ABSTRACT

In many real-life situations the consumption rate for certain types of consumer goods (such as fruits, vegetables, doughnuts, cassava, yams and so on) is sometimes influenced by the stock-level. In this research, five deterministic inventory models for delayed deteriorating items with inventory level dependent demand rate have been developed. The models are (1) an economic order quantity (EOQ) model with demand rate as a polynomial function of instantaneous inventory level and constant rate of deterioration, (2) an EOQ model with demand rate as a linear function of instantaneous inventory level and shortages, (3) an economic production quantity (EPQ) model with demand rate as a linear function of instantaneous inventory level with linear-time dependent holding cost, (4) an EPQ model with demand rate as a linear function of instantaneous inventory level and time-dependent deterioration rate, and (5) an EOQ model with demand rate as a polynomial function of instantaneous inventory level and linear time-dependent holding cost. In all the models, theoretical control model results for the optimal replenishment policy of the inventory systems have been determined in order to minimize the inventory system cost or maximize profit per unit time. Analytic formulation and analysis of the inventory problems were developed on the framework of the models assumptions. The proposed inventory systems were represented by systems of differential equations including initial and boundary conditions and typical differential and integral calculus were used to analyse the inventory problems. Newton-Raphson method has been used to determine the optimal solutions (optimal order or production quantity and the optimal cycle length) of the developed cost minimization models for the first four models and the profit maximization model for the fifth model. Some numerical examples to illustrate the applications of the models developed are also
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presented. The numerical example presented to demonstrate the fifth model was accompanied by sensitivity analysis. It was shown from the results of the models developed that the impact of stock dependent demand rate, holding cost parameter, deterioration and backlogging rate affect the optimal replenishment policy, and hence should not be left out in developing inventory models for delayed deteriorating items with inventory level dependent demand rate. Finally, the effects of errors in the estimation of various model parameters or even changes in the values of model parameters may occur due to uncertainties surrounding the future of the optimal solutions. In order to examine the implications of these errors or changes, sensitivity analysis was studied based on the given example for the fifth model.
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TABLE OF CONTENTS

Cover Page …………………………………………………………………………………………………………. i
Fly Leaf …………………………………………………………………………………………………………….. ii
Title Page ………………………………………………………………………………………………………….. ii
Declaration ……………………………………………………………………………………………………….. iii
Certification ……………………………………………………………………………………………………… iv
Acknowledgement ……………………………………………………………………………………………… v
Dedication ……………………………………………………………………………………………………….. vii
Abstract ………………………………………………………………………………………………………….. viii
Table of Contents ……………………………………………………………………………………………….. x
List of Tables …………………………………………………………………………………………………… xv
List of Figures …………………………………………………………………………………………………. xvi
List of Appendices ………………………………………………………………………………………….. xvii
Notation ……………………………………………………………………………………………………….. xviii
CHAPTER ONE …………………………………………………………………………………………………. 1
INTRODUCTION ………………………………………………………………………………………………. 1
1.1 Background of the Study …………………………………………………………………………….. 1
1.2 Classical Square Root EOQ Model ………………………………………………………………. 3
1.3 Categorization of Inventory Models ……………………………………………………………… 4
1.4 Components of Inventory Models ………………………………………………………………… 5
1.4.1 Relevant Inventory costs ………………………………………………………………….. 6
1.4.2 Demand for inventory items …………………………………………………………….. 8
1.4.3 Order cycle …………………………………………………………………………………….. 9
1.4.4 Lead time ………………………………………………………………………………………. 9
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1.4.5 Inventory replenishment ………………………………………………………………… 10
1.4.6 Time horizon ………………………………………………………………………………… 10
1.4.7 Operating decision rules ………………………………………………………………… 10
1.5 Some Terminologies connected with Inventory Modelling …………………………….. 10
