This document summarizes a mathematical model for an inventory system with the following characteristics:
1. Demand follows an exponential distribution and production rate varies proportionally with demand.
2. Items deteriorate according to a Weibull distribution.
3. Shortages are backordered and permitted.
The model is analyzed using differential equations to describe inventory levels over the production cycle. Solutions to the equations provide expressions for inventory levels during production, non-production, shortage, and backlog periods. The purpose is to investigate a computing schema for the economic production quantity under these conditions.
Modeling Of an Inventory System for Non-Instantaneous decaying Items with Par...iosrjce
We will discuss an inventory model is investigates with variable demand rate and time dependent deteriorating items.In this study, we have taken shortages in inventory are allowed and fully backlogged. This model is studied under the condition for decaying items of permissible delay in payments which is most
important and an outcome of interaction between product and financial markets which arises. This model based on time-dependent, holding cost, shortages cost and the combination of model is unique and practical.
An inventory model for variable demand, constant holding cost and without sho...iosrjce
Deterioration is defined as decay, change, damage, spoilage or obsolescence that results in decreasing usefulness from its original purpose. Some kinds of inventory products (e.g., vegetables, fruit, milk, and others) are subject to deterioration
Perishable Inventory Model Having Weibull Lifetime and Time Dependent DemandIOSR Journals
In this paper we develop and analyse an inventory model for deteriorating items with Weibull rate of decay and time dependent demand. Using the differential equations, the instantaneous state of inventory at time‘t’, the amount of deterioration etc. are derived. With suitable cost considerations the total cost function and profit rate function are also obtained by maximizing the profit rate function, the optimal ordering and pricing policies of the model are derived. The sensitivity of the model with respect to the parameters is discussed through numerical illustration. It is observed that the deteriorating parameters have a tremendous influence on the optimal selling price and ordering quantity.
An Inventory Model for Constant Demand with Shortages under Permissible Delay...IOSR Journals
This paper presents an inventory model for deteriorating products with constant demand and time varying deteriorating rate, under permissible delay in payments. It discussed in two cases whether the permissible periods are less than or equal to or greater than replenishment cycle. During the permissible period both supplier and retailer got some benefits. Shortages are allowed and are completely backlogged. This model is explained with numerical example and sensitivity analysis.
Modeling Of an Inventory System for Non-Instantaneous decaying Items with Par...iosrjce
We will discuss an inventory model is investigates with variable demand rate and time dependent deteriorating items.In this study, we have taken shortages in inventory are allowed and fully backlogged. This model is studied under the condition for decaying items of permissible delay in payments which is most
important and an outcome of interaction between product and financial markets which arises. This model based on time-dependent, holding cost, shortages cost and the combination of model is unique and practical.
An inventory model for variable demand, constant holding cost and without sho...iosrjce
Deterioration is defined as decay, change, damage, spoilage or obsolescence that results in decreasing usefulness from its original purpose. Some kinds of inventory products (e.g., vegetables, fruit, milk, and others) are subject to deterioration
Perishable Inventory Model Having Weibull Lifetime and Time Dependent DemandIOSR Journals
In this paper we develop and analyse an inventory model for deteriorating items with Weibull rate of decay and time dependent demand. Using the differential equations, the instantaneous state of inventory at time‘t’, the amount of deterioration etc. are derived. With suitable cost considerations the total cost function and profit rate function are also obtained by maximizing the profit rate function, the optimal ordering and pricing policies of the model are derived. The sensitivity of the model with respect to the parameters is discussed through numerical illustration. It is observed that the deteriorating parameters have a tremendous influence on the optimal selling price and ordering quantity.
An Inventory Model for Constant Demand with Shortages under Permissible Delay...IOSR Journals
This paper presents an inventory model for deteriorating products with constant demand and time varying deteriorating rate, under permissible delay in payments. It discussed in two cases whether the permissible periods are less than or equal to or greater than replenishment cycle. During the permissible period both supplier and retailer got some benefits. Shortages are allowed and are completely backlogged. This model is explained with numerical example and sensitivity analysis.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
call for paper 2012, hard copy of journal, research paper publishing, where to publish research paper,
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
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Inventory Model with Different Deterioration Rates for Imperfect Quality Item...IJLT EMAS
One of the assumptions for an economic order quantity
model is that all items received in an order are of perfect quality
is not always fulfilled. Some of the items are of defective qual ity
in the lot received. Another assumption is that as soon as items
are received, payments are made. In today’s competitive the
supplier allows certain fixed period known as permissible delay
for payment to the retailer for settling the amount of items
received. Keeping this reality, a deterministic inventory model
with imperfect quality is developed when deterioration rate is
different during a cycle. Here it is assumed that demand is a
function of time and price. Numerical example is taken to
support the model. Sensitivity analysis is also carried out for
parameters.
