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
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.
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.
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.
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.
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.
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.
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.
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.
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.
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
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 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.
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.
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.
A Fuzzy Arithmetic Approach for Perishable Items in Discounted Entropic Order...Waqas Tariq
This paper uses fuzzy arithmetic approach to the system cost for perishable items with instant deterioration for the discounted entropic order quantity model. Traditional crisp system cost observes that some costs may belong to the uncertain factors. It is necessary to extend the system cost to treat also the vague costs. We introduce a new concept which we call entropy and show that the total payoff satisfies the optimization property. We show how special case of this problem reduce to perfect results, and how post deteriorated discounted entropic order quantity model is a generalization of optimization. It has been imperative to demonstrate this model by analysis, which reveals important characteristics of discounted structure. Further numerical experiments are conducted to evaluate the relative performance between the fuzzy and crisp cases in EnOQ and EOQ separately.
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 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.
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 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.
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.
An Examination of Effectuation Dimension as Financing Practice of Small and M...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Does Goods and Services Tax (GST) Leads to Indian Economic Development?iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
More Related Content
Similar to An inventory model for variable demand, constant holding cost and without shortages
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.
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
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 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.
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.
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.
A Fuzzy Arithmetic Approach for Perishable Items in Discounted Entropic Order...Waqas Tariq
This paper uses fuzzy arithmetic approach to the system cost for perishable items with instant deterioration for the discounted entropic order quantity model. Traditional crisp system cost observes that some costs may belong to the uncertain factors. It is necessary to extend the system cost to treat also the vague costs. We introduce a new concept which we call entropy and show that the total payoff satisfies the optimization property. We show how special case of this problem reduce to perfect results, and how post deteriorated discounted entropic order quantity model is a generalization of optimization. It has been imperative to demonstrate this model by analysis, which reveals important characteristics of discounted structure. Further numerical experiments are conducted to evaluate the relative performance between the fuzzy and crisp cases in EnOQ and EOQ separately.
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 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.
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 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.
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.
An Examination of Effectuation Dimension as Financing Practice of Small and M...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Does Goods and Services Tax (GST) Leads to Indian Economic Development?iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Childhood Factors that influence success in later lifeiosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Emotional Intelligence and Work Performance Relationship: A Study on Sales Pe...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Customer’s Acceptance of Internet Banking in Dubaiiosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
A Study of Employee Satisfaction relating to Job Security & Working Hours amo...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Consumer Perspectives on Brand Preference: A Choice Based Model Approachiosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Student`S Approach towards Social Network Sitesiosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Broadcast Management in Nigeria: The systems approach as an imperativeiosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
A Study on Retailer’s Perception on Soya Products with Special Reference to T...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
A Study Factors Influence on Organisation Citizenship Behaviour in Corporate ...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Consumers’ Behaviour on Sony Xperia: A Case Study on Bangladeshiosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Design of a Balanced Scorecard on Nonprofit Organizations (Study on Yayasan P...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Public Sector Reforms and Outsourcing Services in Nigeria: An Empirical Evalu...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Media Innovations and its Impact on Brand awareness & Considerationiosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Customer experience in supermarkets and hypermarkets – A comparative studyiosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Social Media and Small Businesses: A Combinational Strategic Approach under t...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Secretarial Performance and the Gender Question (A Study of Selected Tertiary...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Implementation of Quality Management principles at Zimbabwe Open University (...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Organizational Conflicts Management In Selected Organizaions In Lagos State, ...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Deep Behavioral Phenotyping in Systems Neuroscience for Functional Atlasing a...Ana Luísa Pinho
Functional Magnetic Resonance Imaging (fMRI) provides means to characterize brain activations in response to behavior. However, cognitive neuroscience has been limited to group-level effects referring to the performance of specific tasks. To obtain the functional profile of elementary cognitive mechanisms, the combination of brain responses to many tasks is required. Yet, to date, both structural atlases and parcellation-based activations do not fully account for cognitive function and still present several limitations. Further, they do not adapt overall to individual characteristics. In this talk, I will give an account of deep-behavioral phenotyping strategies, namely data-driven methods in large task-fMRI datasets, to optimize functional brain-data collection and improve inference of effects-of-interest related to mental processes. Key to this approach is the employment of fast multi-functional paradigms rich on features that can be well parametrized and, consequently, facilitate the creation of psycho-physiological constructs to be modelled with imaging data. Particular emphasis will be given to music stimuli when studying high-order cognitive mechanisms, due to their ecological nature and quality to enable complex behavior compounded by discrete entities. I will also discuss how deep-behavioral phenotyping and individualized models applied to neuroimaging data can better account for the subject-specific organization of domain-general cognitive systems in the human brain. Finally, the accumulation of functional brain signatures brings the possibility to clarify relationships among tasks and create a univocal link between brain systems and mental functions through: (1) the development of ontologies proposing an organization of cognitive processes; and (2) brain-network taxonomies describing functional specialization. To this end, tools to improve commensurability in cognitive science are necessary, such as public repositories, ontology-based platforms and automated meta-analysis tools. I will thus discuss some brain-atlasing resources currently under development, and their applicability in cognitive as well as clinical neuroscience.