1.6 Assumptions in Economic Order Quantity (EOQ) Models ……………………………… 12
1.7 The Basic Economic Order Quantity Model …………………………………………………. 12
1.7.1 Assumptions of basic EOQ model …………………………………………………… 13
1.7.2 Derivation of basic EOQ model………………………………………………………. 14
1.8 The Research Problem ……………………………………………………………………………….. 16
1.9 Justification/Significance of the Study …………………………………………………………. 17
1.10 Aim and Objectives of the Study …………………………………………………………………. 18
1.11 Research Methodology ………………………………………………………………………………. 20
1.12 Organization of the Thesis ………………………………………………………………………….. 21
CHAPTER TWO ………………………………………………………………………………………………. 25
LITERATURE REVIEW …………………………………………………………………………………… 25
2.1 Introduction ……………………………………………………………………………………………… 25
2.2 Classical Square Root EOQ Model …………………………………………………………….. 27
2.3 Inventory Models with Constant / Variable Demand …………………………………….. 28
2.4 Deteriorating Inventory Models………………………………………………………………….. 32
2.5 Delayed (Non-instantaneous) Deteriorating Inventory Models……………………….. 34
2.6 Variable Holding Inventory Models …………………………………………………………….. 35
2.7 Production Inventory Models for Deteriorating Items …………………………………… 36
2.8 Deteriorating Inventory Models with Shortages ……………………………………………. 38
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CHAPTER THREE …………………………………………………………………………………………… 42
AN EOQ MODEL FOR DELAYED DETERIORATING ITEM WITH INVENTORY LEVEL – DEPENDENT DEMAND RATE AND CONSTANT DETERIORATION RATE ……………………………………………………………………………………………………………… 42
3.1 Introduction …………………………………………………………………………………………….. 42
3.2 Modelling Assumptions ……………………………………………………………………………. 43
3.3 Mathematical Analysis …………………………………………………………………………….. 45
3.4 Numerical Examples ………………………………………………………………………………… 54
3.5 Concluding Remarks………………………………………………………………………………… 55
CHAPTER FOUR …………………………………………………………………………………………….. 57
AN EOQ MODEL FOR DELAYED DETERIORATING ITEMS WITH INVENTORY LEVEL DEPENDENT DEMAND RATE AND SHORTAGES ………………………………… 57
4.1 Introduction …………………………………………………………………………………………….. 57
4.2 Modelling Assumptions ……………………………………………………………………………. 58
4.3 Mathematical Model and Analysis …………………………………………………………….. 58
4.4 Solution Procedure …………………………………………………………………………………… 68
4.5 Numerical Examples ………………………………………………………………………………… 74
4.6 Results and Discussion …………………………………………………………………………….. 76
4.7 Concluding Remarks………………………………………………………………………………… 77
CHAPTER FIVE ………………………………………………………………………………………………. 79
AN EPQ MODEL FOR DELAYED DETERIORATING ITEMS WITH STOCK-DEPENDENT DEMAND RATE AND LINEAR TIME – DEPENDENT HOLDING COST WITH TIME- DEPENDENT DETERIORATION RATE ……………………………… 79
5.1 Introduction ……………………………………………………………………………………………… 79
5.2 Linear Time Dependent Holding Cost Model ………………………………………………. 79
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5.2.1 Modelling assumptions …………………………………………………………………… 80
5.2.2 Model development ……………………………………………………………………….. 80
5.2.3 Numerical examples ………………………………………………………………………. 89
5.2.4 Concluding Remarks ……………………………………………………………………… 90
5.3 LinearTime Dependent Deterioration Rate Model ………………………………………….. 91
5.3.1 Modelling assumptions ……………………………………………………………………. 91
5.3.2 Model development ……………………………………………………………………….. 92
5.3.3 Numerical examples …………………………………………………………………….. 102
5.3.4 Results and discussion ………………………………………………………………….. 104
5.3.5 Concluding Remarks ……………………………………………………………………. 105
CHAPTER SIX ………………………………………………………………………………………………. 107
AN EOQ MODEL FOR DELAYED DETERIORATING ITEMS WITH INVENTORY LEVEL DEPENDENT DEMAND RATE AND LINEAR TIME – DEPENDENT HOLDING COST ……………………………………………………………………………………………. 107
6.1 Introduction …………………………………………………………………………………………… 107
6. 2 Modelling Assumptions ………………………………………………………………………….. 109
6.3 Mathematical Model and Theoretical Results ……………………………………………. 109
6.4 Selection Criteria …………………………………………………………………………………… 125
6.5 Numerical Examples ………………………………………………………………………………. 126
6.6 Results and Discussion …………………………………………………………………………… 130
6.7 Sensitivity Analysis ……………………………………………………………………………….. 131
6.8 Concluding Remarks………………………………………………………………………………. 134
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CHAPTER SEVEN …………………………………………………………………………………………. 136
SUMMARY, CONCLUSION AND RECOMMENDATIONS ………………………………… 136
7.1 Summary ………………………………………………………………………………………………. 136
7.2 Conclusion ……………………………………………………………………………………………. 138
7.4 Recommendations ………………………………………………………………………………….. 143
REFERENCES ……………………………………………………………………………………………….. 144
APPENDICES ……………………………………………………………. Error! Bookmark not defined.
