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.
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.
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.
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.
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.
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.
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.
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.
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.
"La gestion económica en las explotaciones de vacuno en extensivo de la Comarca de La Janda" Momoria literal de las ponencias desarrolladas por los expertos.
El blog “El Sacapuntas del Chivo” fue creado en 2010 con el objetivo de ser un medio para la expresión libre del psicopedagogo Rafael López Azuaga, con vistas a hablar sobre temas que no estuviesen relacionados con la educación. Por falto de ideas, de apoyos y por el tiempo al que tenía que dedicar a otras ocupaciones, cerró el blog a mediados de 2011. Coincidió con el inicio del blog “Investigación en Educación”, el cual actualmente sigue en activo con nuevas publicaciones del autor sobre temas relacionados con la educación. En este cuaderno, junto con otro denominado “Los Cómics de nuestro tiempo”, la intención es recopilar las mejores entradas, de entre un total de 132, y recordarlas con el objetivo de vivir emociones o, al menos, pasar un buen rato.
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.
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.
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.
"La gestion económica en las explotaciones de vacuno en extensivo de la Comarca de La Janda" Momoria literal de las ponencias desarrolladas por los expertos.
El blog “El Sacapuntas del Chivo” fue creado en 2010 con el objetivo de ser un medio para la expresión libre del psicopedagogo Rafael López Azuaga, con vistas a hablar sobre temas que no estuviesen relacionados con la educación. Por falto de ideas, de apoyos y por el tiempo al que tenía que dedicar a otras ocupaciones, cerró el blog a mediados de 2011. Coincidió con el inicio del blog “Investigación en Educación”, el cual actualmente sigue en activo con nuevas publicaciones del autor sobre temas relacionados con la educación. En este cuaderno, junto con otro denominado “Los Cómics de nuestro tiempo”, la intención es recopilar las mejores entradas, de entre un total de 132, y recordarlas con el objetivo de vivir emociones o, al menos, pasar un buen rato.
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.
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.
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.
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.
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.
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
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.
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.
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.
A Partial Backlogging Two-Warehouse Inventory Models for Decaying Items with ...IOSR Journals
An order level inventory model for decaying items with inventory level dependent demand rate. We have considered two cases: first is, model started with no shortages and second is model started from shortages. We have also taken the concept of inflation in this study. Finally, a numerical example for illustration is provided with sensitivity analysis.
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
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.
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.
UiPath Test Automation using UiPath Test Suite series, part 4DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 4. In this session, we will cover Test Manager overview along with SAP heatmap.
The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
What will you get from this session?
1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
Topics covered:
Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
LF Energy Webinar: Electrical Grid Modelling and Simulation Through PowSyBl -...DanBrown980551
Do you want to learn how to model and simulate an electrical network from scratch in under an hour?
Then welcome to this PowSyBl workshop, hosted by Rte, the French Transmission System Operator (TSO)!
During the webinar, you will discover the PowSyBl ecosystem as well as handle and study an electrical network through an interactive Python notebook.
PowSyBl is an open source project hosted by LF Energy, which offers a comprehensive set of features for electrical grid modelling and simulation. Among other advanced features, PowSyBl provides:
- A fully editable and extendable library for grid component modelling;
- Visualization tools to display your network;
- Grid simulation tools, such as power flows, security analyses (with or without remedial actions) and sensitivity analyses;
The framework is mostly written in Java, with a Python binding so that Python developers can access PowSyBl functionalities as well.
What you will learn during the webinar:
- For beginners: discover PowSyBl's functionalities through a quick general presentation and the notebook, without needing any expert coding skills;
- For advanced developers: master the skills to efficiently apply PowSyBl functionalities to your real-world scenarios.
PHP Frameworks: I want to break free (IPC Berlin 2024)Ralf Eggert
In this presentation, we examine the challenges and limitations of relying too heavily on PHP frameworks in web development. We discuss the history of PHP and its frameworks to understand how this dependence has evolved. The focus will be on providing concrete tips and strategies to reduce reliance on these frameworks, based on real-world examples and practical considerations. The goal is to equip developers with the skills and knowledge to create more flexible and future-proof web applications. We'll explore the importance of maintaining autonomy in a rapidly changing tech landscape and how to make informed decisions in PHP development.
This talk is aimed at encouraging a more independent approach to using PHP frameworks, moving towards a more flexible and future-proof approach to PHP development.
Slack (or Teams) Automation for Bonterra Impact Management (fka Social Soluti...Jeffrey Haguewood
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We believe integration and automation are essential to user experience and the promise of efficient work through technology. Automation is the critical ingredient to realizing that full vision. We develop integration products and services for Bonterra Case Management software to support the deployment of automations for a variety of use cases.
This video focuses on the notifications, alerts, and approval requests using Slack for Bonterra Impact Management. The solutions covered in this webinar can also be deployed for Microsoft Teams.
Interested in deploying notification automations for Bonterra Impact Management? Contact us at sales@sidekicksolutionsllc.com to discuss next steps.
