A Note on an Order Level Inventory Model with Varying Two-Phased Demand and Time-Proportional Deterioration

The main purpose of this paper is to generalize the effect of two-phased demand and variable deterioration within the EOQ (Economic Order Quantity) framework. The rate of deterioration is a linear function of time. The two-phased demand function states the constant function for a certain period and the quadratic function of time for the rest part of the cycle time. No shortages as well as partial backlogging are allowed to occur. The mathematical expressions are derived for determining the optimal cycle time, order quantity and total cost function. An easy-to-use working procedure is provided to calculate the above quantities. A couple of numerical examples are cited to explain the theoretical results and sensitivity analysis of some selected examples is carried out.

Analysis of an Inventory System for Items with Stochastic Demand and Time Dependent Three-Parameter Weibull Deterioration Function

In recent times, mathematical models have been developed to describe various scenarios obtainable in the management of inventories. These models usually have as objective the minimizing of inventory costs. In this research work we propose a mathematical model of an inventory system with time-dependent three-parameter Weibull deterioration and a stochastic type demand in the form of a negative exponential distribution. Explicit expressions for the optimal values of the decision variables are obtained. Numerical examples are provided to illustrate the theoretical development.

An Inventory Model for Deteriorating Items with Generalised Exponential Decreasing Demand, Constant Holding Cost and Time-Varying Deterioration Rate

In this present paper, an inventory model with a generalised exponential decreasing demand is considered. A numerical example is used to illustrate the application of the model. Sensitivity analysis of the optimal solution with respect to various parameters is carried out to see the effect of parameter changes on the solution.

An Inventory Model for Items with Two Parameter Weibull Distribution Deterioration and Backlogging

In this paper an inventory model is developed with time dependent power pattern demand and shortages due to deterioration and demand. The deterioration is assumed to follow a two parameter Weibull distribution. Three different cases with complete, partial, no backlogging are considered. The optimal analytical solution of the model is derived. Suitable numerical example has been discussed to understand the problem. Further sensitivity analysis of the decision variables has been done to examine the effect of changes in the values of the parameters on the optimal inventory policy.

Analysis of an Inventory System for Items with Price-Dependent Demand and Time Dependent Three-Parameter Weibull Deterioration Function

In this research work we propose a mathematical model of an inventory system with time dependent three-parameter Weibull deterioration and price-dependent demand rate. The model incorporates shortages and deteriorating items are considered in which inventory is depleted not only by demand but also by decay, such as, direct spoilage as in fruits, vegetables and food products, or deterioration as in obsolete electronic components. Furthermore, the rate of deterioration is taken to be time-proportional, and a power law form of the price dependence of demand is considered. This price-dependence of the demand function is nonlinear, and is such that when price of a commodity increases, demand decreases and when price of a commodity decreases, demand increases. The objective of the model is to minimize the total inventory costs. From the numerical example presented to illustrate the solution procedure of the model, we obtain meaningful results. We then proceed to perform sensitivity analysis of our model. The sensitivity analysis illustrates the extent to which the optimal solution of the model is affected by slight changes or errors in its input parameter values.

Inventory models for deteriorating items with fixed lifetime, partial backordering and carbon emissions policies

In this study,a sustainable inventory model is devised to obtain the retailer’s optimal pricing and replenishment policies for degrading items having a certain lifetime incorporating partial back order as a shortage and dynamic demand under two different scenarios(a)carbon cap and trade policy(b)carbon tax policy.The primary objective of this study is to maximize the retailer’s annual total profit.The retailer’s profit function has been optimized with the help of convexity/concavity criteria employing classical optimization techniques.Based on a real case study,two different numerical examples and corresponding optimal solutions have been shown for both models with the help of Lingo 17 software.Moreover,the impact of the major inventory parameters and prominent managerial insights are presented for the robustness of the proposed model that can cooperate with industrial managers/decision-makers for the overall improvement of his/her industry to take effective and qualitative decisions.

An inventory model with finite replenishment,probabilistic deterioration and permissible delay in payments

This paper develops an inventory model for deteriorating items with finite replenishment rate under a progressive payment scheme within the cycle time.In this model,the deterioration function follows a probability distribution such as a(1)uniform distribution,(2)triangular distribution or(3)beta distribution.Here,the retailer is allowed a trade-credit offer by the supplier to buy more items.This model aids in minimizing the total inventory cost of the retailer by finding the optimal cycle length,the optimal time length of replenishment and the optimal order quantity.Some theorems have been framed to characterize the optimal solutions.The necessary and sufficient conditions of the existence and uniqueness of the optimal solutions are also provided.The optimal solution of the model is illustrated with the help of numerical examples,and numerical comparisons between the three models are also given.Finally,sensitivity analysis and graphical representations are given to demonstrate the model.