Toxic effects of heavy metals : Lead and Arsenicsanjana502982
Heavy metals are naturally occuring metallic chemical elements that have relatively high density, and are toxic at even low concentrations. All toxic metals are termed as heavy metals irrespective of their atomic mass and density, eg. arsenic, lead, mercury, cadmium, thallium, chromium, etc.
(May 29th, 2024) Advancements in Intravital Microscopy- Insights for Preclini...Scintica Instrumentation
Intravital microscopy (IVM) is a powerful tool utilized to study cellular behavior over time and space in vivo. Much of our understanding of cell biology has been accomplished using various in vitro and ex vivo methods; however, these studies do not necessarily reflect the natural dynamics of biological processes. Unlike traditional cell culture or fixed tissue imaging, IVM allows for the ultra-fast high-resolution imaging of cellular processes over time and space and were studied in its natural environment. Real-time visualization of biological processes in the context of an intact organism helps maintain physiological relevance and provide insights into the progression of disease, response to treatments or developmental processes.
In this webinar we give an overview of advanced applications of the IVM system in preclinical research. IVIM technology is a provider of all-in-one intravital microscopy systems and solutions optimized for in vivo imaging of live animal models at sub-micron resolution. The system’s unique features and user-friendly software enables researchers to probe fast dynamic biological processes such as immune cell tracking, cell-cell interaction as well as vascularization and tumor metastasis with exceptional detail. This webinar will also give an overview of IVM being utilized in drug development, offering a view into the intricate interaction between drugs/nanoparticles and tissues in vivo and allows for the evaluation of therapeutic intervention in a variety of tissues and organs. This interdisciplinary collaboration continues to drive the advancements of novel therapeutic strategies.
What is greenhouse gasses and how many gasses are there to affect the Earth.moosaasad1975
What are greenhouse gasses how they affect the earth and its environment what is the future of the environment and earth how the weather and the climate effects.
Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
Seminar of U.V. Spectroscopy by SAMIR PANDASAMIR PANDA
Spectroscopy is a branch of science dealing the study of interaction of electromagnetic radiation with matter.
Ultraviolet-visible spectroscopy refers to absorption spectroscopy or reflect spectroscopy in the UV-VIS spectral region.
Ultraviolet-visible spectroscopy is an analytical method that can measure the amount of light received by the analyte.
An inventory model for variable demand, constant holding cost and without shortages
1. IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 1 Ver. IV (Jan - Feb. 2015), PP 49-55
www.iosrjournals.org
DOI: 10.9790/5728-11144955 www.iosrjournals.org 49 |Page
An inventory model for variable demand, constant holding cost
and without shortages
Dr.Ravish Kumar Yadav1
,Amit Kumar Vats2,
Ranjeet Kaur3
1,
Associate Professor, Department of mathematics , Hindu College, Moradabad
2,
Assistant Professor, ABES, Institute of technology, Ghazibad
3,
Assistant Professor, ABES, Institute of technology, Ghazibad
I. Introduction
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. Ghare and Schrader (1963) first established an economic order quantity model
having a constant deterioration. Chang gives a Partial Backlogging Inventory Model rate of deterioration and
constant rate of demand over a finite planning horizon. Covert and Philip (1973) extended Ghare and
Schrader‟s constant deterioration rate to a two parameter Weibull distribution. Dave and Patel (1981) discussed
an inventory model for deteriorating items with time-proportional demand when shortages were not allowed.
The related analysis on inventory systems with deterioration have been performed by Balkhi and Benkherouf
(1996), Wee (1997), Mukhopadhyay et al. (2004),etc.