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Kubernetes & AI - Beauty and the Beast !?! @KCD Istanbul 2024Tobias Schneck
As AI technology is pushing into IT I was wondering myself, as an “infrastructure container kubernetes guy”, how get this fancy AI technology get managed from an infrastructure operational view? Is it possible to apply our lovely cloud native principals as well? What benefit’s both technologies could bring to each other?
Let me take this questions and provide you a short journey through existing deployment models and use cases for AI software. On practical examples, we discuss what cloud/on-premise strategy we may need for applying it to our own infrastructure to get it to work from an enterprise perspective. I want to give an overview about infrastructure requirements and technologies, what could be beneficial or limiting your AI use cases in an enterprise environment. An interactive Demo will give you some insides, what approaches I got already working for real.
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All of this illustrated with link prediction over knowledge graphs, but the argument is general.
UiPath Test Automation using UiPath Test Suite series, part 3DianaGray10
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Topics covered:
UI automation Introduction,
UI automation Sample
Desktop automation flow
Pradeep Chinnala, Senior Consultant Automation Developer @WonderBotz and UiPath MVP
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
"Impact of front-end architecture on development cost", Viktor TurskyiFwdays
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Software Delivery At the Speed of AI: Inflectra Invests In AI-Powered QualityInflectra
In this insightful webinar, Inflectra explores how artificial intelligence (AI) is transforming software development and testing. Discover how AI-powered tools are revolutionizing every stage of the software development lifecycle (SDLC), from design and prototyping to testing, deployment, and monitoring.
Learn about:
• The Future of Testing: How AI is shifting testing towards verification, analysis, and higher-level skills, while reducing repetitive tasks.
• Test Automation: How AI-powered test case generation, optimization, and self-healing tests are making testing more efficient and effective.
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• Inflectra's AI Solutions: See demonstrations of Inflectra's cutting-edge AI tools like the ChatGPT plugin and Azure Open AI platform, designed to streamline your testing process.
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Dev Dives: Train smarter, not harder – active learning and UiPath LLMs for do...UiPathCommunity
💥 Speed, accuracy, and scaling – discover the superpowers of GenAI in action with UiPath Document Understanding and Communications Mining™:
See how to accelerate model training and optimize model performance with active learning
Learn about the latest enhancements to out-of-the-box document processing – with little to no training required
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👩🏫 Lenka Dulovicova, Product Program Manager, UiPath
Empowering NextGen Mobility via Large Action Model Infrastructure (LAMI): pav...
F027037047
1. International Journal of Mathematics and Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759
www.ijmsi.org Volume 2 Issue 7 || July. 2014 || PP-37-47
www.ijmsi.org 37 | P a g e
Two warehouses deteriorating items inventory model under partial backlogging, inflation and permissible delay in payments
Shital S. Patel
Department of Statistics, Veer Narmad South Gujarat University, Surat, INDIA ABSTRACT: A deteriorating items inventory model with two warehouses under time varying holding cost and linear demand with inflation and permissible delay in payments is developed. Shortages are allowed and partially backlogged. A rented warehouse (RW) is used to store the excess units over the capacity of the own warehouse. Numerical examples are provided to illustrate the model and sensitivity analysis is also carried out for parameters. KEYWORDS: Deterioration, Two-warehouse, Inventory, partial backlogging, Inflation, Permissible delay in payment
I. INTRODUCTION
In past few decays deteriorating items inventory models were widely studied. An inventory model with constant rate deterioration was developed by Ghare and Schrader [12]. The model was further extended by considering variable rate of deterioration by Covert and Philip [10]. Shah [30] further extended the model by considering shortages. The related work are found in (Nahmias [23], Raffat [25], Goyal and Giri [14], Mandal [21]), Mishra et al. [22]). The classical inventory model generally deal with single storage facility with the assumption that the available warehouse of the organization has unlimited capacity. But in actual practice, to take advantages of price discount for bulk purchases, a rented warehouse (RW) is used to store the excess units over the fixed capacity W of the own warehouse (OW). The cost of storing items at the RW is higher than that at the OW but provides better preserving facility with a lower rate of deterioration. Hartley [15] was the first one to consider a two warehouse model. A two warehouse deterministic inventory model with infinite rate of replenishment was developed by Sarma [29]. Pakkala and Achary [24] extended the two-warehouse inventory model for deteriorating items with finite rate of replenishment and shortages. Yang [31] considered a two warehouse inventory model for deteriorating items with constant rate of demand under inflation in two alternatives when shortages are completely backordered. Dye et al. [11] considered a two warehouse inventory model with partially backlogging. Jaggi et al. [16] developed an inventory model with linear trend in demand under inflationary conditions and partial backlogging rate in a two warehouse system.Related work is also find in (Benkherouf [3], Bhunia and Maiti [4], Kar et al. [19], Rong et al. [26], Sana et al. [28], Agarwal and Banerjee [1], Bhunia et al. [5]). Goyal [13] first considered the economic order quantity model under the condition of permissible delay in payments. Aggarwal and Jaggi [2] extended Goyal’s [13] model to consider the deteriorating items. Aggarwal and Jaggi’s [2] model was further extended by Jamal et al. [17] to consider shortages. The related work are found in (Chung and Dye [7], Jamal et al. [18], Salameh et al. [27], Chung et al. [8], Chang et al. [6]). Chung and Huang [9] proposed a two warehouse inventory model for deteriorating items under permissible delay in payments, but they assumed that the deterioration rate of the two warehouse were same. Liao and Huang [20] considered an order level inventory model for deteriorating items with two warehouse and a permissible delay in payments. In this paper we have developed a two-warehouse inventory model under time varying holding cost and linear demand with inflation and permissible delay in payments. Shortages are allowed and partially backlogged. Numerical examples are provided to illustrate the model and sensitivity analysis of the optimal solutions for major parameters is also carried out.
