Time Dependent Quadratic Demand Research Articles

A production inventory model for ameliorating and deteriorating items with two levels of production under time dependent quadratic demand and preservation technology investment

A production inventory model for ameliorating and deteriorating items is established with time varying quadratic demand under the influence of preservation technology. Two different production rates are considered such that the initial production rate switches to another rate of production after some interval of time. Preservation technology is used to lower the deterioration effect. The main objective is to obtain the optimal production time, cycle length and preservation technology cost to minimize the total cost per unit time of the system. Numerical examples under different situations are presented to illustrate the model. The effect of the parameters on optimal solution is investigated through sensitivity analysis while various observation obtained are discussed accordingly.

Analysis of Inventory Model with Time Dependent Quadratic Demand Function Including Time Variable Deterioration Rate without Shortage

This paper deals with an inventory model for deteriorating items along with time dependent demand which is quadratic function of time. In this model, the deterioration rate follows deterministic deterioration which is quadratic function of time. Here shortages are not allowed. The main purpose of this paper is to investigate minimum total cost per unit time of the inventory model. The result are validated with the help of numerical example. The sensitivity analysis of the optimal solution with respect to various parameters is carried out. Finally the behavior of the relation between parameters and total inventory cost have been shown using figure.

An inventory model for non-instantaneous deteriorating items with time dependent quadratic demand, two storage facilities and shortages under trade credit policy

In this paper, an EOQ model for non-instantaneous deteriorating items with two-phase demand rates, two storage facilities (own warehouse and rented warehouse) and shortages under trade credit policy has been developed. The demand rate before deterioration sets in is assumed to be time dependent quadratic and that is considered as a constant after deterioration sets in. Shortages are allowed and completely backlogged. The purpose of this research is to determine length of time at which the inventory level reaches zero in OW and cycle length simultaneously such that total variable cost has a minimum value. The corresponding economic order quantity will also be determined. The necessary and sufficient conditions for the existence and uniqueness of solutions are provided. Numerical examples are provided to illustrate the theoretical results of model. Sensitivity analyses are carried out to see the effects of changes in some system parameters on optimal solutions. Based on the results, several ways are suggested for the retailer to effectively reduce the total variable cost of the inventory system.

An inventory model for non-instantaneous deteriorating items with time dependent quadratic demand, two storage facilities and shortages under trade credit policy

In this paper, an EOQ model for non-instantaneous deteriorating items with two-phase demand rates, two storage facilities (own warehouse and rented warehouse) and shortages under trade credit policy has been developed. The demand rate before deterioration sets in is assumed to be time dependent quadratic and that is considered as a constant after deterioration sets in. Shortages are allowed and completely backlogged. The purpose of this research is to determine length of time at which the inventory level reaches zero in OW and cycle length simultaneously such that total variable cost has a minimum value. The corresponding economic order quantity will also be determined. The necessary and sufficient conditions for the existence and uniqueness of solutions are provided. Numerical examples are provided to illustrate the theoretical results of model. Sensitivity analyses are carried out to see the effects of changes in some system parameters on optimal solutions. Based on the results, several ways are suggested for the retailer to effectively reduce the total variable cost of the inventory system.

An EOQ model for non-instantaneous deteriorating items with time-dependent quadratic demand and two-level pricing strategies under trade credit policy.

An EOQ model for non-instantaneous deteriorating items with time-dependent quadratic demand and two-level pricing strategies under trade credit policy.

Purchasing Inventory Models for Deteriorating Items with Quadratic Demand

This paper deals with purchasing inventory replenishment policy for deteriorating items, considering the time-dependent quadratic demand and time-dependent backlogging. Two models were formulated and solved. First, it is for deteriorating items with quadratically time-dependent demand for deteriorating items. Second, quadratically time-dependent demand for deteriorating items and shortages. A mathematical model is developed to the fourth-order equation for each model. The optimal production lot size, which minimizes the total cost, is derived. Sensitivity analysis is carried out to demonstrate the effects of changing parameter values on the optimal solution of the system. Numerical examples are taken to illustrate the procedure of finding the optimal inventory cost, cycle time, and optimal lot size. The numerical experiment in this model was coded in Microsoft Visual Basic 6.0.

An EPQ model for delayed deteriorating items with quadratic demand and linear holding cost

In this paper, an EPQ model for delayed deteriorating items is presented, where the demand before deterioration sets in is assumed to be time dependent quadratic demand and the holding (carrying) cost is assumed to be linearly dependent on time. Three stages are considered as follows: (1) production build up period, (2) period before deterioration starts and, (3) period after deterioration sets in. There is no demand during production build up period and the demand before deterioration begins is assumed to be quadratic time dependent while that after deterioration sets in is assumed to be constant. It is also assumed that shortages are not allowed. The purpose of this paper is to investigate the optimal set of production rates that minimizes the total inventory cost per unit time, the best cycle length and the economic production quantity. A numerical example is given to illustrate the applicability of the model and sensitivity analysis carried out on the example to see the effect of changes on some system parameters.

