A Study on Ordering Policy with Time-Varying Demand and Holding Cost for Deteriorating Items with Shortages

Abstract

The present paper focuses on time varying demand and holding cost, taking the same into consideration a finite horizon inventory problem for a deteriorating item has been developed for study. Deterioration rate is considered as a linearly time-dependent. The demand rate varies with the time until the shortage occurs, but during the tenure of shortages it becomes static i.e. constant. Shortages are considered to be a partially backlogged in this study. In the study it is also considered that the backlogging rate is inversely proportionate to the length of the waiting time for next recovery. The main objective of the study is to minimize to the total cost of inventory with optimal order quantities of the products in the system. The graphs in the study show the convexity of the cost function. Finally, numerical results are discussed and presented in the section 5 of the study to analyze the degree of sensitivity of the optimal policies with respect to variations in the economic parameters of the model.

Introduction

I. INTRODUCTION

There are four forms of demand in the literature on inventory models namely, time-dependent, stock dependent, constant demand, time and stock dependent and probabilistic demand. The majority of inventory models are predicated on the premise that an item's demand remains constant over the planning horizon. In reality, however the product demand cannot be fixed over time. It is dependent on inventory levels, selling prices and in some cases, time.

Many scholars have been working on inventory models for decaying items in recent years. Deterioration of objects has become a typical occurrence in everyday life.

Many things, including fruits, vegetables, pharmaceuticals, volatile liquids, blood banks, high-tech products, and others, degrade with time due to evaporation, spoilage, obsolescence, and other factors. Inventory systems suffer from shortages as a result of deterioration, as well as a loss of goodwill or profit. As a result, degradation must be factored into inventory control models. Deteriorating products are seasonal things such as warm clothing, dairy products, green veggies, and so on. Shortages in the inventory system occur as a result of deterioration, affecting total inventory costs as well as total profit. As a result, another important component in the analysis of declining things inventory is the degrading rate, which indicates the nature of the items' deterioration.

The study of deterministic inventory models is having a finite planning horizon, time-varying demand, and holding cost are worth noting.

The rate of deterioration is considered to be linear in time. It is permissible to have a shortage that is partially backlogged. The main aim is to build an appropriate inventory models to non-instantaneous deteriorating items over a finite planning horizon is the goal of this research.

Shortages and partial backlogs are allowed under the paradigm. The backlog rate varies depending on how long it takes for the next replenishment to arrive. The main goal is to calculate the best total cost and order quantity at the same time. We have demonstrated some numerical examples are presented. The sensitivity analysis is used to investigate the consequences of changing parameters. The rest of our paper is laid out as follows: The literature on time-varying demand, holding cost, deterioration rate, and backorder is reviewed in Section II. The assumptions and notation used throughout the paper are described in Section III.

In Section IV, we established the main mathematical model. In Section V, focused on the solution to obtain total inventory cost and order quantity.

We have illustrated some numerical examples with sensitivity analysis and observations. Finally, we conclude in the section VI with the future work prospects.

II. LITERATURE REVIEW

The first economic order quantity (EOQ) model was initiated by Harris (1915) for constant and known demand. Later on many researchers considered the variable demand such as, constant demand, time-dependent, stock dependent, price dependent and probabilistic demand. Further, some researchers were worked on inventory with deteriorating items. Initially ‘A model for exponentially decaying inventory ‘was proposed by Ghare. Nahmias (1975) suggested a perishable inventory model with constant demand for optimum ordering. Later Abad P (2001), Chung K.J. and Huwang Y.F. (2009), Shah N.H. AND Shukla (2009) Inventory models with time dependent demand rate were considered by Sivazlian and Stanfel (1976), Chung and Ting (1993) for replenishment of deteriorating items.

Due to shortages, the backlogging happens for which several researchers considered partial backlogging while some assumed fully backlogged inventory models. In practice, it is observed that at the time of the shortages either consumer wait for the arrival of next order (entirely backlogged) or they leave the system (entirely lacked). However, it is sensible to consider that, some consumers can wait to fulfill their demand during the stock- out period (a case of full back order), whereas others don't wish to wait and fulfill their demand from another sources (a case of partial back order).