2. Two warehouses deteriorating Items inventory model…
www.ijmsi.org 38 | P a g e
II. ASSUMPTIONS AND NOTATIONS
NOTATIONS: The following notations are used for the development of the model: D(t) : Demand rate is a linear function of time t (a+bt, a>0, 0<b<1) A : Replenishment cost per order for two warehouse system c : Purchasing cost per unit p : Selling price per unit c2 : Shortage cost per unit c3 : Cost of lost sales per unit HC(OW): Holding cost per unit time is a linear function of time t (x1+y1t, x1>0, 0<y1<1) in OW HC(RW): Holding cost per unit time is a linear function of time t (x2+y2t, x2>0, 0<y2<1) in RW Ie : Interest earned per year Ip : Interest charged per year M : Permissible period of delay in settling the accounts with the supplier T : Length of inventory cycle I(t) : Inventory level at any instant of time t, 0 ≤ t ≤ T W : Capacity of owned warehouse I0(t) : Inventory level in OW at time t Ir(t) : Inventory level in RW at time t Q1 : Inventory level initially Q2 : Shortage of inventory Q : Order quantity R : Inflation rate tr : Time at which the inventory level reaches zero in RW in two warehouse system θ1t : Deterioration rate in OW, 0< θ1<1 θ2t : Deterioration rate in RW, 0< θ2<1 TCi : Total relevant cost per unit time (i=1,2,3) ASSUMPTIONS: The following assumptions are considered for the development of two warehouse model.
The demand of the product is declining as a linear function of time.
Replenishment rate is infinite and instantaneous.
Lead time is zero.
Shortages are allowed and partially backlogged.
OW has a fixed capacity W units and the RW has unlimited capacity.
The goods of OW are consumed only after consuming the goods kept in RW.
The unit inventory costs per unit in the RW are higher than those in the OW.
During the time, the account is not settled; generated sales revenue is deposited in an interest bearing account. At the end of the credit period, the account is settled as well as the buyer pays off all units sold and starts paying for the interest charges on the items in stocks.
III. THE MATHEMATICAL MODEL AND ANALYSIS
At time t=0, a lot size of certain units enter the system. W units are kept in OW and the rest is stored in RW. The items of OW are consumed only after consuming the goods kept in RW. In the interval [0,tr], the inventory in RW gradually decreases due to demand and deterioration and it reaches to zero at t=tr. In OW, however, the inventory W decreases during the interval [0,tr] due to deterioration only, but during [tr, t0], the inventory is depleted due to both demand and deterioration. By the time to t0, both warehouses are empty. Shortages occur during (t0,T) of size Q2 units. The figure describes the behaviour of inventory system.
3. Two warehouses deteriorating Items inventory model…
www.ijmsi.org 39 | P a g e
Figure 1
. Hence, the inventory level at time t at RW and OW are governed by the following differential
equations:
r
2 r
dI (t)
+ θ tI (t) = - (a+bt),
dt
0 t tr
(1)
with boundary conditions Ir(tr) = 0 and
0
1 0
dI (t)
+ θ tI (t) = 0,
dt
r 0 t t
(2)
with initial condition I0(0) = W, respectively.
While during the interval (tr, t0), the inventory in OW reduces to zero due to the combined effect of
demand and deterioration both. So the inventory level at time t at OW, I0(t), is governed by the following
differential equation:
0
1 0
dI (t)
+ θ tI (t) = -(a+bt),
dt
r 0 t t t
(3)
with the boundary condition I0(t0)=0.
Similarly during (t0, T) the shortage level at time t, Is(t) is governed by the following differential equation:
s -δ(T - t) dI (t)
= - e (a+bt),
dt
t0≤t≤T, (4)
with the boundary condition Is(t0)=0.
The solutions to equations (1) to (4) are given by:
2 2 3 3
r r 2 r
r
4 4 2 2 2 2
2 r 2 r 2 r
1 1
a t - t + b t - t + aθ t - t
2 6
I (t) =
1 1 1
+ bθ t - t - aθ t t - t - bθ t t - t
8 2 4
r 0 t t (5)
2
o 1 I (t) = W 1 - θ t , r 0 t t (6)
2 2 3 3
0 0 1 0
o
4 4 2 2 2 2
1 0 1 0 1 0
1 1
a t - t + b t - t + aθ t - t
2 6
I (t) =
1 1 1
+ bθ t - t - aθ t t - t - bθ t t - t
8 2 4
r 0 t t t (7)
3 2
s
3 2 2 2
0 0 0 0 0 0
1 1
- bδt - [aδ + b(1-δT)]t - at(1-δT)
3 2
I (t) =
1 1 1 1
+ bδt + aδt + bt - bδTt - aδt T + at
3 2 2 2
, t0≤t≤T
(8)
(by neglecting higher powers of θ1, θ2)
Using the condition Ir(t) = Q1 – W at t=0 in equation (5), we have
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2 3 4
1 r r 2 r 2 r
1 1 1
Q - W = at + bt + aθ t + bθ t ,
2 6 8
2 3 4
1 r r 2 r 2 r
1 1 1
Q = W + at + bt + aθ t + bθ t .
