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.
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 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.
Fast Algorithm for Computing the Discrete Hartley Transform of Type-IIijeei-iaes
The generalized discrete Hartley transforms (GDHTs) have proved to be an efficient alternative to the generalized discrete Fourier transforms (GDFTs) for real-valued data applications. In this paper, the development of direct computation of radix-2 decimation-in-time (DIT) algorithm for the fast calculation of the GDHT of type-II (DHT-II) is presented. The mathematical analysis and the implementation of the developed algorithm are derived, showing that this algorithm possesses a regular structure and can be implemented in-place for efficient memory utilization.The performance of the proposed algorithm is analyzed and the computational complexity is calculated for different transform lengths. A comparison between this algorithm and existing DHT-II algorithms shows that it can be considered as a good compromise between the structural and computational complexities.
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.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
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 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.
Fast Algorithm for Computing the Discrete Hartley Transform of Type-IIijeei-iaes
The generalized discrete Hartley transforms (GDHTs) have proved to be an efficient alternative to the generalized discrete Fourier transforms (GDFTs) for real-valued data applications. In this paper, the development of direct computation of radix-2 decimation-in-time (DIT) algorithm for the fast calculation of the GDHT of type-II (DHT-II) is presented. The mathematical analysis and the implementation of the developed algorithm are derived, showing that this algorithm possesses a regular structure and can be implemented in-place for efficient memory utilization.The performance of the proposed algorithm is analyzed and the computational complexity is calculated for different transform lengths. A comparison between this algorithm and existing DHT-II algorithms shows that it can be considered as a good compromise between the structural and computational complexities.
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.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Efficient Analysis of high-dimensional data in tensor formatsAlexander Litvinenko
We solve a PDE with uncertain coefficients. The solution is approximated in the Karhunen Loeve/PCE basis. How to compute maximum ? frequency? probability density function? with almost linear complexity? We offer various methods.
We have implemented a multiple precision ODE solver based on high-order fully implicit Runge-Kutta(IRK) methods. This ODE solver uses any order Gauss type formulas, and can be accelerated by using (1) MPFR as multiple precision floating-point arithmetic library, (2) real tridiagonalization supported in SPARK3, of linear equations to be solved in simplified Newton method as inner iteration, (3) mixed precision iterative refinement method\cite{mixed_prec_iterative_ref}, (4) parallelization with OpenMP, and (5) embedded formulas for IRK methods. In this talk, we describe the reason why we adopt such accelerations, and show the efficiency of the ODE solver through numerical experiments such as Kuramoto-Sivashinsky equation.
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.
Stochastic methods for uncertainty quantification in numerical aerodynamicsAlexander Litvinenko
We developed a gPCE based surrogate. gPCE coefficients were computed with sparse Gauss-Hermite quadrature and compared with coefficients computed via MC and QMC methods
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.
Efficient Analysis of high-dimensional data in tensor formatsAlexander Litvinenko
We solve a PDE with uncertain coefficients. The solution is approximated in the Karhunen Loeve/PCE basis. How to compute maximum ? frequency? probability density function? with almost linear complexity? We offer various methods.
We have implemented a multiple precision ODE solver based on high-order fully implicit Runge-Kutta(IRK) methods. This ODE solver uses any order Gauss type formulas, and can be accelerated by using (1) MPFR as multiple precision floating-point arithmetic library, (2) real tridiagonalization supported in SPARK3, of linear equations to be solved in simplified Newton method as inner iteration, (3) mixed precision iterative refinement method\cite{mixed_prec_iterative_ref}, (4) parallelization with OpenMP, and (5) embedded formulas for IRK methods. In this talk, we describe the reason why we adopt such accelerations, and show the efficiency of the ODE solver through numerical experiments such as Kuramoto-Sivashinsky equation.
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.
Stochastic methods for uncertainty quantification in numerical aerodynamicsAlexander Litvinenko
We developed a gPCE based surrogate. gPCE coefficients were computed with sparse Gauss-Hermite quadrature and compared with coefficients computed via MC and QMC methods
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.
