCBZ
PROPERTY 60
3.6.1 Numerical illustration 62
3.7 OPTIMALITY IN CASE OF PLANNED SHORTAGE
USING PARTIAL BACKLOGGING 64
3.7.1 Inference 67
3.8 GENERALIZATION OF NEWSBOY PROBLEM
WITH DEMAND DISTRIBUTION SATISFYING THE
SCBZ PROPERTY 68
3.8.1 Basic Model 68
3.8.2 Numerical illustration 74
3.8.3 Inference 76
3.9 CONCLUSION 77
4 TRUNCATED DEMAND DISTRIBUTION AND
RENEWAL REWARD THEORY IN SINGLE
PERIOD MODEL 78
x
CHAPTER NO. TITLE PAGE NO.
4.1 INTRODUCTION 78
4.2 OPTIMAL HOLDING COST USING THE
TRUNCATED EXPONENTIAL DISTRIBUTION 79
4.3 RENEWAL REWARD SHORTAGE AND PARTIAL
BACKORDERING 82
4.3.1 Numerical illustration 88
4.3.2 Inference 88
4.3.3 Numerical illustration 87
4.3.4 Inference 89
4.4 CONCLUSION 90
5 BASE-STOCK SYSTEM FOR PATIENT
CUSTOMER WITH DEMAND DISTRIBUTION
UNDERGOING A CHANGE 91
5.1 INTRODUCTION 91
5.2 ASSUMPTIONS 94
5.3 NOTATIONS 94
5.4 ERLANG2 DISTRIBUTION FOR OPTIMAL BASE
STOCK 94
5.4.1 Numerical illustration 97
5.4.2 Inference 98
xi
CHAPTER NO. TITLE PAGE NO.
5.4.3 Numerical illustration 98
5.4.4 Inference 99
5.5 TRUNCATED EXPONENTIAL DISTRIBUTION FOR
PATIENT CUSTOMER 100
5.5.1 Numerical illustration 104
5.5.2 Inference 104
5.6 CONCLUSION 104
6 BASE STOCK IMPATIENT CUSTOMER USING
FINITE-HORIZON MODEL 105
6.1 INTRODUCTION 105
6.2 ASSUMPTIONS 105
6.3 OPTIMIZING THE NUMBER OF BEDS 106
6.3.1 BASIC MODEL 107
6.3.2 Numerical illustration 110
6.3.3 Cost model 110
6.3.4 Inference 112
6.4 BASE STOCK MODEL FOR IMPATIENT
CUSTOMERS WITH VARYING DEMAND
DISTRIBUTION 113
6.5 CONCLUSION 116
xii
CHAPTER NO. TITLE PAGE NO.
7 THE MULTI-PERIOD MODEL WITH TWO
VARYING DEMANDS 118
7.1 INTRODUCTION 118
7.2 BASIC MODEL 119
7.3 NOTATIONS AND ASSUMPTIONS 121
7.4 THE MULTI-DEMAND TRUNCATED
EXPONENTIAL DISTRIBUTION 122
7.4.1 Numerical illustration 124
7.4.2 Inference 123
7.4.3 Numerical illustration 126
7.4.4 Inference 127
7.5 NTH EPOCH TWO COMMODITY MODEL 128
7.5.1 Conclusion 131
7.6 GENERALIZED GAMMA BESSEL MODEL 131
7.6.1 Basic model 132
7.6.2 Numerical illustration 135
7.6.3 Inference 136
7.6.4 Numerical illustration 136
xiii
CHAPTER NO. TITLE PAGE NO.
7.6.5 Inference 136
7.7 A MULTI-COMMODITY EXPONENTIAL ORDER
STATISTICS 137
7.8 CONCLUSION 140
8 CONCLUSION 141
8.1 SUMMARY 141
8.2 SCOPE FOR FURTHER WORK 144
REFERENCES 146
TECHNICAL BIOGRAPHY 153
xiv
LIST OF TABLES
TABLE NO. TITLE PAGE NO.
1.1 Classification of inventories 10
3.1 Numerical tabulation for obtaining optimal
supply 57
3.2 Comparative result for supply size 59
3.3 Load of data for 1 to 6 tonnes for finding 63
3.4 variation for obtaining 74
3.5 variation for obtaining 75
3.6 variation for obtaining 75
4.1 Optimal profit for increasing value 87
4.2 Optimal profit for decreasing value 89
5.1 Shortage variability for base stock 98
5.2 Holding variability for base stock 99
5.3 Optimal base stock case with varying 103
6.1 Number of beds and queue characteristics
corresponding to 110
6.2 The value of average cost per unit time 111
7.1 Numerical value for for obtaining 124
7.2 Tabulation for obtaining
127
xv
LIST OF FIGURES
FIGURE NO. TITLE PAGE NO.
1.1 Lot-size model with shortages allowed 5
1.2 A sample path of the (environment–inventory)
process 6
1.3 Classification of inventory control model 8
3.1 Shortage curve under negative inventory level 48
3.2 Supply size against the time t 52
3.3 Expected profit curve 57
3.4 Optimal supply vs truncation point 58
3.5 Comparative graph 59
3.6 Supply against holding cost 63
3.7 State space for the expected total profit 64
3.8 Supply against curve 74
3.9 Curve for supply against 75
3.10 Curve for supply and truncation point 76
4.1 Truncation point with respect to the optimal cost 88
4.2 Supply curve when is varied with respect to
89
5.1 Base-stock with the shortage cost 98
5.2 Base-stock with holding cost 99
5.3 Base-stock curve for truncation point 104
xvi
FIGURE NO. TITLE PAGE NO.
6.1 Actual cost per bed 112
6.2 Indifference curve for the optimal number of
beds 112
7.1 Optimal expected ordering when the truncation
occurs 119
7.2 Optimal supply against the truncation point
when 124
7.3 Optimal supply against the truncation point
when 127
7.4 Optimal profit curve with respect to arrival of
demand 135
7.5 Lead time with optimal supply size 136
xvii
LIST OF SYMBOLS
- The cost of each unit produced but not sold called holding
cost.
- The shortage cost arising due to unsatisfied demand.
- Random variables denoting the demand
- Truncation Point
- Supply level and is the optimal value of .
- Base Stock
T, t - Total Time interval
- Time interval with respect to the shortage and holding cost
- The probability density function.
- Probability density functions when .
- Probability density function when > .
- Parameter prior to the truncation point
- Parameter posterior to the truncation Point .
- Inventory level at the time t
- Total expected cost
- Optimal expected cost
- n-fold convolution of , Cumulative distribution
L - Mean lead time
xviii
LIST OF ABBREVATIONS
SCBZ - Setting the Clock Back to Zero property
PDF - Probability Distribution Function
CDF - Cumulative Distribution Function
LMP - Lack of Memory Property
GLD - Generalised Lead time Demand
CTMC - Continuous Time Markov Chain
EOQ - Economic Order Quantity
PH - PHase type distribution
NPV - Net Present Value
KKT - Karush Kuhn Tucker(KKT)
IFR - Increasing Failure Rate
PR - Protection lost sale
1
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.
2
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
3
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 ( ): 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 c , 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 ( ): 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,
4
and the latter is called a backorder. Although lost sales are often important in
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 ( ): 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 and its function is given as .
4. Setup cost ( ): 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 ( ): 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
is the unit cost and is the amount ordered.
6. Demand rate ( ): This is the constant rate at which the product is
withdrawn from inventory.
7. Order level ( ): The maximum level reached by the inventory is the order
level. When backorders are not allowed, this quantity is the same as .
When backorders are allowed, it is less than .
8. Cycle time ( ): The time between consecutive inventory replenishments is
the cycle time.
9. Cost per time ( ): This is the total of all costs related to the inventory
system that are affected by the decision under consideration.
10. Optimal Quantities ( ): 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 and occurs when the order arrives. The maximum
backorder level is – and backorder is represented in the figure 1.1 by a
negative inventory level.
5
Figure 1.1 Lot-size model with shortages allowed
12. Random Variable: A random variable, usually written as , 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 ( ): 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 ( ): When
demand is assumed to be a discrete random variable, ) gives the
probability that the demand equals .
6
17. Discrete Cumulative Distribution Function ( ): The probability that
demand is less than or equal to b is when demand is discrete then
(1.1)
18. Continuous Demand Probability Density Function ( ): When
demand is assumed to be continuous, is its density function. The
probability that the demand is between and is
(1.2)
When the demand is assumed to be nonnegative, then is zero for
negative values.
19. Continuous Cumulative Distribution Function ( ): The probability
that demand is less than or equal to when demand is continuous then
(1.3)
20. Standard Normal Distribution Function and : 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
A sample path of the environment-inventory level process of K. Yan et.al
[79] is illustrated in figure 1.2, where is the production rate and is
7
demand rate are associated with each state of the inventory system. ‘ ’ is
taken as the supply in the interval . 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
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
8
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
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
9
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
The classification of the class I, II, III, IV and V of inventories are
discussed in form of Table 1.1.
10
Table 1.1 Classification of inventories
Class Inventory Supply process Demand
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
(1.4)
Where
= supply rate / unit time
= demand rate / unit time
= 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 and 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.
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
11
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
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
12
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
may be represented as a function of the mean inventory level. This mean
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
13
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
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.
14
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.
1.8 SELECTING 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
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.
15
1.9 STOCHASTIC 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 . During some interval of time, the demand
is a random variable with PDF and CDF . The mean and
standard deviation of this distribution are and respectively. With the given
distribution, the probability of a shortage and the probability of excess
are computed. For a continuous distribution, and is given as
(1.5)
(1.6)
In some cases it may be interesting to obtain expected
shortage . This depend on whether the demand is greater or less than
Items short =
(1.7)
Then is the expected shortage and is
(1.8)
Similarly for excess, the expected excess is
(1.9)
Also the expected excess can be represented in terms of
-
(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.
16
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 variable is denoted as . For every
probability distribution there are corresponding one or more parameters. The
corresponding distribution function is denoted as , and
is called the survivor function and it gives the probability that a random
variable .
For example, if a random variable is distributed as exponential with
parameter then . 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 , (where is the space parameter) is
said to have the ‘Setting the Clock Back to Zero’ (SCBZ) property if
remains unchanged except for the value of the parameters under the
following three cases,
(i) Truncating the original distribution at some point
(ii) Considering the observable distribution for inventory control
and
(iii) Let be a truncation point and be fixed. If
, then
When
When (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.
17
2. CHANGE OF DISTRIBUTION AT A CHANGE POINT: The concept of
SCBZ property indicates that a random variable with density function
undergoes a parametric change after a certain value of say 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 is a random variable denoting the life time of the
component and 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 has a PDF with CDF , whenever
and it has PDF with CDF if . Here is called the
change point. It can be noted that
(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 is a
random variable with exponential Probability Density Function (PDF) of mean
(
) then the PDF of the random variable is truncated on the right at is
given by Deemer W.L et.al [18] and the maximum likelihood estimator of the
parameter is derived in the form of truncated exponential distribution as
(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
18
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
(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
(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
. In stock level is given as 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
19
statistics is independent and the key to model when there is joint PDF
is
(1.16)
Suppose are the order statistics of a random variable of
size n arising from along with the distribution of the form
(1.17)
Then will constitute the renewal process. Considering the joint probability
density function of all order to be given by
(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.
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
20
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.
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.
21
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.
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 has the form
where each of it is continuous and differentiable. The function
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
( an inventory model is discussed under the following
assumptions regarding the model.
22
(i) There is a onetime supply at the start of the period .
(ii) The demands occur at random epochs in and the
magnitudes of the demands are random variables denoted as
(iii) If the cumulative demand , then salvage occurs and
if then stock-out occur during . The random variable
representing demand namely has PDF and CDF
is identically independently distributed random variables. This chapter the
demand and lead time is considered a constant and a random variable. By
assuming exactly demand epochs in , and using renewal theory the
optimal value of is obtained. Another extension discussed in this chapter is
by the assuming that the random variable has a distribution initially but
there a change of distribution after a truncation. The optimal one time supply
during the interval 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.
In Chapter 8, a brief summary of the results and conclusions drawn
hereby are furnished.
23
2. LITERATURE OVERVIEW
2.1 INTRODUCTION
In any research study, the work done in the past is of great importance
since it forms the foundation of the work to be carried out in future. It is
rather a continuous process. While carrying out any research the different
types of problems taken for investigation and the various approaches to solve
the problem are all quite important. Hence, a brief idea of the work done in
the past should all be clearly stated. In this chapter, the development of
inventory control theory is reviewed through stochastic techniques. Since the
development of inventory control has been over many decades, it is
necessary to cover relevant research papers. Some selected research
papers which are of greater importance with possibility of practical
applications are taken up for review.
Application of mathematics, statistics and a stochastic process has
contributed to the development of many models, which has real life
applications. Also separate set of models has been developed for the
determination of optimal re-order size for perishable products such as
vegetables, fruits, eatables, and drugs. Proceeding below is few of the
literature related to the models under study.
2.2 EOQ MODELS
The Inventory control is a major discipline of operations research and
the concept of optimization forms the basis for inventory control theory. The
overall aim in most of the problems in the existing literature was to determine
the optimal reorder quantity and the optimal lot size etc. The famous EOQ
formula from Whitin T.M [77] has given an insight on the basic model for the
determination of the optimal reorder quantity. This formula provided the base
for many of the inventory models which have been developed subsequently
and during the subsequent years, different authors have contributed many
relative versions of EOQ model.
24
It is interesting to note that Goh M [23] has discussed the concept of
EOQ models in demand and holding cost. EOQ formula in the conventional
model was derived with the assumption that the holding cost was fixed. In
this model, the holding cost was assumed to be a variable and the demand
rate depended upon the inventory level and the demand rate was considered
as deterministic and known function of the level of inventory. The concave
polynomial function defined in this model is
(2.1)
is considered as constant, as the inventory level and also as a shape
parameter and is considered as on hand inventory level. Under these
assumptions the two cases discussed in this model are instantaneous
replenishment with non linear time dependent holding cost and instantaneous
replenishment with non linear stock dependent carrying cost.
From the past few decades, researchers have attempted many
variations of EOQ models. Brill Percy H et.al [10] has developed an EOQ
model with random variations in demand and adopted a system point level
crossing theory for the formulation of system of equations in this model. The
objective of this model was to explore the implications of demand disruptions.
Demand rates assumed in this model were and 0. Here was
considered to be the difference between the supplies and demand where
demand and was the supply size. The markov
process was continuous time markov chain and was
considered as sojourn times for the three states 1, 2 and 3 and exponential
with parameters 1, 2 and 3. Salameh M.K et.al [61] developed an
economic order quantity model for the case where a random proportion of the
items in a lot are defective.
The concept of uncertainty was first introduced in EOQ model by
Arrow K.J et.al [6]. It was a generalized model and many other inventory
models were proved as a special case of this model. In this model, n -
periods were taken and the reordering decisions were considered at n -
different points called the checking points. Here the demand was assumed to
be a random variable ' 'r and these random variables for the n -periods were
25
taken to be identically Independent distributed random variables. The lead-
time was taken to be zero and shortages occurred in one period were taken
to the next period. Under these assumptions the expressions for the
expected cost was given in the form of where is
stock on hand before ordering, is the quantity to be ordered and is the
demand. The optimal policy was given as .
The base of inventory was founded through the model of EOQ. Hence
one of the seminal work on EOQ model using policy was by Axsater S
[9]. The model was considered a single-item continuous review inventory
system with stationary stochastic demand and when the inventory position
were dropped down to or below, a number of lot size were ordered so
that the inventory position balances . The author has discussed the case
where the lead time demand is deterministic and has taken up an
improvement over this model using the time homogeneous markov process.
This model was solved by approximating its stochastic demand by finding its
mean and the order quantity using the conventional EOQ formula.
Yan K et.al [79] considered a single stage production inventory system
whose production and demand rates were modulated by an environment
process modeled as a finite state Continuous Time Markov Chain (CTMC).
When the inventory level reached zero, an order was placed from an external
supplier, and it arrived instantaneously. The authors derived an Economic
Order Quantity (EOQ) policy that minimizes the long run average cost, if one
replaces the deterministic demand rate by the expected demand production
rate in steady state and extended the model with backlogging.
2.3 HIDDEN MARKOV MODELS (HMMS)
Following the above discussed policy, the next model in review was
developed by Metin Cakanyyildirim et.al [43]. In this model, the author
considered a continuous review inventory model with random lead times
which depended upon the lot size and a policy in which stands for
the order quantity and stands for the reorder point. Also the demand rate
26
was taken to be a constant and lead-time was assumed to be a random
variable. Here the decision variable namely and were derived in a closed
form and the expression for the expected cost of holding and shortage was
obtained using the renewal reward theorem. This concept of closed form is
studied in chapter 3 and chapter 4 of this thesis and the renewal reward
theorem is studied in chapter 4.