A Deterministic Inventory Model for Perishable Items with Price Sensitive Qua...IJMERJOURNAL
ABSTRACT: An inventory system is considered with trended demand which is assumed to be a function of price and time dependent quadratic demand. For minimizing the total cost of the inventory system, it is assumed that the deterioration rate is constant and the supplier offers the retailer a credit period to settle the account of the procurement units. To solve the model it is further assumed that shortages are not allowed. Salvage value is also considered and observed its effect on the total cost. A numerical example is given to test the strength of the model. Critical parameters are identified by studying the sensitivity of the system.
A Retail Category Inventory Management Model Integrating Entropic Order Quant...Waqas Tariq
A retail category inventory management model that considers the interplay of entropic product assortment and trade credit financing is presented. Specifically, the proposed model takes into consideration of key factors like discounted cash flow. Therefore, to incorporate the concept of supplier-retailer integration and order size dependent trade credit, we established a stylized model to determine the optimal strategy for an integrated supplier-retailer inventory system under the condition of trade credit financing and is entropy type demand. This paper demonstrates that the optimal solution is flexible enough to accommodate the supplier retail relationship under a perfect trade credit understanding without significantly deviating from the main goal at increasing sales revenue of supplier and retailer. Finally numerical examples are used to illustrate the theoretical results followed by the insights into a lot of managerial inputs.
5-S is a Japanese technique basically for proper housekeeping for easy and better performance. 5-S has been derived from five Japanese words namely Seiri, Seiton, Seiso, Seiketsu & Shitsuke. This presentation consists:
a. What is 5-S
b. Why 5-S
c. Explanation of 5-Ss
d. Types of workplaces based on 5-S
e. Examples of 5-S
Would like to know more about how to balance your PM program with your PdM Program? Then this course is for you. I thought I knew all about PdM until I attended this course 4 years ago and I have to say it changed my life. Check it out and then join me for this awesome workshop.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
call for paper 2012, hard copy of journal, research paper publishing, where to publish research paper,
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
Inventory Model with Different Deterioration Rates for Imperfect Quality Item...IJLT EMAS
One of the assumptions for an economic order quantity
model is that all items received in an order are of perfect quality
is not always fulfilled. Some of the items are of defective qual ity
in the lot received. Another assumption is that as soon as items
are received, payments are made. In today’s competitive the
supplier allows certain fixed period known as permissible delay
for payment to the retailer for settling the amount of items
received. Keeping this reality, a deterministic inventory model
with imperfect quality is developed when deterioration rate is
different during a cycle. Here it is assumed that demand is a
function of time and price. Numerical example is taken to
support the model. Sensitivity analysis is also carried out for
parameters.
A Deterministic Inventory Model for Perishable Items with Price Sensitive Qua...IJMERJOURNAL
ABSTRACT: An inventory system is considered with trended demand which is assumed to be a function of price and time dependent quadratic demand. For minimizing the total cost of the inventory system, it is assumed that the deterioration rate is constant and the supplier offers the retailer a credit period to settle the account of the procurement units. To solve the model it is further assumed that shortages are not allowed. Salvage value is also considered and observed its effect on the total cost. A numerical example is given to test the strength of the model. Critical parameters are identified by studying the sensitivity of the system.
A Retail Category Inventory Management Model Integrating Entropic Order Quant...Waqas Tariq
A retail category inventory management model that considers the interplay of entropic product assortment and trade credit financing is presented. Specifically, the proposed model takes into consideration of key factors like discounted cash flow. Therefore, to incorporate the concept of supplier-retailer integration and order size dependent trade credit, we established a stylized model to determine the optimal strategy for an integrated supplier-retailer inventory system under the condition of trade credit financing and is entropy type demand. This paper demonstrates that the optimal solution is flexible enough to accommodate the supplier retail relationship under a perfect trade credit understanding without significantly deviating from the main goal at increasing sales revenue of supplier and retailer. Finally numerical examples are used to illustrate the theoretical results followed by the insights into a lot of managerial inputs.