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.
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
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.
A Marketing-Oriented Inventory Model with Three-Component Demand Rate and Tim...IJAEMSJORNAL
This paper, an attempt has been made to extend the model of “An EOQ model for perishable items under stock-dependent selling rate and time-dependent partial backlogging” with a view to making the model more flexible, realistic and applicable in practice. Here, objectives are to maximize the profit and minimize the total shortage cost. In this model, fuzzy goals are used by linear membership functions and after fuzzification, it is solved by weighted fuzzy non-linear programming technique. The model is illustrated with a numerical example adopted partially from “An EOQ model for perishable items under stock-dependent selling rate and time-dependent partial backlogging”.
The smile calibration problem is a mathematical conundrum in finance that has challenged quantitative analysts for decades. Through his research, Aitor Muguruza has discovered a novel resolution to this classic problem.
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.
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.
Bayesian Estimation For Modulated Claim HedgingIJERA Editor
The purpose of this paper is to establish a general super hedging formula under a pricing set Q. We will compute
the price and the strategies for hedging an European claim and simulate that using different approaches including
Dirichlet priors. We study Dirichlet processes centered around the distribution of continuous-time stochastic
processes such as a continuous time Markov chain. We assume that the prior distribution of the unobserved
Markov chain driving by the drift and volatility parameters of the geometric Brownian motion (GBM) is a
Dirichlet process. We propose an estimation method based on Gibbs sampling.
Genetic Algorithm for solving Dynamic Supply Chain Problem AI Publications
The solution of dynamic supply chain problem is studied using both genetic algorithms and multistage dynamic programming. This is possible by employing Euler approximation method to approximate the differential of the variables. The problem is reformulated as unconstrained optimization problem which solved by genetic Algorithm and Multi Stages Dynamic Programming. The solution evaluation results from using dynamic programming and genetic algorithm is performed
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.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
CFD Simulation of By-pass Flow in a HRSG module by R&R Consult.pptxR&R Consult
CFD analysis is incredibly effective at solving mysteries and improving the performance of complex systems!
Here's a great example: At a large natural gas-fired power plant, where they use waste heat to generate steam and energy, they were puzzled that their boiler wasn't producing as much steam as expected.
R&R and Tetra Engineering Group Inc. were asked to solve the issue with reduced steam production.
An inspection had shown that a significant amount of hot flue gas was bypassing the boiler tubes, where the heat was supposed to be transferred.
R&R Consult conducted a CFD analysis, which revealed that 6.3% of the flue gas was bypassing the boiler tubes without transferring heat. The analysis also showed that the flue gas was instead being directed along the sides of the boiler and between the modules that were supposed to capture the heat. This was the cause of the reduced performance.
Based on our results, Tetra Engineering installed covering plates to reduce the bypass flow. This improved the boiler's performance and increased electricity production.
It is always satisfying when we can help solve complex challenges like this. Do your systems also need a check-up or optimization? Give us a call!
Work done in cooperation with James Malloy and David Moelling from Tetra Engineering.
More examples of our work https://www.r-r-consult.dk/en/cases-en/
Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
NO1 Uk best vashikaran specialist in delhi vashikaran baba near me online vas...Amil Baba Dawood bangali
Contact with Dawood Bhai Just call on +92322-6382012 and we'll help you. We'll solve all your problems within 12 to 24 hours and with 101% guarantee and with astrology systematic. If you want to take any personal or professional advice then also you can call us on +92322-6382012 , ONLINE LOVE PROBLEM & Other all types of Daily Life Problem's.Then CALL or WHATSAPP us on +92322-6382012 and Get all these problems solutions here by Amil Baba DAWOOD BANGALI
#vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore#blackmagicformarriage #aamilbaba #kalajadu #kalailam #taweez #wazifaexpert #jadumantar #vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore #blackmagicforlove #blackmagicformarriage #aamilbaba #kalajadu #kalailam #taweez #wazifaexpert #jadumantar #vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore #Amilbabainuk #amilbabainspain #amilbabaindubai #Amilbabainnorway #amilbabainkrachi #amilbabainlahore #amilbabaingujranwalan #amilbabainislamabad
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
Learn about the cost savings, reduced environmental impact, and minimal disruption associated with trenchless technology. Discover detailed explanations of popular techniques such as pipe bursting, cured-in-place pipe (CIPP) lining, and directional drilling. Understand how these methods can be applied to various types of infrastructure, from residential plumbing to large-scale municipal systems.
Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
Student information management system project report ii.pdf
An Inventory Management System for Deteriorating Items with Ramp Type and Quadratic Demand: A Structural Comparative Study
1. International Journal on Soft Computing (IJSC) Vol.11, No.1/2/3/4, November 2020
DOI: 10.5121/ijsc.2020.11401 1
AN INVENTORY MANAGEMENT SYSTEM FOR
DETERIORATING ITEMS WITH RAMP TYPE AND
QUADRATIC DEMAND: A STRUCTURAL
COMPARATIVE STUDY
Biswaranjan Mandal
Associate Professor of Mathematics
Acharya Jagadish Chandra Bose College, Kolkata
ABSTRACT
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.
KEYWORDS
Inventory, ramp type, quadratic, deterioration.
1. INTRODUCTION
The economic order quantity model is the oldest inventory management model. An extensive
research work has already been done by many researchers like Donaldson W.A.[1], Silver
E.A.[2], Mandal and Pal[3]etc on inventory model assuming mostly on two types of time
dependent demands – linear and exponential. Linear time dependent demand implies a steady
increase or decrease in demand which is not realistic in real market. Again an exponential time
varying demand is not realistic because in the real market situations, demand is unlikely to vary
with a rate which is so high as exponential. Therefore we developed here two types of inventory
models. First, effort has been made to analyze an inventory model for anitem that deteriorates at a
constant rate, assuming the demand rate a ramp typefunction of time. Such type of demand
pattern is generally seen in the case ofany new brand of consumer goods coming to the market.
The demand rate forsuch items increases with time upto a certain time and then ultimately
stabilizes and becomes constant. It is believed that such type of demand rate is quiterealistic,
vide, Hill [61]. Second, the quadratic time dependent demand seems to be the better
representation of time-varying market demand.In this context few researchers like
BiswaranjanMandal[4], M Cheng et al [ 5] and Ghosh et al [6] are noteworthy.
Finally numerical examples are done to illustrate the theory. The sensitivity of the optimal
solution to change in the parameter values is examined and discussed. Also the comparative study
between two types of inventory models are done along with finding its conclusion.
2. ASSUMPTIONS AND NOTATIONS
The mathematical models are developed under the following assumptions and notations:
2. International Journal on Soft Computing (IJSC) Vol.11, No.1/2/3/4, November 2020
2
(i) Replenishment size is constant and replenishment rate is infinite.
(ii) Lead time is zero.
(iii) T is the fixed length of each production cycle.
(iv) hC is the inventory holding cost per unit per unit time.
(v) 0C is the ordering cost/order.
(vi) dC is the cost of each deteriorated unit.
(vii) T is the cycle time.
(viii) TC is the average total cost per unit time.
(ix) A constant fraction of the on-hand inventory deteriorates per unit time.
A deteriorated item is lost.
(x) Shortages are not allowed.
(xi) The demand rate R(t) is assumed in the model
(xii) There is no repair or replacement of the deteriorated items.
3. MATHEMATICAL MODELS
Model 1: An inventory model with ramp type demand rate.