However, Das C [17] analyzed these models using the quadratic
approximation procedure. This model was based on the (Q, r) policy which
also takes into account the time weighted backorders. The author considered
the lead time as known constant and shorter than the time between
successive orders. Also the reorder point was assumed to be non-negative.
In this model demand during the stock out period was completely backlogged
and the stock out cost was assumed to be directly proportional to the amount
as well as the duration of backorders. Under these assumptions, the average
annual cost of the HMMS model II was given as
(2.2)
The exact development of Hidden Markov Models (HMMS) part II for
single item and its extension to multiple items was proposed by Holt C.C et
al. [34]. In single station models, the concept of dynamic models is very
interesting. Several authors have formulated some ordering rules of very
general nature.
2.4 ORDER STATISTICS
The inventory problem in which the replenishment of inventory takes
place from two sources was constituted by Ramasesh R.V et al [56]. In this
model, the concept of macro studies and micro studies has been discussed.
While in the macro studies, the examination of merits of the procurement
policies with cost benefit analysis of dual source competition was considered.
Where else in micro studies, the modeling and optimization of total cost
arising due to reordering inventory holdings and shortages was considered.
The solutions for sole and dual sourcing models were developed in this
27
model and the lead time was considered as stochastic along with the concept
of uniform distribution and order statistics.
Srinivasan Rao S [69] have discussed an inventory model in which the
demand over the time interval (0, t) was taken to be a random variable and a
onetime supply denoted as S was considered. In this model, the demands at
N random epochs in (0, t) were denoted as random variables X1, X2… XN.
It was assumed that the random sample of N observations on demands were
taken and arranged in increasing order of magnitude. It is to be noted that
X(1) was the first order statistic and X(n), the nth order statistic. Using the
distribution of X(1) and X(n), the optimal size of the supply has been
determined.
2.5 SINGLE-PERIOD MODELS
In case of single-period model, which is discussed above the role of
inventory problem by the lead time is important factor. Hence Cawdery M.N
[11] have discussed the role of time inventory control problem by the
assuming the lead time, lead time demand and influenced many of the result
of inventory control. In many inventory problems the lead time was taken to
be random variable and the lead time demand were also considered a
random variable.
The authors in their work compare the lead time demand to the
number of customers arising in a single server queuing system during the
service time and hence considered the correlation between the lead-time
distributions and the lead- time demand distributions. Assuming that the lead-
time was not correlated with the consumption of the stock, the authors have
derived the expression for the variance of demand during the lead-time. They
have also considered a model called the stock control model in which the
expression for the cost was derived by the suitable determination of
economic batch quantity denoted as . The optimal re-ordering policy or the
optimal stock size was determined accordingly. Chapter 6 of this thesis is a
motivated work from this model.
28
It is interesting to note that a very general model to form the ordering
rule has been formulated by Dvoretsky A et.al [20]. In this model, the authors
have considered dynamic single station model with finite number of decision
intervals. The demand in any period ' 'i was given by a conditional probability
distribution where the vector was a
summary description of the history of demands and the stock level before
and after ordering , including the stock levels and of the present period
. Consequently, the expected cost in period was taken as a function of the
form , where . An
ordering policy was defined by a set function , = 1, 2, 3, …, n
which obeyed the restrictions , = 1, 2, 3, …, n and the expression for
the optimal supply size was obtained.
Many inventory models of classical nature have been under the
assumption that the demand and lead time are of deterministic nature.
Considerable changes in the models have been introduced only by assuming
that the variables like demand, lead time are stochastic in nature. In this
context Newsboy problem and Base-stock system inventory models have
been developed which are known as the stochastic models. Also this
approach is discussed as chapter 6.
2.5.1 Newsboy problem
An interesting inventory model is the so-called Newsboy problem in
which a somewhat different concept is introduced. In most of the models, the
inventory on hand is such that it can be kept as a stock for any period of time
and hence it is called an infinite process. However, a somewhat different
type of problem that arises is that, the inventory processes are terminated
after a finite period and several models have been developed using this
concept.
A typical example of this type of model is the so called newsboy
problem in which the newspaper supplied at the beginning of the day is sold
and the demand for the same is a random variable. The unsold papers will
be called the wastage and there is an associated cost for the same. If the
29
supply is less than the demand, again there is shortage cost. The
determination of the optimal one time supply is to be determined. Based on
this concept several authors have attempted these inventory models of
similar type which are discussed in succeeding part of this chapter.
The application of the newsboy problem to quality control and
container fill was studied by John S. Rose [36]. The conventional newsboy
problem was considered as a one, in which there is a onetime supply with the
demand for the product being a random variable, and the holding cost and
salvage cost are known quantities. This type of model is known as finite
process model and chapter 3 is on the study of this concept. The reversal of
the conventional newsboy problem is taken up where the demand is
assumed to be known but the replenishment quantity is a random variable
which is absolutely continuous. Assuming the material cost ( ), shortage cost
( ) and inventory holding cost ( ) and also the average of the
replenishment quantity the expected cost denoted in this model is follows
(2.3)
Where was taken as support of , , , as the quantity
of demand and was considered the quantity of replenishment. This model
was a complete reverse of the conventional newsboy problem as discussed
by Hanssman F [33]. The author has attempted to determine the optimal
value of . By considering the distribution of control variable which is denoted
as and the family of absolutely continuous distributions, the optimal value
of the replenishment has been derived and in doing so the cost function was
taken to be normally distributed with mean and standard deviation . The
expected cost function was given as
(2.4)
In order to minimize , the author attempts to find and such that
is minima. This approach differ from the conventional newsboy problem in
the sense that in the conventional model the optimal supply size was
30
determined and in there model, the optimal demand size was determined.
The author has also discussed the asymptotic behaviour of the optimal
solution and has obtained a special character for the determination of optimal
demand distribution based on the mean and variance
In the category of newsboy or newsvendor problem an intergraded
model was developed by Liang-Yuh Ouyang et.al [39]. The authors have
discussed an integrated Vendor Buyer inventory model with quality
improvement and lead time reduction and also have discussed the
advantages of the just – in – time (JIT) production. The concept of JIT is
directed towards the shortening of the lead time and improving the quality of
the product. In the previous model of classical inventory theory, it has been
implicitly assumed that the quality level of the product is fixed at an optimal
level and all the items are assumed to have perfect quality. But in real
production environment it can be observed that there may be defective items
and these items are rejected, repaired and reworked or refunded to the
customers. In all such cases substantial costs were incurred. Therefore
investing capital on quality improvement will reduce this kind of cost. Hence,
the authors have formulated a single vendor, single buyer inventory model
with quality issue and lead-time reduction. It was a non-linear programming
model in which minimizing the total cost was attempted using algorithms.
Numerical example by assuming specific values of the cost component was
also provided.
The motivational base work on newsboy was from Sehik Uduman P.S
et.al [65]. In this work it was shown that the newsboy inventory model with
demand satisfied the so called SCBZ property. Since the newsboy problem
is an inventory model with a finite process, this implies that the product in
question can be sold only for a finite duration, after which the product cannot
be used and so it has only a salvage cost.
The similar situation exists in the case of newspapers. The newspaper
of the day should be sold within the same day itself. It cannot be sold the
next day as it has only the value of a waste paper. If the supply is
inadequate, the shortage cost arises. So the determination of the optimal
supply size is important. The authors have taken up this problem under the
31
assumption that the random variable denoting the demand for newspaper is
such that it satisfies the SCBZ property. Under these assumptions, the
optimal supply size has been determined. Hence this thesis involves this
concept in chapter 3.
Grubbstrom R.W [25] provided a compound variation of the newsboy
problem. Instead of demand simply being known as to its distribution, here
demand was generated by customers arriving at different points in time
requiring amounts of varying size. Customer arrival followed a renewal
process, and an amount required was taken from a second independent
distribution. It was shown, how the optimal purchase quantity in explicit form
depends on properties of the two distributions, maximising the expected net
present value (NPV) of the payments involved. The development was to use
this relation between the NPV and the Laplace transform and also
simultaneously using the Laplace transform as a moment-generating
function.
The work on generalization was reviewed in Kumaran M et.al [38] and
the lead time was considered to be the random variable on the basis of the
generalized (- type) (GLD). The concept of GLD has been introduced by
Ranboy Schmeiser in the year 1974.The GLD distribution can be applied
whenever a complete and precise knowledge of the distribution of random
variable is not available. The Pth quantile denoted as R(p) was considered
and it was based on 1 and 2 called the location and scale parameters. By
taking into account the setup cost, purchase cost, salvage cost, penalty cost,
the selling price, and the expression for the expected profit and loss have
been constituted. The optimal size to be produced has been determined.
2.5.2 Base-stock systems
Gaver D.P [21] developed a model on the so-called base-stock level
inventory. In this model, a given period was taken and subdivided into
smaller intervals of equal length and the demand during each sub-interval of
time was taken to be a random variable. The optimal value of the base-stock
was derived and in doing so, the author has considered the stationary
32
distribution of available inventory when the customers wait. It is also
interesting to note that the author has considered the stationary distribution
function of available inventory when the customers are impatient.
The motivating contribution on the base-stock system for patient
customers is studied by Ramanarayanan R [54, 55]. In this model the inter-
arrival times between successive demand epochs were taken to be random
variables which were identically distributed but not independent and were
shown to be constantly correlated random variables. In this model, under
these assumptions the optimal base-stock levels have been derived by using
the distribution of sum of correlated random variables which was discussed
by Gurland J [28]. Markus Ettl et.al [42] had modelled a supply network with
base-Stock Control and Service Requirements. Dong-Ping Song [19] had
discussed the stability and optimization of a production inventory system
under prioritized base-stock control.
It is common for suppliers operating in batch production mode to deal
with patient and impatient customers. Haifeng Wang et.al [31] considered the
inventory models in which a supplier provides alternative lead time to its
customers, a short or a long term lead time. In this model orders from patient
customers were taken by the supplier and included in the next production
cycle while orders from impatient customers were satisfied from the on-hand
inventory. In their model, the action to commit one unit of on-hand inventory
to patient or impatient customers was denoted as the inventory commitment
decision and the initial inventory stocking as the inventory replenishment
decision. They first characterized the optimal inventory commitment policy as
a threshold type and then proved that the optimal inventory replenishment
policy to be a base-stock type.
This model was extended to analysis a multiple cycle setting, a supply
capacity constraint and the online charged inventory holding costs. Haifeng
Wang et.al [31] also evaluated and compared the performance of the optimal
inventory commitment policy and the inventory rationing policy. Finally, they
further investigated the benefit and pitfall of introducing an alternative lead
time choice and they used the customer choice model to study the demand
33
gains and losses known as demand induction and demand cannibalization
effects.
The analysis of the base-stock control production inventory system
using queuing theory was discussed by Sandeep Jain et.al [62].They have
considered a production inventory system which consists of a manufacturing
plant and a warehouse. The demands from the customers were supplied
from the inventory in the warehouse and the demand orders from the
customer arrival accordingly as Poisson process.
In this model the finished goods inventory was considered as the
base-stock and its level fixed at K. The finished goods inventory was well
defined and each finished goods inventory was attached with production
authorization card. The expression for total cost K which is the base-stock
level at the warehouse has been obtained. Assuming the arrival process to
be Poisson the optimal value of K has been determined.
Optimal reorder size is an important parameter of interest. The model
on the determination of optimal reorder quantity has been discussed by
Hanssman F [33]. Ramanarayanan R [55] has discussed an inventory model
based on the Markov processes. The essential difference between this model
and the conventional model is that, it uses the phase type (PH) distribution
for the representation of the lead time distributions. In this model, it was
assumed that the demands occurred according to a Poisson process with
parameter and rate of demand was one unit at a time. The inventory
capacity was denoted as ‘S’ and the reorder level as ‘s’. In this model the
explicit steady state solution has been derived and it gave a reordering rule
at different points of the demand epochs. The concept of phase type
distribution and its applications have been studied by Neuts.M.F [48] and a
variation of phase type distribution is attempted in chapter 6.
Chenniappan P.K et.al [13] have considered a new type of an
inventory situation. In this model, the inventories are kept as two different
stocks i.e., when a demand occurs one unit from each of the two inventories
is sold. The model was such that the order for the first product is supplied
along with the second product. Sometimes the first product alone is supplied
34
without the second product. For example, computers are sold with or without
a printer. The following distributions of the inter-arrival times between
demands which were considered are exponential distribution and general
distributions and the steady state probabilities for the inventory levels were
derived. They have used the matrix geometric method as proposed by Neuts
M.F [48]. It may be noted that the scope of this problem was to find out
probability of different inventory levels.
So far the review of literature discussed in this chapter involved the
single period models and its real time applications. Now the proceeding
literature review involves the multi-period models and their real time
application.
2.6 MULTI-PERIOD DEMAND MODELS
Ata Allah Taleizadeh et.al [8] have considered a multi-product
inventory control problem in which, the periods between two replenishments
of the products were assumed independent random variables. The increasing
and decreasing functions were assumed to model the dynamic demands of
each product. Furthermore, the quantities of the orders were assumed
integer-type, space and budget as constraints, the service-level was
considered as a chance-constraint, and that the partial back-ordering policy
was taken into account for the shortages. This model was an integer
nonlinear programming type and to solve it, a harmony search approach was
used. At the end, three numerical examples of different sizes are given to
demonstrate the applicability of the proposed methodology in real world
inventory control problems, to validate the results obtained, and to compare
its performances with the ones of both a genetic and a particle swarm
optimization algorithms.
Guray Guler M [27] analyzed a periodic review inventory system in
which the random demand was contingent on the current price and the
reference price. The randomness was considered due to additive and
multiplicative random terms. The objective of the model was to maximize the
discounted expected profit over the selling horizon by dynamically deciding
35
on the optimal pricing and replenishment policy for each period. The author
studied three key issues using numerical computation and simulation. First
was the study on the effects of reference price mechanism and the total
expected profit. It was shown that high dependence on a good history
increases the profit. Second was the investigation on the value of dynamic
programming and it was shown that the firm that ignores the dynamic
structure suffers from the revenue. Third was the analysis on the value of
estimating the correct demand model with reference effects. It was observed
that this value is significant when the inventory related costs are low.
Mirzazadeh A [44] has analyzed a complex inventory system under
uncertain situations. In this model, the item deterioration has been
considered and the shortages were allowable. The objectives of the model
were the minimization of the total present value of costs over time horizon
and decreasing the total quantity of goods in the warehouse over time
horizon. In this model the inventory system was considered in a bi-criteria
situation and lead time was negligible. Also, the initial and final inventory
level was zero and the demand rate was known and constant. Where else
the Shortages were allowed and fully backlogged except for the final cycle.
The replenishment was instantaneous and lead time was zero and the
system was operated for prescribed time-horizon of length H and finally a
constant fraction of the on-hand inventory deteriorated per unit time. Hence
the solution obtained from this model is as follows
(2.5)
This article presented inspection scenarios for the multi-objective
multi-constraint mixed backorder and lost sales inventory model with
imperfect items. There were two inspection scenarios which are the imperfect
items observed during inspection and screenings are either all reworked or
all discarded.
36
In order to fit some real environment, this study assumed the
maximum permissible storage space and available budget were limited.
Backorder rate was considered as a function of expected shortages at the
end of cycle. Stochastic inflationary conditions with a probability density
function were also considered in the presented model. This study assumed
that the purchasing cost is paid when an order arrives at the beginning of the
cycle, and the ordering cost is paid at the time of the order placing. The
aggregate demand followed a normal distribution function. Finally, a solution
procedure was proposed in order to solve the discussed multi-objective
model. In addition, numerical examples were presented to illustrate the multi-
objective model and its solution procedure for different inspection scenarios,
and a sensitivity analysis is conducted with respect to the important system
parameters. The objective of this model was to minimize expected annual
cost and variance of shortages.
Roger D.H et.al [57] has discussed a multi-echelon (multilevel)
inventory model and newsboy problem for obtaining the optimal solution. It is
very common that the inventory may be at different levels of a production
system and the centralized decisions for the location and control of
inventories is an important aspect. The inventories that are to be maintained
at different levels of a production oriented system are very important. The
determination of the optimal inventory arises at different locations and at
each level the demand may be different. The demand function for the
common component was following normal distribution N (u, 2). The optimal
values of the decision variables were obtained by taking a Hessian matrix (H)
and using the Lagrangian Multiplier technique.