2 6 8
(9)
Using the condition Is(t) = Q - Q1 at t=T in equation (8), we have
2 2
1 0 0
1
Q - Q = - a(T - t ) + b(T - t )
2
2 2
1 0 0
1
Q = Q - a(T - t ) + b(T - t ) .
2
(10)
Using the continuity of I0(t) at t=tr in equations (6) and (7), we have
2 2 3 3
0 r 0 r 1 0 r
2
o r 1
4 4 2 2 2 2
1 0 r 1 r 0 1 r 0 r
1 1
a t - t + b t - t + aθ t - t
2 6
I (t ) = W 1 - θ t =
1 1 1
+ bθ t - t - aθ t t - t - bθ t t - t
8 2 4
(11)
which implies that
2 2 2 2
1 r r r
0
- a + a + 2bW - bWθ t + b t + 2abt
t =
b
(12)
(by neglecting higher powers of tr and t0)
From equation (12), we note that t0 is a function of tr, therefore t0 is not a decision variable.
Based on the assumptions and descriptions of the model, the total annual relevant costs TCi, include the
following elements:
(i) Ordering cost (OC) = A
(13)
(ii)
tr
-Rt
2 2 r
0
HC(RW) = (x +y t)I (t) e dt
r
2 2 3 3
t r r 2 r
-Rt
2 2
0 4 4 2 2 2 2
2 r 2 r 2 r
1 1
a t - t + b t - t + aθ t - t
2 6
= (x +y t) e dt
1 1 1
+ bθ t - t - aθ t t - t - bθ t t - t
8 2 4
7 6 2 5
2 2 r 2 2 2 2 2 r 2 2 2 2 2 2 2 r r r
2 4
2 2 2 2 2 r r 2 r
2 2
1 1 1 1 1 1 1 1 1 1
= - y Rθ bt + y -x R θ b- y Rθ a t + x θ b+ y -x R θ a-y R - θ bt +at - b t
56 6 8 3 5 8 3 2 2 2
1 1 1 1 1
+ x θ a + y -x R - θ bt + at - b + y Ra t
4 3 2 2 2
1 1 1
+ x - θ b
3 2 2
2 4 3 2 3
r r 2 2 2 2 r 2 r r r r
4 3 2 2 4 3 2
2 2 2 2 r 2 r r r r 2 2 r 2 r r r r
1 1 1 1
t + at - b - y -x R a - y R bθ t + aθ t + bt + at t
2 8 6 2
1 1 1 1 1 1 1
+ -x a + y -x R bθ t + aθ t + bt +at t + x bθ t + aθ t + bt +at t
2 8 6 2 8 6 2
(14)
(by neglecting higher powers of R)
(iii)
t0
-Rt
1 1 0
0
HC(OW) = (x +y t)I (t) e dt
r 0
r
t t
-Rt -Rt
1 1 0 1 1 0
0 t
= (x +y t)I (t)e dt + (x +y t)I (t)e dt
tr
2 -Rt
1 1 1
0
= (x +y t)W 1 - θ t e dt
0
r
2 2 3 3
t 0 0 1 0
-Rt
1 1
t 4 4 2 2 2 2
1 0 1 0 1 0
1 1
a t - t + b t - t + aθ t - t
2 6
+ (x +y t) e dt
1 1 1
+ bθ t - t - aθ t t - t - bθ t t - t
8 2 4
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5 4 3 2
1 1 r 1 1 1 r 1 1 1 0 1 1 0 1 r
1 1 1 1 1
= W y Rbθ t - y -x R θ t + - x θ - y R t + y -x R t x t
10 8 3 2 2
7 6
1 1 0 1 1 1 1 1 0
2 5
1 1 1 1 1 1 1 0 0 0
2 4
1 1 1 1 1 0 0 1 0
1
1 1 1 1
- y Rbθ t + y -x R bθ - y Raθ t
56 6 8 3
1 1 1 1 1 1
+ x bθ + y -x R aθ -y R - θ bt +at - b t
5 8 3 2 2 2
1 1 1 1 1
+ + x aθ + y -x R - θ bt +at - b y Ra t
4 3 2 2 2
1 1
+ x - θ
3 2
2 4 3 2 3
1 0 0 1 1 1 1 0 1 0 0 0 0
4 3 2 2 4 3 2
1 1 1 1 0 1 0 0 0 0 1 1 0 1 0 0 0 0
1 1 1 1 1
bt +at - b - y -x R a y R bθ t + aθ t + bt +at t
2 2 8 6 2
1 1 1 1 1 1 1
+ -x a - y -x R bθ t + aθ t + bt +at t + x bθ t + aθ t + bt +at t
2 8 6 2 8 6 2
7 6