Cold supply chain inventory models for agricultural products with multi-stage stochastic deterioration rates

In India,majority of the families depend upon agricultural business.Also,40%of agricultural food and vegetables are rotted due to improper planning.It is predicted in 2023 that US$293.27 billions will be wastage of food.That’s why,the cost of these items are high in the retail market and farmers get least price for their produce.It is essential to avoid the loss by adopting cold supply chain and its impact will help to increase the GDP of country.Here,we proposed the supply chain model which includes one warehouse and multi-retailers.Cold storage facility is available at warehouse,transport facility and retailers.To support the proposed model,we have collected the numerical data of coriander(highly perishable)product from the market yard,Pune,India.The proposed model solved by using real world numerical case study.The life of this product is very less hence it needs to develop a cold supply chain for this item.Also,we solved multiple inventory differential equations by using boundary value problem.To solve the proposed models for single retailer and multiple retailers,we have developed a solution methodology based on evolutionary algorithm.We have observed that,optimum profit is highly sensitive to the warehouse demand and purchase cost.Hence it is essential to reduce the deteriorated items at storage point.The warehouse,retailer and customer demand significantly affect on the deteriorated items.In single and multiple retailer,for low warehouse and retailer demand,we get maximum profit.Thus,less number of deteriorated items of the supply chain will helps to the society,as well as country to increase the profit of the supply chain.

Fuzzy Inventory Model for Deteriorating Items with Time Dependent Demand and Partial Backlogging

In this paper we developed a fuzzy inventory model for deteriorating items with time dependent demand rate. Shortages are allowed and completely backlogged. The backlogging rate of unsatisfied demand is assumed to be a decreasing exponential function of waiting time. The demand rate, deterioration rate and backlogging rate are assumed as a triangular fuzzy numbers. The purpose of our study is to defuzzify the total profit function by signed distance method and centroid method. Further a numerical example is also given to demonstrate the developed crisp and fuzzy models. A sensitivity analysis is also given to show the effect of change of the parameters.

An Inventory Model for Deteriorating Items with Exponential Declining Demand and Time-Varying Holding Cost

In the present paper, a total optimal cost of an inventory model with exponential declining demand and constant deterioration is considered. The time-varying holding cost is a linear function of time. Shortages are not allowed. The items (like food grains, fashion apparels and electronic equipments) have fixed shelf-life which decreases with time during the end of the season. A numerical example is presented to demonstrate the model and the sensitivity analysis of various parameters is carried out.

An Inventory Model for Deteriorating Items under Conditionally Permissible Delay in Payments Depending on the Order Quantity

The purpose of this inventory model is to investigate the retailer’s optimal replenishment policy under permissible delay in payments. In this paper, we assume that the supplier would offer the retailer partially permissible delay in payments when the order quantity is smaller than a predetermined quantity (W). The most inventory systems are usually formed without considering the effect of deterioration of items which deteriorate continuously like fresh fruits, vegetables etc. Here we consider the loss due to deterioration. In real world situation, the demand of some items varies with change of seasons and occasions. So it is more significant if the loss of deterioration is time dependent. Considering all these facts, this inventory model has been developed to make more realistic and flexible marketing policy to the retailer, also establish the result by ANOVA analysis by treating different model parameters as factors.

An Inventory Model for Perishable Items with Time Varying Stock Dependent Demand and Trade Credit under Inflation

In the classical inventory models, it is assumed that the retailer pays to the supplier as soon as he received the items and in such cases the supplier offers a cash discount or credit period (permis-sible delay) to the retailer. In this paper we presented an inventory model for perishable items with time varying stock dependent demand under inflation. It is assumed that the supplier offers a credit period to the retailer and the length of credit period is dependent on the order quantity. The purpose of our study is to minimize the present value of retailer’s total cost. Numerical examples are also given to demonstrate the presented mode.

Pricing and inventory decisions for non-instantaneous deteriorating items with price and promotional effort stochastic demand

This study investigates an inventory model for a non-instantaneous deteriorating item with partial backlogging wherein the demand is stochastic in nature and depends on price and promotional effort whereas the deterioration rate is time proportional.Under these settings,a mathematical model is developed with the objective to maximise the expected profit per unit time by determining the optimal price and the length of replenishment cycle.Some useful theoretical results are established to deduce the optimal replenishment schedule.An effective algorithmic procedure is developed to find the optimal solutions to the proposed model.The applicability of the proposed model is illustrated by numerical example.Sensitivity analysis of the optimal solution with respect to key parameters has been carried out and the implications are discussed.The study indicates that though promotional effort stimulates the market demand,it is beneficial in an economic sense when it applies in a conservative manner.

AN OPTIMAL DYNAMIC ADVERTISING MODEL WITH INVERSE INVENTORY SENSITIVE DEMAND EFFECT FOR DETERIORATING ITEMS

This paper proposes a dynamic advertising model for deteriorating items, and the demand is influenced by goodwill and inventory level. The goodwill affected by advertising effort has a positive effect on demand while the inventory level has a negative effect on demand, which is named as inverse inventory sensitive demand effect. We assume that the deteriorating rate could be influenced by the inventory level and we determine the deteriorating rate formulation based on this assumption. The optimal advertising effort and inventory level are obtained by solving the optimization problem based on Pontryagin＇s maximum principle. Furthermore, numerical studies are provided and some managerial insights are presented.