Haifeng Wang et.al [30] have considered a multi-period newsvendor
problem with partially observed supply capacity information which evolved as
a Markovian Process. The supply capacity was fully observed by the buyer
when the capacity was smaller than the buyer's ordering quantity. Otherwise,
the buyer knew the current-period supply capacity was greater than its
ordering quantity. The buyer updates the future supply capacity forecasting
accordingly and it was observed that the optimal order quantity was greater
37
than the myopic order quantity. Using dynamic programming formulation the
existence of an optimal ordering policy was derived.
Zohar M.A. Strinka et.al [80] have studied a class of selective
newsvendor problems, where a decision maker has a set of raw materials
each of which can be customized shortly before satisfying demand. The goal
was then to select which subset of customizations maximizes expected profit.
It was shown that certain multi-period and multi-product selective
newsvendor problems fall within this problem class. Under the assumption
that the demands were independent and normally, but not necessarily
identically distributed, it was shown that some problem instances from this
class can be solved efficiently using an attractive sorting property that was
also established in the literature for some related problems.
For a general model, the Karush-Kuhn-Tucker (KKT) condition was
used to develop an exact algorithm that is efficient in the number of raw
materials. In addition, a class of heuristic algorithms was developed. From
the numerical study, the performance of the algorithms was evaluated and it
was shown that the heuristic have excellent performance and running times
as compared to available commercial solvers. A considerably more limited
case, not including any stochastic intensity, has been reported by the current
author Grubbstrom R.W. [25].
Newsboy models have wide applications in solving real-world
inventory problems. Shih-Pin Chen et.al [66] analyzed the optimal inventory
policy for the single-order newsboy problem with fuzzy demand and quantity
discounts. The availability of the quantity discount caused the analysis of the
associated model to be more complex, and the proposed solution was based
on the ranking of fuzzy numbers and optimization theory. By applying the
Yager ranking method, the fuzzy total cost functions with different unit
purchasing costs were transformed into convex, piecewise nonlinear
functions. In this model by proving certain properties of the ranking index of
the fuzzy total cost, several possible cases were identified for investigation.
After analyzing the relative positions between the price break and the
minimum of these nonlinear functions, the optimal inventory policy was
38
provided and closed-form solutions to the optimal order quantities was
derived. Several cases of a numerical example were solved to demonstrate
the validity of the proposed analysis method. The advantage of using the
proposed approach was also demonstrated by comparing it to the classic
stochastic approach. It was clear that the proposed methodology is
applicable to other cases with different types of quantity discounts and more
complicated cases.
In Valentín Pando et.al [73], a generalization of the newsboy problem
was presented, where an emergency lot can be ordered to provide a certain
fraction of shortage. This fraction was described by a general backorder rate
function which was non-increasing with respect to the unsatisfied demand.
An exponential distribution for the demand during the selling season was
assumed and an expression in a closed form for the optimal lot size and the
maximum expected profit was obtained. A general sensitivity analysis of the
optimal policy with respect to the backorder rate function and the parameters
of the inventory system were developed. When the backorder rate function
was described by some particular functions, its behaviour was analyzed with
respect to changes in the parameters. To illustrate the theoretical results,
some numerical examples were also given in this model.
Nicholas A.Nechval et.al [49] had shown how the statistical inference
equivalence principle could be employed in a particular case of finding the
effective statistical solution for the multiproduct newsboy problem with
constraints. Snyder L.V et.al [68] had simulated inventory systems with
supply disruptions and demand uncertainty. Also, this model showed a study
on how the two sources of uncertainty can cause different inventory designs
to be optimal. Dada M et.al [16] had extended the stochastic demand
newsboy model to include multiple unreliable suppliers. Guiqing Zhang et.al
[26] had considered the newsboy problem with range information. In Jixan
Xiao et.al [35] a stochastic newsboy inventory control model was considered
and it was solved on multivariate product order and pricing.
39
2.7 GENERAL OVERVIEW
Ozalp Ozer et.al [51] have discussed the problem of dual purchase
contract systems in which a new contract form with the manufacturer can:
i) Push inventory to the retailer, known also as channel stuffing.
ii) Create a strict Pareto improvement over the whole sale price contract
while inheriting the whole sale price contract’s simplicity, and
iii) Reduce the manufacturer’s profit variability.
To do so, the authors have proposed a dual purchase contract that
induces a retailer to place two consecutive orders which is before and after
obtaining the final forecast update. This was essentially a supply chain
problem in which the manufacturer and retailer were in series. The authors
have formulated a demand model where is the forecast of
demand after a market research, is a random error. Assuming that the
random variable has a PDF (.)g and CDF (.)G with Increasing Failure
Rate (IFR), the authors have discussed the maximization of profit of the
retailer and determined the optimal order quantity. Several variations of this
model have been taken up and theorems have been established.
Another interesting model is by Covert R.P et.al [15]. In this model,
the authors have assumed a variable rate of deterioration of the items. A two
parameter Weibull distribution has been used to represent the distribution of
time to deterioration. Using this model, they have derived the optimum cycle
time for reordering with the assumption of associated costs. A generalization
of this model has been attempted by Philip G.C [52].
This model was a generalized version of the Covert R.P et.al [15]
model, where a three parameter Weibull distribution was used to represent
the distribution of time to deterioration. Here three parameters namely =
scale parameter, = shape parameter and = location parameter was
included. The demand was taken to be deterministic and three costs namely
(i) cost of the unit (ii) cost of holding per unit time and (iii) reordering cost was
incorporated into the model. The authors have obtained the optimal values
40
of the optimum cycle time, the economic order quantity and also the total
deterioration during the cycle time.
In Vijaya [74], a study on Greenhouse effect was discussed as one of
the important aspects of global warming relating to increase of temperature.
In this model, it was also discussed that CO2, CO and Nitrogen etc is said to
plays a vital role to hasten the process of increase in global temperature and
the only source of global warming is CO2 emission. The stochastic models
are widely used in the study of global warming and its consequences but in
this model it was shown that, if the global temperature crosses the threshold
level it will in turn leads to greenhouse effect. The threshold itself was
considered to be a random variable. In this model, the threshold was
considered to satisfy the property known as Setting the Clock Back to Zero
(SCBZ) property and the expected time to sero-conversion and its variance
were derived.
In Murthy S et.al [46], an analysis on (s,S) inventory system was
carried out. In this model, the demand process was assumed to be a single
and bulk demand for entire inventory were the rate of demand had SCBZ
property. Also the lead times and intervals of time between successive
demand were identically independent random variables. Here, the
exponential case of 2 models was discussed. In the first model the unit
demand rate were varying and in second model the bulk demand was
varying. Also in this model the steady state probability vector of inventory
level was obtained through NEUTS matrix.
Sathyamoorthy R et.al [63] have obtained the expected time to recruit
when the loss of manpower is a continuous random variable and the
threshold for loss of manpower is a continuous random variable having SCBZ
property. Here SCBZ property was used instead of exponential distribution
which has lack of memory property and the inter decision times form a
sequence of independent and identically distributed random variables was
derived.
In the area of manpower planning, research has been enormous with
the result that a large number of research works have been published since
41
1970. An interesting paper by Abodunde T.T et.al [3] contains the
discussions about the model were the manpower system with a constant
level of recruitment is considered. It was related to the production planning in
the development of telephone services and linking the same to the workforce.
In this condition the constant level of recruitment was necessary to bring the
number of installations eventually up to their final levels. Also a stochastic
model was developed which evaluated the effect of implementing the
recruitment policies in terms of changing distribution of staff members, and
the changing number of installations with time. Numerical results were
provided.
Friedrich Wilhelm Bessel (1784 – 1846) studied disturbances in
planetary motion, which led him in 1824 to make the first systematic analysis
of solutions of this equation. The solutions became known as Bessel
functions. The solution of bessels function of order zero was given as
(2.6)
This concept of Gamma bessels function is discussed in chapter 7.
Motivated from the model of Nicy Sebastian [50] who introduced a
new probability density function associated with a Bessel function, which is
the generalization of a gamma-type distribution led to the study of chapter 7.
Some of the special cases of this model were also discussed. Multivariate
analogue, conditional density, best predictor function, Bayesian analysis,
etc., connected with this new density were introduced. Suitability of this
density as a good model in Bayesian inference and regression theory was
also discussed. This model involved a different concept of the Bessel
functions along with the gamma distribution which was new approach. The
Bessel function appears in many diverse scenarios, particularly situations
involving cylindrical symmetry.
Recently, the work on the estimation of maximum likelihood of
truncated exponential distributions was carried out by George Lominashvili
et.al [22]. In this model, it was shown that the maximum likelihood equation
for truncated exponential distribution has a unique solution which gives an
asymptotic effective estimator of the parameter. However the applications of
42
the modified generalized gamma distribution in inventory control was studied
by Abd El-Fatah I.M et.al [2].
In their model, the protection lost sale and was determined when
the lead time demand had the modified generalized gamma distribution. By
using the maximum likelihood method, the five unknown parameters were
estimated. Also in this model the protection and the complement of protection
lost sales, the mean and the variance of potential lost sales for the modified
generalised gamma distribution and its special cases were estimated.
Although renewal processes have been related to following models
but the particular treatment given still appears to be untouched until now. To
insert our current contribution into the context of recent literature, it is need to
mention the interest in developments related to the newsboy problem in the
last few years have increasingly focused on aspects of risk. A thorough
literature survey of the newsboy problem was given in Khouja M [37],
containing 92 references. However, the current type of renewal demand and
the use of truncated exponential distribution in the model process appear to
be lacking. Hence, in order to fill this gap many variations of the newsboy
model and base-stock model is studied in this thesis, which are yet
untouched in the literature. Also in this thesis, the gap between literatures of
the SCBZ property is bridged using inventory systems having periodic order
moments in single periodic and multi-period models. This property plays a
crucial role due to its application to real life in both theoretical and applied
work.
43
3. SINGLE PERIOD NEWSBOY PROBLEM WITH STOCHASTIC DEMAND
AND PARTIAL BACKLOGGING
3.1 INTRODUCTION
In today’s highly competitive business environment and inventory
management, the ability to plan and control inventories to meet the
competitive priorities is becoming increasingly important in many types of
organizations. Depending on this, the type of inventory problem frequently
encountered with seasonal or customized products is the newsboy problem,
also called the newsvendor problem or single period stochastic inventory
problem because only a single procurement is made. The typical examples
are the dilemmas of making a one-period decision on the quantity of
newspapers that a newsboy should buy on a given day or the quantity of
seasonal goods that a retailer should purchase for the current year or goods
that cannot be sold the next year because of style changes.
The single period inventory model has wide application in the real world in
assisting the decision maker to determine the optimal quantity to order. This
is one type of inventory problem frequently discussed in the literature. A wide
variety of real world problems including the stocking of spare parts,
perishable items, style goods and special season items offer practical
example of this sort of situation. These types of problems are referred to as
the newsboy problem. Since it can be phrased as a problem of deciding how
many newspapers a boy should buy on a given day for his corner news
stand.
In some real life situation there is a part of the demand which cannot
be satisfied from the inventory and it leaves the system stock-out. If the order
quantity is larger than the realised demand, the items which are left over at
the end of period are sold at a salvage value or disposed off. Hence, both the
factors such as salvage and stock-out situations are equally important. The
basic Newsboy inventory model has been discussed in Hanssman F [33]. If
the demand is uncertain then it must be predicted and the continuous
44
sources of uncertainty or stochastic demand, has a different impact on
optimal inventory settings and prevents optimal solutions from being found in
closed form. Notably, there are cases in which the probability distribution of
the demand for new products is typically unknown because of a lack of
historical information, and the use of linguistic expressions by experts for
demand forecasting is often employed. The Assorted level of demand is
viewed in form of a special class of inventory evolution known as finite
inventory process.
In this chapter, a variation of the finite Inventory process model i.e.,
the classical Newsboy problem is attempted. This variation of the Newsboy
problem discussed has not been investigated earlier in the literature,
although compound demand processes have been studied for a long time. A
closed form is introduced and there are many benefits of having a closed-
form approximate solution. The objective is to obtain the optimal solution in
which the demand is varied according to the SCBZ property. Appropriate
Numerical illustrations provide a justification for its unique existence.
3.2 ASSUMPTIONS AND NOTATIONS
- The cost of each unit produced but not sold called holding
cost.
- The shortage cost arising due to each unit of unsatisfied
demand.
- Random variable denoting the demand.
- The truncation point.
- Total cost per unit time.
- Supply level and is the optimal value.
- A random variables denoting the demand for both the case
45
when demand is less and more than the production, its PDF
is given by , .
- The probability distribution function when .
- The probability distribution function when .
- The cost of each unit of Newspapers purchased for several
individual demand but not sold called salvage loss.
- The shortage cost arising due to each unit of unsatisfied
individual demand of Newspapers.
- Optimal supply size.
- The probability distribution function when .
- The probability distribution function when
- The expected total cost
- The optimal expected cost
- Total holding cost
- Total shortage cost
- Random variable denoting the several individual demands at
the th location where
- The expected several individual demand before the
truncation point
and after the truncation point
is
46
3.3 BASIC MODEL
Many researchers have suggested that the probability of achieving a
target profit level is a realistic managerial objective in the Newsboy problem.
However Hanssman F [33] has given a different perspective of the Newsboy
model. In this model, somewhat different problem arises when the salvage
loss for the left-over units is negligible but a significant holding cost per unit
time is incurred. It is assumed that the demand materializes in a linear
fashion during a given planning interval . The shortage cost is assumed
proportional to the area under the negative part of the inventory curve. If the
total cost per unit time is , then the cost incurred during the interval is
given as
(3.1)
The expected total cost given in Hanssman.F [33] is as follows
(3.2)
Motivated from Newsboy model discussed above and the concept of
the setting the clock back to zero property, this chapter follows the different
variation of the probability distribution function. A brief discussion on the
SCBZ property can be followed from chapter 1 and in present chapter this
property is slightly modified with respect to the model and it is defined as a
random variable is said to satisfy the SCBZ property if
where denotes the survivor function
which is . Here is called the truncation point of the
random variable . SCBZ property means that the probability distribution of
the random variable undergoes a parametric change after the truncation
47
point . Similar definition of SCBZ property was discussed in Raja Rao et.al
[53]. Accordingly SCBZ property is defined by the pdf as
(3.3)
where is constant denoting truncation point. The probability distribution
function is denoted as if and if (3.4)
Similar model is discussed by Sathiyamoorthy R et.al [63]. Use of this
property is throughout the thesis.
3.4 FINITE PROCESS INVENTORY MODEL USING SCBZ PROPERTY
During certain situations the inventory process gets terminated after a
finite duration. In order to control this, an inventory model is studied. The
Newsboy problem is under the category of finite inventory process. There is a
onetime supply of the items per day and the demand is probabilistic. The
classical model assumes that if the order quantity is larger than the realized
demand, the items which are left over at the end of period are sold at a
salvage value or are disposed of. Further in cases of stock-out unsatisfied
demand is lost. From Hanssman F [33], the total cost incurred during the
interval is modified as given
(3.5)
Where and are defined as in Figure 3.1. Hence the expected total cost
function per unit time is given in the form
(3.6)
48
To find optimal ,
is considered. Since the limit of the integral
involves which is also in the integrand, the differential of integral is applied
(3.7)
which can be proved to result in the following equation
(3.8)
Given the probability distribution of the demand and using the
expression for , the optimal is determined. This basic Newsboy
problem is quite similar to the one discussed in Hanssman F [33].
Figure 3.1: Shortage curve under negative inventory level
The probability distribution function defined above satisfies the SCBZ
property and the optimal is to be derived. Now using the equation 3.4 and
3.6, the total expected cost is given as
(3.9)
49
Case i) when the holding cost before and after the truncation point is
considered and using equation 3.3, the result obtained is as follows
(3.10)
(3.11)
Now to find
the following substitution is carried
Let
(3.12)
(3.13)
(3.14)
(3.15)
Now solving the equation the above equation for
,
using the
geometry
and
and by the rule given by equation 3.7. Hence
the following result is obtained
50
(3.16)
(3.17)
(3.18)
(3.19)
Hence the total expected cost is given by
(3.20)
Case ii) When the shortage cost before and after the truncation point is
considered and using equation 3.3 the following result is obtained
(3.21)
51
(3.22)
Similarly to find
the above procedure is followed and hence taking
and
the result is as follows
(3.23)
Considering the case when the supply and the level of inventory are
same and all the other cases are considered negligible. On substitution the
equation 3.20 and 3.23 becomes
(3.24)
Therefore after substituting the value for in , the optimal
solution of the expected cost is obtained as =19. In contrary to the
above model 3.4 another model 3.5 is developed in order to the test the
hypothesis in case of single individual demand which will be a base for the
following model 3.5.