In reality, not all kinds of inventory items deteriorated as soon as they received by the retailer. In the
fresh product time, the product has no deterioration and keeps their original quality. Ouyang et al. (2006)
named this phenomenon as “non-instantaneous deterioration”, and they established an inventory model for non-
instantaneous deteriorating items with permissible delay in payments. In some fashionable products, some
customers would like to wait for backlogging during the shortage period. But the willingness is diminishing with
the length of the waiting time for the next replenishment. The longer the waiting time is, the smaller the
backlogging rate would be. The opportunity cost due to lost sales should be considered. Chang and Dye (1999)
developed an inventory model in which the demand rate is a time-continuous function and items deteriorate at a
constant rate with partial backlogging rate which is the reciprocal of a linear function of the waiting time.
Papachristos and Skouri (2000) developed an EOQ inventory model with time-dependent partial backlogging.
They supposed the rate of backlogged demand increases exponentially with the waiting time for the next
replenishment decreases. Teng et al. (2002) then extended the backlogged demand to any decreasing function of
the waiting time up to the next replenishment. The related analysis on inventory systems with artial backlogging
have been performed by Teng and Yang (2004), Yang (2005), Dye et al. (2006), San José et al. (2006), Teng
et al. (2007), etc.
Many articles assume that the demand is constant during the sale period. It needs to be discussed. In
real life, the requirements may be stimulated if there is a large pile of goods displayed on shelf.Gupta and Vrat
(1986) presented an inventory model for stock-dependent consumption rate on initial stock level rather than
instantaneous inventory level. Baker and Urban (1988) established a deterministic inventory system in which
the demand rate depended on the inventory level is described by a polynomial function. Wu et al. (2006)
presented an inventory model for non-instantaneous deteriorating items with stock-dependent. The related
analysis on inventory systems with stock-dependent consumption rate have been performed by Datta and Paul
(2001), Balkhi and Benkherouf (2004), Chang et al. (2007),etc.
In all of the above mentioned models, the influences of the inflation and time value of money were not
discussed. Buzacott (1975) first established an EOQ model with inflation subject to different types of pricing
policies. Chung and Lin (2001) followed the discounted cash flow approach to investigate inventory model
with constant demand rate for deteriorating items taking account of time value of money. Hou (2006)
established an inventory model with stock-dependent consumption rate simultaneously considered the inflation
and time value of money when shortages are allowed over a finite planning horizon.
In this article, inventory model for deteriorating items with stock-dependent demand rate, along with
the effects of inflation are considered.
The rest of this paper is organized as follows. In Section 2, we described the assumptions and notations
used through out this paper. In Section 3, we establish the mathematical model and boundary condition to find
the minimum total relevant cost and the optimal order quantity. In Section 6, we use numerical examples to
illustrate the results we proposed and we make a sensitivity analysis to study the effects of changes in the
system parameters on the inventory model.
2. An inventory model for variable demand, constant holding cost and without shortages
DOI: 10.9790/5728-11144955 www.iosrjournals.org 50 |Page
II. Notation and Assumptions
The mathematical model is based on the following notations and assumptions:
Notations:-
Co: the ordering cost per order;
CP: the purchase cost per unit;
θ : the deterioration rate; 0<θ<1.
CH (t): the inventory holding cost per unit per time unit;
CB: the backordered cost per unit short per time unit;
CL: the cost of lost sales per unit;
T: the time at which the inventory level reaches zero .
I max: the maximum inventory level during [0, T];
Q o: I max the order quantity during a cycle of length T;
IP(t): the positive inventory level at time t;
IN(t): the negative inventory level at time t;
TC( T ): the total cost per time unit.
Assumptions:-
1. The demand rate is time dependent that is if „a‟ is fix fraction of demand and „b‟ is that fraction of demand
which is vary with time then demand function is D (t) = (a + bt )√I, where a > 0, b > 0.
2. Holding cost is h(t) = h is constant
3. Shortages are not allowed.
4. The lead time is zero.
5. The replenishment rate is infinite.
6. The planning horizon is finite.
7. The deterioration rate is θ is constant.
8. Inflation rate r
III. Mathematical Formulation of the Model
The rate of change of inventory during positive stock period [0, T] is governed by the differential equations.