5-S is a Japanese technique basically for proper housekeeping for easy and better performance. 5-S has been derived from five Japanese words namely Seiri, Seiton, Seiso, Seiketsu & Shitsuke. This presentation consists:
a. What is 5-S
b. Why 5-S
c. Explanation of 5-Ss
d. Types of workplaces based on 5-S
e. Examples of 5-S
Would like to know more about how to balance your PM program with your PdM Program? Then this course is for you. I thought I knew all about PdM until I attended this course 4 years ago and I have to say it changed my life. Check it out and then join me for this awesome workshop.
tOTAL QUALITY MANAGEMENT - FMEA, FINAL YEAR B.E.CS- PRESENTED BY DR. K. BARANIDHARAN, SAIRAM INSTITUTE OF MANAGEMENT STUDIES (sims) SRI SAI RAM INSTITUTE OF TECHNOLOGY (sit) CHENNAI
9 different deterioration rates of two warehouse inventory model with time a...BIOLOGICAL FORUM
ABSTRACT: A two warehouse inventory model with different deterioration rates is developed. Demand is considered as function of price and time. Holding cost is considered as linear function of time. Inflation factor is also considered with permissible delay. Shortages are not allowed. Numerical case is given to represent the model. Affectability investigation is likewise done for parameters.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Inventory Model with Different Deterioration Rates under Exponential Demand, ...inventionjournals
An inventory model with different deterioration rates under exponential demand with inflation and permissible delay in payments is developed. Holding cost is taken as linear function of time. Shortages are allowed. Numerical examples are provided to illustrate the model and sensitivity analysis is also carried out for parameters.
Inventory Management Models with Variable Holding Cost and Salvage valueIOSR Journals
Inventory management models are developed for deteriorating items when the demand rate is
assumed to be linear function of time and the deterioration rate is proportional to time. The model is solved
when shortages are allowed. The salvage value is used for deteriorated items in the system. A numerical
example is taken to discuss the sensitivity of the models.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
EOQ Inventory Models for Deteriorating Item with Weibull Deterioration and Ti...ijceronline
This paper develops an EOQ inventory model for deterioratingitemswith two parameters Weibull deterioration. Shortages are permissible andpartially backlogged. In this model we consider time varying quadratic holding cost and ramp-type demand. The model is developed under two different replenishment policies: (i) Starting with no shortages (ii) Starting with shortages.The aim of this study is to find the optimal solutiontominimizing the total inventory costs for both the above mentioned strategies. To elevate the model a numerical example has been carried out and a sensitivity analysis occurred to study the result of parameters on essential variables and the entire cost of this model.
Perishable Inventory Model with Time Dependent Demand and Partial BackloggingIJERA Editor
An Inventory Model For Decaying Items Under Inflation Has Been Developed. Demand Rate Is Taken As Linear Time Dependent. Holding Cost Is Also Taken As Time Dependent. We Have Developed The Two Cases: In The First Case Shortages Are Not Allowed And In The Second Case Partially Backlogged Shortages Are Allowed. Cost Minimization Technique Is Used In This Study.
An EOQ Model for Weibull Deteriorating Items With Price Dependent DemandIOSR Journals
In the present paper we developed an economic order quantity model for Weibull deteriorating items with price dependent demand rate together with a replenishment policy for profit maximization. The demand rate is a continuous and differentiable function of price. The variable items deteriorate with time shortages are allowed and completely back-ordered. Further it is illustrated with the help of numerical examples.
11 two warehouse production inventory model with different deterioration rate...BIOLOGICAL FORUM
ABSTRACT: A two warehouse production inventory model with different deterioration rates under linear demand is developed. Holding cost is considered as linear function of time. Shortages are not allowed. Numerical case is given to represent the model. Affectability investigation is likewise done for parameters.
Keywords: Two warehouse, Production, deterioration, Linear demand, Time varying holding costs.
An Inventory Management System for Deteriorating Items with Ramp Type and Qua...ijsc
The present paper deals with an inventory management system with ramp type and quadratic demand rates. A constant deterioration rate is considered into the model. In the two types models, the optimum time and total cost are derived when demand is ramp type and quadratic. A structural comparative study is demonstrated here by illustrating the model with sensitivity analysis.