3.5 OPTIMALITY OF TOTAL EXPECTED COST USING SCBZ PROPERTY
The case discussed in the model 3.4 is modified and a study over a
single demand is carried out. Here the assumptions are as given in the model
3.4. Now a change to the model in form of cost function is that here in this
model the cost function is denoted as .Total cost incurred during the
interval T given is given as
(3.25)
Where and are defined as in Figure 3.2
52
Figure 3.2: Supply size against the time t
Therefore the expected total cost function per unit time is given in the form
(3.26)
From the geometry it follows that
and
(3.27)
The uncertainty here is related to a well known property called as SCBZ
Property. SCBZ property which is defined in model 3.4 is applied in this
model 3.5. Accordingly SCBZ property is defined by the PDF as
(3.28)
where is constant denoting truncation point. The probability distribution
function is denoted as if and if (3.29)
Sathiyamoorthy R et.al [63] introduced the concept of SCBZ property in
inventory. The probability distribution function defined above satisfies the
SCBZ property under the above assumptions and the optimal is derived.
Now, the total expected cost given in equation 3.26 is written as
(3.30)
53
where
(3.31)
The solution of equation 3.31 is dealt in form two model i.e., model 3.6
and model 3.7 due to the complexity involved while using the limit. In solving
model 3.5 following is study carried out. Using equation 3.7
(3.32)
(3.33)
54
(3.34)
using equation 3.27, the following equation are obtained
(3.35)
55
(3.36)
(3.37)
(3.38)
56
(3.39)
The following assumption is considered while solving equation 3.39
that the lead time is zero and single period inventory model will be used with
the time horizon considered to as finite. When the supply and the level of
inventory are same and all the other cases are considered zero. Since
analytical solutions to the problem are difficult to obtain. Equation3.39 is
solved using Maple 13 and using equation 3.7. Hence,
(3.40)
The optimal solution is obtained using the numerical illustration by
substituting the value for is obtained in form of numerical
illustration 3.5.1 shown below. The Table 3.1 and Figure 3.3, shows the
numerical illustration for model 3.5 when the shortage cost is permitted in the
interval .
3.5.1 Numerical illustration
In this Numerical illustration the value for .
and is evaluated and the graph representing these
values are given below which is obtained to get the optimal expected cost
. This numerical illustration provides a clear idea of the increased profit
form curve.
57
Table 3.1 Numerical tabulation for obtaining optimal supply
Figure 3.3 Expected profit curve
1 0.5 5 2 1.648 12.18249 0.1213 0.16417 0.410425 -0.54
2 1 10 1 2.718 22026.47 0.0735 4.54E-05 0.000454 0.926
3 1.5 15 0.6 4.481 5.91E+09 0.0446 1.13E-10 2.54E-09 2.288
4 2 20 0.5 7.389 2.35E+17 0.0270 2.12E-18 8.5E-17 3.472
5 2.5 25 0.4 12.182 1.39E+27 0.0164 2.88E-28 1.8E-26 4.583
6 3 30 0.3 20.085 1.14E+26 0.0149 2.92E-27 1.75E-25 5.651
58
Figure 3.4 Optimal supply vs truncation point
3.5.2 Inference:
Supply against is shown in the Figure 3.3. When the supply size is
increased according to the demand then there a profit or otherwise
instantaneous increase in is noted. This model shows a sharp increase
in the cost curve is obtained. Figure 3.4 shows a instantaneous decrease in
optimal supply.
3.5.3 Numerical illustration
A comparative study was carried out with the data value available from
Sehik Uduman P.S et.al [65] to check the optimality if the cost curve unique.
Table 3.2 and Figure 3.5 shows the comparative result for supply size and
the shortage graph. When there is shortage in the supply size then there is a
decrease in the expected cost which leads to the profit loss for the company.
59
Table 3.2 Comparative result for supply size
3.5.4 Inference
A comparative study on the optimal expected profit curve with that of a
Sehik Uduman P.S et.al [65] is shown in form of the Figure 3.5 where curve
[1] is a curve as in the existing model Sehik Uduman P.S et.al [65] and curve
[2] is a new curve for model 3.5. It observed that there is an increase in the
0.5 0.5 5 1.648 12.18249 2 0.0606 0.164169997 0.410425 -0.986
0.7 1 10 2.718 22026.47 1 0.0257 4.53999E-05 0.000454 -0.325
0.9 1.5 15 4.481 5.91E+09 0.66 0.0133 1.12793E-10 2.54E-09 0.2199
1.1 2 20 7.389 2.35E+17 0.5 0.0074 2.12418E-18 8.5E-17 0.5925
1.3 2.5 25 12.18 1.39E+27 0.4 0.0042 2.87511E-28 1.8E-26 0.8957
1.5 3 20 20.08 1.14E+26 0.33 0.0037 2.91884E-27 1.75E-25 1.1629
Figure 3.5 Comparative graph
0.4
0.6
0.8
1.0
1.2
1.41.6
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
B
C
[1]
[2]
EX
PE
CTE
D C
OS
T
SUPPLY SIZE
60
profit from negative to positive value leading to an instantaneous increase in
supply size and increased profit. The curve shows the optimality and validity
of this model.
3.6 OPTIMALITY FOR HOLDING COST USING SCBZ PROPERTY
A model is developed to study the optimality for holding cost using
SCBZ property. In such case the units of items unsold at the end of the
season if any are removed from the retail shop to the outlet discount store
and are sold at a lowest price than the cost price of the item which is known
as the salvage loss. A situation is discussed when there is a holding cost
occurred and there is no shortage allowed. In this case the attention can be
restricted to the consideration of the part when the holding cost 1 is
involved, in which case there is an immense loss to the organisation leading
to the setup cost and the cost of holding the item. From Hanssman F [33], the
expected holding cost is given as follows and this cost is truncated before
and after the particular event in the interval
(3.41)
Using the rule given in equation 3.7 and substituting equation 3.28, the
equation 3.41 changes as follows
61
(3.42)
(3.43)
(3.44)
(3.45)
62
(3.46)
(3.47)
(3.48)
(3.49)
Hence the expected cost obtained is given as equation 3.49.
3.6.1 Numerical illustration
A load of items from 1 tonnes to 6 tonnes is varied accordingly and the
result shows an increase of the expected profit curve which is given by Table
3.3 and Figure 3.6.
63
Table 3.3 Load of data for 1 to 6 tonnes for finding
Figure 3.6 Supply against holding cost
0
1
2
3
4
5
6
7
1 2 3 4 5 6
S
U
P
P
L
Y
S
I
Z
E
EXPECTED HOLDING COST
(
1 0.05 20 5 1.28 0.2840 0.2840 5.6808 -5.3968
2 0.03 33.33 10 1.34 0.3498 0.6997 11.661 -11.312
3 0.07 14.28 15 2.85 1.8576 5.5729 26.537 -24.680
4 0.09 11.11 20 6.04 5.0496 20.198 56.107 -51.057
5 0.08 12.5 25 7.38 6.3890 31.945 79.863 -73.474
6 0.09 11.11 20 6.04 5.0496 30.297 56.107 -51.057
64
A state space representation when the supply size is excess then the
total expected cost rules out and this state of condition is shown in the figure
3.7
Figure 3.7 State space for the expected total profit
3.7 OPTIMALITY IN CASE OF PLANNED SHORTAGE USING PARTIAL
BACKLOGGING
Planned Shortages or backordering model is illustrated in very few text
books (Anderson et.al [5] and Vora N.D [75]). In literature, few authors use
term "back ordering" while many authors prefer "planned shortages" to
describe this model. Notable work is observed in partial backordering. The
backlogging phenomenon is modelled without using the backorder cost and
the lost sale cost as these costs are not easy to estimate in practice. Abad. P
[1] had studied a continuous review inventory control system over an infinite-
horizon with deterministic demand where shortage is partially backlogged.
Khouja M [37] had discussed the state of condition in which a single
period imperfect inventory model with price dependent stochastic demand
and partial backlogging was considered. Mainly there are two types of
shortages, inventory followed by shortages and shortages followed by
inventory. Occurrence of shortage may be either due to the presence of the
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
1 2 3 4 5 6
s
E ( C )
EXPECTED COST
SUPPLY SIZE
65
defective items in the ordered lot or due to the uncertainty of demand. The
shortage cost is assumed proportional to the area under the negative part of
the inventory curve.
The following are the assumptions which are relative to Vora N.D [75]
used in this model.
a) The demand for the item is taken to be constant and continuous.
b) The replenishment for order quantity is done when shortage level
reaches planned shortage level.
c) Stock outs are permitted and shortage or backordering cost per unit is
known and is constant.
From (a) and (b), the limit of integral is considered to be . Now
from equation 3.32, the expected holding cost is given as follows and this
cost is truncated before and after the particular event in the interval
(3.50)
Using equation 3.28 in equation 3.50, the following equation is obtained
(3.51)
Using equation 3.27 in equation 3.51, the following observation is carried out
(3.52)
66
(3.53)
Using the equation 3.7, the equation 3.53 changes as follows
(3.54)
From equation 3.49 and equation 3.54 the following result is obtained
(3.55)
The solution of equation 3.55 requires the basic property and
from equation 3.32 it suggests a general principle of balancing the shortage
and overage which shall have an occasion to be applied repeatedly. By
recalling the standard notations generally a control variable and a random
variable with known density which was earlier introduced and two functions
and which may be interpreted as overage and
shortage levels respectively. Assuming the fundamental property of linear
control:
(3.56)
where denotes the expected value and is a constant.
For minimizing a cost of the form using equation
3.56 by differentiated with respect to , thus following equation is obtained
(3.57)
The following condition for the optimal valve is obtained
(3.58)
67
In other words the derivative of the expected overage must be equal to the
characteristics cost ratio in the equation 3.58. Accordingly, the SCBZ
Property satisfies the existence of the solution hence
(3.59)
when and by computing the general principle of balancing shortage
and overage the following is computed
(3.60)
Thus the optimal solution is given by the following
(3.61)
by considering a linear control with =1 equation 3.61 to the following result
(3.62)
3.7.1 Inference
The necessity of storage of items cannot be ignored and emphasis
should be given whether the storage is needed or not in the context of
deteriorating items and allowing shortages.
So far in the previous models of this thesis the newsboy problem for
single and double demand was considered, now the generalisation of
newsboy problem is discussed in model 3.8 using SCBZ property.
68
3.8 GENERALIZATION OF NEWSBOY PROBLEM WITH DEMEND
DISTRIBUTION SATISFYING THE SCBZ PROPERTY
The basic Newsboy model has been discussed in Hanssman F [33].
According to the review, the researchers have followed two approaches to
solve the newsboy problems. In the first approach, the expected cost is
overestimated and demand was underestimated. In the second approach,
the expected profit is maximized. But, both the approach yields the same
result. In this chapter, the first approach is used to solve the newsboy
problem. By an appropriate demand decision, the expected cost due to lost
sale could be minimised.
Amy Lau et.al [4] studied the price dependent demand in the Newsboy
problem, Chin Tsai et.al [14] studied the generalisation of Chang and Lin’s
model in a multi location Newsboy problem in which the actual model of
Chang and Lin’s model was extended by adding the delay supply product
cost. In Chin Tsai et.al [14], the Newsboy problem was solved in the
centralised and decentralised system. Nicholas A. Nechval et.al [49] showed
how the statistical inference equivalence principle could be employed in a
particular case of finding the effective statistical solution for the multiproduct
Newsboy problem with constraints.
In this chapter, a generalisation of the actual problem as discussed in
Sehik Uduman P.S [65] is derived using the demand distribution which
satisfies the SCBZ property. This chapter aims to show how SCBZ property
is applied in case of single period, single product inventory model with
several individual source of demand. The objective is to derive the optimal
stock level or the optimal reorder level. Hence the optimal order quantity is
derived. Numerical illustrations are also provided as an example for the
validation of this model.
3.8.1 Basic model
The concept of decentralized inventory system is introduced. The
decentralized inventory system is a system in which a separate inventory is
69
kept to satisfy the several individual source of demand and there is no
reinforcement between locations of demands. Its aim is to minimize the
expected total cost and hence the expected total cost function is given
in the form
(3.63)
where
The basic newsboy problem is derived in Hanssman F [33] and hence
adopting the expected total cost given in this model, which is as follows
(3.64)
To find optimal ,
is considered. Since the limit of the integral
involves which is also in the integrand, the differential of integral is applied
as given in equation 3.7. Hence it is proved to result in the equation 3.8.
Given the probability distribution of the demand using the expression
for , the optimal was determined. This was the basic Newsboy problem
discussed by Hanssman F [33]. In this model, the SCBZ property is
reformulated as
(3.65)
where is constant denoting truncation point. The probability distribution
function is denoted as
if
if (3.66)
70
The probability distribution function defined above satisfies the SCBZ
property under the above assumptions and the optimal is to be derived.
Now, the total expected cost from equation 3.63 is given by
(3.67)
Using the PDF given of equation 3.65 for and in equation
3.67, the following is obtained
(3.68)
Using the equation 3.7, the equation 3.68 is solved as follows
71
(3.69)
To find
. Assuming
(3.70)
(3.71)
To find
(3.72)
72
(3.73)
Adding equation 3.72 and equation 3.73
(3.74)
Similarly to find
(3.75)
(3.76)
73
(3.77)
Hence by the equation 3.76 and equation 3.77,
is obtained as follows,
(3.78)
Hence
Therefore
(3.79)
Taking log on both sides
(3.80)
(3.81)
74
By substituting the values of , , and , the value of is obtained
which satisfies equation 3.81. The value of is also evaluated using a
suitable computer program.
3.8.2 Numerical illustration
In the numerical example is fixed and is varied
accordingly and these value are substituted in equation 3.81 to obtain the
value of
Case i) , ,
Table 3.4: variation for obtaining
1.0 1.5 2.0
5.7 2.6 0.17
Figure 3.8: Supply against curve
75
Case ii) , ,
Table 3.5: variation for obtaining
1.5 2.0 2.5
0.23 3.56 6.89
Figure 3.9: Curve for supply against
Case iii) , ,
Table 3.6: variation for obtaining
10 15 20
1.4 2.04 2.67
76
Figure 3.10: Curve for supply and truncation point
3.8.3 Inference
From the numerical illustrations and corresponding figures the
following conclusions may be drawn.
Case i) If the parameter of the demand distribution prior to the truncation
point is varied exponentially then the expected demand will decrease
because
this implies that whenever increases the demand will
decrease and hence a smaller supply size for several individual demand is
suggested.
Case ii): When the parameter is fixed and which denotes the parameter
of the demand distribution posterior to the truncation point is increased,
then a corresponding increase in supply size for several individual demand
is suggested.
Case iii): If both and are fixed and if the truncation point increases
then there will be an increase in the supply size. The demand after is
smaller. As the truncation point increases then for several individual
demand increases and the demand is dominated by .Therefore an
increased inventory is suggested.
77
3.9 CONCLUSION
In the most realistic setting, the variability of benefit in stochastic
inventory models cannot be ignored. This model is examined in form of time
point occurring before the truncation point, and the time point occurring after
the truncation point. In which case, the SCBZ property seems to be a useful
concept and needs further attention. Thus the demand may be stock
dependent up to certain time after that it is constant due to some good will of
the retailer. This model can be considered in future with deteriorating items.
Hence the optimal order level or Supply is less than the risk neutral
counterpart is applied as base stock policy which is discussed in chapter 5.
This model framework can be extended in several ways. An obvious
extension would be to consider this as newsvendor model for two products or
multiproduct, in which case the expression would be more complex and it will
have complex probability functions and integrations. So an attempt is made
to solve these models in form of chapter 7.
78
4. TRUNCATED DEMAND DISTRIBUTION AND RENEWAL REWARD
THEORY IN SINGLE PERIOD MODEL
4.1 INTRODUCTION
The single period Newsboy problem discussed in chapter 3 is
extended using Truncated Exponential Distribution and Renewal Reward
Theory. In 4.2, the salvage cost alone undergoes a change using the
Truncated Exponential Distribution. Truncated distributions can be used to
simplify the asymptotic theory of robust estimators of location and regression.
