An Inventory Model for both Variable Holding and Sales Revenue Cost

 

Satish Kumar1, Yashveer Singh2, A. K. Malik3

1Department of Mathematics, D.N. (PG) College, Meerut, U.P., India

2Department of Computer Science, GRD Institute of Management and Technology Engineering College, Dehradun, U.K., India

3Department of Mathematics, B K Birla Institute of Engineering and Technology, Pilani, Rajasthan, India

*Corresponding Author E-mail: skg22967@gmail.com, yashveersinghrke@gmail.com, ajendermalik@gmail.com

 

ABSTRACT:

This paper discusses the inventory models for non-instantaneously deteriorating items with stock dependent demand. The holding cost is the increasing function of time and sales revenue cost is taken as decreasing linear function of time. This consideration has enhanced developing mathematical model for optimal order quantity and the total profits value with respect to major parameters is approved out with the facilitate of numerical example.

 

KEY WORDS: Inventory, time dependant increasing holding cost, time dependant decreasing sales revenue cost, stock-dependent demand.

 

 


INTRODUCTION:

Today’s for any malls, supermarket and store; demonstration of the buyer goods in big quantities attracts the advanced demand which creates more customers.  Ghare (1963) formulated an inventory model considered the deteriorating rate for products. First Gupta and Vrat (1986) developed for multi –items inventory models with stock-dependent demand. Wu et al. (2006) suggested a mathematical model for non-instantaneous deteriorating items under the stock dependent demand and partial backlogging rate.

 

The most understandable holding costs which take in rent for the apparatus, raw materials, space, and worker to control/work at space, security, insurance, interest on money invested in the inventory product and space, and other direct everyday expenditure.

 

Roy (2008) formulated a mathematical model with price dependent demand and time varying holding cost for deteriorating items. Malik et al (2008) derived an inventory model for deteriorating items under the time dependent demand. Geetha and Uthayakumar (2010) designed an economic inventory policy for non-instantaneous deteriorating items under the permissible delay in payments. Chang et al (2010) developed an optimal replenishment policy with stock-dependent demand. Sarkar et al (2010) derived a finite replenishment mathematical model with time increasing demand under the inflation rate. Singh, and Malik (2010) formulated an model for decaying items with time varying holding cost under the two shops. Sana, S.S. (2010) investigated an optimal selling price and lot size with time varying deterioration under the partial backlogging rate. Singh and Malik (2010) studied an optimal ordering policy with linear deterioration, exponential demand and two storage capacities. Malik and Sharma (2011) proposed an inventory model with multi-variate demand and partial backlogging under the inflation rate. Gupta et al (2013) suggested an optimal ordering policy for stock-dependent demand inventory model with non-instantaneous deteriorating items.

Sarkar and Sarkar (2013) developed an improved inventory model under the partial backlogging rate and stock-dependent demand. The deterioration rate is taken as time dependent. Singh et al (2014) studied a model with stock-dependent demand inventory model and permissible delay in payment under inflation rate. Yang et al (2014) investigated the model with supply chain coordination and stock-dependent demand rate under the credit incentives. Chang et al (2015) proposed an optimal pricing and ordering policies for the developed model in which considered the order-size-dependent delay in payments for non-instantaneously deteriorating items. Vashisth et al (2015) formulated a mathematical model in which considered the price decreasing and stock dependent demand under the maximum life time products. Kumar et al (2016) developed an inventory model with linear holding cost and stock-dependent demand for non-instantaneous deteriorating items. Malik et al (2016aandb) demonstrate the mathematical model with non-instantaneous and time-varying holding cost and demand for deteriorating products. Vashisth et al (2016) gave a model using multivariate demand for non-instantaneous decaying products under the trade credit policy.

 

This paper deals with a mathematical model with time dependent sales revenue cost and stock-dependent demand. The holding cost is taken as a linear increasing function of time. The developed model seeks to maximize the total profit. Finally in order to analyze the optimal solution with the behaviour of different parameters has been discussed.

 

Notation and Assumptions:

The following assumptions and notations are used in this paper:

1)       The demand rate D(t) at time t is where a, b are positive constants and I(t) is the inventory level at time t.

2)       The inventory holding cost per unit time is

3)       The sales revenue cost per unit time is

4)       t1 is the length of time in which the product has no deterioration (i.e., fresh product time).

5)       a is the deterioration rate.

6)       CO denotes the ordering cost per order, Cp, the purchasing cost per unit, and Cd , the deteriorating cost per unit. All of the parameters are positive.

7)       I1 is the inventory level at time [0, t1] in which the product has no deterioration. I2 denotes the inventory level at time [t1, T] in which the product has deterioration.

8)       TP (t1, T) is the total present value of profit per unit time of inventory system (T=t1+t2).


 

Mathematical Model:

For developing this mathematical model, the inventory level I1, during [0, t1] in which the product has no decline. The inventory level I2, during [t1, T] in which the product has decay and inventory level decrease due to demand also. The inventory levels for the proposed model are dominated by the following differential equations:

 

Numerical Example:

To illustrate the above results, we consider the following example: CO=540, s1=20, s2=0.2, Cp=12, h1=0.4, h2=0.02, Cd = 0.3, a=0.3 and a=150, b=0.30, t1=0.20. The inventory system cost TP=720) is Maximum when t2=1.4 and optimal order quantity is Q*=400.

 

The following graphs show the relation between total profit and time period t1 and t2.