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CHAPTER ONE

INTRODUCTION
1.1 Background of the Study
This section and the following sections in this chapter will introduce the more commonly used basic inventory terms and concepts (such as definition of inventory, terms connected with inventory systems, some of the features of inventory system and so on) which authors use to develop inventory policy. Decision-makers are in many instances in charge of assigning widely varied type of resources such as manpower, material, or money. In a relatively uncomplicated form of decision-making problem, the decision-maker can imagine the best possible course of action leading to most effective use of the resources. If, however, the problem is complex in nature, the decision-maker cannot estimate the future as clearly as he/she intends by using intuition, common sense or experience. For this reason there is the need to find a procedure which can be effectively used by a decision-maker to identify the various courses of action available to him/her and then recommending the best course of action. A common analytical technique can be used to find the solution of decision-making problems belonging to the same general category. These are commonly referred to as Operations Research Techniques. One of the most prominent Operations research techniques is inventory control. Inventory control (management) has to do with determining when and how much of needed resource should be kept as stock which minimizes the total cost or maximizes the profit over a given period of time.
2
The word inventory refers to any kind of stock of resource, such as unused labour (manpower), materials and money having economic value and is being held for future use or sale by an organization. The resources may be physical resources such as raw materials, semi-finished goods, finished goods, spare parts, lubricants, and so on, or human resources such as unused labour (manpower) and financial resources such as working capital. Maintaining inventory is considered a necessary investment in order to achieve a smooth system of production, distribution and marketing of physical goods. Some of the important reasons for carrying inventory are to: improve customer service, reduce costs, satisfy irregular supply and demand, provide quantity discounts, and avoid stock-outs (shortages). Inventory management is a common problem in businesses. In a small business, the manager may control his/her inventory and take these decisions by experience, common sense or intuition. However, for large businesses, scientific inventory management is required by way of formulating a mathematical model describing the characteristics of the inventory system and consequently deriving an inventory control model (cost minimization model or profit maximization model equation) for optimal inventory replenishment policy for the model.
In particular, for each order of items, a businessman must incur cost to cover handling and transportation. When large orders (batches) are made infrequently, it leads to low ordering (set-up) cost with high carrying cost. On the other hand, when small orders (batches) are made frequently, it leads to high ordering (set-up) cost with low carrying
3
cost. There are always conflicting costs to be incurred with having too little or too much
inventory on-hand and so there is need to control the inventory by seeking an inventory
level that minimizes the sum of these costs, and other costs such as shortage costs,
deterioration costs and so on. The costs are conflicting, in the sense that an increase in
either ordering cost or holding cost may result in the reduction of the other. The
problem reduces to controlling the inventory level by developing an inventory policy
which answers two questions of when and how much to order (replenish inventory) so as
to minimize the sum of all costs per unit time. The quantity to order, so that the sum of
ordering cost, carrying cost and other costs such as shortage costs, deterioration cost and
so on, are minimized is called Economic Order Quantity (EOQ).
1.2 Classical Square Root EOQ Model
The concept of EOQ was first developed by Harris (1913) for finding the optimum
order quantity in order to balance the holding cost and ordering cost. The model
developed by the author to determine this order quantity is often called classic square
root EOQ model and it is given as follows.
EOQ =
2 Demand rate  Ordering cost per Order 
Cost of holding a unit for a unit time
This is the optimal (economic) ordering quantity which minimizes the total inventory
cost comprising of holding cost and ordering cost.
An application of the EOQ concept to production is the Economic Lot Size or Economic
Production Quantity (EPQ). In this model, an inventory is produced or manufactured
instead of being ordered or purchased in the company itself and subsequently used or
4
sold. An EPQ model is an inventory control model that determines the amount of product to be produced to minimize the total inventory cost so as to meet a deterministic demand.