The truncated distributions have found many applications. Several examples
have been given employing the truncated distributions in fitting rainfall data
and animal population studies where observations usually begin after
migration has commenced or concluded before it has stopped. Similar
situations arise with regard to aiming errors i.e., range, deflection, etc., in
gunnery and other bombing accuracy studies. For example, in gun camera
missions, the view angle of the camera defines a known truncation point for
an exponentially distributed random variable, observable as some function of
the radial error or the distance from the aiming point to the point of impact.
Muhammad Aslam et.al [45] have studied Time-Truncated
acceptance sampling plans for generalized exponential distribution and also
they studied Double acceptance sampling based on truncated life tests in
Rayleigh distribution. Where else the work on Truncated Exponential
Distribution satisfying the Base stock policy for the patient customer as a
continuous model will be discussed in chapter 5.
The use of Renewal Reward Theory for obtaining the solution
involving the occurrence of partial backlogging due to stock-out is studied in
4.3. The expected cost is derived in its existence form in a way by taking the
log as negligible. The objective is to derive the optimal stock level and
appropriate numerical illustration is provided.
79
4.2 OPTIMAL HOLDING COST USING THE TRUNCATED EXPONENTIAL
DISTRIBUTION
The basic model of Hanssman F [33] discussed in chapter 3 is
considered in this chapter with the following variations as discussed below.
Definition 1: Let be a (one sided) truncated exponential where
be a random variable, then its PDF is given as
for
0 where is the same as the usual definition for expectation if
is a continuous random variable.
Deemer W.L et.al [18] derived the maximum likelihood estimator of the
parameter in the truncated exponential distribution as
(4.1)
Since the truncated exponential distribution constitute an exponential
family. In this case the attention can be restricted to the consideration of the
part when the holding cost is occurred. Now the expected cost
satisfies equation 4.1. Hence from Hanssman F [33]
(4.2)
using the concept of truncation the equation 4.2 is given as
(4.3)
80
Using equation 4.1 in equation 4.3, the following equation is obtained,
(4.4)
(4.5)
81
(4.6)
Equation 4.6 is solved using the equation 3.7 in chapter 3
(4.7)
Hence after due simplification the following equation is obtained
(4.8)
82
To find optimal , it is needed to formulate the well known result
, then
(4.9)
Now equation 4.9 satisfies the condition for a single period model in the limit
. Where else in model 4.3 which is discussed below, the renewal reward
and partial backordering concept is initiated.
4.3 RENEWAL REWARD SHORTAGE AND PARTIAL BACKORDERING
In this model the close form introduced in the chapter 3 is considered
for study. There are many benefits of having a closed-form approximate
solution. A closed-form solution clearly demonstrates the sensitivity of
solutions to input parameters. It can also be embedded into more
complicated models to add- tractability. Closed-form approximations are also
useful tools in practice, since they are easier to implement and use on an
ongoing basis.
When the price increase, its components is anticipated hence the
companies purchase large amounts of items without considering related
costs. However ordering large quantities would not be economical if the items
in the inventory system deteriorate and demand depends on the stock level.
A situation is modelled by considering the partial backordering in a
mathematical formulation of inventory model.
83
Exact closed form solution was derived for the optimal solution for
order of existence of backordering quantity and maximum expected profit.
The usage of annual profit function in its simplified way into model gives a
wide range of change and hence the expected cost was derived in its
existence form. Also, the concept of backorder as discussed in chapter 3 is
also adopted in this chapter.
During the determination of the optimality, the shortage of item is
subject to backordering. The backlogging rate was considered as random
variable and depended on the length of waiting time for the next replacement.
The backlogging rate is assumed as
where was taken to be the
nonnegative constant backlogging.
Many authors in the literature used the Renewal Reward Theorem to
derive the expected profit per unit time for their model. Exact closed form
solution was derived for the optimal lot size, backordering quantity and
maximum expected profit. The annual profit function in their simplified model
was given by
(4.10)
The above is discussed in Salameh M.K et.al [61] also they discussed
the case when buyer’s cycle starts with shortage that may have occurred due
to lead time or labour problems. The fraction of the demand in this stock out
period was varied according to time and the items were backordered. Where
else the time invariant demand was left as a lost sale.
Considering the shortage cost from equation 3.26 of chapter 3 is given by
(4.11)
Equation 4.10 satisfies the equation 4.11 and hence the following equation is
obtained
84
(4.12)
This model is solved in a view to obtain an annual profit function. Equation
4.13 involves the differential of integral as discussed in equation 3.7 of
chapter 3
(4.13)
Where
85
(4.14)
Now to find
following calculation is carried
(4.15)
Hence
(4.16)
86
(4.17)
Similarly
can be calculated as
(4.18)
The following result is obtained
Therefore the expected cost is given by
(4.19)
Hence after due simplification the total expected cost is calculated, as
(4.20)
87
Hence the Optimal Expected Cost from equation 4.20 and equation
4.9 is given by
(4.21)
Thus the above result is proved using the numerical illustration.
Equation 4.21 is evaluated numerically by substituting the values in
ascending and descending order of its initial values for
4.3.1 Numerical illustration
Considering the initial value of the cost function to be ,
, , and these values are set in an ascending order to
obtain the profit curve as follows
Table 4.1: Optimal profit for increasing value
1.0 1.5 10 27.3125
1.2 2 12 60.1344
1.4 2.5 14 139.7725
1.6 3 16 317.5424
1.8 3.5 18 679.5981
2 4 20 1360
88
Figure 4.1: Truncation point with respect to the optimal cost
4.3.2 Inference
From Table 4.1 and the corresponding figure 4.1, it is seen that as
namely the inventory holding cost increases then smaller size of inventory is
suggested. Similarly if the shortage cost increases then it is desirable to
have a larger stock size.
4.3.3 Numerical illustration
Considering the initial value of the cost function to be
, , , and these values are set in decreasing
order to obtain the profit curve as follows
89
Table 4.2: Optimal profit for decreasing value
2 1 30 140
1.8 0.8 25 98.04557
1.6 0.6 20 66.69107
1.4 0.4 15 42.66483
1.2 0.2 10 24.08525
1 0 5 10
Figure 4.2: Supply curve when is varied with respect to
4.3.4 Inference
From Table 4.2.and the corresponding figure 4.2 it is seen that as
namely the inventory holding cost decreases then larger size of inventory is
0
20
40
60
80
100
120
140
160
1 2 3 4 5 6
Z
0Q
90
suggested. Similarly if the shortage cost decreases then it is desirable to
have a smaller stock size.
4.4 CONCLUSION
In the most realistic setting the variability of benefit in stochastic
inventory models cannot be ignored. Closed form approximations are also
useful tools in practice, since they are easier to implement and are used on
an ongoing basis. A closed form solution clearly demonstrates the sensitivity
of solutions to input parameters. It can also be embedded into more
complicated models to add tractability. When the price increases, its
component is anticipated. In this situation companies may purchase large
amounts of items without considering related costs. However ordering large
quantities would not be economical if the items in the inventory system
deteriorate. Also demand depends on the stock level.
The newsboy problem is treated in this chapter which involves the
use of truncated exponential distribution and the renewal reward theory for
the optimal expected cost. This model is also illustrated numerically in order
to prove its uniqueness. Also, this model can be extended when in
case of the truncated exponential distribution.
So far, the previous chapter is dealt with the single period model using
different variation of newsboy problem. The preceding chapter 5 deals on the
base stock system for patient customer.
91
5. BASE-STOCK SYSTEM FOR PATIENT CUSTOMER WITH DEMAND
DISTRIBUTION UNDERGOING A CHANGE
5.1 INTRODUCTION
In this chapter, the base-stock for patient customer is studied. The
base-stock system for patient customer is a different type of inventory policy
in which an ordering mechanism of a new type is introduced. Under the base-
stock systems, 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. In
this model, the inventory process starts with initial inventory of size .
Whenever a customer order is received, it is supplied immediately and
at the same time a replenishment order is placed immediately. The
replenishment takes place after a lead time . If the demand exceeds this
stock level on hand then customer do not leave, but they wait till supply is
received. For this reason the customer are called patient customer. In this
case there is no shortage cost, but some concession is shown to the
customer and it is a denoted as a shortage cost. The total inventory is
denoted as , which is the sum of the inventory on hand, and inventory on
order. This is called Base-Stock. Here the demand during the period is
taken to be a random variable.
In this model it is assumed that, the distribution of the random variable
denoting the demand undergoes a change in the distribution after a change
or truncation point. The demand distributions in this model is distributed as
an exponential before the truncation point and distributed as Erlang2 after the
truncation point. Truncated exponential distribution discussed in chapter 4 is
used to obtain the optimal expected cost of base-stock system for patient
customer.
The objective is to derive the expression for optimal base-stock and
also numerical illustration is provided. If lead-time demand is denoted as
then is the probability density of this random variable. Under these
circumstances the equation for the expected cost can be written in the form
92
(5.1)
The optimal value of is to be determined by taking
and can
be shown that the optimal value is one such that
where is
optimal and is the cumulative distribution of the random variable .
This model has been discussed in Hanssman F [33]. The base stock system
for patient customer has been initially discussed by Gaver D.P [21] and a
modification of this model has been attempted by Ramanarayanan R [54]
In this chapter, a new model is developed by assuming that during the
lead time which is deterministic, there are different demand epochs and
the demand during these epochs are denoted as , which are
identically independent random variables. The inter-arrival times between the
demand epochs are also random variables which are identically independent
with the density function and the distribution function . The
probability there will be exactly n demands denoting the lead-time is given
as by the renewal theory which is discussed in
chapter 4 where is -fold convolution of G with itself. Therefore the
probability of total demand is utmost during is
(5.2)
where denotes the number of demand epochs during
(5.3)
is the –fold convolution of with itself.
The expression for the expected cost is given as
93
(5.4)
where the assumption are followed below. To find the optimal level
satisfies the following equation in accordance with the equation for the
optimal base-stock. Now
(5.5)
Using the equation 3.7 of chapter 3, following equation is obtained
(5.6)
Hence
(5.7)
on simplification since , then equation 5.7
becomes
. From the model discussed above, a new model is
developed by considering the fact that the demand distribution undergoes a
change after a change point. This assumption of the demand distribution
undergoing a change is valid, since the demand distribution has the very
basic nature that the probability that a random variable denoting the demand
taking a value beyond a certain level may undergo change in its structure.
94
5.2 ASSUMPTIONS
i) The total demand is a constant, which under goes a change of
distribution after a change point .
ii) The distribution of the total demand follows exponential with
parameter and becomes Erlang2 with parameter after the change point.
iii) Also the demand is considered to be truncated exponential
distribution.
5.3 NOTATIONS
= Inventory holding cost /unit
= Shortage cost/unit
= k fold convolution of
= The total demand
5.4 ERLANG2 DISTRIBUTION FOR OPTIMAL BASE STOCK
Assuming that the distribution of of the random variable
denoting the demand undergoes a change of distribution in the sense that
if
if
Where is called the change point. Let the change of distribution in the
expression for expected total cost is incorporated. In doing so, becomes
less than base stock. Hence considering the model when
95
If
(5.8)
by the formulation of rule discussed in chapter 3 as equation 3.7, the
following result is obtained
(5.9)
(5.10)
To find
96
(5.11)
To find
(5.12)
97
(5.13)
Substituting equation 5.11, 5.12 and 5.13 in equation 5.10, the following
result is obtained
(5.14)
Any value of which satisfies equation 5.14 is the optimal base stock
namely .
5.4.1 Numerical illustration
Considering the value , , ,
98
Table 5.1: Shortage variability for base stock
Figure 5.1: Base-stock with the shortage cost
5.4.2 Inference
In fig.5.1, as the value of the shortage cost ‘ ’ increases, a larger
inventory size is suggested as in the case of all other models discussed
earlier by many authors the above curve obtained in the figure is valid and is
similar to the one obtained earlier by the other authors.
5.4.3 Numerical illustration
Considering the value , , ,
10 20 30 40 50
7.7 7.9 8.0 8.1 8.2
99
Table 5.2: Holding variability for base stock
Figure 5.2: Base-stock with holding cost
5.4.4 Inference
In figure 5.2, as the inventory holding cost ‘ ’ increases then this
model suggests a smaller inventory size to be stocked, which is common to
all inventory models.
5 10 15 20
7.7 7.4 7.0 6.8
100
5.5 TRUNCATED EXPONENTIAL DISTRIBUTION FOR PATIENT
CUSTOMER
Maintaining inventories is necessary in order to meet the demand of
stocks for a given period of time which may be either finite or infinite. An
optimal base-stock inventory policies using finite horizon is examined. In
Hanssman F [33] the basic model for the base-stock systems is discussed.
Sachithanantham S et.al [58] had discussed the model of base stock system
for patient customers with lead time distribution undergoing a parametric
change. Suresh Kumar R [72] showed how by applying the threshold, a
Shock model had a change of distribution after a change point. A modified
model had been attempted by Sachithanantham S et.al [58].
The base-stock for Patient customer model discussed in model 5.2 is
evaluated using the Truncated Exponential Distribution. Since among the
parametric models, the exponential distribution is perhaps the most widely
applied statistical approach in several fields. Hence, it is justified to apply the
truncated exponential distribution approach. In this model demand during the
period [0, t] is taken to be a random variable and truncated exponential
distribution satisfies the base-stock policy for the patient customer as a
continuous model. Whenever a customer orders for units is received it is
supplied immediately and at the same time a replenishment order for units
is placed immediately. The replenishment takes place after a lead-time L.
From Hanssman F [33]
(5.15)
Assuming the PDF of the random variable denoting the demand
undergoes a change of distribution in the sense that
(5.16)
101
Where is called the change point. The following equation is obtained while
incorporating the change of distribution in equation 5.15,
(5.17)
Equation 5.17 is differentiated by using the differential of integral method as
discussed in equation 3.7 of chapter 3 and is solved as follows
(5.18)
Now
(5.19)
Hence
(5.20)
102
Deemer W.L et.al [18] derived the maximum likelihood estimator of
the parameter in the truncated exponential distribution as
(5.21)
Applying the equation 5.21, the equation 5.20 becomes as follows,
(5.22)
(5.23)
103
(5.24)
Hence substituting the equation 5.22, 5.23 and 5.24 in equation 5.20, the
result obtained is
(5.25)
The model discussed above is formulated numerically. Therefore by
substituting the value for the truncation point, holding cost, base stock and
time variant which is denoted as follows , the optimal base stock in
case of the patient customer is obtained. Also, this result is compared with
that of earlier models.
5.5.1 Numerical illustration
The following table 5.3 and figure 5.3 shows the numerical existence
of the model developed.
Table 5.3: Optimal base stock case with varying
1 10 6 0.5 -1.049 0.393 -4.130
1.5 20 6.2 0.7 -1.013 0.650 -13.17
2 30 6.4 0.9 -1.003 0.834 -25.11
104
Figure 5.3: Base-stock curve for truncation point.
5.5.2 Inference
In figure 5.3, as the inventory holding cost is monotonically
increasing, then this model demands for stocking of a limited or very fewer
inventory.
5.6 CONCLUSION
In this chapter, the demand during the period [0, t] is taken to be a
constant. In theoretical and applied work, the truncated exponential
distribution plays a crucial role due to its application to real life. The idea of
inventory decisions could be applied to production systems with several
machines and impatient customers. The model presented in this chapter can
be extended to system with both customer impatience and allocation of
hospital bed. This direction of research is taken into study in the next chapter
6.
105
6. BASE STOCK IMPATIENT CUSTOMER USING FINITE HORIZON MODEL
6.1 INTRODUCTION
The initial work in the field of queuing theory was carried out by Erlang in
1909. Queuing model have been proved to be very useful in practical
applications such as inventory systems, production system and communication
system. In this chapter, the base-stock impatient customer using finite-horizon
models is studied. So far the base-stock for impatient customer leads to a
discrete case, but in this work it is extended for a continuous case. A way of
optimizing the average cost per day by balancing cost of empty beds against
cost of delay patients is discussed. If the hospital beds are unavailable,
impatient customers have no option but to get admitted in another hospital in
order to get the immediate health care facilities. Previous work related to this
problem was discussed by Gorunescu F [24] and a theoretical model along with
optimization of the number of beds was presented in this model.
Upper and lower echelon case of the impatient customer in base-stock
policy is discussed. The base-stock is viewed as the number of initial inventory
facility in stock. The objective is to derive the optimal stock level. Finally the
expression for optimal base-stock is derived. This approach is justified
mathematically and also numerically.