Fuzzy Inventory Model of Deteriorating Items under Power Dependent Demand and...orajjournal
The present paper deals with the development of a fuzzy inventory model of deteriorating items
under power demand rate and inventory level dependent holding cost function. The deterioration
rate, demand rate, holding cost and unit cost are considered as trapezoidal fuzzy numbers. Both
the crisp model and fuzzy model are developed in this paper. The graded mean integration
method(GM) and signed distance method(SD) are used to defuzzify the total cost of the present
model. Both the models are illustrated by suitable numerical examples and a sensitivity analysis
for the optimal solution towards changes in the system parameters are discussed. Lastly a
graphical presentation is furnished to compare the total costs under the above two mentioned
methods in the fuzzy model.
Volume Flexibility in Production Model with Cubic Demand Rate and Weibull Det...IOSR Journals
In the present paper a volume flexible production inventory model is developed for deteriorating items with time dependent demand rate. The demand rate is taken as cubic function of time and production rate is decision variable. Production cost becomes a function of production rate. Unit production cost is depending upon material cost, Labor cost and tool or die cost. The deteriorating of unit in an inventory system is taken to weibull distribution. Shortage with partially backlogged are allowed a very natural phenomenon in inventory model.
An Inventory Model with Variable Demand Rate for Deteriorating Items under Pe...Editor IJCATR
A continuous production control inventory model for deteriorating items with variable demand rate is developed. Demand
rate is the linear function of time. In this paper we have done all work in the environment of permissible delay of payments. A number
of structural properties of the inventory system are studied analytically. We have discussed the minimum total system cost under the
condition of permissible delay is relaxed to that at the end of the credit period, the retailer will make a partial payment on total
purchasing cost to the supplier and pay off the remaining balance by loan from the bank. Numerical examples are taken to illustrate the
procedure of finding the optimal total inventory cost and production cycle time. Sensitivity analysis is carried out to demonstrate the
effects of changing parameter values on the optimal solution of the system.
Modelling of repairable items for production inventory with random deteriorationiosrjce
Keeping in view the concern about environmental protection, the study incorporate the concept of repairing in a production inventory model consisting of production system and repairing system over infinite
planning horizon. This study presents a forward production and reverse repairing system inventory model with a time dependent random deterioration function and increasing exponentially demand with the finite production rate is proportional to the demand rate at any instant. The shortages allow and excess demand is backlogged. Expressions for optimal parameter are obtained .We also obtained Production and repairing scheduling period, maximum inventory level and total average cost. Using calculus, optimum production policy is derived, which minimizes the total cost incurred
Optimal Pricing Policy for a Manufacturing Inventory Model with Two Productio...IJERDJOURNAL
ABSTRACT: When a new product is launched, a manufacturer applies the strategy of offering a quantity incentive initially for some time to boost up the demand of the product. The present paper describes a manufacturing inventory model with price sensitive demand enhanced by a quantity incentive. Later on demand becomes time increasing also. Inventory cycle starts with low production rate which is followed by higher production rate when demand is boosted up. Shortages are not allowed in this model. Presentation of numerical examples, tables, graphs and sensitivity analysis describes the model very well. Lastly case without incentive illustrates that usually the quantity incentive offered initially is beneficial.
Similar to An epq model having weibull distribution deterioration with (20)
An epq model having weibull distribution deterioration with
1. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.1, 2013
An Epq Model Having Weibull Distribution Deterioration With
Exponential Demand and Production With Shortages Under
Permissible *Delay In Payments
S.Sarkar And T.Chakrabarti
University of Calcutta Department of Applied Mathematics , 92, APC Road, Kolkata-700009,India.
*Email of the corresponding author sanchita771@rediffmail.com
Abstract
In the fundamental production inventory model, in order to solve the economic production quantity (EPQ) we always
fix both the demand quantity and the production quantity per day. But, in the real situation, production is usually
dependend on demand. This paper derives a production model for the lot-size inventory system with finite
production rate, taking into consideration the effect of decay and the condition of permissible delay in payments.
Usually no interest is charged if the outstanding amount is settled within the permitted fixed settlement period.