Backlog shortages that occur will depend on the length of waiting time for the replenishment and is therefore the main factor for determining whether the backlogging will be accepted or not. For commodities with short life cycles like fashionable and high-tech products, the rate of backorder is declining with the length of waiting time. The longer the waiting time is lower the backlogging rate, this results to a greater fraction of loss sales and hence a less profit as a consequence taking into the consideration of partial backlogging is necessary.   Customers who have experienced stock-out will be less likely to buy again from the suppliers, they might turn to another store to purchase the goods. The sales for the product might decline due to the introduction of more competitive product or the change in consumers’ preferences. The longer the waiting time, the lower the backlogging rate is. This leads to a larger fraction of lost sales and a less profit. As a result, taking the factor of partial backlogging into account is necessary.

Several inventory models under the conditions of perishability and partial backordering for constant demand for optimal ordering policy were studied by Abad, P. (2001), Shah and Shukla (2009). Chung and Huang (2009), Hu and Liu (2010) developed optimal replenishment policy for deteriorating items with constant demand under condition of permissible delay in payment. An inventory model with time proportional demand for deteriorating items were developed by Dave & Patel (1981), Chung and Ting (1993), Chang and Dye (1999), Ouyang et al. (2005) with exponential declining demand and partial backlogging. Teng et al. (2007) studied the comparison between two pricing and lot-sizing models with partial backlogging and deteriorating items. Alamri and Balkhi (2007) proposed a model for the effects of learning and forgetting on the optimal production lot size for deteriorating items with time varying demand.

Some inventory models with ramp type demand rate, partial backlogging and weibull distributed deteriorating items are proposed by Jalan et al. (1996), Skouri et al. (2009), Mandal (2010), Hung (2011), Sana (2010). Kumar et al. (2012) proposed EOQ model with time dependent deterioration rate under fuzzy environment for ramp type demand and partial backlogging. In real-life inventory model, the holding cost is also time dependant. Mishra (2014), Dutta and Kumar (2015), karmakar (2016), Chandra (2017) developed an inventory model for deteriorating items with time dependent demand and the time varying holding cost. Pervin et al. (2018) developed an EOQ model for time-varying holding cost including stochastic deterioration.

Hou (2006) developed an inventory model for deteriorating items with stock-dependent consumption rate and shortages under inflation and time discounting Roy et al. (2011a, 2011b) proposed inventory models for imperfect items in a stock-out situation with partial backlogging. Choudhary et al. (2015) developed inventory model for deteriorating items with stock-dependent demand, time-varying holding cost and shortages. Pervin et al. 2017) developed two-echelon inventory model with stock-dependent demand and variable holding cost for deteriorating items.

Abad (1996) developed an EOQ model under conditions of perishability and partial backordering with price dependent demand. Dye (2007) developed an EOQ model with a varying rate of deterioration and exponential partial backlogging with price-sensitive demand. Roy (2008) developed an inventory model for deteriorating items with price dependent demand and time varying holding cost. Sana (2010) proposed an EOQ model with time varying deterioration and partial backlogging. Shah et al. (2012) proposed an inventory model for optimisation and marketing policy for non-instantaneous deteriorating items with generalised type deterioration and holding cost rates. Rastogi et al. (2017) developed an EOQ model with variable holding cost and partial backlogging under credit limit policy and cash discount. Sharma et al. (2018) developed an inventory model for deteriorating items with expiry date and time varying holding.

Some products like fashionable and seasonal products are characterized by unpredictable demand.

The demand of such items is low at the beginning of the season and increases as the season progresses, i.e., it changes with time, especially seasonal products like seasonable fruits, garments, shoes, etc. Most of these products have a lifespan of very short window in which to make the right calls on demand and procurement and achieve profits. There are three fundamental questions to answer such situations. How do sellers plan at firms selling such products for a season? How do they decide how much to order, when to schedule shipments and how much to mark down prices to reduce season-ending inventory as much as possible?