In this mode, the demand rate R(t) is assumed to be a ramp type function of time :
R(t) = 0D [t – (t - )H(t - )], 0D > 0
where H(t - ) is the well-known Heaviside’s function defined as follows :
( ) 1,H t t − =
= 0, t<
Let I(t) be the on-hand inventory at any time t. The differential equations which the on-hand
inventory I(t) must satisfy during the cycle time T is the following
( )
( ) ( ),0
dI t
I t R t t T
dt
+ = − (1)
In this model, we assume < t1 and therefore the above governing equation becomes
0
( )
( ) ,0
dI t
I t D t t
dt
+ = − (2)
and 0
( )
( ) ,
dI t
I t D t T
dt
+ = − (3)
Solutions of the equations (2) and (3) are the following:
0
0 2 2
1
( ) ( ) ( ),0t Dt
I t D e Q t
−
= − − + − (4)
3. International Journal on Soft Computing (IJSC) Vol.11, No.1/2/3/4, November 2020
3
And ( )0
( ) ( 1),T tD
I t e t T
−
= − (5)
From (4) and (5), we get
0 0
2
(1 ) (1 )TD D
Q e e e
= − + + −
Or 2 2
0
3
( )
2 2 2
Q D T T
= + − − ( by using
2
1
2
e
= + + as O( 3
)<<1) (6)
Total Cost: The total cost comprises of the sum of the setup cost, holding cost and deteriorating
cost.
1. Setup Cost = oC
T
(7)
2. Holding cost =
0
( )
T
hC
I t dt
T =
0
[ ( ) ( ) ]
T
hC
I t dt I t dt
T
+
=
2
( )0 0 0 0
2 2
1
[ ( ) ( 1)]
2
ThC D D T D D
e Q e
T
− −
− + − − −
=
3 4 2 2 2 2 2 2 2
20
( ]
4 4 2 2 2 4
hC D T T T T
T
− + + − + + −
(by using
2
1
2
e
= + + , as O( 3
)<<1 ) (8)
3. Deteriorating cost =
0
( )
T
dC
I t dt
T
=
3 4 2 2 2 2 2 2 2
20
( ]
4 4 2 2 2 4
dC D T T T T
T
− + + − + + − (9)
Therefore the average total cost per unit time is given by
TC(T) = Setup cost + holding cost + deteriorating cost
= oC
T
+
3 4 2 2 2 2 2 2 2
20( )
( ]
4 4 2 2 2 4
h dC C D T T T T
T
+
− + + − + + − (10)
For minimum, the necessary condition is 0
dTC
dT
=
4. International Journal on Soft Computing (IJSC) Vol.11, No.1/2/3/4, November 2020
4
Or,
2 2 2 2
2 30 0
0(1 ) (1 ) 0
2 2 4 4 h d
D C
T D
C C
+ − + − − − =
+
(11)
Which is the equation for optimum solution.
Let *
T be the positive real root of the above equation (11), then *
T is the optimum cycle
time.
The optimum average total cost of TC(T) is TC( *
T ).
Note: If there be no deterioration i.e. = 0, the equation (10) becomes
2 30 0
0 0
2 h
D C
T D
C
+ − = or 2
0
2
( 2 )o
h
C
T
D C
= − (12)
4. NUMERICAL EXAMPLE
For an inventory system, let the values of parameters be as follows:
oC = $100; hC = $10; dC = $100; 0D =100 units; = 0.12 year; = 0.01
Solving the quadratic equation (11) with the above numerical values, we find the optimum values
of T as *
T = 1.22 year.
The optimum values of Q, setup cost, holding cost and deteriorating cost are *
Q = 12.57 units,
Setup cost* = Rs 81.97, Holding cost*= Rs 71.86 and Deteriorating cost* = Rs 7.19.
The minimum average total cost per unit time is found to be *
( )TC T = Rs 161.02.
5. SENSITIVITY ANALYSIS
We now study the effects of increasing rate of deteriorating items on the optimal average costs
and cycle time. The results of this analysis are shown in the following table.