1.3 Categorization of Inventory Models
The amount of stock to be maintained will naturally depend upon the consumption or usage of the commodity. It is, therefore, necessary to make a study of the demand pattern of the item under consideration. Mathematical inventory models can be divided into two broad categories; deterministic models and stochastic models. If the demand in future periods can be predicted with some degree of accuracy, then, this is the case of known demand where a deterministic inventory model would be used. This is usually expressed over equal periods (commonly annual) and referred to as demand rate. The demand may vary or remains fixed. However, when demand cannot be predicted well, then this is the case where the demand in any period is a random variable rather than a known constant, and its pattern can be described by a known probability distribution where a stochastic inventory model would be used.
The other categorization refers to whether the current inventory level is being monitored continuously or periodically. An inventory system may review its inventory continuously or periodically. In continuous review an order is placed as soon as the stock level falls below the prescribed reorder point, whereas in the periodic review case the inventory is replenished periodically (e.g., ordering at the start of every week or every month). This is usually expressed over equal periods (commonly annual) and referred to as demand rate, the rate at which the product is taken out from inventory in (units / time period). This categorization of demand pattern assumes the availability of reliable data to estimate demand in the future.
5
One of the ways of analyzing the nature of demand of an item is by using some
statistical tools such as evaluating the mean and standard deviation of demand of the
item for a given period of time (e.g., monthly).
Letting coefficient of variation as  ,
Standard deviation
V = 100%
Mean
 
 
 
Taha (2011)
suggested that the nature of demand of an item can be estimated using the following
guidelines.
1. If the average monthly demand (taken over a number of years) is “approximately”
constant and V is reasonably small (< 20%), then the demand may be considered
deterministic and constant.
2. If the average monthly demand varies appreciably among the different months but
V remains reasonably small for all months, then the demand may be considered
deterministic but variable.
3. If in case 1, V is high (> 20%) but approximately constant, then the demand is
probabilistic and stationary.
4. When the averages and coefficients of variation vary appreciably month to month,
the case is probabilistic and dynamic.
1.4 Components of Inventory Models
The following are some of the basic considerations involved in determining an
inventory policy (when and how much to replenish inventory) which are connected with
mathematical inventory modeling.
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1.4.1 Relevant Inventory costs
It is generally acknowledged that costs have effects on profit of any business organization. There are a number of costs that are associated with inventory policy and some of the basic costs considered in an inventory system are (1) replenishment (production/purchase) cost, (2) holding (carrying) cost, and (3) shortage cost. Other relevant factors include (4) revenue, (5) salvage cost, and (6) discount rate. These six factors are described in turn below.
(i) Ordering or set up cost (in production environment) This is the cost of placing an order to an outside supplier or releasing a production order to a manufacturing business. An ordering cost is the cost incurred whenever an order is placed. It is independent of the quantity being placed. It is primarily clerical and administrative in nature. Typical elements of this cost include the cost associated with processing, labour, overhead (telephone, postage, e-mails and so on.), and transport (delivery charges). In other environments, the setup cost could be clerical and administrative cost. The cost of purchasing or producing one unit of a material or product is called acquisition cost.
(ii) Product or purchasing cost This is the actual price paid for the procurement of the item. It is the variable associated with purchasing a single unit. Typically, the unit purchasing cost includes the cost of raw materials associated with purchasing or producing a single unit. This cost may be a constant for all quantities, or it may vary with quantity purchased or produced.
7
(iii) Holding or carrying cost This cost is the cost of holding an item in inventory for some given unit of time. It generally includes the cost of capital tied up in the inventory, salary of warehouse employees, insurance cost, pilferage cost and many other costs which are proportional to the amount kept in inventory. It is given in say (N / unit time).
(iv) Shortage or stockout 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, and the latter is called a backorder in (N /unit/unit time). The shortage cost is the cost incurred when the amount of the inventory demanded exceeds the available stock. This cost depends on whether backordering is allowed or not. For instance, if a customer demands a product and the demand is not met on time, a stockout, or shortage, is said to occur. If customers will accept delivery at later days, no matter how long it takes them waiting, that demand is said to be backordered. The case in which back-ordering is allowed is often referred to as the backlogged demand. If no customer will allow late delivery, then that is a lost-sales case. The process of holding customer orders to be filled later when they cannot be settled immediately because of stockouts is called backlogging. The total backorder cost is assumed to be proportional to the number of units backordered and the time the customer must wait.
8
(v) Backorder cost This is the cost of handling the backorder (special handling, follow-ups etc) plus whatever loss of goodwill occurs as a result of having to backorder the item.
(vi) Salvage cost The salvage value of an item is the value of a remaining item beyond which no inventory is needed.