1 1 r 1 1 1 1 1 r
2 5
1 1 1 1 1 1 1 0 0 r
2 4
1 1 1 1 1 0 0 1 r
1 1 1 1
y Rbθ t - y -x R bθ - y Raθ t
56 6 8 3
1 1 1 1 1 1
+ - x bθ + y -x R aθ - y R - θ bt +at - b t
5 8 3 2 2 2
1 1 1 1 1
- x aθ + y -x R - θ bt +at - b + y Ra t
4 3 2 2 2
2 4 3 2 3
1 1 0 0 1 1 1 1 0 1 0 0 0 r
4 3 2 2 4 3 2
1 1 1 1 0 1 0 0 0 r 1 1 0 1 0 0 0 r
1 1 1 1 1 1 1
- x - θ bt +at - b - y -x R a - y R bθ t aθ t bt +at t
3 2 2 2 8 6 2
+
1 1 1 1 1 1 1
- -x a + y -x R bθ t aθ t bt +at t - x bθ t aθ t bt +at t
2 8 6 2 8 6 2
(15)
(iv) Deterioration cost:
The amount of deterioration in both RW and OW during [0,t0] are:
tr
2 r
0
θ tI (t)dt and
t0
1 0
0
θ tI (t)dt
So deterioration cost
r 0 t t
-Rt -Rt
2 r 1 0
0 0
DC = c θ tI (t)e dt + θ tI (t)e dt
r r 0
r
t t t
-Rt -Rt -Rt
2 r 1 0 1 0
0 0 t
= c θ tI (t)e dt + θ tI (t)e dt + θ tI (t)e dt
7 6 2 5
2 r 2 2 r 2 2 r r r
2 4 4 3 2 3
2 2 r r r 2 r 2 r r r r
1 1 1 1 1 1 1 1 1
- Rθ bt + bθ - Raθ t + aθ -R - θ bt +at - b t
56 6 8 3 5 3 2 2 2
1 1 1 1 1 1 1 1
= cθ + - θ bt + at - b Ra t + -a - R bθ t + aθ t + bt + at t
4 2 2 2 3 8 6 2
+
4 3 2 2 5 4 3 2
2 r 2 r r r r 1 1 r 1 r r r
1 1 1 1 1 1 1 1
bθ t + aθ t + bt + at t + cθ W Rθ t - θ t - Rt + t
2 8 6 2 10 8 3 2
7 6 2 5
1 0 1 1 0 1 1 0 0 0
2 4 4 3 2 3
1 1 0 0 0 1 0 1 0 0 0 0
1 1 1 1 1 1 1 1 1
- Rθ bt + bθ - Raθ t + aθ -R - θ bt +at - b t
56 6 8 3 5 3 2 2 2
1 1 1 1 1 1 1 1
+ cθ + - θ bt + at - b Ra t + -a - R bθ t + aθ t + bt + at t
4 2 2 2 3 8 6 2
4 3 2 2
1 0 1 0 0 0 0
1 1 1 1
+ bθ t + aθ t + bt + at t
2 8 6 2
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7 6 2 5
1 r 1 1 r 1 1 0 0 r
2 4 4 3 2
1 1 0 0 r 1 0 1 0 0 0 r
1 1 1 1 1 1 1 1 1
- Rθ bt + bθ - Raθ t + aθ -R - θ bt +at - b t
56 6 8 3 5 3 2 2 2
1 1 1 1 1 1 1 1
- cθ + - θ bt + at - b Ra t + -a - R bθ t + aθ t + bt + at t
4 2 2 2 3 8 6 2
3
4 3 2 2
1 0 1 0 0 0 r
1 1 1 1
+ bθ t + aθ t + bt + at t
2 8 6 2
(16)
(v) Shortage cost:
0
T
-Rt
2
t
SC = - c I(t)e dt
0
3 2
T
-Rt
2
t 3 2 2 2
0 0 0 0 0 0
1 1
- bδt - [aδ + b(1-δT)]t - at(1-δT)
3 2
= - c e dt
1 1 1 1
+ bδt + aδt + bt - bδTt - aδt T + at
3 2 2 2
5 4
3
2
2 2 2 3 2
0 0 0 0 0 0
2 2
0 0
1 1 1 1 1
bδRT + - - aδ - b(1-δT) R - bδ T
15 4 2 2 3
1 1 1
+ a(1-δT)R - aδ - b(1-δT) T
3 2 2
= c
1 1 1 1 1
+ - aδt + bt - bδTt + bδt - aδTt + at R -a(1-δT) T
2 2 2 2 3
1 1
+ aδTt + bTt -
2 2
2 2 3 2
0 0 0 0
1 1
bδT t + bδTt - aδT t + aTt
2 3
5 4
0 0
3
0
2
2 2 2 3 2
0 0 0 0 0 0 0
3 3
0 0
1 1 1 1 1
bδRt + - - aδ - b(1-δT) R - bδ t
15 4 2 2 3
1 1 1
+ a(1-δT)R - aδ - b(1-δT) t
3 2 2
+ c
1 1 1 1 1
+ - aδt + bt - bδTt + bδt - aδTt + at R -a(1-δT) t
2 2 2 2 3
1 1
+ aδt + bt
2 2
3 4 2 2
0 0 0 0
.