6.2 ASSUMPTIONS
= Patient demand for bed
= The number of phases / compartments
= Mixing proportions
106
= Transition rate
= Total inventory cost
= Cumulative distribution
= Number of demand per unit time in case of the idle channels
L = Mean lead time
6.3 OPTIMIZING THE NUMBER OF BEDS
In this model, the demand for beds in a hospital is optimized using the
queuing model of the base-stock system. If all the beds are occupied, then it
leads to unsatisfied demand and loss of revenue to the hospital management. In
this chapter, a base-stock queuing model is adopted.
In emergency situation, the hospital system that is described can result in
a patient being turned away because all beds are occupied such a patient may
not receive the necessary care. Here, a discrete demand model for slow moving
item called queuing model for base-stock system discussed in Hanssman F [33]
is studied. In this model, the number of beds B in hospital is viewed as the
number of channels of queuing facility. Number of beds in hospital which are
vacant is idle channel and the number of beds in need by the patient or demand
for beds is busy channel. Here the demand is considered as the Poisson arrival
i.e., one demand at a time.
The probability lead time for reordered beds corresponds to the service
time. Here the probability distribution is assumed to be of Erlang type. If the
patient demand for bed cannot be satisfied is lost. The basic model of inventory
problem of finding the optimal base-stock was brought into the form of well
known queuing problem of optimizing the number of channel discussed in
107
Hanssman F [33]. The actual problem here is to optimize the number of idle
channel.
6.3.1 Basic model
From Gorunescu F [24], an M/PH/C Queue was used in which the
number of beds was fixed and no queuing was allowed. Patient who finds all
the beds occupied, would admit themselves in another specialties. The general
problem was rather complex. So, a focus of simpler model was adopted in
which proof clarification was not involved. In this model, patient arrival was
considered as a Poisson process with rate and the service time as phase type
with probability density function as
(6.1)
The corresponding mean was given as
(6.2)
The average number of arrivals occurring during an interval of length is and
therefore, the average number of arrivals during an average length of stay is
known as offered load.
Using the standard results from queuing theory, the probability of having
‘ ’ occupied beds was given by
(6.3)
From above formula, it was deduced that probability of there being ‘ ’ beds
occupied is given by Erlang’s loss formula
(6.4)
108
Another useful quantity is represented by the mean number of occupied beds
also known as carried load. It is easy to see that the offered
load ‘ ’ is the load that would be carried, if the number of beds was infinite and
the carried load ‘ ’ was just that portion of the offered load that was not cleared
(lost) from the system. If the bed occupancy was given by , then ,
otherwise the system cannot be in steady state.
Case i): system erlang loss model
In this case, the model discussed above is derived using the
system Erlang loss model and hence it is given by
(6.5)
Consider a ‘ ’ server model with Poisson input and exponential service time
such that, when all the ‘ ’-channels are busy an arrival leaves the system
without waiting for service. This is called a ‘ ’-channel loss system. This is
similar to the birth and death queuing model with
, ,
, , using
and
Then
;
(6.6)
and
(6.7)
109
An arriving unit is lost to the system, when it is found that on arrival all the
channels are busy. The possibility of this event is
(6.8)
The above formula is known as Erlang loss (blocking, or overflow) formula or B
formula is denoted by
.
Case ii): Inventory queuing model to optimize number of bed
From Hanssman F [33], the probability that there are idle channel is
(6.9)
Where
By simplify the expected number of idle channels is given by
(6.10)
Where,
(6.11)
Finally, the expected number of service performed per unit time is
(6.12)
In inventory interpretation, the quantities and represents the
expected level of inventory on the ground and the expected number of sales per
unit time respectively. This quantities is viewed as functions of the only control
variable . Let be the profit per unit sold, not considering ordering and
inventory charges. Here is the profit if the patients get admitted. Further, it is
assumed that the cost of ordering beds and cost of holding bed empty is per
unit time. The expected profit per unit time will be given as
110
(6.13)
The above equation 6.13 is evaluated for different values of and the profit
maximizing value may be selected directly. The first difference of is set
to zero then is obtained.
6.3.2 Numerical illustration
To optimize number of beds, some typical values for the delay probability
is considered and a suitable value of along with corresponding number of
beds needed to maintain this level of service is shown in table 6.1.
For example, to ensure that at most 10% of patients are turned away, in this
case they must have at least 130 beds in hospital.
Table 6.1: Number of beds and queue characteristics corresponding to
Delay probability 0.1% 1% 5% 10%
Minimum number of beds 179 166 150 130
6.3.3 Cost model
In order to illustrate the optimal base-stock level i.e., the average cost per
unit time for holding and shortage cost as a function of the number of beds ‘ ’.
The Table 6.1 corresponds to different ratios of the shortage cost and holding
cost. The total cost per patient per day is considered as Rs.168 where Rs.50 is
incurred with respect to the bed and Rs.118 with respect to treatment. Then
estimate holding cost is =Rs.50 per day and the penalty cost as 25% of the
total cost of turning away the patient. In this case it is taken as the cost per day
multiplied by the expected length of stay i.e., S = 168 x 24.9 x .25 = 1046.
111
This approach is meant to be indicative of a ballpark for cost and is based
on an assumption that shortage may be regarded in some sense as lost revenue
incurred, when a patient is turned away due to there being no empty beds
available. Hence total cost revenue per patient turned away is then cost per day
multiplied by expected number of day that has been lost. In the figure 6.1, an
account of the fact that a proportion of revenue must balance the cost when
profit occurs it may not affect the lost patients.
Table 6.2: The value of average cost per unit time
BEDS S/h=10 s/h=20 s/h=30 s/h=40
120 781 1390 1999 2608
125 723 1244 1765 2286
130 676 1112 1548 1984
135 643 998 1353 1708
140 629* 908 1187 1466
145 638 848 1058 1268
150 677 827* 976 1126
155 752 851 951* 1050
160 867 927 988 1049*
165 1022 1055 1089 1122
112
Figure 6.1: Actual cost per bed
6.3.4 Inference:
In figure 6.1, If shortage to holding cost ratio s/h is four times from 10 to
40, then the corresponding feedback of the number of beds needed to obtain
minimal costs indicates an increase of only 14% from 140 to 160, suggesting
that ratio s/h has no significant influence on the optimal number of beds.
Figure 6.2: Indifference curve for the optimal number of beds
0 50 100 150 200 250 3000
1000
2000
3000
4000
5000
6000Actual Cost per day
Number of beds
Pou
nds
140 145 150 155 1600
1
2
3
4
5
6
7
8
9
The offered Load a
Cos
t ra
tio
The indifference curve
c = 145
c = 150
c = 155
c = 160
113
Finally in figure 6.2, the indifferent curves for different inventory level
c=145, 150, 155 and 160 is shown. This figure suggests that it may be
indifferent to the ratio s/h, if the number of beds is 145, 150, and even 155 i.e.,
the lower curve, but when the number of beds exceeds 160, then the cost
changes dramatically. This is a reflection of the rapidly increasing costs for more
than 155 beds.
6.4 BASE-STOCK MODEL FOR IMPATIENT CUSTOMERS WITH VARYING
DEMAND DISTRIBUTION
An optimal base-stock inventory policy for impatient customers using
finite-horizon models is examined. The base-stock system for impatient
customer is a different type of inventory policy and in case of the impatient
customer, they are likely to bark. Hence their demand is to be satisfied
immediately. The basic model of inventory problem of finding the optimal base-
stock was brought into the form of well known queuing problem of optimizing the
number of channel discussed in Hanssman F [33].
In this model, the upper and lower echelon in applied in the impatient
customer base-stock policy. At the upper echelon is a supplier with a single
production facility which manufactures to order with a fixed production time and
the order are received from retailers on a first-come first-served basis. Where
else the numbers of non-identical, independent retailer are considered at the
lower echelon. The step function in this model is given as
(6.14)
Where
(6.15)
114
The result in this model is considered as continuous demand which was so far
discussed in the form of discrete case. Hence in order to prove this the following
result is given,
Theorem 1: If , is bounded on [ , is continuous at and
then
(6.16)
Proof: let us consider the partition where and
, then where denotes the independent retailers whose stock
is replenished from a single supplier and L where is the number
of idle channels. Since is continuous at . It is seen that and converges
to as
By simplifying the expected number of idle channels, from Hanssman F
[33], it is seen that from equation 6.10 and equation 6.11, the following, equation
is obtained,
(6.17)
(6.18)
(6.19)
115
(6.20)
(6.21)
(6.22)
(6.23)
Using the equation 3.7,
(6.24)
116
Here
Hence the following result is obtained after due simplification
(6.25)
The above equation 6.25 can be evaluated for different values of and the
profit maximizing value may be selected directly. The first difference of
is set to zero then can be obtained.
6.5 CONCLUSION
In model 6.3, it enables the hospital department to balance the cost of
empty beds against the cost of turning patients away, thus facilitating a good
choice of bed provision in order to have low cost and high access to service.
Thus, in this model a means for calculating optimal bed numbers with an
acceptable level of impatient customer in comparison to the model discussed by
Gorunescu F [24] is provided.
However, more generally the queuing theory results used in this model is
valid for any length of stay. So, all it is needed is to use the results for an
estimate of the arrival rate and the average length of stay or the mean lead time.
The advantage of using the phase-type model for length of stay is that, it
provides a useful description of the data and estimation of arrival rates but, the
Poisson arrival and steady state distribution is convenient. Hence the above
discussed methodology is valid.
117
Hanssman F [33] studied the case of model 6.4 in discrete form. However
in model 6.4, a continuous approach is adopted in order to get the optimal base-
stock. This model can be extended to systems with both customer impatience
and perishable inventory. These are two directions of research. Current
research is underway on coordinating the above decisions in the context of
multistage production systems which is discussed in chapter 7.
118
7. THE MULTI-PERIOD MODEL WITH TWO VARYING DEMANDS
7.1 INTRODUCTION
So far in the above chapters, the single period stochastic models are
discussed. In this chapter, the multi-period stochastic model is discussed. The
key difference between single-period model and multi-period model is that, in
single period stock left over will not be carried over to the next period which
means profit is loss. In case of the multi-period, the model may involve stock
leftovers from previous periods, which makes the optimal choice of order
quantities more complicated. It may be observed that in many situations, the
demand for a product cannot be below a particular level.
Another aspect of consideration in the representation of the demand with
a suitable probability distribution is that, the demand size with past has impact or
influence over the demand at a future points of time. Hence, it should be
represented as not having the Lack of Memory Property (LMP). Hence, it is
proposed to use the random variable which follows Erlang 2 distribution that
does not satisfy the LMP property. The random variable follows
exponential distribution with parameter prior to the truncation point and it
follows truncated exponential distribution with parameter after the truncation
point.
Sakaguchi M et.al [59] studied the probabilistic inventory models of multi-
period in which some conditions are reviewed to help getting an optimal policy
provided that the total cost function of single period is known very well. A model
with exponential demand is studied in Sakaguchi M et.al [60], since it is easy
when demand subjects to an exponential distribution.
In this Chapter, two varying demand model is discussed. Under model
7.4, demand and lead time is a constant. In model 7.5, demand and lead time is
considered as random variable. In obtaining the expected total cost, the
probability of having exactly ‘Nth’ demand epochs in the interval is taken
119
under consideration. In model 7.6, the optimal one time supply during the
interval using the generalized gamma distribution with bessel’s function is
discussed, where else model 7.7 deals with a multi-commodity inventory system
with periodic review operating under a stationary policy using the exponential
order statistics.
The expected optimal ordering shown in figure 7.1 indicates a
point at which there is a requirement of reorder. Hence this point is considered
to be the truncation point.
Figure 7.1: Optimal expected ordering when the truncation occurs
The optimal inventory level or the reorder point is determined form the
Figure 7.1 for the multi-period demands. Also adequate numerical analysis
shows its effectiveness.
7.2 BASIC MODEL
In this chapter, a modified version of the model discussed in Sehik
Uduman P.S et.al [64] is considered under the assumption that the random
120
variable denoting demand undergoes a change in the distribution after a
change point or truncation point denoted as . Hence, the use of change of the
distribution after a change point is justified by the fact that demand for any
product over the time interval is not fixed. If the demand for the product is
according to some probability distribution initially and it is very likely that after a
certain point the demand may undergo some changes and the increase in
demand or decrease in demand will undergo considerable change. Hence, to
depict the demand as a random variable undergoing a change of distribution
after the particular magnitude is quite reasonable. The concept of change of
distribution at a change point was discussed by Suresh Kumar R [72]. In the
present model, the expected total cost is given as
(7.1)
Since, equation 7.1 is in form of the differentiation of an integral with respect to
the variable , i.e., is as the integrand, as well as in the limits of integration.
Hence, differential of integral formula is used to solve the result which is
discussed as equation 3.7 of chapter 3. This implies that, the optimal value of
is one which satisfies the equation
(7.2)
Given the values of the inventory holding cost , the shortage cost and the
probability distribution of the random variables denoting Multi-
product demands, the optimal can be determined. This was a basic procedure
for solving the model in Hanssman F [33].
121
7.3 NOTATIONS AND ASSUMPTIONS
- A continuous identically independent random variable denoting
the demand at the Nth epoch, N = 1,2,…, and has PDF
with CDF
- Inventory holding cost / unit
- Shortage cost / unit
- Time variable constant before the truncation point
- Time variable constant after the truncation point
- The supply size or initial stock level
- The change point or truncation point
- Total lead time
- Optimal value of Z
- The inter arrival times between successive demand epochs.
- All are nonnegative, and their inequality is
- the stock level
- Location parameter
122
7.4 THE MULTI-DEMAND TRUNCATED EXPONENTIAL DISTRIBUTION
In this model an extension of the work done by Deemer W.L et.al [18]
form of the truncated exponential distribution considered under the two
parameters
(7.3)
When is a constant, then following cases arises. (i) and (ii)
Case i):
(7.4)
(7.5)
Applying differential of integral equation 3.7 of chapter 3, the equation 7.5 is given
as
(7.6)
123
Hence
(7.7)
Using the equation 7.3, the expression for expected cost is written as
(7.8)
(7.9)
(7.10)
Hence using equation 7.8, 7.9 and 7.10, the following result is obtained
(7.11)
Any value of Z, which satisfies equation 7.11 is the optimal
7.4.1 Numerical illustration
Considering the numerical example when the value of are fixed
and the values of are varied accordingly. Let and
then the following Table7.1 is obtained.
124
Table 7.1: Numerical value for for obtaining
1.5 2.0 2.5 3.0 3.5
0.7348 0.5511 0.4359 0.3649 0.3109
Figure 7.2: Optimal supply z against the truncation point when
7.4.2 Inference
For the case, when is a constant, the condition is independent
of the functions of the parameter . It is observed that as
increases, the value of decreases. This is due to the fact that is the
parameter of the exponential distribution that denotes the demand. Also various
points at which there is a fluctuation in demand is shown in the figure 7.2.
125
Case ii): When , then from equation 7.4
(7.12)
Applying differential of integral equation 3.7 of chapter 3, the equation 7.12 is
given as,
(7.13)
126
(7.14)
(7.15)
Now,
gives,
(7.16)
Any value of which satisfies equation 7.16 is the optimal value ‘ Z ’.
7.4.3 Numerical Illustration
Let us take the numerical example when the value of and
are fixed and the values of are varied accordingly. Let us assume that
127
, , , . Hence, Table 7.2 is obtained as follows,
Table 7.2: Tabulation for obtaining
25 30 35 40 45
4.4804 7.3137 10.0035 12.6383 15.2433
Figure 7.3: Optimal supply z against the truncation point when
7.4.4 Inference
It is observed that as the value of the truncation point increases, the
size of the optimal inventory also increases. This is due to fact that prior to the
truncation point the demand is distributed as exponential with parameter .
After the truncation point the demand is distributed as truncated exponential
distribution with parameter . Also the average of demand before is
exponential and after varies according to the situation. Hence in figure 7.3,
128
the variations of the demand at three points are depicted with different colour
arrows. Therefore the optimal inventory is also increasing.
7.5 Nth EPOCH TWO COMMODITY MODEL
In real life situation demand is always assumed to be random. In this
model, demand and the lead time is considered to be random Nth epoch. Let
there be Nth demand epochs in , i.e., be the random
demands and , i = 1, 2…N are identically independent random variables.
It may be noted that, if , then inventory holding occurs and if
then shortage occurs.