PP
P
IbtaI
dt
dI )( , 0 t T ....(1)
With boundary conditions IP(t)= IN(t) = 0 at t =T and IP(t) = Imax at t = 0.
IV. Analytical Solution and Graphical Representation of Inventory System
Inventory level I(t)
Fig 1
The Inventory level have not shortages,in this period [0, T], the inventory depletes due to the
deterioration and demand. So the inventory level at the time during [0, t1] forms the differential equation:
3. An inventory model for variable demand, constant holding cost and without shortages
DOI: 10.9790/5728-11144955 www.iosrjournals.org 51 |Page
PP
P
IbtaI
dt
dI )( , 0 t T … (2)
Where θ is the constant detritions rate with with boundary conditions IP(t1)=0 and IP(0)=Q
With boundary conditions IP(t)= 0 at t = T.
The system gives the solution:
1
1
2 2
14
2
1
1
2 2 2 2
2
t t
P
t t
a bt b a bt b a bt b
a bt b
eI
e
…(3)
Or
2 2
1 12 3
23 2 3 2 2
1 1 1 1
2
2
1 1 4
1 4
( ) ( ) 2 ( ) 2 ( )( )
( ) 2 ( ) 6 ( ) 4 4
4
( ) 8 ( ) 8
P
b
a bt a bt a bt b a bt t t
I a bt b t t a bt b a bt b b
b
a bt b a bt b
…(4)
Now of we consider a = 4, b = 5, θ = 0.3 in the respective proper units. Then we get inventory label as given
in graph and table computed by silab.
Fig 2. Representation of inventory
Table 1. Representation of inventory
T IP
0 12893.86
0.54 10585.12
1.08 8504.85
1.62 6653.05
2.16 5029.72
2.7 3634.86
3.24 2468.47
3.78 1530.54
4.32 821.09
4.86 340.11
5.4 87.59
V. Cost Associated in Inventory
Putting IP(0)=Q in (5), then
4. An inventory model for variable demand, constant holding cost and without shortages
DOI: 10.9790/5728-11144955 www.iosrjournals.org 52 |Page
2 2 3 2
2 3
23 2 2
2
2
4
1 4
2 ( ) 2 ( ) 2
6 ( ) 4 4
4
( ) 8 ( ) 8
b
a a a bT b a bT b
Q T a bT b a bT b b
b
a bT b a bT b
…(5)
Purchase cost per cycle :-
PC = (Purchase cost per unit) * (Order quantity in one cycle)
=CP*Q
4
2
1
2
1
2
1
22
1
3
1
2
1
3
131
22
2
4
8)(8)(
44)(6
2)(
4
2)(2
1
b
bbtabbta
bbbtabbtat
bbtaat
b
bbtaaa
CPC P
… (6)
Sales revenue=
selling cost
T
22
2T a bT
=
2
)2( bTapT … (7)
Cost of placing order =
T
Co
... (8)
Inventory holding cost per cycle:-
ICH (t) = h(t)IP
t1
0
(t)dt
=
1
0
P
t
rth t dte I
=
1
0
rt
P
t
h e t dtI
1 1 1
1 1
1 1 11
2
2
1 13 2 3 2 3
2
1 13 2 2
2
1 1 1
1 1 2
3 2
1
1 2 2 2
( ) 2( )
4
2 ( ) 2 ( )
( ) ( )
( ) 2
rt rt rt
rt rt
rt rt rtrt
H
a a b e e be
a bt a bt
r r r r r r
b e be a b
a b t b a bt
r r r r
t e bt e at ee
a bt a bt
h r r r r
IC a bt b
T
1 1
1
1
1 1 1
2 3 2 3
2 2 2 1
1 1 2 2
2
2
1 1 4
2 ( )2 2
1
( ) 6 ( ) 4 4
4 1
( ) 8 ( ) 8
rt rt
rt
rt
bt e a bt atbe b
r r r r r
te
a bt b a bt b b
r r r
b e
a bt b a bt b
r r
… (9)
5. An inventory model for variable demand, constant holding cost and without shortages
DOI: 10.9790/5728-11144955 www.iosrjournals.org 53 |Page
The total annual profit is given by
TC (T, p) =Sales revenue-Cost of placing order –Cost of purchasing –
holding Cost
=
4
2
2
2223
23
3
22
2
4
8)(8)(
44)(6
2)(
4
2)(2
1
b
bbTabbTa
bbbTabbTaT
bbTaaT
b
bbTaaa
T
CP +
2
)2( bTapT
rr
eb
bbTabbTa
rr
T
r
e
bbbTabbTa
r
aT
r
b
r
bTa
r
be
r
bTe
r
aTe
r
ebT
r
e
bTa
r
Te
bTa
bbTa
r
b
r
a
r
be
r
e
bTa
b
bbTa
r
be
r
e
bTa
r
e
bTa
r
b
r
a
r
a
T
h
rT
rT
rTrT
rTrTrTrT
rTrT
rTrTrT
14
8)(8)(
1
44)(6)(
2)(22
)()(
2)(
)(
4
2)(2
2
)(2)(
221
4
2
2
22
222
3232
2
2
23
223
2
32
2
32
2
3
oC
T
... (10)
We have e-rT
=1-rT+r2
T2
-……………….and r is very small which is inflation, so we use only e-rT
=1-rT.