In this paper, we develop a deteriorating inventory model with the time dependent demand rate and deterioration rate.  The unsatisfied demands will be partially backlogged. The backlogging rate is a variable and is inversely proportional to the length of the waiting time for next replenishment. So our aim is to minimize the total cost of inventory with optimal order quantity and optimal time of inventory exhausting. This paper is organized as follows: Section 2 describes the notations and assumptions, Section 3 presents the mathematical formulation of the problem, Section 4 provides a numerical example to illustrate the proposed inventory model, Section 5 discuss the sensitivity analysis, and finally Section 6 proposes the conclusions.

Conclusion

We proposed a deteriorating inventory model with time-dependent demand rate and varying holding cost under partial backlogging along with time dependent deterioration rate. Shortages are allowed and partially backlogged. The classical optimisation technique is used to derive the optimal order quantity and optimal average total cost. For the practical use of this model, we consider a numerical example with sensitivity analysis. Both the numerical example and sensitivity analysis are implemented the help of MATHEMATICA-8.0. The proposed model can assist the manufacturer and retailer in accurately determining the optimal order quantity, cycle time and total cost. Moreover, in market there are certain items where during the season period, the demand increases with time, and when the season is off, the demand sharply decreases and then becomes constant. Thereby, the proposed model can also be used in inventory control of seasonal items. This paper can be further extended by considering ramp type demand function. Further, the fuzzy or stochastic uncertainty in inventory parameters may be considered.