Since is sum of the identically independent random variable and
its PDF is given by , which is the Nth convolution of Hence
(7.17)
The probability of having exactly ‘N’ demand epochs in (0,T) is given
by renewal theory as and this statement is
proved in chapter 5 and is the inter arrival times between successive demand
epochs. The determination of , which is the optimal value of is possible
using equation 7.17, provided the number of demand epochs in which is
namely ‘N’ is known. But, in practice the value of N is not a predetermined
constant. It is also of random character. But from Nabil S Faour [47], it is
possible to have an approximate value for N. The author discussed that, If N is
taken to be a particular value then using incomplete gamma function values the
optimal can be obtained. If > 0 units are ordered, the fixed cost will be a
function of , say . In general, will take as many different values as
the number of alternative different ordering decisions.
129
For the two commodity problem
(7.18)
Where are all nonnegative, and the inequality
is satisfied. It may be observed and are obtained on the basis of
an average which is taken by variable of occurrences of demand value before
and after respectively. It is assumed that there are random epochs at
and at epochs and . Using the property of 7.18 in
7.17, the expected optimal profit is given by
(7.19)
Let be continuous and twice differentiable. The function is the cost
charged over a given period of time excluding the ordering cost and in general it
is the holding and shortage costs for each item in a linear form.
From Nabil S Faour [47], considering the case of the two commodity, the optimal
expected cost is obtained for the th demand epoch,
(7.20)
Considering and by solving the equation 7.19 with respect to equation
7.20
130
(7.21)
(7.22)
(7.23)
(7.24)
131
Any value of which satisfies the equation 7.24 for the given values of
gives the optimal namely .
7.5.1 Conclusion
Thus this model provides an insight on the optimal supply using the
truncated exponential distribution when the demand over is a constant and
the demand over is a Nth demand random epoch. Future work may involve
the use of truncation demand distribution when demand over is a Nth
demand random epoch.
7.6 GENERALIZED GAMMA BESSEL MODEL
In Nicy Sebastian [50], a new probability density function associated with
a Bessel function was introduced, which is the generalization of a gamma-type
distribution. Some of its special cases were mentioned. Multivariate analogue,
conditional density, best predictor function, Bayesian analysis, etc., connected
with this new density are also introduced and suitability of this density as a good
model in Bayesian inference and regression theory was also discussed in their
work.
Hence in this model, generalized gamma distribution with Bessel function
is used and the optimal supply at is determined using this function. Under
these assumptions two different approaches are used in analyzing this model. In
model 7.6 using generalized gamma distribution with Bessel function, the
probability density function is derived and hence optimal supply size is obtained.
In model 7.7, a multi commodity inventory system with periodic review operating
under a stationary policy is considered using the exponential order statistics and
this methodology is applied in the well known Hanssman F [33] model. Hence
132
the optimal is obtained for both the cases and adequate numerical example is
provided.
7.6.1 Basic model
The basic model is adopted from the Hanssman F [33] and the notation of
the model is as follows, is considered a continuous random variable
representing demand and ~ generalized gamma distribution with Bessel
function and truncated at ‘ ’ in the left and at ‘ ’ in the right, with parameter ‘ ’.
The general form for the total expected cost given in Hanssman F [33] is
(7.25)
To find the optimal inventory, equation 7.25 is evaluated for
The solution of Involves the concept differentiation of an integral given by
equation 3.7 of chapter 3, because variable ‘ ’ is in limit as well as in the
integrand. Hence
. This implies that quantity Z is determined using
the demand distribution function such that
A different variation of this basic model is attempted in above chapters. But in
this chapter, an attempt to solve the above model using generalized gamma
distribution with bessel function is made.
In model 7.6, the ordering decision in each period is affected by a single
setup cost ‘k’ and expected holding and shortage cost function . Condition
for being in stock level is given as , at the beginning of a period,
is assumed to be twice differentiable. Demand for the item over a
sequence of period is assumed to be independently and identically
distributed, continuous non-negative random variable with continuous joint
density function . Let some constraints be placed on the limits of the
133
demand distribution. Hence the PDF of the generalized gamma distribution
with bessel function is given in the form
(7.26)
Now to prove the validity for the equation 7.26 to be the probability density
function and hence the aim is to find
(7.27)
Consider
(7.28)
Using equation 7.28 in equation 7.27
(7.29)
Hence the value of is obtained as
The PDF for the above model is
obtained as
(7.30)
Hence the expected total cost in this case can be written as
134
(7.31)
To obtain optimal ,
(7.32)
Hence
. So, the Laplace transform for the above model is given as
(7.33)
Now
(7.34)
135
On simplification
(7.35)
To demonstrate that the objective function is convex, the second order for
equation 7.35 is carried out. Hence
(7.36)
Any value of , which satisfies equation 7.35 for the given is
the optimal supply size.
7.6.2 Numerical Illustration
When the demand is increased accordingly to the fixed values ,
, , , then from equation 7.35 the following Figure 7.4
is obtained.
Figure 7.4: Optimal profit curve with respect to arrival of demand
136
7.6.3 Inference
When the demand is increased, the optimal profit decreases. Hence
increases in demand, corresponds to increase in supply size.
7.6.4 Numerical Illustration
When the lead time is increased accordingly for the fixed values , ,
,… then from equation 7.36 the following Figure7.5 is
obtained.
Figure7.5: Lead time with optimal supply size
7.6.5 Inference
When the lead time is increased, the optimal profit increases. Hence
increases in lead time, corresponds to increase in supply size.
137
7.7 A MULTI-COMMODITY EXPONENTIAL ORDER STATISTICS
A multi commodity inventory system with periodic review operating under
a stationary policy is considered. The ordering decision in each period is
affected by a single setup cost k and a linear variable ordering cost
At the beginning of a period, the stock level is . An
inventory system with time to shortage and holding of the items is our prime
interest. A single new component at time zero be started and replace it upon
loss by a new component and so on. This time to loss which is represented by
exponential order statistics is independent and the key to model when
there is joint PDF is
(7.37)
Suppose are the order statistics of a random variable of
size n arising from interested with the distribution of
(7.38)
Then will constitute the renewal process. Let us consider the joint PDF of all
order to be given by
(7.39)
Let us define as the length of time measured backwards from 1 to the last
renewal at or before n
138
(7.40)
where ,
hence
(7.41)
This proves that are all independently and exponentially distributed
and is distributed with an exponential distributed with scale parameter
. From the earlier literature of exponential order statistics, following
equation is obtained
(7.42)
where
(7.43)
139
(7.44)
Using equation 7.30 in equation 7.44, the following equation 7.45 is obtained
(7.45)
Any value of , which satisfies equation 7.45 is the optimal . It may be noted
that the value of depends upon a number of parameter such as ,
. But for the use of this model in practical situations it
becomes necessary to estimate the value of on the basis of sample data.
,
and are of deterministic character and hence they are
fixed. However in the determination of , most vital values are obtained.
140
7.8 CONCLUSION
Under a stationary policy as discussed before either all items will be
ordered to bring to the inventory level to , if is the inventory level at the
beginning of a period prior to making a decision, then after nothing is ordered.
The primary concern of this study is to find the optimality condition for
stationary policy. This is done by minimizing the expression for the stationary
total expected cost per period with respect to the decision variable that
characterizes the policy being used.
141
8. CONCLUSION
8.1 SUMMARY
From the investigation taken up on the various types of inventory
models, it is quite interesting to observe the changes in the optimal solutions
when the models are suitably modified by incorporating some changes in the
models. The conceptualization of the models is by incorporating some real
life-situations, which are acceptable. For example, the demand for any
product or commodity can undergo changes with the passage of times. Then
there are two possibilities analysed in the study i) If the demand is
considered as random variable then this random variable is undergoing a
parametric change. ii) Alternatively it is also seen that the demand
distribution itself is undergoing a change both in its form and the parameters.
Hence, the aim is to study the changes in the optimal inventory size due to
the changes out lined above. These solutions are of practical use and
importance since they provide the optimal size of the inventory to be
maintained.
Motivated from Hanssman F [33], various distributions such as SCBZ
property, truncated exponential distribution, renewal reward, generalized
gamma distribution with Bessel function and exponential order statistics are
analyzed with respect to its stochastic behaviour. This study is not only useful
for computations, but it is also a basic tool for the theoretical investigation of
inventory control problems. The following broad conclusions can be given on
the basis of the models developed in this thesis.
In the case of single-period newsboy inventory model with stochastic
demand and partial backlogging as discussed in chapter 3, it may be
concluded that the excess demand is partially backlogged. It is observed that
when the parameter of the demand distribution prior to the truncation point
increases then the actual demand decreases and so reduction in the supply
size is desirable. If the parameter of the demand distribution after the
truncation point increases, an increase in supply is suggested. So, the
142
behaviour of the supply prior to truncation and after truncation is well defined.
The properties and numerical results that are derived show that there is
structure for an easy-to-understand optimal replenishment policy which can
be implemented in real-life applications. This is an interesting aspect to
investigate in the future.
In case of a truncated exponential distribution and renewal reward concept
used in single period model discussed in chapter 3 and chapter 4, it is noted
that, as the value of parameter prior to the truncation point increases, it
results in a decline in the optimal inventory. As the value of inventory holding
cost increases a fewer size of inventory is recommended. The truncated
distributions have found many applications such as the study involving fitting
rainfall data, animal population studies and to aiming errors i.e., range,
deflection, etc., in gunnery and other bombing accuracy studies.
As per the findings in chapter 5, in which a base-stock system for patient
customer with demand distribution undergoing a change, it is seen that a
base-stock is necessary for a system before sales, because the customer
request may vary accordingly with respect to the demand. In this model it is
seen that when the value increases, a decrease in the base-stock is
suggested and similarly as the value of shortage cost increases, a higher
level of inventory base-stock is desirable as indicated in the numerical
illustration in the case when . The dependency between customer
satisfaction and availability was hardly studied in the literature. Hence, in
order to study this situation, a model of base-stock system for patient
customer is analysed using Erlang2 and truncation exponential distribution.
In case of base stock for impatient customer model discussed in
Chapter 6, a study on two models is carried out. This model enables the
hospital managers to balance the cost of empty beds against the cost of
turning patients away, thus facilitating a good choice of bed provision in order
to have low cost and high access to service. In particular, a case study is
analysed on what the level of inventory has to be considered. This is the
study in which customers do not accept partially fulfilled requests.
143
The purpose of Chapter 7 is to consider multi-period versions of the
single period models. The primary concern of this study is to find the
optimality condition for stationary policy. When is a constant, the condition
is independent of the functions of the parameter . It is
observed that as increases, the value of decreases. This is due to the
fact that is the parameter of the exponential distribution that denotes
the demand. Where else for the second case, it is observed that as the value
of the truncation point increases, the size of the optimal inventory also
increases. This is due to fact that prior to the truncation point the demand is
distributed as exponential with parameter . After the truncation point the
demand is distributed as truncated exponential distribution with parameter .
Also the average of demand before is exponential and after it is varies
according to the situation.
In case of the generalised gamma distribution with Bessel function,
when the demand is increased, the optimal profit decreases. Hence, an
increase in demand corresponds to increase in supply size. Also in this
model when the lead time is increased, the optimal profit increases. Hence
increases in lead time, corresponds to increase in supply size. Finally, the
optimal cost is obtained for all the models.
The overall objective of this thesis is to analysis the stochastic
inventory models. The stochastic inventory model is related to model of
Hanssman F [33]. Accordingly, Newsboy and Base-stock models in general
have a stochastic behaviour. Hence, the aim of selecting these models is
justified. So far these models were studied using SCBZ property, Erlang2
and order statistics, but the concept of truncation needed wide attention.
Therefore, considering a prime motive to contribute on this topic of
truncation, an analysis on the stochastic behaviour of the models is analysed.
Earlier work on truncation was studied for Newsboy models with SCBZ
property and Erlang2 distribution, but in this thesis a generalised newsboy
model is studied using SCBZ property and also newsboy model is studied
using renewal reward theory. The role of SCBZ property was limited in case
144
on generalisation. Hence, a change of distribution for change point needed. A
switch over of distribution from SCBZ property to truncated exponential
distribution is carried out. But when using truncated exponential distribution,
the role of exponential distribution is important. Hence in order to analysis
this, a new distribution is developed. Finally, adding to this analysis a study
on generalised gamma distribution with Bessel function is carried out. The
overall optimal supply and stock level is justified using the numerical
illustrations.
8.2 SCOPE FOR FUTURE WORK
Numerous variations are possible which aim at capturing different real-
world situations. It is quite interesting to observe the changes in the optimal
solution of the inventory models, which are revised taking into consideration
the changes in the demand structure and demand distribution. Also it is quite
reasonable to expect the changes if the supply is taken to be random
variable and in this case the amount of supply has to be decided only by
taking a probability distribution of the random variable denoting the supply. It
is an open area for further studies. More over the models will have greater
utility and real life applications provided the distribution of the random
variable involves in the model are approximately chosen.
Similarly the demand distribution must be suitably formulated on the
basis of real life data. Statistical tests for goodness of fit of the distribution
must be carried out and the approximation of the distribution should be
decided. The exact data on holding cost and shortage cost is another
important point in consideration. If these considerations are carried out
perfectly, the models become more application oriented and hence the utility
of the optimal solution will be appropriate.
The application of the methodologies and techniques developed in this
study can be applied to any inventory environment where attrition reduces
available cost of inventory level. Examples of potential applications include
145
perishable products such as medicines, tenure based organizations and
organizations with retirement eligibility based on years of service. With minor
modifications to account for the characteristics of the asset and the
operational conditions, this model can be applied and can provide
management with useful information.
The probability distribution developed in this model can be used in
cancer study. Tumour growth in initial stage is unknown, which is
exponentially growing. But, the growth of tumour in latter half can be studied
using truncated exponential distribution. This concept is helpful for the
doctors to give an appropriate treatment to the patient.
146
REFERENCE
[1] Abad P, “Optimal price and order size for a reseller under partial
backordering”, Computers & Operations Research, Vol.28, Issue 1, pp.53-
65, 2001.
[2] Abd El-Fatah I.M, Mead M.E and Semary H.E, “The application of the
modified generalized gamma distribution in inventory control”, Int. J.
Contemp. Math. Sciences, Vol.6, Issue 35, pp.1699-1712, 2011.
[3] Abodunde T.T and McClean S.I, “Production planning for manpower system
with a constant level of recruitment”, Applied Statistics, Vol.29, pp.43 – 49,
1980.
[4] Amy Lau and Hon-Shiang Lau, “Newsboy problem with price distribution
demand distribution”, IIE Transaction, Vol.20, pp.168-175, 1988.
[5] Anderson, Sweeney and Williams, “Inventory models-quantitative methods
for business”, 1st ed. Kundali, India, Thompson Southwest, pp.135-185,
2009.
[6] Arrow K.J, Harris and Marschak, “Optimal inventory policy”, Econometrica,
Vol.19, pp.250-272, 1951.
[7] Arrow K.J, Karlin S and Scarf H, “Studies in the mathematical theory of
inventory production”, Stanford Univ. Press, California, 1958.
[8] Ata Allah Taleizadeh, Seyed Taghi Akhavan, Niaki Seyed and Mohammad
Haji Seyedjavadi, “Multi-product multi-chance-constraint stochastic inventory
control problem with dynamic demand and partial back-ordering: A harmony
search algorithm”, Journal of Manufacturing Systems, Vol.31, pp.204– 213,
2012.
[9] Axsater S, “Continuous review policies for multi-level inventory systems with
stochastic demand”, Handbook in operations research and management
Science, Vol. 4, 1993.
[10] Brill Percy H and Ben A.C, “An EOQ model with random variables in
demand”, Management Science, Vol.41, Issue 5, pp.927–936, 1995.
[11] Cawdery M.N, “The lead time and Inventory control problem”, Operational
Research Quarterly, Vol. 27, pp.971–982, 1976.
147
[12] Chang H.C and HO C.H, “Exact closed-form solutions for optimal inventory
model for items with imperfect quality and shortage backordering’’. Omega,
Vol.38, pp.233–237, 2010.
[13] Chenniappan P.K and Ramanarayanan R, “Inventories in series”,
International Journal of Information and Management Sciences, Vol. 7, Issue
1, pp.45–52, 1996.
[14] Chin Tsai and Shiuh-Nan Hwang, “A Generalization of Chang and Lin’s
model in a multi location Newsboy problem”, Information and Management
Sciences, Vol.9, Issue 2, pp.11-18, 1998.
[15] Covert R.P and Philip G.C, “An EOQ model for items with weibull distribution
deterioration”, AIIE Transactions, Vol.5, pp.323–326, 1973.
[16] Dada M, Petruzzi N, and Schwarz L, “A newsvendor's procurement problem
when suppliers are unreliable”, Manufacturing & Service Operations
Management, Vol.9, pp.19-32, 2007.