1 1
- bδTt + bδt - aδTt + at
2 3
(17)
(vi) Cost due to lost sales:
LS =
2
T
-Rt
3
t
c (a + bt)[1- (1 - δ(T - t)]e dt
2 2
0 0
3
3 3 4 4
0 2
1
aT(T - t ) + (bT - a -aRT)(T - t )
2
= cδ .
1 1
+ (- b -bRT + aR )(T - t ) + bR(T - t )
3 4
(18)
(vii) Interest Earned: There are two cases:
Case I : (M ≤ tr ≤ T):
In this case interest earned is:
M
-Rt
1 e
0
IE = pI a + bt te dt 4 3 2
e
1 1 1
pI - bRM + - Ra + b M + aM
4 3 2
(19)
Case II : (t0 ≤ M ≤ T):
In this case interest earned is:
t0
-Rt
2 e 0 0 0
0
IE = pI a+bt te dt + a + bt t M - t
2 3 2
e 0 0 0 0 0 0
1 1 1
= pI - bRt + - Ra + b t + at + a+bt t M-t
4 3 2
(20)
(viii) Interest Payable: There are three cases described as in figure:
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Case I : (M ≤ tr ≤ T):
In this case, annual interest payable is:
r r 0
r
t t t
-Rt -Rt -Rt
1 p r 0 0
M M t
IP = cI I (t)e dt + I (t)e dt + I (t)e dt
6 5 2 4
2 r 2 2 r 2 2 r r r
2 3 4 3 2
p 2 r r r 2 r 2 r r r
1 1 1 1 1 1 1 1 1
- Rθ bt + θ b - Rθ a t + θ a - R - θ bt +at - b t
48 5 8 3 4 3 2 2 2
1 1 1 1 1 1 1 1
= cI + - θ bt +at - b+Ra t + - a - R bθ t + aθ t + bt +at
3 2 2 2 2 8 6 2
2
r
5 4 3 2
2 r 2 r r r
t
1 1 1
+ θ bt + aθ t + bt + at
8 6 2
6 5 2 4
2 2 2 2 2 r r
2 3 4 3 2 2
p 2 r r 2 r 2 r r r
1 1 1 1 1 1 1 1 1
- Rθ bM + θ b - Rθ a M + θ a - R - θ bt +at - b M
48 5 8 3 4 3 2 2 2
1 1 1 1 1 1 1 1
- cI + - θ bt +at - b+Ra M + - a - R bθ t + aθ t + bt +at M
3 2 2 2 2 8 6 2
+
4 3 2
2 r 2 r r r
1 1 1
θ bt M + aθ t M + bt M + at M
8 6 2
4 3 2 4 3 2
p r 1 r 1 r r p 1 1
1 1 1 1 1 1
+ cI W t + Rθ t - θ t - Rt - cI W M + Rθ M - θ M - RM
8 6 2 8 6 2
6 5 2 4
1 0 1 1 0 1 1 0 0 0
2 3 4 3 2
p 1 0 0 0 1 0 1 0 0 0
1 1 1 1 1 1 1 1 1
- Rθ bt + θ b - Rθ a t + θ a - R - θ bt +at - b t
48 5 8 3 4 3 2 2 2
1 1 1 1 1 1 1 1
+ cI + - θ bt +at - b+Ra t + - a - R bθ t + aθ t + bt +at
3 2 2 2 2 8 6 2
2
0
5 4 3 2
1 0 1 0 0 0
t
1 1 1
+ θ bt + aθ t + bt + at
8 6 2
6 5 2 4
1 r 1 1 r 1 1 0 0 r
2 3 4 3 2
p 1 0 0 r 1 0 1 0 0 0
1 1 1 1 1 1 1 1 1
- Rθ bt + θ b - Rθ a t + θ a - R - θ bt +at - b t
48 5 8 3 4 3 2 2 2
1 1 1 1 1 1 1 1
- cI + - θ bt +at - b+Ra t + - a - R bθ t + aθ t + bt +at
3 2 2 2 2 8 6 2
2
r
4 3 2
1 0 r 1 0 r 0 r 0 r
t
1 1 1
+ θ bt t + aθ t t + bt t + at t
8 6 2
(21)
Case II : (tr ≤ M ≤ T):
In this case interest payable is:
t0
-Rt
2 p 0
M
IP = cI I (t)e dt
6 5 2 4
1 0 1 1 0 1 1 0 0 0
2 3 4 3 2
p 1 0 0 0 1 0 1 0 0
1 1 1 1 1 1 1 1 1
- Rθ bt + θ b - Rθ a t + θ a - R - θ bt +at - b t
48 5 8 3 4 3 2 2 2
1 1 1 1 1 1 1 1
= cI + - θ bt +at - b+Ra t + - a - R bθ t + aθ t + bt +aT t
3 2 2 2 2 8 6 2
2
0
5 4 3 2
1 0 1 0 0 0
1 1 1
+ θ bt + aθ t + bt + at
8 6 2
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6 5 2 4
1 1 1 1 1 0 0
2 3 4 3 2 2
p 1 0 0 1 0 1 0 0 0
1 1 1 1 1 1 1 1 1
- Rθ bM + θ b - Rθ a M + θ a - R - θ bt +at - b M
48 5 8 3 4 3 2 2 2
1 1 1 1 1 1 1 1
- cI + - θ bt +at - b+Ra M + - a - R bθ t + aθ t + bt +at M
3 2 2 2 2 8 6 2
+
4 3 2
1 0 1 0 0 0
1 1 1
θ bt M + aθ t M + bt M + at M
8 6 2
(22)
Case III : (t0 ≤ M ≤ T):
In this case, no interest charges are paid for the item. So,
IP3 = 0. (23)
The retailer’s total cost during a cycle, TCi(tr,T), i=1,2,3 consisted of the following:
i i i
1
TC = A + HC(OW) + HC(RW) + DC+ SC + LS + IP - IE
T
(24)
and t0 is approximately related to tr through equation (12).