An Ordering Policy for Deteriorating Items with Time-Dependent Quadratic Demand and Salvage Value under Permissible Delay in Payment

The article deals with an economic order quantity (EOQ) inventory model for deteriorating items in which the supplier provides the purchaser a permissible delay in payment. This is so when deterioration of units in the inventory is subject to constant deterioration rate, demand rate is quadratic function of time and salvage value is associated with the deteriorated units. Shortages in the system are not allowed to occur. A mathematical formulation is developed when the supplier offers a permissible delay period to the customers under two circumstances: 1) when delay period is less than the cycle of time; and 2) when delay period is greater than the cycle of time. The method is suitable for the items like state-of-the-art aircrafts, super computers, laptops, android mobiles, seasonal items and machines and their spare parts. A solution procedure algorithm is given for finding the optimal order quantity which minimizes the total cost of an inventory system. The article includes numerical examples to support the effectiveness of the developed model. Finally, sensitivity analysis on some parameters on optimal solution is provided.

Inventory policies for maximum fixed life-time product with quadratic demand under permissible delay in payments for a single supplier-two retailers

In this article, we study inventory policies with permissible delay in payment from a single supplier to two retailers. We consider the product with maximum fixed-life time deterioration as loss of utility is the real scenario of products like fruits, vegetables, juices, etc. Here, time dependent quadratic demand is discussed which is suitable for the products whose demand increases primarily and afterward it starts to decrease. First, we discuss the centralised inventory system in which supplier and retailers willingly take joint decision. Next we address the model with decentralised policy in which supplier and both retailers take individual decisions. The objective is to minimise the total cost in centralised and decentralised policy with respect to cycle times of supplier and two retailers. The model is supported with numerical examples. Sensitivity analysis is carried out to derive insights for decision-maker.

Study of Imperfect Manufacturing System with Preservation Technology Investment Under Inflationary Environment for Quadratic Demand: A Reverse Logistic Approach

This paper focuses on imperfect manufacturing system with remanufacturing, manufacturing and product returned cycle for a single item which deteriorates with respect to time. The product has maximum fixed lifetime. To decrease deterioration of the product, manufacturer spends capitals on preservation technology to preserve the item. In this paper, the effect of inflation is also considered. Here, time-dependent quadratic demand is debated which is suitable for the products whose demand increases initially and afterward it starts to decrease. The objective is to minimize the total cost of manufacturer with respect to cycle time and investment for preservation technology. The model is supported with numerical example. Sensitivity analysis is done to derive insights for decision makers. Graphical result, in three dimensions, is exhibited with supervisory decision.

An Inventory Model for Increasing Demand with Probabilistic Deterioration, Permissible Delay and Partial Backlogging

In this paper an inventory model for deteriorating items with time dependent quadratic demand and permissible delay in payments is developed. Shortages are allowed and are partially backlogged. An optimal policy that minimizes the total cost is developed. The objective of this study is to consider three different types of continuous probabilistic deterio-ration functions and to find the associated total cost. To illustrate the proposed model some numerical examples are given. Sensitivity analysis of the optimal solutions with respect to major parameters are carried out and comparison is made between the three models.

An economic lot sizing production model for deteriorating items under two level trade credit

This paper develops an economic lot-sizing production model for deteriorating items with time dependent quadratic demand and delay in payments under two levels of trade credit policy. The trade credit policy adopted here is a two-level trade credit policy in which the supplier offers the retailer a permissible delay period M, and the retailer in turn provides customers a permissible delay period N. Cases where M < N and are investigated. In addition, the demand rate is assumed to be time and deterioration rate dependent quadratic demand. This paper deals with the problem of determining the optimal replenishment cycle length, optimal total cost and economic order quantity for an integrated inventory system under both two-level trade credit and time dependent quadratic demand rate. Numerical examples are presented to illustrate the developed model.

An optimal replenishment policy for EOQ models with time-varying demand and shortages

While considering time dependent demand rate, most of the researchers developed their inventory models considering time dependent linear demand rate. Some models have been developed also by considering time dependent quadratic demand because by considering quadratic demand increasing, decreasing or constant all types of situations can be explained and hence it is more realistic than that of linear demand rate. But in the present paper, we have developed an EOQ model to describe an optimal replenishment policy with shortages and general time-varying demand instead of time dependent linear or time dependent quadratic demand. The successive replenishment cycle lengths are assumed to be reduced in arithmetic progression. The model is extended to the no shortage case also. The model is solved analytically to obtain the optimal solution of the problem. It is then illustrated with the help of numerical examples. The sensitivity analysis of the optimal solution towards changes in the values of different system parameters is also studied.

An EOQ model for a deteriorating item with time dependent quadratic demand and variable deterioration under permissible delay in payment

In a recent paper, Khanra, Ghosh and Chaudhuri’s (2011) presented an EOQ model for a deteriorating item with time dependent quadratic demand under permissible delay in payment. Deterioration considered in most of the EOQ models is constant, while in most of the practical cases the deterioration rate increases with time. This work is motivated by Khanra, Ghosh and Chaudhuri’s (2011) paper extending their model to allow for a variable rate of deterioration when delay in payment is permissible. The time varying demand rate is taken to be a quadratic function of time. For settling the account, the model is developed under two circumstances: case-1: The credit period is less than or equal to the cycle time and case-2: the credit period is greater than the cycle time. A numerical example is provided to illustrate the model. Sensitivity analysis has also been conducted to study the effect of the parameters.