(vii) Discount rate This is the interest rate one needs to earn on a given amount of money today with a given amount of money in the future. The discount rate considers with other factors such as the time value of money before reaching a decision. When a business ties up its capital in inventory, the business is prevented from using this money elsewhere to earn some kind of return.
1.4.2 Demand for inventory items
The nature of demand (i.e. its rate, size and pattern) for inventory items is one of the most basic features to determine an optimal inventory policy. (i) Size of demand This is the number of items required in each period. The size of demand may be either deterministic or probabilistic. This can be fixed or can vary from period to period
9
(ii) Demand rate This is the amount of stock that is required for sale, production or other uses. It is expressed as a weekly, monthly, yearly, and so on, rate of demand. The demand may be constant (static) or variable (dynamic). (iii) Demand pattern This is the manner in which inventory items are obtained. The demand for a given period of time may be satisfied instantaneously at the beginning of the period, or uniformly during that period.
1.4.3 Order cycle
This is made up of the activities of sensing a need for ordering materials, placing an order, lead time for getting the material delivered, receiving the material and using it. This process is continuously repeated for a material and is thus cyclical. The ordering cycle may be determined in either of the two ways: continuous review or periodic review.
1.4.4 Lead time
This is the time between ordering an inventory item and actual receipt of the item into stock. The lead-time can be either deterministic (constant or variable) or probabilistic. Constant lead time is when the lead time for each order is a known constant. If the lead time is zero, then we have a special case of instantaneous delivery where there is no need for placing an order in advance.
10
1.4.5 Inventory replenishment
This simply refers to the actual replenishment of stock at specified intervals which may occur instantaneously or uniformly. Instantaneous replenishment occurs when inventory is received or obtained at one point in time, while uniform replenishment inventory is received over a period of time.
1.4.6 Time horizon
The period over which the inventory level will be controlled is called the time horizon. It can be finite or infinite depending on the nature of demand.
1.4.7 Operating decision rules
There are two types of decisions need to be made in developing inventory models: (i) How much (size) is to order (or produce) for each replenishment and (ii) When (timing) it is necessary to place an order (or setup production) to replenish inventory.
1.5 Some Terminologies connected with Inventory Modelling
The following are the basic definitions of the inventory terms commonly used by authors of inventory modelling and used in the thesis.
(i) Inventory level This refers to the amount of materials on hand in inventory that is ready for use. It is also called the stock level.
11
(ii) Inventory depletion This is the removal of inventory items from the stock through demand, deterioration or other factors.
(iii) Buffer stock This is also called the minimum or safety stock. It is acquired in order to prepare for uncertainties in demand or to correct errors in forecasting the lead time.
(iv) Order Quantity or Lot Size This is the fixed quantity received in the inventory every inventory replenishment.
(v) Optimal production lot size It is that particular quantity, which if produced in one production run, will minimize the total cost of setting up and carrying finished goods in inventory.
(vi) Product Cost This is the unit cost of purchasing the product as part of an order.
(vii) Backlogging The process of holding customer orders to be filled later when they cannot be settled immediately during stockout period.
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(viii) Backorder
This is the item ordered but unable to be delivered on which the customer will wait for
delivery without canceling the order.
(ix) Deterministic Inventory Model
This is the inventory model in which demand is assumed to be fixed and known.
1.6 Assumptions in Economic Order Quantity (EOQ) Models
The following are the assumptions made in developing EOQ model.
1. Repetitive ordering: This is the system by which orders are placed often at uniform
intervals.
2. Constant demand: The exact amount of items required during any time period is
known at constant rate.
3. Uniform lead time: The lead time for each order is known and constant.
4. Continuous/periodic ordering: This is a situation in which inventory system is
reviewed continuously or periodically.
1.7 The Basic Economic Order Quantity Model
Consider an order of size Q has just arrived at initial time t = 0 and demand occurs at a
rate of D per cycle, it will take
Q
D
time units for inventory to reach zero again. Since
demand during any period of length t is D t , the inventory level over any time interval
will decline along a straight line of slope (- D). At which time the inventory is depleted
down to zero, an order of amount Q is received instantly, which raises the inventory
level back to Q as illustrated pictorially and shown in Figure 1.1.
13
Figure 1.1: Behaviour of Inventory Level I(t) in Basic EOQ Model
1.7.1 Assumptions of basic EOQ model
The basic EOQ model makes the following assumptions.