Therefore, it makes economic sense for the customer to delay the settlement of the replenishment account up to the
last moment of the permissible period allowed by the supplier. In this model shortages are permitted and fully
backordered . The purpose of this paper is to investigate a computing schema for the EPQ. The model is illustrated
with a numerical example.
Keywords Economic production quantity, permissible delay, weibull distribution, deterioration.
1. Introduction
For solving the EPQ for each cycle, we always fix both the demand quantity and production quantity per day in the
crisp model. But, in the real situation, both of them probably will have some little disturbances per day. In recent
years, many researchers have studied inventory models for deteriorating items such as electronic components, food
items, drugs and fashion goods. Deterioration is defined as decay, change or spoilage that prevent the items from
being used for its original purpose. There are many items in which appreciable deterioration can take place during
the normal storage period of the units and consequently this loss must be taken into account when analyzing the
model. Therefore, many authors have considered Economic order quantity models for deteriorating items.Acting as
the driving force of the whole inventory system, demand is a key factor that should be taken into consideration in an
inventory study. There are mainly two categories demands in the present studies, one is deterministic demand and the
other is stochastic demand. Some noteworthy work on deterministic demand are : Chung and Lin [1,2] have
considered constant demand, Giri and Chakrabarty[3] and Teng and Chang[4] have considered time-dependent
demand where as Giri and Chaudhuri[6], Bhattacharya[7] and Wu.et.al[8] have worked on inventory level-
dependent demand and Wee and Law [9] have considered price-dependent demand . Among them, ramp type
demand is a special type of time-dependent demand. Hill [10] was the first to introduce the ramp type demand to the
inventory study. Then Mandal and Pal [11] introduced the ramp type demand to the inventory study of the
deteriorating items. Deng.et.al [12] and Shah and Jaiswal[13] have extensively studied this type of demand.
In the classical inventory model depletion of inventory is caused by a constant demand rate alone. But subsequently,
it was noticed that depletion of inventory may take place due to deterioration also. In the early stage of the study,
most of the deteriorating rates in the models are constant, Padmanabhana and Vratb [14], and Bhunia and Maiti [15]
worked on constant deterioration rate. In recent research, more and more studies have begun to consider the
relationship between time and deteriorating rate. Wee[18], and Mahapatra[19], considered deterioration rate as linear
increasing function of time. Chakrabarty.et.al[20] have considered three-parameter Weibull distribution. In this
connection, studies of many researchers like Ghare and Schrader [21], Goyel et al. [22] are very important. Misra [23]
developed a two parameter Weibull distribution deterioration for an inventory model. In recent research, the
extensive use of trade credit as an alternative has been addressed by Goyal[4] who developed an economic order
quantity(EOQ) model under the conditions of permissible delay in payments. Chung[5] then developed an alternative
approach to the problem. Chand and Ward [6] analyzed Goyal’s problem under assumptions of the classical
economic order quantity model, obtaining different results. Next Aggarwal and Jaggi’s [7] model to shortages.There
are several interesting and relevant papers related to the delay of payments such as Chu et al.[9], Chung [10], Hwang
1
2. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.1, 2013
and Shinn[11], Jamal et al[12], Sarker et al[13,14] , Shah[15], Shinn[16], Khouja and Mehrez[17] and there
references. However , these studies were developed under the assumption that the items obtained from an outside
supplier and the entire lot size is delivered at the same time. In fact, when an item can be produced in-house, the
replenishment rate is also the production rate and is hence finite . Hence we amen Goyal’s model by considering the
replenishment rate is finite, the difference between purchasing price and selling cost and taking into consideration
decay and shortage.
In the present paper, efforts have been made to analyze an EPQ model that have weibull distribution
deterioration assuming demand rate to be exponential. Here production is demand dependent.
2. Assumptions and Notations
2.2 Assumptions:
• The demand is taken as exponential ,R (t)= aebt.
• Rate of production varies with demand i.e. P = kR (t) where k is constant.
• Replenishment is instantaneous.
• Lead-time (i.e. the length between making of a decision to replenish an item and its actual addition to
stock) is assumed to be zero. The assumption is made so that the period of shortage is not affected.
