An Inventory Model for Deteriorating Items with Stock Dependent Demand and Partial Backlogging

Abstract

In this paper, we formulate a deteriorating inventory model with stock-dependent demand Moreover, it is assumed that the shortages are allowed and partially backlogged, depending on the length of the waiting time for the next replenishment. The objective is to find the optimal replenishment to maximizing the total profit per unit time. We then provide a simple algorithm to find the optimal replenishment schedule for the proposed model. Finally, we use some numerical examples to illustrate the model.

Introduction

I. INTRODUCTION

Most researchers have considered demand in their models as fixed or varies as time whether it is linear or quadratic or other pattern. But one more parameter effect the demand and that is stock of a particular inventory. We see that demand may be increase or decrease due to stock level.  The big stock level of a certain product some time increases the rate of demand and low stock level of such kind of products reduce the demand because the customers made a perception that the product is in last stage and will not be fresh. The vegetables, eggs, sweets etc. are the examples of that kind of products.

The fundamental result in the development of economic order quantity model with deterioration is that of Ghare and Schrader [1963] who considered an exponentially decaying inventory for a constant demand. However, as evident by chemical and basic sciences, the rate of deterioration especially with regard to perishable food items is seldom constant. Goyal and Giri [2001] presented several tendencies of the modeling of deteriorating inventory. Zauberman et al. [1989] presented a method for color retention of Litchi fruits with SO2 fumigation. In order to reduce the deterioration rate and to extend the expiration date of the product, preservation technologies like procedural changes and specialized equipment acquisition have been mathematically modeled by many researchers. In recent years, deteriorating inventory problems have been widely studied by many researchers. As presented by Wee [1995], deterioration was defined as decay, damage, spoilage, evaporation, obsolescence, pilferage, loss of utility or loss of marginal value of commodities that result in decreasing usefulness. Yang et al. [2015] developed the trade-off between preservation technology investment and the optimal dynamic trade credit for a deteriorating inventory model. Hsu et al. [2010] designed a deteriorating inventory model considering constants deterioration and demand rates, where in preservation technology also included. Optimization of the product portfolio has been recognized as a critical problem in industry, management, economy and so on. It aims at the selection of an optimal mix of the products to offer in the target market. Vipin Kumar [2020] developed an inventory model for deteriorating items with multivariate demand. Sanjay Sharma et al. developed a production inventory model for deteriorating Items with effect of price discount under the stock dependent demand, So while solving some problems a multi objective integer non-linear constraint model was developed by Ahmadi and Nikabadi [2019]. Having taken some realistic problem many researcher as Nadjafikhah [2017]; Ezzati et al. [2017]; Kazemi and Asl [2015] presented some special model. Wu et al. [2006] derived an optimal replenishment policy for items with non-instantaneous deterioration, stock-dependent demand and partial backlogging. Zhang et al. [1995] proposed a pricing policies for deteriorating items with preservation technology investment without shortage and stock. Pal et al. [2014] derived a deteriorating inventory model with stock and price-sensitive demand, where they assumed inflation and delay in payment. Shah and Shah [2014] attempt the same problem. However, they could not prove the existence of the optimal solution analytically. Moreover, they considered deterioration to start from the very beginning of replenishment time. Mishra [2014] developed an inventory model with controllable deterioration rate under time-dependent demand and time-varying holding cost. Liu et al. [2015] provided joint dynamic pricing and investment strategy for foods perishing at a constant rate with price and quality dependent demand. Zhang et al. [20] studied the integrated supply chain model for deteriorating items, in this model both manufacturer and retailer cooperatively invest in preservation technology in order to reduce their deterioration cost under different realistic scenarios.

Thereafter, Lu et al. [2016] presented an inventory model, in which they suggested the joint dynamic pricing and replenishment policy for a deteriorating item under limited capacity. Khedlekar et al. [2016] established an inventory model with declining demand under preservation technology investment. The chapter is organised after the introduction part as, we describe the fundamental notations and assumptions for the proposed model in Section 2. In Section 3, a mathematical model is established and solution procedure is discussed for maximizing the total profit, based on which an algorithm for finding the optimal policy is suggested. To illustrate the proposed model, numerical examples are provided in Section 4. In Section 5, sensitivity analysis of the optimal solution with respect to major parameters is carried out. Lastly, in Section 6, we draw the conclusions and give suggestions for future research.

II. ASSUMPTIONS AND NOTATIONS

III. MATHEMATICAL FORMULATION AND SOLUTION

IV. SOLUTION PROCEDURE

VI. SENSITIVITY ANALYSIS

In this part, the sensitivity analysis has been discussed by taking the different values of the used parameter as -50%, -25%,0, +25%,50% with respect to the optimal value of the total profit and keeping remaining parameter unchanged.  

VII. OBSERVATIONS

From tables (3), the following facts are apparent

1. With increment of a,  b and δ , the total profit Ps,v  shows increasing Behaviour
2. If α,  β,  h,  A,  and T  are increases then the total profit function Ps,v  decreases.  

Conclusion

In this chapter, the model is profit maximize policy for deteriorating products by taking the multivariate demand which is price, stock and time dependent. The rate of deterioration is two parameter Weibull function. In this model shortages is also considered and partial backlogged with variable rate. The model is solved analytically to check the optimality. Numerical examples and graphs are also demonstrated to validate the policy along with sensitivity with respect to different parameters used in the model. Some other extension can be made by assuming more realistic assumptions like as non-zero lead time, stochastic demand rate etc.

References

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Copyright

Copyright © 2022 Vikas Kumar, Vipin Kumar, Anupama Sharma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.