Substituting values from equations (13) to (18) and equations (19) to (23) in equation (24), total costs for the
three cases will be as under:
1 1 1
1
TC = A + HC(OW) + HC(RW) + DC+ SC + LS + IP - IE
T
(25)
2 2 1
1
TC = A + HC(OW) + HC(RW) + DC+ SC + LS + IP - IE
T
(26)
3 3 2
1
TC = A + HC(OW) + HC(RW) + DC+ SC + LS + IP - IE
T
(27)
The optimal value of tr = tr*, T=T* (say), which minimizes TCi can be obtained by solving equation (25), (26)
and (27) by differentiating it with respect to tr and T and equate it to zero i.e.
i.e. i r i r
r
TC (t ,T) TC (t ,T)
= 0, = 0,
t T
i=1,2,3, (28)
provided it satisfies the condition
2 2
i r i r
2 2
r
C (t ,T) C (t ,T)
>0, >0
t T
and
2
2 2 2
i r i r i r
2 2
r r
C (t ,T) C (t ,T) C (t ,T)
- > 0, i=1,2,3.
t T t T
(29)
IV. NUMERICAL EXAMPLES
Case I: Considering A= Rs.150, W = 100, a = 200, b=0.05, c=Rs. 10, p= Rs. 15, θ1=0.1, θ2 =0.06, x1 = Rs. 1,
y1=0.05, x2= Rs. 3, y2=0.06, Ip= Rs. 0.15, Ie= Rs. 0.12, R = 0.06, c2 = Rs. 8, c3 = Rs. 2, δ = 0.8, M=0.01 year, in
appropriate units. The optimal value of
*
r t =0.0831, T*=0.8081 and
*
1 TC = Rs. 349.8128.
Case II: Considering A= Rs.150, W = 100, a = 200, b=0.05, c = Rs. 10, p= Rs. 15, θ1=0.1, θ2 =0.06, x1= Rs. 1,
y1=0.05, x2= Rs. 3, y2=0.06, Ip= Rs. 0.15, Ie = Rs. 0.12, c2 = Rs. 8, c3 = Rs. 2, δ = 0.8, M=0.55 year, in
appropriate units. The optimal value of
*
r t =0.0867, T*=0.7151 and
*
2 TC = Rs. 215.9081.
Case III: Considering A= Rs.150, W = 100, a = 200, b=0.05, c = Rs. 10, p= Rs. 15, θ1=0.1, θ2 =0.06, x1= Rs. 1,
y1=0.05, x2= Rs. 3, y2=0.06, Ip= Rs. 0.15, Ie= Rs. 0.12, c2 = Rs. 8, c3 = Rs. 2, δ = 0.8, M = 0.65 year, in
appropriate units. The optimal value of
*
r t =0.1014, T*=0.7101 and
*
1 TC = Rs. 185.5066.
The second order conditions given in equation (29) are also satisfied. The graphical representation of
the convexity of the cost functions for the three cases are also given.
Case I
tr and cost T and cost tr, T and cost
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Graph 1 Graph 2 Graph 3
Case II
tr and cost
T and cost
tr, T and cost Graph 4 Graph 5 Graph 6
Case III
tr and cost
T and cost
tr, T and cost Graph 7 Graph 8 Graph 9
V. SENSITIVITY ANALYSIS
On the basis of the data given in example above we have studied the sensitivity analysis by changing the following parameters one at a time and keeping the rest fixed.