• The rate of deterioration at any time t>0 follow the two parameter Weibull distribution: θ(t) = γβt β−1 ,
where γ (0 < γ < 1) is the scale parameter and β (> 0) is the shape parameter. The implication of the
Weibull rate (two parameter) is that the items in inventory start deteriorating the instant they are received
into inventory.
• Shortages are allowed and are fully backlogged.
2.2 Notations:
• Replenishment rate is finite and it is demand dependent.
• Lead time is zero.
• T the cycle time.
• I(t) inventory level at time t.
• C1 is the holding cost per unit time.
• C2 is the shortage cost per unit time.
• C3 is the unit purchase cost .
• C4 is the fixed ordering cost of inventory.
• θ d e te r io ra tio n ra te o f f in is h e d ite m s .
• M trade credit period.
• I1(t) the inventory level that changes with time t during production period.
• I2(t) the inventory level that changes with time t during non-production period.
• I3(t) the inventory level that changes with time t during shortage period.
2
3. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.1, 2013
• I4(t) the inventory level that changes with time t during the period when shortages are fully backlogged
3. Mathematical Modeling and Analysis
Here we assume production starts at t=0 at the rate K and the stock attains a level Q at t= t1. The production stops at
t= t1 and the inventory gradually depletes to zero at t=t2 mainly to meet the demands and partly for deterioration
.Now shortages occur and accumulate to the level S at time t=t3.The production starts again at a rate K at t=t3 and the
backlog is cleared at time t=T when the stock is again zero. The cycle then repeats itself after time T.
The model is represented by the following diagram:
Figure: Graphical representation of inventory situation
Let I(t) be the inventory level at any time t(0 ≤ t ≤ T) and demand rate R(t) is assumed to be deterministic and is
increasing exponentially with time. Further let R(t)= a ebt , 0 ≤ b<1, a>0.
The differential equations describing instantaneous state of I(t) in the interval[0,T] are
d I 1 (t)
+ β γ t β − 1 I 1 (t) = ae b t (k - 1 ) (1)
dt
d I 2 (t)
+ β γ t β − 1 I 2 (t) = -ae b t
dt (2)
dI 3 (t)
= -ae bt
dt (3)
d I 4 (t)
= -(k - 1 )ae b t
dt (4)
with the initial conditions I1(t0)=0 ,I1(t1)=Q,I2(t1)=Q, I2(t2 )=0,I3 (t2)=0, I3(t3)=S, I4(t3)=S, I4(T)=0.
Now solving the above differential equations we get
β+2 β +1 β+2
β +1 bt2 bγt γt b γt (5)
I ( t ) = a ( k − 1) { t − γ t + − + + } 0 ≤ t ≤ t
1 2 2 β +1 β + 2 1
β +1 β+ 2 β +1
b t 2 γt b γt β +1 b γ t
I (t) = -a(t+ + + -γ t - )
2 2 β +1 β+2 2
β +1 β+ 2 β 2
bt 2 γ t b γt β b γt t 2
+ a(t + 2 + 2 + 2 -γ t t - ) t ≤ t≤ t
2 2 β +1 β+2 2 2 1 2
(6)
3
4. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.1, 2013
bt
a (e 2 − ebt )
I (t) = t ≤ t≤ t (7)
3 b 2 3
bt
( β − 1) a ( e 3 − ebt )
I (t) = +S t ≤ t ≤ T (8)
4 b 3
t1 t2
H o ld in g c o s t o v e r th e p e r io d [ 0 ,T ]= ∫
0
I 1 ( t )d t + ∫
t1
I2 (t)d t
t2 γt1 + 2
β 3
b t1 b γ t1 +3
β
γ t1 + 2
β
b γt1 +3
β
= C a ( k − 1)[ 1 − + - + + ]
1 2 β + 2 6 2 (β + 3 ) (β + 1)(β + 2 ) (β + 2 )(β + 3 )
( t 2 − t 12 ) b ( t 3 − t1 )
3
γ ( t β + 2 − t1 + 2 )
β
b γ ( tβ+ 2 − t1 +3 )
β
γ ( t β + 2 − t1 + 2 )
β
-a C 1[ 2
+ 2
+ 2
+ 2
− 2
2 6 (β + 1)(β + 2 ) (β + 2 )(β + 3 ) β + 2
b γ ( t β + 3 − t1 + 3 )
β
b t 2 ( t 2 − t1 ) γ t β +1 ( t 2 − t1 )
− 2
] + a C 1[ t 2 ( t 2 − t1 ) + 2