Optimum values of
T Q Setup
cost
Holding
cost
Deteriorating
cost
Total cost
0.01 1.22 12.57 81.97 71.86 7.19 161.02
0.02 1.16 11.92 86.21 68.26 13.65 168.12
0.03 1.12 11.50 89.29 65.87 19.76 174.92
0.04 1.08 11.08 92.59 63.48 25.39 181.46
0.05 1.04 10.96 96.15 61.07 30.54 187.76
0.06 1.00 10.19 100.00 58.65 35.19 193.84
0.07 0.97 9.87 103.09 56.85 39.79 199.73
0.08 0.94 9.54 106.38 44.05 44.03 205.45
0.09 0.92 9.33 108.70 53.84 48.46 211.00
0.10 0.89 8.99 112.36 52.01 52.01 216.38
Analyzing the results given in the above table, the following observations are made:
5. International Journal on Soft Computing (IJSC) Vol.11, No.1/2/3/4, November 2020
5
(i) When the rate of deterioration increases, the optimum values Setup cost,
Deteriorating cost and Total cost increase.
(ii) The optimum values of T,Q and Holding cost decrease with increases in the
values of rate of deterioration .
Model 2: An inventory model with quadratic demand rate.
In the present model, we discussed a deterministic inventory model having a quadratic demand
function with a constant deteriorating items. The demand rate function is considered as
2
( )R t a bt ct= + + , where a>0, 0b , 0c at time t and “a” is initial demand.
The differential equation which the on-hand inventory I(t) must satisfies in the cycle
time T is the following:
2( )
( ) ( ),0
dI t
I t a bt ct t T
dt
+ = − + + (13)
The boundary conditions are I(0) = Q , I(T) = 0.
The solution of the equation (13) is
2
( ) [ ( ) ]
T
t t
t
I t e a bt ct e dt −
= + +
=
2 2
( )
2 3 2 3
2 2 2 2
( ) T ta bT cT b ct c a bt ct b ct c
e
−+ + + + + +
− + − + − (14)
Using I(0) = Q , we get
Q =
2
2 3 2 3
2 2
( ) Ta bT cT b c a b c
e
+ +
− + − + −
= 2 3 4
2
2 2
( ) ( ) ( )
2 2 2 2
c a b c b c
a T T c T T
+ + + + + + + (15)
(by using
2
1
2
e
= + + as O( 3
)<<1 )
Total Cost: The total cost comprises of the sum of the setup cost, holding cost and deteriorating
cost.
1. Setup cost = oC
T
(16)
2. Holding cost =
0
( )
T
hC
I t dt
T
6. International Journal on Soft Computing (IJSC) Vol.11, No.1/2/3/4, November 2020
6
=
2 2
( )
2 3 2 3
0
2 2 2 2
[( ) ]
T
T thC a bT cT b cT c a bt ct b ct c
e dt
T
−+ + + + + +
− + − + −
=
2 3 4
2 3
[ ] [ ]
2 2 2 2
h hC CaT bT cT
aT bT cT
T
+ + = + + (by using
2
1
2
e
= + + , as O( 3
)<<1 )
(17)
3. Deteriorating cost =
0
( )
T
dC
I t dt
T
= 2 3
[ ]
2
dC
aT bT cT
+ + (18)
Therefore the average total cost per unit time is given by
TC(T) = Setup cost + holding cost + deteriorating cost
= oC
T
+ 2 3( )
[ ]
2
h dC C
aT bT cT
+
+ + (19)
For minimum, the necessary condition is 0
dTC
dT
=
Or, 4 3 2 02
3 2 0
h d
C
cT bT aT
C C
+ + − =
+
(20)
Which is the equation for optimum solution.
6. NUMERICAL EXAMPLE
For an inventory system, let the values of parameters be as follows:
oC = $100; hC = $10; dC = $100; 0D = 100 units; = 0.01; a = 5;b = 3; c = 2.
Solving the fourth order equation (20) with the above numerical values, we find the optimum
values of T as *
T = 1.02 year.
The optimum values of Q, setup cost, holding cost and deteriorating cost are *
Q = 41225 units,
Setup cost* = Rs 98.04, Holding cost*= Rs 51.72 and Deteriorating cost* = Rs5.17.