1. Demand is constant. The exact number of items needed during any time period
is known.
2. An ordering or setup cost K is fixed and the model does not allow quantity
discount.
3. The lead time for each order is zero. Each order arrives in the inventory as soon
as it is placed.
4. No shortages are allowed. That is all demands are met on time; a negative
inventory is not allowed either.
5. The holding cost per unit per unit time held in inventory is a constant, say h.
Q
Q/D 2Q/D 3Q/D 4Q/D
t
Q
Inventory
Level
14
1.7.2 Derivation of basic EOQ model
The total cost per cycle TC is obtained from the following cost components.
From the above assumption 3, for the reason that the lead time is zero, when an order is
placed, it is instantly received in the inventory. If the amount of order is of fixed rate,
then there is need for the same quantity Q, every time an order is placed.
Production or ordering cost including purchasing cost per cycle is given by
PC  K  cQ (1.7.2.1)
where c is the unit purchase cost and Q is the order quantity that enters the inventory at
the beginning of the cycle length.
The average inventory level during a cycle is
 0
2 2
Q  Q
 units, where Q and 0 are
regarded as initial and terminal inventory levels respectively. Therefore, the
corresponding holding cost is
2
h Q
per unit time. Since the cycle length is
Q
D
, then
holding cost per cycle is given by
2
2 2
h Q
HC h
Q Q
D D
 
  
 

(1.7.2.2)
Then the total cost per cycle for ordering Q units is given by
CT (Q) = Ordering cost/cycle + Holding cost/cycle
2
2
hQ
K CQ
D
   (1.7.2.3)
which leads to the total cost per unit time to be
15
 
2
2
2
T
h Q
K cQ
C D KD h Q TC Q cD
T Q Q
D
 
     (1.7.2.4)
where T is the cycle length.
The necessary condition for finding the value of Q, say Q*, that minimizes cost function
TC (Q) from equation (1.7.2.4) is when the first derivative with respect to Q of this
function is zero.
That is
   2 0
2
d KD h
TC Q
dQ Q
    , (1.7.2.5)
Solving for Q gives the Q* (the optimal order quantity), which is
* 2KD
Q
h
 
Since we have said from assumption 4 that no shortages are allowed then we do not
desire a negative inventory. Hence we get rid of the negative value and take on the
positive value as the value that minimizes the total cost function. We now take our EOQ
to be,
* 2KD
Q
h
 , (1.7.2.6)
The second derivative   
2
2 3
2
0
d KD
TC Q
dQ Q
  implying Q* is indeed a minimizer,
providing optimal solution to equation (1.7.2.5).
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which is the well known basic square root EOQ model. The corresponding cycle time,
say t*, is
*
* 2
.
Q K
t
D hD
  (1.7.2.7)
We see that Q* is independent of c. It is a function of only k, h and D. This implies that
purchasing cost being constant is not included, as is the case in most inventory models,
when determining the EOQ.
1.8 The Research Problem
There are several items which do not start deteriorating as soon as they arrive in stock,
but they maintain quality or the original condition for some period (fresh product time),
before they begin to deteriorate. Consumer goods such as fruits, vegetables, meat, bread,
cassava, yams and so on, exhibit this property.
In many real-inventory situations, the demand for these items depends on the amount of
inventory items displayed on shelves in a shop. It is usually observed that a large pile of
goods displayed on shelves in a shop will influence the customers to buy more and more
which will in turn cause sales to rise.
The holding cost and deterioration rate of these types of inventory items increase
linearly with time. These inventory systems represent many real-life inventory
situations and this is particularly true in the storage of deteriorating and perishable items
such as food products. This is because, the longer these deteriorating items are kept in
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storage, the higher the holding cost and deterioration cost. These are justified since the cost of holding an item in stock increases with each passing time which eventually will affect total system cost. Deteriorating items will also affect total system cost most especially if the rate of deterioration is significant. It is assumed that shortages are allowed and the unsatisfied demand is partially backlogged at a fixed fraction of demand rate during the shortage period as in Wee (1995). It is therefore very desirable to have models that address the problems above. In this thesis, we investigate the inventory policies of delayed deteriorating items with inventory level dependent demand rate which answer some of these problems. Specifically, we attempt to develop models for inventory systems where the demand rate depends on the current stock level and items have constant/variable rate of delayed deterioration and constant/variable holding cost with/without shortages.