+ 2
2 (β + 3 ) 2 β + 1
b γ t β+ 2 ( t 2 − t1 ) γ t 2 ( t β +1 − t1 +1 )
β
b γ t 2 ( t β +1 − t1 +1 )
2 β
+ 2
− 2
− 2
] (9 )
(β + 2 ) β + 1 2 (β + 1)
S h o r ta g e C o s t( S .C ) o v e r th e p e r io d [ 0 ,T ]
T
2 t∫
= C I(t)d t
2
t T
3
= C 2 ∫ I(t)d t + C 2 ∫ I ( t ) d t
t t
2 3
bt
a bt bt C a bt C ( k − 1) a e 3 ( T − t )
= C (e 2 − e 3 ) + 2 e 2 (t − t ) + 2 3
2 2 b 3 2 b
b
bt
C ( k − 1) a ( e b T − e 3 )
- 2 + C S (T − t ) (1 0 )
2 3
b2
t t
1 2
D e te rio ra tin g C o s t = C 3 ∫ (P -D )d t -C 3 ∫ D dt
0 t
1
bt bt bt
1 − 1) aC (e 2 − e 1)
( k − 1) ( e 3
= aC 3 −
b b (11)
Set up Cost = C4 (12)
Case I:M ≤ t2
In this situation since the length of period with positive stock is larger than the credit period, the buyer can use the
sale revenue to earn interest at an annual rate Ie in (0,T2) .The interest earned IE1, is
t
2 b t d t
I.E 1 = C Ie ∫ a e (13)
0
b t
C Ie a (e 2 - 1 )
=
b
However beyond the credit period, the unsold stock is supposed to be financed with an annual rate Ic and the interest
payable I.P . is given by
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5. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.1, 2013
t
2 btdt
I.P = C I c ∫ ae (14)
M
bt
C Ic a (e 2 - ebM )
=
b
Total average Cost (TVC)= (Holding Cost+Deteoration Cost+ Shortage Cost+Setup Cost+Interest payable-
Interest earned)/T
TVC=(C4+H.C.+S.C.+I.P-I.E1)/T (15)
Case II: M > T2
Since M > T2 , the buyer pays no interest but earns interest at an annual rate Ie during the period (0,M). Interest
earned in this case , denoted by I.E2 , is given by
t M
2 b t d t + C I b t d t
I.E
2
= C I e ∫ a e e ∫ a e
0 t
2
C I e a (e b M - 1 )
= (16)
b
Total average Cost (TVC)= (Holding Cost+Deteoration Cost+ Shortage Cost+Setup Cost-Interest earned)/T
TVC=(C4+H.C.+D.C.+S.C.-I.E2)/T (17)
Using the initial conditions the total average cost becomes function of t1 and T. Hence we find the global optimal
solution of total average cost by using LINGO 12.
4. Numerical
Case (i): When permissible delay period is less than the cycle time i.e 0<M ≤ T then interest is earnerd for the period
[0,M] but beyond the permissible delay period [M,T] interest Ic is charged on the outstanding amount. So total
variable cost is calculated considering both interest charged and interest earned.
Let C1=$2 per unit; C3= $.02 per unit ; C4= $100 per order ; b=.2; beta=.6; Ic =12% , IE=10% ; M=.3year ; a=3
units, gamma= .03; t0=.7; t1=7.
Optimal Value $ 24.7525.
Case (ii): When permissible delay period is greater than the cycle time then interest charged Ic =0. So the total
variable cost is calculated considering only the interest earned.
Let C1=$2 per unit; C3= $.02 per unit ; C4= $100 per order ; b=.4; beta=1.6; IE=10% ; M=.3year ; a=3 units,
gamma= .03; t0=.7; t1=7.
Optimal Value $ 34.952.
5. Conclusion
In the present paper an EPQ model of deteriorating items having weibull distribution deterioration has been studied
with permissible delay in payments. The paper , however considers only one break in the delay period.A natural
extension of the model would be to study the case of N breaks in the permissible delay period , i.e. to assume M=Mi,
if qi-1 ≤ q ≤ qi ,i=1,2,…..,N, Where M1 < M2<……<MN, and q0=0<q1<….<qN =∞.
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6. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.1, 2013
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