References

[1] Abad, P. ‘Optimal pricing and lot-sizing under conditions of perishability and partial backordering’, Management Science, Vol. 42, No. 8, pp.1093–1104 (1996) [2] Abad, P. ‘Optimal price and order-size for a reseller under partial backlogging’, Computers and Operation Research, Vol. 28, No. 1, pp.53–65 (2001) [3] Chang, H. and Dye, C. ‘An EOQ model for deteriorating items with time varying demand and partial backlogging’, Journal of the Operational Research Society, 50, 11, 1176–1182 (1999) [4] Chung, K.J. and Ting, P.S. ‘A heuristic for replenishment of deteriorating items with a linear trend in demand’, Journal of Operational Research Society, Vol. 44, No. 12, pp.1235–1241(1993) [5] Dave, U. and Patel, L. ‘(T, si)-policy inventory model for deteriorating items with time proportional demand’, Journal of Operational Research Society, 32, 2, 137–142 (1981) [6] Dye, C. ‘Determining optimal selling price and lot size with a varying rate of deterioration and exponential partial backlogging’, European Journal of Operational Research, Vol. 181, No. 2, pp.668–678 (2007) [7] Ghare, P.M. ‘A model for exponentially decaying inventory’, The Journal of Industrial Engineering, Vol. 5, No. 14, pp.238–243 (1963) [8] Goyal, S. and Giri, B. ‘Recent trends in modeling of deteriorating inventory’, European Journal of Operational Research, Vol. 134, No. 1, pp.1–16 (2001) [9] Hung, K. ‘An inventory model with generalized type demand, deterioration and backorder rates’, European Journal of Operational Research, Vol. 208, No. 3, pp.239–242 (2011) [10] Jalan, A., Giri, R. and Chaudhary, K. ‘EOQ model for items with weibull distribution deterioration shortages and trended demand’, International Journal of System Science, Vol. 27, No. 9, pp.851–855 (1996) [11] Kumar, R.S., De, S.K. and Goswami, A. ‘Fuzzy EOQ models with ramp type demand, partial backlogging and time dependent deterioration rate’, International Journal of Mathematics in Operational Research, Vol. 4, No. 5, pp.473–502 (2012) [12] Roy, A. ‘An inventory model for deteriorating items with price dependent demand and time varying holding cost’, Advanced Modelling and Optimization, Vol. 10, No. 1, pp.25–37 (2008) [13] Ouyang, L.Y., Wu, K.H. and Cheng, M.C. ‘An inventory model for deteriorating items with exponential declining demand and partial backlogging’, Yugoslav Journal of Operations Research, Vol. 15, No. 2, pp.277–288 (2005) [14] Mandal, B. ‘An EOQ inventory model for Weibull distributed deteriorating items under ramp type demand and shortages’, OPSEARCH, Vol. 47, No. 2, pp.158–165 (2010) [15] Mishra, V.K.: Controllable deterioration rate for time-dependent demand and time-varying holding cost. Yugosl. J. Oper. Res. 24, 87–98 (2014) [16] Palanivel, M.; Uthayakumar, R.: An EPQ model for deteriorating items with variable production cost, time dependent holding cost and partial backlogging under inflation. OPSEARCH 52, 1–17 (2015) [17] Choudhury, K.D.; Karmakar, B.; Das, M.; Datta, T.K.: An inventory model for deteriorating items with stock-dependent demand, time-varying holding cost and shortages. OPSEARCH 52, 55–74 (2015) [18] Rangarajan, K.; Karthikeyan, K.: Analysis of an EOQ Inventory Model for Deteriorating Items with Different Demand Rates. Appl. Math. Sci. 9, 2255–2264 (2015) [19] Karmakar, B.: Inventory models with ramp-type demand for deteriorating items with partial backlogging and time-varing holding cost. Yugosl. J. Oper. Res. 24, 249–266 (2016) [20] Alfares, H.K.; Ghaithan, A.M.: Inventory and pricing model with price-dependent demand, time-varying holding cost, and quantity discounts. Comput. Ind. Eng. 94, 170–177 (2016) [21] Giri, B.C.; Bardhan, S.: Coordinating a two-echelon supply chain with price and inventory level dependent demand, time dependent holding cost, and partial backlogging. Int. J. Math. Oper. Res. 8, 406–423 (2016) [22] Pervin, M.; Roy, S.K.; Weber, G.W.: A Two-echelon inventory model with stock-dependent demand and variable holding cost for deteriorating items. Numer. Algebra Control Optim. 7, 21–50 (2017) [23] Chandra, S.: An inventory model with ramp type demand, time varying holding cost and price discount on backorders. Uncertain Supply Chain Manag. 5, 51–58 (2017) [24] Rastogi, M.; Singh, S.; Kushwah, P.; Tayal, S.: An EOQ model with variable holding cost and partial backlogging under credit limit policy and cash discount. Uncertain Supply Chain Manag. 5, 27–42 (2017) [25] Pervin, M.; Roy, S.K.; Weber, G.-W.: Analysis of inventory control model with shortage under time-dependent demand and time-varying holding cost including stochastic deterioration. Ann. Oper. Res. 260, 437–460 (2018a) [26] Pervin, M.; Roy, S.K.; Weber, G.W.: An integrated inventory model with variable holding cost under two levels of trade-credit policy. Numer. Algebra Control Optim. 8, 169–191 (2018b) [27] Sen, N.; Saha, S.: An inventory model for deteriorating items with time dependent holding cost and shortages under permissible delay in payment. Int. J. Procure. Manag. 11, 518–531 (2018) [28] Sharma, S.; Singh, S.; Singh, S.R.: An inventory model for deteriorating items with expiry date and time varying holding cost. Int. J. Procure. Manag. 11, 650–666 (2018) [29] Geetha, K.V.; Naidu S.R.; Economic ordering policy for deteriorating items with inflation induced time dependent demand under infinite time horizon. Int. Journal of Operational Research. 39(1):69-94 (2020) [30] Khatri, P.D.; Gothi, U.B.; An EPQ Model for Non-Instantaneous Weibully Decaying Items with Ramp Type Demand and Partially Backlogged Shortages. Int. J. of Math. Trends and Tech. (IJMTT) (66)3 (2020) [31] Adak, S.; Mahapatra, G.S.; Effect of reliability on varying demand and holding cost on inventory system incorporating probabilistic deterioration. American Institute of Mathematical Sciences 18(1): 173-193 (2022)

Copyright

Copyright © 2022 Dr. Mamta Sharma, Dr Raveendra Babu A.. 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.