Optimal Decision Rule of the inventory Problem for Deteriorating items with Shortages and Time－varying Demand

In this paper, we consider the general inventory problem for deteriorating items having time-varying demand and shortages. An optimal order policy is obtained for determining the number of reorders and the time of the reorder points and shortage points in order to minimize the total cost of system. Several fundamental theoretical results are established for this model. The corresponding cases for no-deterioration and no-shortage is also discussed. To illustrate the solution procedure,numerical examples are shown for the special case of linear increasing demand.

Inventory Policy for a Deteriorating Item with Time-Varying Demand Under Trade Credit and Inflation

It’s often the case that the supplier will provide the retailer with a permissible delay period in payments, during which the supplier charges the retailer no interest and the retailer accumulates interest earned from investment return. As a type of price reduction and an alternative to price discount,trade credit helps the supplier encourage the retailer’s ordering. This paper develops an inventory replenishment model for a deteriorating item with time-varying demand and shortages, taking account of trade credit and time value of money under inflation over a finite time horizon. This model is an extension and development of the existing studies related to the inventory system considering trade credit and time value of money and offers a more general model with more flexibility and resilience to handle the situation where demand of the end market is non-decreasing with regard to time.

Two-warehouse inventory model for deteriorating items with demand influenced by innovation criterion in growing technology market

One of the major concerns for the technology market is the demand volatility andits impact on inventory policies. Demand volatility in the technology sector mayarise due to many factors namely customer choices, competition, growingmarket size, and so on. Often companies use rented warehouses to absorb anyfluctuations in demand. Unfortunately, warehouse and inventory researchesignore the phenomenon of growing market size to formulate policy decisions. Inthis paper, we proposed a two-warehouse inventory model with deterioration fortechnology products with linearly increasing market size where demand followsinnovation diffusion criterion. The model is based on the assumption that theholding costs in the rented warehouse are more than the own warehouse. Asimple solution procedure also discussed to solve nonlinear cost function.Numerical example and sensitivity analysis are also used to describe the utilityof the model.

Sustainable Integrated Inventory Model for Substitutable Deteriorating Items Considering Both Transport and Industry Carbon Emissions

Sustainable integrated inventory management policy focuses on reducing the environmental impact of the manufacturing and transport sector.This study develops an integrated single-manufacturer single-buyer inventory model for substitutable deteriorating items considering transport and industry carbon emissions.This article categorizes carbon emissions into two parts:direct and indirect emissions.We also assume two mutually substitutable items,and substitution is considered on the buyer's side.If one of the items is out of stock,the demand for that item can be partially satisfied by the stock of the other item;else,demand will be lost.The purpose of this study is to minimize the integrated cost of the supply chain with total carbon emissions produced by industry and transport sectors and to determine the optimal number of deliveries,order level and carbon emission quantity.A solution procedure is developed to find the optimal values of the policy.Numerical experiments and sensitivity analyses are presented to demonstrate the proposed model.

Optimal pricing, replenishment scheduling, and preservation technology investment policy for multi-item deteriorating inventory model under shortages

In a wholesale shop,the retailer is required to keep an inventory of various ranges of a product due to uncertainty in consumer preferences and behavior.Different Economic Order Quantity(EOQ)models for each variety of a product are needed for optimum level of inventory.EOQ model decides how much stock a shopkeeper has to store,according to demand and deterioration of different items to optimize the inventory policy of multiple products.Items like fashion,food,vegetables,pharmaceuticals,etc.,deteriorate in the retailer’s warehouse during storage.An investment is required in preservation technology to significantly reduce the deterioration rate,reduce economic losses,and increase business competitiveness.Here,a multi-item deteriorating EOQ model with a fixed cycle duration has been developed incorporating the preservation technology.The demands for the items are a quadratic function of the selling price,and shortages are allowed.The main objective is to determine the optimal selling price,investment under preservation,the time length up to zero inventory,order quantity of the proposed inventory model such that the overall profit will be maximized.First,we prove that the optimal selling price and investment in preservation exist and are unique.Next,the total profit per unit time is a concave function of selling price and preservation technology investment.An algorithm is designed to obtain the optimum joint policy.A numerical example has been given to illustrate the model,and sensitivity analysis is conducted,which gives some managerial insights.

Decaying items inventory models with partial linked-to-order upstream trade credit and downstream full trade credit

In practice,any member of supply chain may offer full or partial trade credit contract to his downstream level.Full trade credit is the case that the latter is allowed to defer whole payment to the end of credit period.In partial trade credit,however,the downstream supply chain member must pay for a proportion of the purchased goods at first,and can delay paying for the rest until the end of credit period.This paper considers a two-level trade credit,where the supplier offers order-quantity-dependent partial trade credit to a retailer,who suggests full trade credit to his customers.An economic order quantity(EOQ)inventory model of a deteriorating item with expiration dates is formulated here.Theoretical results are developed to obtain the optimal solutions to the problem.Numerical examples and sensitivity analysis are performed to justify the proposed models and theoretical results and managerial insights are provided.