1.9 Justification/Significance of the Study
Inventory management is a common problem in business. An inventory consists of usable but idle resources such as men, machines, materials or money. Maintenance of inventory absorbs fluctuations in demand and supply. However, keeping too much of an item on hand increases cost of capital and storage, and keeping too little adversely disrupts production and/or sales. Thus there is a need to seek an inventory level that balances the two extreme situations by minimizing these costs. How to determine “when and how much” of needed resource should be kept as stock that minimizes the total cost or maximizes the net profit can be achieved by modelling the inventory system in question.
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We find from the results of the models developed that the effects of stock dependent demand rate, holding cost parameter, deterioration and backlogging rate have impact on the optimal replenishment policy, and hence should not be ignored in developing inventory models. The models will help to determine the inventory policies that minimize the system costs and/or maximize net profit. The proposed models can be used in inventory control of certain delayed deteriorating items such as food items, electronic components, fashionable commodities, vegetables, fruits, yams, potatoes, meat, and so on.
1.10 Aim and Objectives of the Study
The main aim of this research is to develop Deterministic Inventory Models for Delayed Deteriorating Items with Inventory Level Dependent Demand Rate. This aim will be achieved through the following objectives. (i) To develop an EOQ model for delayed deteriorating items with inventory level dependent demand rate and constant deterioration rate. (ii) To develop an EOQ model for delayed deteriorating items with inventory level- dependent demand rate and shortages. (iii) To develop an EPQ model for delayed deteriorating items with stock-dependent and linear time dependent holding cost.
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(iv) To develop an EPQ model for delayed deteriorating items with stock- dependent demand rate and time-dependent deterioration rate. (v) To develop an EOQ model for delayed deteriorating items with inventory level dependent demand rate and linear time-dependent holding cost. The purpose of the first model (i) is to modify the constant demand rate function of Musa and Sani (2009), by considering the demand rate to be a power-form function of instantaneous inventory level. In the initial stage, inventory depletes down to a certain level of the inventory due to market demand only. In the second stage the inventory level gets depleted due to the effect of both market demand and deterioration but still dependent on stock until the inventory level falls to zero at the end of the cycle. In the second model (ii), we modify the power- form demand rate function in the first model (i) by considering linear-form demand function and allow shortages during stock-out period. The unsatisfied demand is partially backlogged at a rate which is a fixed fraction of demand rate during the shortage period as in Wee (1995).
In the third model (iii), the problem considered has finite replenishment rate rather than infinite replenishment rate as in the case of the first model. Considering same demand rate function as in the second model, we propose an EPQ model for a single item with delayed deterioration, i.e. non- instantaneous deterioration, in which the production rate is constant. The demand rate is linear during and after production run and the holding cost is also a linear function of time. Sugapriya and Jeyarama (2008a) established an inventory model for non-instantaneous deteriorating item with constant demand rate for
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both in- production run and out-production run, while in this model we have assumed the demand rate to be inventory level dependent for both in- production run and out-production run. In the fourth model (iv), we modify the third model (iii) by considering the rate of deterioration to be a linear increasing function of time rather than constant deterioration rate. In the fifth model (v), we extended the constant holding cost of the first model (i) by considering a linear time dependent holding cost. However, the objective is to determine the optimal replenishment policy by maximizing average net profit per unit time. Unlike in the first four models in which the optimal replenishment policies are obtained by minimizing the average total variable costs per unit time. The reason for using maximization criterion in this case is because of the consideration of stock dependent selling rate r which indicates our inventory system is a sales environment as opposed to non sales environment as in the first four models. In the literature of inventory models for delayed deteriorating items with stock dependent demand rate, we have not come across any model that studied any of the five models investigated in this thesis.