For profit maximization we calculate
,CdT T p
dT
2
)2()2(2
,
bTabpbTapT
dT
pTdTC
+ 2
T
Co
+
)
4
2)(2( 2
2
2
2
b
bbTaa
a
T
CP
2222
)]2([)2( aTbbbTaaab
T
CP
]
2223
2223
8)(2]6)(2[
44)(6
bbTaTbbTabT
bbbTabbTa
6. An inventory model for variable demand, constant holding cost and without shortages
DOI: 10.9790/5728-11144955 www.iosrjournals.org 54 |Page
4
2
2
32232
2
2
23
223
2
232
2
32
2
3
2
4
8)(8)(
2)(1
2
1
2
111
)(
1
)(
2)(
11
)(
4
2)(2
1
2
1
)(2
1
)(
221
b
bbTabbTaT
r
aT
r
b
r
bTa
r
T
r
b
r
T
r
bT
T
r
aTT
r
bT
r
T
r
bTaT
r
TbTa
bbTa
r
b
r
a
r
T
r
bT
r
bTa
b
bbTa
r
T
r
b
r
T
r
bTaT
r
bTa
r
b
r
a
r
a
T
h
2
3 2 2
2 2
3
2 2
3
1 1 1 2 1
2 ( ) ( ) 2 ( ) 2
4 1
2 ( ) 2 ( ) 2
1 1
1
( ) (
T
b a bT T a bT b a bT b
r r r r r
b b
a bT b b T a bT b
r r
T a b
a bT T b
r r r r r
a bT T T a bT
r
b
h
T
2
2
2 3 2 2 3
2
3 2
2
1 1 1
)
1 1 ( ) 2
2 2
1 1 1
( ) ( )
( ) 1
[ ( ) 2 ] 2
T
bT T aT T
r r r r
T T a bT b aT
bT b
r r r r r r r
T
a bT T bT T T a bT b
r r r r
a bT
a bT b bT T
r r
bT aT
2 2
2
2 2
4
1 1 2
2
4
( ) 8 ( ) 8 2 ( ) 8
T bT b a
a T b
r r r r r r
b
a bT b a bT b T b a bT b
… (11)
Putting,
,
0.CdT T p
dT
for the condition
2
2
,Cd T T p
dT
<0 we can find the optimal replenishment cycle.
VI. Numerical Example And Sensitivity Analysis
We consider example for computation no shortages that is t1=T and t2=0 , Co = 25, a = 4, b = 5, CP=10, p = 100,
r = 0.5, θ = 0.3 in the respective proper units. Then we get T=5.9. Then Q = 14356.34 and total profit TC =
823547.40
7. An inventory model for variable demand, constant holding cost and without shortages
DOI: 10.9790/5728-11144955 www.iosrjournals.org 55 |Page
Table 2
Constant
Parameter
Changing
Parameter Variation T Q TC
a = 4 0.1 0.54 12423.62 378457.66
b =5 0.1 1.08 12465.4 243975.86
p=100 r 0.2 2.75 12650.68 176002.58
0.3 3.78 12801.04 287549.40
θ = 0.3 0.4 4.32 12889.03 366557.07
a = 4 1.62 12516.51 10717.87
b =5 All variables 2.75 12650.68 123957.99
p=100 are fixed 3.27 12723.54 187167.44
r = 0.5 3.78 12801.04 261297.57
θ = 0.3 4.32 12889.03 355846.46
4.86 12912.77 383395.77
5.4 13080.76 605362.30
References
[1]. Balkhi,Z. T., and Benkherouf, L. (2004), “On an inventory model for deteriorating items with stock dependent and time-varying
demand rates [J],” Computers & Operations Research, Vol. 31, pp. 223–240.