[17] Das C, “(Q, r) Inventory models with time-weighted backorders”, Journal of
Operational Research Society, Vol. 34, Issue 5, pp.401-412, 1983.
[18] Deemer W.L and David F.V, “Estimation of parameters of truncated or
censored exponential distributions”, Anal. Math. Stat, Vol.26, pp.498-504,
1955.
[19] Dong-Ping Song, “Stability and optimization of a production inventory system
under prioritised base-stock control”, IMA journal of management
mathematics, Vol. 20, Issue 1, pp. 59-79, 2009.
[20] Dvoretzky A, Kiefer J and Wolfowitz J, “The inventory problem”,
Econometrica, Vol. 20, Issue 2, pp.187 – 222, 1952.
[21] Gaver D.P, “On base-stock level inventory control”, Operations Research,
Vol. 7, pp.689–703, 1959.
[22] George Lominashvili and Mzevinar Patsatsia, “On the estimation of a
maximum likelihood of truncated exponential distributions”, Bulletin of the
Georgian National Academy of Sciences, Vol.7, Issue 1, pp.21-24, 2013.
[23] Goh M, “EOQ models with general demand and holding cost functions”,
European Journal of Operational Research, Vol.73, pp. 50 – 54, 1994.
[24] Gorunescu F, “A queuing model for bed occupancy management and
planning of hospitals”, Journal of Operation Research Society, Vol.53, pp.19-
24, 2002.
148
[25] Grubbstrom R.W, “The newsboy problem when customer demand is a
compound renewal process”, European Journal of Operational Research,
Vol.203, pp.134-142, 2010.
[26] Guiqing Zhang and Yinfeng Xu, “Risk reward competitive analysis for the
newsboy problem with range information”, COCOA, lncs 5573, Springer-
Verlang Berlin Herdelberg, pp.334-345, 2009.
[27] Guray Guler M, “The value of modeling with reference effects in stochastic
inventory and pricing problems”, Expert Systems with Applications, Vol.40,
pp.6593–6600, 2013.
[28] Gurland J, “Distribution of the maximum arithmetic mean of correlated
random variables”, AMS, Vol.26, Issue 2, pp.294-300, 1955.
[29] Hadley G and Whitin T.M, “An optimal final inventory model”, Management
Science, Vol.7, Issue 2, pp.179-183, 1961.
[30] Haifeng Wang and Houmin Yan, “Inventory management for customer with
alternative lead times”, Production and Operation management, Vol.18,
Issue 6, pp.705-720, 2009.
[31] Haifeng Wang, Bocheng Chen and Houmin Yan, “Optimal inventory
decisions in a multi-period newsvendor problem with partially observed
markovian supply capacities”, European Journal of Operational Research,
Vol. 202, Issue 2, pp. 502-517, 2010.
[32] Hanssman F and Rivett B.H.P, “Competitive bidding”, Operational Research
Quarterly, Vol.10, Issue 1, pp.49–55, 1959.
[33] Hanssman F, “Operations research in production and inventory control”,
John Wiley and Sons Inc, New York, 1962.
[34] Holt C.C, Modigliani F, Muth J. F and Simon H. A, “Planning, Production”,
Inventories and Work force, Prentice-Hall, NJ, 1960.
[35] Jixan Xiao and Fangling Lu, “Stochastic newsboy inventory control model its
solving on multiple product order pricing”, Proceedings of the First
International Conference on Information Computing and Applications
(ICICA), pp.65-72, 2010.
[36] John S. Rose, “The newsboy with known demand and uncertain
replenishment applications to quality control and container fills”, Operations
Research Letters, Vol.11, pp.111–117, 1992.
149
[37] Khouja M, “The single-period (news-vendor) problem: literature review and
suggestions for future research” Omega, Vol.27, pp.537-553, 1999.
[38] Kumaran M and Achary K.K, “On approximating lead time demand
distributions using the generalized type distribution”, Journal of the
Operational Research Society, Vol.47, pp.395–404, 1996.
[39] LiangYuh Ouyang, KunShan Wu and ChiaHuei Ho, “An integrated vendor
buyer inventory model with quality improvement and lead-time reduction”,
International Journal of Production Economic, Vol.108, pp.349–358, 2007.
[40] Little J.D.C, “The use of storage water in a hydroelectric system”, Journal of
Operations Research Society of America, Vol.3, Issue 2, pp.187-197, 1955.
[41] Maddah B, Jaber M.Y, “Economic order quantity for items with imperfect
quality: revisited”, International Journal of Production Economics, Vol.112,
Issue.2, pp.808–815, 2008.
[42] Markus Ettl, Gerald E.F, Grace Y. Lin, and David D. Yao, “A supply network
model with base-stock control and service requirements”, Operations
Research, Vol. 48, Issue 2, March-April, pp. 216-232, 2000.
[43] Metin Cakanyyildirim, James H.Bookbinder and Yigal Gerchak, “Continuous
review inventory models where random lead time depends on lot-size and
reserved capacity”, International Journal of Production Economics, Vol.
68, Issue 3, pp. 217-228, 2000.
[44] Mirzazadeh A, “Inventory management under stochastic conditions with
multiple objectives”, Artificial Intelligence Research, Vol. 2, Issue 3, pp.16-26,
2013.
[45] Muhammad Aslam, Debasis Kunda and Munir Ahmad, “Time truncated
acceptance sampling plans for generalized exponential distribution”, Journal
of Applied Statistics, Vol. 37, Issue 4, pp.555-566, 2010.
[46] Murthy S and Ramanarayanan R, “Inventory system exposed to calamities
with SCBZ arrival property”, Journal of Morden Mathematics and Statistics,
Vol.No:2 Issue 3, pp.109-119, 2008.
[47] Nabil S. Faour, “Stationary analysis and optimality conditions for
policies in multi-commodity inventory control problems”, Ph.D thesis,
University of Florida, 1972.
[48] Neuts M.F, “A versatile markovian point process”, Journal of Applied
Probability, Vol.16, Issue 4, pp.764-779, 1979.
150
[49] Nicholas A. Nechval and Konstantin N. Nechval, “Constrained optimization
in newsboy problems under uncertainty via statistical inference equivalence
principle”, Proceeding 12th International Conference, ASMTA, pp. 1-5, 2005.
[50] Nicy Sebastian, “A generalized gamma model associated with a Bessel
function”, Integral Transform and special functions, Vol. 22, Issue 9, pp.631-
645, 2011.
[51] Ozalp Ozer, Onur Uncua and Wei Wei, “Production, manufacturing and
logistics selling to the newsvendor with a forecast update: Analysis of dual
purchase contract”, European Journal of Operational Research, Vol.182, pp.
1150 – 1176, 2007.
[52] Philip G. C, “A generalized EOQ model for items with Weibull distribution
deterioration”, AIIE Transactions, Vol. 6, pp.159-162, 1974.
[53] Raja Rao and Talwakar S, “Setting the clock back to zero property of a family
of life distribution”, Journal of Statistical Planning and Inference, Vol.24,
pp.347-352, 1990.
[54] Ramanarayanan R, “Distribution of the maximum arithmetic mean of
correlated random variable”, AMS Vol.26, pp.1-6, 1988.
[55] Ramanarayanan R, “Explicit steady state solution for a markovian inventory
system”, Statistica neerlandica, Vol.35, Issue 4, pp.215-219, 1981.
[56] Ramasesh R. V, Ord J. K and Hayya J. C, “Note: Dual Sourcing with non-
identical Vendors”, Naval Research Logistics, Vol.40, pp. 279-288, 1993.
[57] Roger D.H and Warburton, “An exact analytical solution to the production
inventory control problem”, International Journal Production Economics,
Vol.92, pp.81–96, 2004.
[58] Sachithanantham S, Ganesan V, R Sathyamoorthy, “A model of base stock
system for patient customers with lead time distribution undergoing a
parametric change”, Ultrascience: phy sci, Vol.20, no.3 (M), pp.651-658,
2008.
[59] Sakaguchi M and Kodama M, “Inventory models with a varying demand
distribution”, In Proceedings of International Workshop on Recent Advances
in Stochastic Operations Research, Canmore, pp.237-244, August 25-26
2005.
[60] Sakaguchi M and Kodama M, “On the dynamic probabilistic inventory
problems With Piece wise cost functions which may not be piecewise
151
smooth”, Bulletin of Informatics and Cybernetics, Research Association of
Statistical Sciences, Kyushu University, Vol. 34, pp.75-90, 2002.
[61] Salameh M.K and Jaber M.Y, “Economic production quantity model for items
with imperfect quality”, International Journal of Production Economics,
Vol.64, pp.59-64, 2000.
[62] Sandeep Jain and Srinivasa Raghavan N.R, “Analysis of base-stock
controlled production inventory system using discrete time queuing models”,
Proceedings of IEEE International Conference on Automation Science and
Engineering, Edmonton, Canada, August 12, 2005.
[63] Sathiyamoorthy R and Parthasarthy, “On the expected time to recruitment
when threshold distribution has SCBZ property”, International Journal of
Management and system, Vol.19, No.3, pp.233-240, 2003.
[64] Sehik Uduman P.S, “A study on inventory control models using SCBZ
property”, Ph.D Thesis, Bhartidasan University, 2009.
[65] Sehik Uduman P.S, Sulaiman A and Sathiyamoorthy R, “Newsboy inventory
model with demand satisfying SCBZ Property”, Bulletin of pure and applied
Science,Vol.26E, Issue 1, pp.145-150, 2007.
[66] Shih-Pin Chen and Yann-Horng Ho, “Optimal inventory policy for the fuzzy
newsboy problem with quantity discounts”, Information Sciences, Vol. 228,
pp.75-89, 2013.
[67] Silver E. A, Pyke D. F and Peterson R, “Inventory management and
production planning and scheduling”, John Wiley & Sons, 1998.
[68] Snyder L.V and Shen Z.J.M, “Supply and demand uncertainty in multi-
echelon supply chains”, INFORMS, pp.1-46, Pittsburgh, PA, 2006.
[69] Srinivasa Rao S, “Double acceptance sampling plans based on truncated life
tests for the Marshall – Olkin extended exponential distribution”, Austrian
Journal of Statistics, Vol.40, Issue 3, pp.169 – 176, 2011.
[70] Srinivasan S, Sulaiman A and Sathiyamoorthi R, “On the determination of
optimal inventory with demand distribution as order statistics”, Bulletin of
Pure and Applied Sciences, pp. 25 – 32, 2007.
[71] Stagnl D. K, “Exponential change point model”, Statistics in Medicine, Vol.14,
pp.2173-2190, 1995.
[72] Suresh Kumar R, “Shock model when the threshold has a change of
distribution after a change point”, Journal of IAMS, Vol.28, pp.73-89, 2006.
152
[73] Valentin Pando, Luis A. San-Jose, Juan Garcia-Laguna and Joaquin Sicilia,
“A newsboy problem with an emergency order under a general backorder
rate function”, Omega, Vol.41, Issue.6, pp.1020–1028, 2013.
[74] Vijaya, “A stochastic model with SCBZ property in global warming, IOSR-
Journal of Mathematics, Vol.1, Issue 4, pp.37-40, 2012.
[75] Vora N.D, “Inventory management”, Quantitative Techniques in
Management, New Delhi, India, Tata McGraw-Hill, pp.386-451, 2011.
[76] Wee H.M, Yu J, Chen M.C, “Optimal inventory model for items with imperfect
quality and shortage backordering”, Omega Vol.35 Issue 1, pp. 7–11, 2007.
[77] Whitin T.M, “The Theory of Inventory Management”, 2nd edition, Princeton
University Press, Princeton, New Jersey, 1957.
[78] Xiao T and Qi X, “Price competition cost and demand disruption and
coordination of a supply chain with one manufacturer and two competing
retailers”, Omega36, pp.741-753, 2008.
[79] Yan K and Kulkarni V, “Optimal inventory policies under stochastic
production and demand rates”, Stochastic Models, Vol.24, pp.173–190,
2008.
[80] Zohar M.A. Strinka, H. Edwin Romeijn and Jingchen Wu, “Exact and
heuristic methods for a class of selective newsvendor problems with normally
distributed demands”, Omega, Vol.41, Issue 2, Pages 250–258, April 2013.
153
TECHNICAL BIOGRAPHY
Mrs. DOWLATH FATHIMA (RRN. 0989205) was born in Chennai, Tamil Nadu
on 11th April 1984. She finished her schooling from P.N Dhawan Adarsh
Vidyalaya Matriculation higher secondary school, Chennai. She received her
B.Sc. degree in Mathematics from Justice Baheer Ahmed Syeed College for
Women (Affiliated to Madras University) in the year 2004. During her school
days she obtained N.C.C ‘A’ certificate and during her college days she
obtained N.C.C ‘B’ and ‘C’ certificate with ‘A’ grade in N.C.C ‘C’ certificate. She
was selected as department secretary during 2003-2004. She completed her
M.Sc. in Mathematics from Stella Maris College (Affiliated to Madras University)
in the year 2006. She secured 87% in her M.Sc and was awarded with
proficiency price for securing centum in Complex Analysis during the year 2005-
2006. Further, she received her M. Phil Mathematics from Allagappa University,
Karaikudi in the year 2009. She is currently pursuing her Ph.D. Mathematics in
the Department of Mathematics, B. S. Abdur Rahman University, Chennai.
During her Ph.D. trajectory, she was awarded Maulana Azad National
Fellowship for minority community for the year 2010-2011. She has six papers
under her credit out of which four papers are published in peer reviewed
journals and two papers are communicated in International Journals. She has
presented three papers in the International Conferences and one paper in
National Conference. She has attended a workshop organized by
Bhaskarachariya pratiniya and UGC, Pune. Also, she has attended SERB
school on multivariable conducted by CMS Pala, Kerala.
154
List of publications based on the research work
[1] Dowlath Fathima, P.S Sehik Uduman and S Srinivasan, “Generalization
of Newsboy problem with demand distribution satisfying the SCBZ
property”, International Journal of Contemporary Mathematical sciences,
Vol. 6, Issue 40, pp.1989 – 2000, 2011.
[2] P.S Sehik Uduman, S Srinivasan, Dowlath Fathima and R.
Sathyamoorthy, “Inventory model with change in demand distribution”,
Australian Journal of Basic and Applied sciences,Vol.5, Issue 8, pp. 468-
478, 2011.
[3] Dowlath Fathima and P.S Sehik Uduman, “Single period inventory model
with stochastic demand and partial backlogging”, International Journal of
Management, Vol. 4, Issue 1, pp.95-111, 2013.
[4] Dowlath Fathima and P.S Sehik Uduman, “Truncated distribution and
renewal reward theory in single period model”, International Journal of
Applied Mathematics, Vol. 15, Issue 1, pp.1110-1114, 2013.
Papers communicated based on the research work
[1] Dowlath Fathima and P.S Sehik Uduman, “A multi-period inventory
model with change in demand distribution using truncated exponential
distribution”, to European Journal of Operation Research.
[2] Dowlath Fathima and P.S Sehik Uduman, “On the determination of
optimal supply size with truncated generalized Gamma Bessel model”, to
Appl Math Inf Sci.
Presentation in National and International conferences
[1] Dowlath Fathima and P.S Sehik Uduman, “Base stock queuing model to
optimize the demand of beds in a hospital”, National seminar on Graph
theory Algorithms and Modeling (GAM 2010), organized by Jamal
Mohamed College, Trichy, 2010.
155
[2] Dowlath Fathima and P.S Sehik Uduman, “Base stock systems for
patient customers with demand distribution undergoing a change with
constant coefficient”, International Conference on Emerging Trends in
Mathematics and Computer Applications (ICETMCA 2010), organized by
Mepco schlenk engineering college, Sivakasi, 2010.
[3] Dowlath Fathima and P.S Sehik Uduman, “A Finite process inventory
model using SCBZ property”, International Conference on Mathematics
in Engineering and Business Management (ICMEB2012), organized by
Stella Maris College and Loyola Institute of Business Administration,
Chennai, 2012.
[4] Dowlath Fathima and P.S Sehik Uduman, “Base stock systems for
Patient VS Impatient customers with varying demand distribution”,
International Conference on Mathematical Sciences and Statistics
(ICMSS 2013), Kuala Lumpur, Malaysia, AIP conference proceedings,
Vol.1557, pp.529-533, 2013.
![New Approach for Stochastic Modelling of Microgrid ... · involves various stochastic modelling and simulation methods [4, 6, 7]. The survey of stochastic modelling of microgrid is](https://static.documents.pub/doc/80x56/5f4190048356da16412b2f00/new-approach-for-stochastic-modelling-of-microgrid-involves-various-stochastic.jpg)