Optimal pricing and ordering policy for an integrated inventory model with quadratic demand when trade credit linked to order quantity

PurposeThe purpose of this paper is to study integrated inventory system and pricing and ordering strategy for vendor‐buyer supply chain system. Here, the vendor offers a trade credit to the buyer when the buyer's order quantity exceeds a given pre‐specified quantity. Therefore, to incorporate the concept of vendor‐buyer integration and trade credit linked, the authors analyze the model to determine the optimal strategy for an integrated vendor‐buyer inventory system under the condition of credit linked to the order quantity when demand is quadratic.Design/methodology/approachA mathematical model for integrated inventory system is developed when demand rate is increasing function of the time and decreasing function of the retail price. By analyzing the total channel profit function, the authors developed some useful results to characterize the optimal solution and provide an iterative algorithm to find the retail price, buyer's order quantity and the number of shipments per production run from the vendor to the buyer.FindingsBy developing a solution algorithm, the optimal retail price, order quantity and number of shipments from the vendor to the buyer are provided. Numerical examples and sensitivity analyses are presented to validate the proposed model. Through extensive numerical analyses, it is observed that a longer credit term increases profits of the player for the entire supply chain. The vendor should establish the threshold for allowing trade credit comprehensively to ensure the greatest benefit for both players.Originality/valueMost of the research articles available in the literature considered the constant demand or linearly changing demand. In this paper, a mathematical model is developed considering time dependent quadratic demand. Very few researchers have investigated joint optimal policy in vendor‐buyer supply chain system, considering trade credit is linked to order quantity, and still there are not many findings on the benefit of integrated policy and trade credit.

An EOQ model for a deteriorating item with time dependent quadratic demand under permissible delay in payment

Abstract In this paper, an EOQ (Economic Order Quantity) model is developed for a deteriorating item having time dependent demand when delay in payment is permissible. The deterioration rate is assumed to be constant and the time varying demand rate is taken to be a quadratic function of time. Mathematical models are also derived under two different circumstances, i.e. Case I: The credit period is less than or equal to the cycle time for settling the account and Case II: The credit period is greater than the cycle time for settling the account. The results are illustrated with numerical examples. Justification for considering a time quadratic demand and permissible delay in payment are discussed.

A single cycle perishable inventory model with time dependent quadratic ramp-type demand and partial backlogging

In order to analyse the behaviour of inventory level of a product level after it's introduction to the market a single item perishable inventory model is developed in this paper, where it is assumed that the demand is time dependent accelerated growth-effect of accelerated growth-steady type. The deterioration of inventory starts after a certain time. Shortages are allowed and are partially backlogged. The profit function is derived and unique existence of the solution is established. A numerical example is presented and sensitivity of the model is examined. The rationale of time dependent quadratic ramp-type demand is discussed.

Replenishment policy for EOQ models with time-dependent quadratic demand and shortages

In this paper, we develop two Economic Order Quantity (EOQ) models for quadratic time-varying demand. It is assumed that successive replenishment cycle lengths are in arithmetic progression and γ is the rate of reduction of successive cycle lengths. The model with shortages is studied first. The corresponding results for the model without shortages are derived next. Decision variables are determined numerically for given input parameters. Optimum costs for both the models are also calculated. Computational results show that the model with shortages allowed is economically beneficial. Finally, sensitivity analysis is performed to study the effect of changes in input parameters on the optimal cost.

A note on an order-level inventory model for a deteriorating item with time-dependent quadratic demand

An order-level inventory problem is discussed with the demand rate being represented by a continuous, quadratic function of time. It is assumed that a constant fraction of the on-hand inventory deteriorates per unit of time. The solution of the model is discussed both for infinite and finite time-horizon. A numerical example is taken up to illustrate the solution procedure and sensitivity analysis is also carried out. The rationale for the time-dependent quadratic demand is discussed. Scope and purpose The purpose of the present paper is to give a new dimension to the inventory literature on time-varying demand patterns. Researchers have extensively discussed various types of inventory models with linear trend (positive or negative) in demand. The main limitation in linear time-varying demand rate is that it implies a uniform change in the demand rate per unit time. This rarely happens in the case of any commodity in the market. In recent years, some models have been developed with a demand rate that changes exponentially with time. Demands for spare parts of new aeroplanes, computer chips of advanced computer machines, etc. increase very rapidly while the demands for spares of the obsolete aeroplanes, computers etc. decrease very rapidly with time. Some modellers suggest that this type of rapid change in demand can be represented by an exponential function of time. The present authors feel that an exponential rate of change in demand is extraordinarily high and the demand fluctuation of any commodity in the real market cannot be so high. A realistic approach is to think of accelerated growth (or decline) in the demand rate in the situations cited above and it can be best represented by a quadratic function of time. Thus, this paper has the scope of direct application in the very practical situations noted above.