[2]. Balkhi, Z.T., Benkherouf, L., (1996) ,A production lot-size inventory model for deteriorating items and arbitrary production and
demand rate. European Journal of Operational Research, Vol. 92, pp. 302-309.
[3]. Buzacott, J.A., (1975),
[4]. Economic order quantities with inflation. Operational Research Quarterly,26: 553-558.
[5]. Chang, H. and Dye, C. (1999), An EOQ model for deteriorating items with time varying demand and partial backlogging.Journal
of the Operational Research Society, 50,1176–1182..
[6]. Chung K.J. and Lin, C.N. (2001), “Optimal inventory replenishment models for deteriorating items taking account of time
discounting ” Computers & Operations Research,Vol. 28, No. 1, pp. 67–83.
[7]. Covert, R.B., and Philip, G.S.,(1973), “An EOQ model with Weibull distribution deterioration“, AIIE Transactions, 5 (1973) 323-
326.
[8]. Dave, U. and Patel, L. (1981). (T, si)-policy inventory model for deteriorating items with time proportional demand. Journal of
Operational Research Society, 32,137–142..
[9]. Dutta, T.K., Pal, A.K., (1988), Order level inventory system with power demand pattern for items with variable rate of
deterioration. Indian Journal of Pure and Applied Mathematics, Vol. 19, No. 11, pp. 1043-1053.
[10]. Dye, C. ,Chang,H, and Teng, J. (2006), “A deteriorating inventory model with time-varying demand and shortage- dependent
partial backlogging ” European Journal of Operational Research, Vol. 172, pp. 417–429.
[11]. Ghare, P.M., and Schrader, G.H.,(1963) , “A model for exponentially decaying inventory system“, International Journal of
Production Research, 21 , 449-460.
[12]. Gupta R. and Vrat, P. (1986) “Inventory model with multi-items under constraint systems for stock dependent consumption rate ”
Operations Research, Vol. 24, pp. 41–42.
[13]. Hou, K.L., (2006): An inventory model for deteriorating items with stock dependent consumption rate and shortages under
inflation and time discounting, European Journal of Operational Research, 168 (2), 463–474.
[14]. Mukhopadhyay, S. R. ,Mukherjee, N. and. Chaudhuri, K. S.(2004) “Joint pricing and ordering policy for a deteriorating
inventory,” Computers & Industrial Engineering, Vol.47, pp. 339–349.
[15]. Ouyang, L.Y.,Wu, . K.S. and Yang, C.T. (2006) “A study on an inventory model for non-instantaneous deteriorating items with
permissible delay in payments ” Computers & Industrial Engineering, Vol. 51, pp. 637–651.
[16]. Papachristos S. and K. Skouri,(2000), “An optimal replenishment policy for deteriorating items with time-varying demand and
partial-exponential type-backlogging [J],” Operations Researh Letters, Vol. 27, pp. 175–184.
[17]. Teng, J., Ouyang, L., and Chen, L. (2007) A comparison between two pricing and lot-sizing models with partial backlogging and
deteriorated items. International Journal of Production Economics, 105:190–203.
[18]. Teng, J.T. ,Chang, H.-J. , C.Y. Dye, and C.H. Hung, (2002), “An optimal replenishment policy for deteriorating items with time-
varying demand and partial backlogging ” Operations Research Letters, Vol. 30, pp. 387–393.
[19]. Teng, J.T., Chang, C.T., (2005) Economic production quantity models for deteriorating items with price and stock-dependent
demand. Computers and Operational Research Vol. 32, pp. 297-308.
[20]. Wu, K.S. Ouyang, L.Y. and Yang, C.T.(2006) “An optimal replenishment policy for non-instantaneous deteriorating items with
stock-dependent demand and partial backlogging [J],” International Journal of Production Economics, Vol. 101, pp. 369–384.
[21]. Yang, H.L., 2005. a comparison among various partial backlogging inventory lot-size models for deteriorating items on the basis of
maximum profit. International Journal of Production Economics, Vol. 96, pp. 119-128.