The minimum average total cost per unit time is found to be *
( )TC T = Rs 154.93.
7. SENSITIVITY ANALYSIS
We now study the effects of increasing rate of deteriorating items on the optimal average costs
and cycle time. The results of this analysis are shown in the following table.
Optimum values of
T Q Setup
cost
Holding
cost
Deteriorating
cost
Total cost
0.01 1.02 41225.00 98.04 51.72 5.17 154.93
0.02 0.99 10104.48 101.01 49.15 9.83 159.99
7. International Journal on Soft Computing (IJSC) Vol.11, No.1/2/3/4, November 2020
7
0.03 0.97 4444.80 103.10 47.49 14.25 164.84
0.04 0.94 2446.21 106.38 45.06 18.02 169.46
0.05 0.92 1559.88 108.70 43.48 21.74 173.92
0.06 0.90 1061.40 111.11 41.94 25.16 178.21
0.07 0.88 769.79 113.64 40.43 28.30 182.37
0.08 0.87 588.68 114.94 39.69 31.75 186.38
0.09 0.85 458.72 117.65 38.23 34.41 190.29
0.10 0.84 370.98 119.05 37.51 37.51 194.07
Analyzing the results given in the above table, the following observations are made:
(i) When the rate of deterioration increases, the optimum values Setup cost,
Deteriorating cost and Total cost increase.
(ii) The optimum values of T, Q and Holding cost decrease with increases in the
values of rate of deterioration .
(iii) The similar nature is observed for both the model 1 and model 2.
8. COMPARATIVE STUDY BETWEEN TWO MODELS
The comparative study in numerical and graphical is performed here on the basis of the above
data.
Items Ramp type demand Quadratic demand
Holding cost 71.86 51.72
Setup cost 81.97 98.04
Total cost 161.02 154.93
8. International Journal on Soft Computing (IJSC) Vol.11, No.1/2/3/4, November 2020
8
CONCLUDING REMARKS
In this study, we have carried out two types of inventory models for deteriorating items with
ramptype demand and quadratic demand in nature. The models are developed analytically as well
as computationally. Numerical examples and sensitivity analysis of the solutions have been
performed separately.
We have also done comparative study on holding cost, setup cost and total cost of the two types
of models. It is observed from the analytical and graphical presentation that holding cost and total
cost for model having quadratic demand rate are less than that of model having ramptype demand
rate. On the other hand setup cost behaviour is opposite. So, holding cost and total cost in
quadratic function demand are better to compare of ramptype demand and special attention is
made on the inventory model having quadratic function of demand rate.
REFERENCES
(1) Donaldson W.A.,(1977) “Inventory replenishment policy for a linear trend in demand – An
analytical solution”, Operations Research Quarterly,Vol. 28, pp663-670.
(2) Silver E.A.,(1979) “A simple inventory decision rule for a linear trend in Demand”,J. Operational
Research Society, Vol. 30, pp71-75.
(3) Mandal B.& Pal A.K., (1988) “Order Level inventory system with ramp type demand rate”, J.
International Mathematics, Vol 1, No. 1, pp49-66.
(4) BiswaranjanMandal, (2010) “An EOQ inventory model for weibull distributed deteriorated items
under ramp type demand and shortages”, OPSEARCH, Indian J. of Operations Research, Vol 47, No. 2,
pp158-165.
(5) Minghao Cheng, Bixi Zhang &Guoquing Wang, (2011) “Optimal policy for deteriorating items
with trapezoidal type demand and partial backlogging”, Applied Mathematical Modelling,Vol. 35, pp3552-
3560.
(6) Ghosh S. K., Sarkar T. &Chaudhuri K.S., (2012) “ An optimum inventory replenishment policy
for a deteriorating item with time-quadratic demand and time dependent partial backlogging with shortages
in all cycles”, Applied Mathematics and Computation, Vol. 218, pp 9147-9155.