Case I
Case II
Case III
(M ≤ tr ≤ T)
(tr ≤ M ≤ T)
(t0 ≤ M ≤ T)
Para- meter
%
tr
T
Cost
tr
T
Cost
tr
T
Cost
a
+10%
0.0951
0.7620
367.8316
0.1068
0.6783
218.6641
0.1149
0.6672
185.5466
+5%
0.0895
0.7841
358.9034
0.0973
0.6959
217.3366
0.1086
0.6878
185.5075
-5%
0.0758
0.8344
340.5512
0.0749
0.7362
214.3719
0.0932
0.7345
185.3120
-10%
0.0674
0.8633
331.1092
0.0616
0.7596
212.7207
0.0838
0.7613
184.9143
x1
+10%
0.0774
0.8055
353.8748
0.0830
0.7144
220.5564
0.0969
0.7087
190.3883
+5%
0.0802
0.8068
351.8509
0.0848
0.7148
218.2381
0.0991
0.7094
187.9540
-5%
0.0860
0.8094
347.7605
0.0885
0.7154
213.5663
0.1036
0.7108
183.0461
-10%
0.0888
0.8107
345.6942
0.0904
0.7157
211.2129
0.1058
0.7115
180.5725
x2
+10%
0.0786
0.8037
350.0558
0.0833
0.7119
216.2113
0.0965
0.7054
185.9206
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+5% 0.0808 0.8059 349.9375 0.0850 0.7134 216.0626 0.0989 0.7077 185.7184
-5% 0.0856 0.8105 349.6812 0.0885 0.7168 215.7475 0.1040 0.7127 185.2846
-10% 0.0882 0.8131 349.5419 0.0904 0.7186 215.5806 0.1068 0.7153 185.0514
θ1
+10% 0.0806 0.8062 350.6601 0.0850 0.7140 216.8554 0.0991 0.7085 186.5307
+5% 0.0819 0.8072 350.2374 0.0858 0.7145 216.3824 0.1003 0.7093 186.0197
-5% 0.0844 0.8091 349.3862 0.0876 0.7157 215.4322 0.1025 0.7109 184.9916
-10% 0.0857 0.8101 348.9576 0.0885 0.7162 214.9549 0.1037 0.7118 184.4746
θ2
+10% 0.0831 0.8081 349.8143 0.0867 0.7151 215.9099 0.1013 0.7101 185.5096
+5% 0.0831 0.8081 349.8135 0.0867 0.7151 215.9090 0.1014 0.7101 185.5081
-5% 0.0831 0.8082 349.8121 0.0867 0.7151 215.9071 0.1014 0.7102 185.5051
-10% 0.0832 0.8082 349.8114 0.0867 0.7151 215.9062 0.1014 0.7102 185.5037
δ
+10% 0.0835 0.8083 350.2002 0.0869 0.7141 216.1404 0.1016 0.7092 185.6874
+5% 0.0833 0.8082 350.0065 0.0868 0.7146 216.0246 0.1015 0.7097 185.5973
-5% 0.0829 0.8081 349.6190 0.0866 0.7156 215.7909 0.1013 0.7106 185.4152
-10% 0.0827 0.8080 349.4251 0.0865 0.7160 215.6730 0.1012 0.7111 185.3232
R
+10% 0.0836 0.8098 349.4392 0.0865 0.7156 215.9521 0.1011 0.7104 185.5842
+5% 0.0834 0.8090 349.6263 0.0866 0.7153 215.9302 0.1013 0.7103 185.5455
-5% 0.0829 0.8073 349.9988 0.0868 0.7149 215.8857 0.1015 0.7100 185.4675
-10% 0.0826 0.8065 350.1843 0.0869 0.7146 215.8630 0.1017 0.7098 185.4282
A
+10% 0.0832 0.8080 349.5957 0.1004 0.7179 199.1456 0.1044 0.7005 165.5004
+5% 0.0832 0.8081 349.7043 0.0935 0.7164 207.4508 0.1029 0.7053 175.5502
-5% 0.0831 0.8082 349.9214 0.0800 0.7141 224.5186 0.0998 0.7148 195.3707
-10% 0.0830 0.8082 350.0298 0.0734 0.7133 233.2833 0.0981 0.7195 205.1433
M
+10% 0.1018 0.8418 367.9956 0.1017 0.7440 236.4685 0.1205 0.7427 206.1560
+5% 0.0926 0.8251 358.9970 0.0943 0.7297 226.2903 0.1110 0.7266 195.9471
-5% 0.0735 0.7910 340.4325 0.0789 0.7003 205.3104 0.0915 0.6934 174.8192
-10% 0.0636 0.7735 332.2985 0.0710 0.6853 194.4848 0.0814 0.6763 163.8678
From the table we observe that as parameter a increases/ decreases average total cost increases/
decreases in case I and case II and there is very slight increase/ decrease in case III with respective increase/
decrease in parameter a..
From the table we observe that with increase/ decrease in parameters x1 and θ1, there is corresponding increase/
decrease in total cost for case I, case II and case III respectively.
Also, we observe that with increase and decrease in the value of x2, θ2, δ and R, there is corresponding
very slight increase/ decrease in total cost for case I, case II and case III. From the table we observe that with
increase/ decrease in M, there is corresponding decrease/ increase in total cost for case I, case II and case III
respectively. Moreover, we observe that with increase/ decrease in the value of A, there is corresponding
increase/ decrease in total cost in cases I, II and III.
VI. CONCLUSION
We have developed a two warehouse inventory model for deteriorating items with linear demand and
partial backlogging under inflationary conditions and permissible delay in payments in this model. It is assumed
that rented warehouse holding cost is greater than own warehouse holding cost but provides a better storage
facility and there by deterioration rate is low in rented warehouse. Sensitivity with respect to parameters have
been carried out. The results show that there is increase/ decrease in cost when there is increase/ decrease in the
parameter values.
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