1.11 Research Methodology
This research being a modelling research has been carried out mainly using literature on (i) Classic square root EOQ model, (ii) Inventory models with constant/variable demand, (iii) Deteriorating inventory models, (iv) delayed (non instantaneous) deteriorating inventory models, (v) Variable holding deteriorating inventory models,
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(vi) Production inventory models for deteriorating items, and (vii) Deteriorating inventory models with shortages. Analytic formulation and analysis of the inventory problems are developed on the framework of the model assumptions aimed at determining the inventory policies of the inventory systems. The proposed inventory systems are represented by systems of differential equations including initial and boundary conditions. We then develop a mathematical model for each of the five different inventory systems for non-instantaneous deteriorating items with demand rate depending on the instantaneous inventory level. In all the models, the main objective function to be optimized (minimized) is the function of the total cost for each of the first four models; and the objective function to be optimized (maximized) is the function of net profit per unit time for the fifth model. Differential and integral calculus methods are used to analyze the cost- minimization models and the profit maximization model. We use the approximation method of Newton-Raphson to find the optimal solution (optimal order quantity and the optimal cycle length) of the developed cost minimization model and profit maximization model. Furthermore, we present some numerical examples to illustrate the applications of the models developed. We also carried out a sensitivity analysis on the decision variables with regard to changes in the model parameters for the fifth model. Finally, specific conclusions were given at the end of each of the inventory models developed. However, general conclusions are drawn across the entire research work and future directions for research are presented in the last chapter.
1.12 Organization of the Thesis
The contents of the thesis have been divided into seven chapters.
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Chapter One: Introduction. This chapter starts with the introduction of the research, the basic EOQ model, basic classification of inventory models, components of inventory models and basic terms we use in the work. Also included are the research problem, aim and objectives of the study and research methodology. Chapter Two: Literature Review. This chapter reviews the published literature related to mathematical modelling of deteriorating items with inventory level dependent demand pattern; and also where deterioration is a function of the on-hand level of inventory. The models have been classified and grouped under specific sections and organized according to: classical square root EOQ Model, inventory models with constant / variable demand, delayed (non-instantaneous) deteriorating inventory models, variable holding deteriorating inventory models, production inventory models for deteriorating items, and deteriorating inventory models with shortages. Chapter Three: An EOQ Model for Delayed Deteriorating Items with Inventory Level – Dependent Demand Rate and Constant Deterioration Rate . This chapter is concerned with the development of an EOQ model for delayed deteriorating items with stock-dependent demand rate in polynomial functional form, and it is organized as follows. The chapter begins with defining the problem and then building the model assumptions. This is followed by developing the inventory model. Subsequently, the model is analyzed and numerical solution using Newton-Raphson method is presented. Moreover, some numerical examples are presented to illustrate the application of the model developed and, finally concluding remarks are given at the end of the chapter.
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Chapter Four: An EOQ Model for Delayed Deteriorating Items with Inventory Level Dependent Demand Rate and Shortages. This chapter presents EOQ model for delayed deteriorating items with stock-dependent demand, allowing shortages and fixed partial backlogging rate, and it is organized as follows. The chapter begins with defining the problem and then building the model assumptions. This is followed by developing the inventory model. Subsequently, the model is analyzed and numerical solution using Newton-Raphson method is presented. Moreover, some numerical examples are presented to illustrate the application of the model developed and, finally concluding remarks are given at the end of the chapter. Chapter Five: An EPQ Model for Delayed Deteriorating Items with Stock Dependent Demand Rate and Linear Time Dependent Holding Cost with Time Dependent Deterioration Rate. This chapter presents an EPQ model for delayed deteriorating items with stock-dependent demand rate and linear time – dependent holding cost with time – dependent deterioration rate, and it is organized as follows. The chapter presents two models, each having basic introduction to the problems they are related to and each of these problems begins with defining the problem and then building the model assumptions. This is followed by developing the inventory model. Subsequently, the model is analyzed and numerical solution using Newton-Raphson method is reported. Moreover, some numerical examples are presented to illustrate the application of the model developed and, finally concluding remarks are given at the end of the chapter.
Chapter Six: An EOQ Model for Delayed Deteriorating Items with Inventory Level Dependent Demand Rate and Linear Time – Dependent Holding Cost. This chapter is concerned with an inventory replenishment model for delayed deteriorating items
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with inventory level dependent demand rate and linear time-dependent holding cost for profit maximization criteria, and it is organized as follows: the chapter begins with defining the problem and then building the model assumptions. This is followed by developing two models: (i) deterioration starts before reorder point occurs and (ii) deterioration sets in after reorder point occurs. Subsequently, the model is analyzed and numerical solution method using Newton-Raphson method is presented. Moreover, some numerical examples are presented to illustrate the application of the model developed and, finally concluding remarks are given at the end of the chapter. Chapter Seven: Summary, Conclusion and Recommendations. The summary of the entire work, the conclusion with contributions of the research and suggestions on the direction of the related future research is presented in chapter seven. References and Appendices The thesis closes with references and appendices in that order.
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