 

 


CONCLUSION:

In this article, an inventory model has been proposed both having linearly holding cost and sales revenue cost with stock-dependent demand for non-instantaneous deteriorating items. The necessary and sufficient conditions of the existence and uniqueness of the optimal solution are shown for this inventory model. Numerical example is provided to illustrate the above solution procedure to obtained total profit and investigate the effect of changes in the parameter values on the TP in our inventory models. Further, a future research direction is the study of a multi-item inventory model for multivariable demand, inflation, production model, shortages, partial backlogging, two warehouses and trade credit etc.

 

REFERENCES:

1.        Ghare, P.M. Schrader, G.P. (1963) A model for an exponentially decaying inventory. Journal of Industrial Engineering. 14, 5, 238-243.

2.        Gupta, R. and Vrat, P. (1986) Inventory model with multi-items under constraint systems for stock dependent consumption rate, Operations Research 24, 41–42.

3.        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, International Journal of Production Economics, 101, 369–384

4.        Roy, A.  (2008) An inventory model for deteriorating items with price dependent demand and time varying holding cost, Advanced Modeling and Optimization. 10, 1, 25-37.

5.        Malik, A. K., Singh, S. R. and Gupta, C. B. (2008). An inventory model for deteriorating items under FIFO dispatching policy with two warehouse and time dependent demand, Ganita Sandesh Vol. 22, No. 1, 47-62.

6.        Geetha, K.V. and Uthayakumar R. (2010) Economic design of an inventory policy for non-instantaneous deteriorating items under permissible delay in payments, J. Comput. Appl. Math. 233, 2492–2505.

7.        Chang, C.T., J. T. and Goyal S. K. (2010) Optimal replenishment policies for non-instantaneous deteriorating items with stock-dependent demand, International Journal of Production Economics, Volume 123, 62–68.

8.        Sarkar, S., Sana, S.S. and Chaudhuri, K. (2010). A finite replenishment model with increasing demand under inflation, Int. J. Math. Oper. Res., 2(3), 347–385.

9.        Singh, S.R., Malik, A.K., (2010). Inventory system for decaying items with variable holding cost and two shops, International Journal of Mathematical Sciences, Vol. 9, No. 3-4, 489-511.

10.     Sana, S.S. (2010). Optimal selling price and lot size with time varying deterioration and partial backlogging, Appl. Math. Comput., 217, 185–194.

11.     Singh, S.R. and Malik, A.K. (2010). Optimal ordering policy with linear deterioration, exponential demand and two storage capacity, Int. J. Math. Sci., 9(3-4), 513–528.

12.     Malik, A.K., and Sharma, A. (2011).An Inventory Model for Deteriorating Items with Multi-Variate Demand and Partial Backlogging Under Inflation, International Journal of Mathematical Sciences, Vol. 10, No. 3-4, 315-321.

13.     Gupta K. K., Sharma, A., Singh, P. R. and Malik, A. K.(2013) Optimal Ordering Policy for Stock-dependent Demand Inventory Model with Non-Instantaneous Deteriorating Items, International Journal of Soft Computing and Engineering 3, 279-281.

14.     Sarkar B. and Sarkar, S. (2013) An improved inventory model with partial backlogging, time varying deterioration and stock-dependent demand, Economic Modelling, 30, 924–932. 

15.     Singh, Y., Malik, A. K. and Kumar S. (2014). An Inflation Induced Stock-Dependent Demand Inventory Model with Permissible delay in Payment, International Journal of Computer Applications, Vol. 96, No., 25, 14-18.

16.     Yang, S., Hong, K. and Lee C. (2014) Supply chain coordination with stock-dependent demand rate and credit incentives, International Journal of Production Economics, 157,105–111.

17.     Chang, C. T, Cheng, M.C. and Ouyang, L. Y. (2015) Optimal pricing and ordering policies for non-instantaneously deteriorating items under order-size-dependent delay in payments, Applied Mathematical Modelling, 39, 747–763.

18.     Vashisth, V., Tomar, Ajay, Soni, R. and Malik, A. K. (2015). An Inventory Model for Maximum Life Time Products under the Price and Stock Dependent Demand Rate, International Journal of Computer Applications 132 (15), 32-36.

19.     Vashisth, V., Tomar, Ajay, Shekhar, C. and Malik, A. K. (2016) A Trade Credit Inventory Model with Multivariate Demand for Non-Instantaneous Decaying products,  Indian Journal of Science and Technology, Vol. 9, No., 15, 1-6.

20.     Malik, A. K., Tomar, A. and Chakraborty D. (2016).  Mathematical Modelling of an inventory model with linear decreasing holding cost and stock dependent demand rate,  International Transactions in Mathematical Sciences and Computers, Vol. 9, 97-104.

21.     Kumar, S., Malik, A. K., Sharma, A., Yadav,  S. K. and Singh, Y. (2016) An Inventory Model with linear holding cost and Stock-Dependent Demand for Non-Instantaneous Deteriorating Items, AIP Conference Proceedings 1715, 020058 (2016); doi: 10.1063/1.4942740.

22.     Malik, A. K., Shekhar, C., Vashisth, V., Chaudhary, A.K., and Singh, S. R. (2016) Sensitivity analysis of an inventory model with non-instantaneous and time-varying deteriorating Items, AIP Conference Proceedings 1715, 020059, doi: 10.1063/1.4942741.

23.     Malik, A. K. Malik, Singh, P. R., Tomar, A., Kumar, S. and Yadav, S. K.  (2016) Analysis of an Inventory Model for Both Linearly Decreasing Demand and Holding Cost, AIP Conference Proceedings 1715, 020063; doi: 10.1063/1.4942745.

 

 

 

 

 

Received on 02.06.2017                Modified on 10.07.2017

Accepted on 10.08.2017          © A&V Publications all right reserved

Asian J. Management; 2017; 8(4):1111-1114.

DOI:  10.5958/2321-5763.2017.00169.X