An inventory model for Pareto deterioration with parabolic time varying holding cost and inflation under trade credit

Introduction

Inventory modeling is a prerequisite for developing the optimal ordering and pricing policies in several places such as ware houses, market yards production process, cargo handling. When one is concerned with the inventory control, it is essential to identity two important factors namely the nature of the demand. When we consider the nature of procurement, we have to consider life time of the commodity along with the demand pattern. The Pareto life time distribution is capable of charactering age dependent rate of deterioration. The Pareto distribution is extensively used in reliability and life testing experiments. Recently much work has been reported in inventory models for deteriorating items. In enveloping the inventory model it is needed to ascribe a probability distribution to the life time of the commodity. K. Srinivas Rao & J.K. Begum (2007) [1,2] , Viv. Vs kesavara, K. Srivinas a rao, Y. Sirinivas rao (2012) [14] , assumed that the lifetime of commodity following Pareto distribution. Recently G.C Mahanta, P Mahanta (2009) [7] , R.P. Tripathy, S.S. Mishra (2012) [8] , Vipin Kumar, Gopal Pathak and C. B. Gupta (2013) [3] developed on Deterministic inventory model under trade credit. Trade credit is an important economic phenomenon. It is most effective tool for the supplier more and to attract more retailers. A supplier will allow a certain period for settling the amount owed to him for the items supplied. Usually there is no charge, if the outstanding amount is settled within the permitted fixed settlement period. Beyond this period interest in charged. Secondly due to high inflation and consequent sharp decline in the purchasing power of money, especially in the developing countries the financial situation has been charged, so it has been impossible to ignore the effect of inflation. C.K. Jaggi & A. Khana (2008) [11] developed on impact of inflation & credit policies on a production lot size model. H.C. Liao, C.H. T Sai, C.T. Su (2000) [5] developed on an inventory model for deteriorating items under inflation when a delay in payments is permissible. R.P. Tripathi & Neha Sang (2012) [6] developed on EOQ model for constant demand rate with completely backlogged shortage. Vipin Kumar, Gopal Pathak, C.B. Gupta (2013) [3] developed on a Deterministic inventory model for deteriorating items with selling price dependent demand and parabolic time varying holding cost under trade credit. In this paper we have developed an inventory model for Deteriorating items in which the deterioration rate is Pareto distribution and demand is constant but shortages are allowed which is fully backlogged and holding cost is parabolic time varying. The effects of inflation of delay in payments have been discussed. Using the differential equal the instantaneous state of inventory is derived. A numerical illustration is also discussed. The sensitivity of the model with respect to the costs and par aments is also discussed.

Assumptions and Notations:

A. The demand rate is assumed to be constant which is

The lead time is zero C. Shortages allowed which is completely backlogged D. Life time of the commodity is random and follows a Pareto distribution having probability density function of the form 0 , , ) (

The instantaneous rate of deterioration is

Mathematical formulation of the model

Let I(t) be the inventory level at time "t" (0≤t≤T). The differential equations governing the system in the cycle time [0,T] are 1) and ( 2)

The expected Deterioration cost per unit time is

Expected inventory holding cost per unit per unit time is

CASE-I:

For M≤t 1 , the buyer has stock on hand beyond M and so he can use the sale revenue to earn interest at an annual rate e I up to t 1 .

The interest earned denoted by IE 1 , is therefore

However beyond the fixed credit period M, the unsold stock is assumed to be financed with annual charge I C and the interest charge, denoted by IC 1 , is given by

Therefore the total average cost in this case comes out to be

Substituting the equations of HC, DC, SC, IC 1 and IE 1 in equation ( 11), we have

The optimal values of t 1 and T which minimize total average cost per unit per time, can be obtained by solving following equations 0 ) , (

Since M>t 1 , the retailer pays no interest but earns interest at an annual rate I e during the period (0,M).But [0,T], the retailer sells product at selling price P/unit and deposits the revenue into interest earning account at the rate of I e /year. In the period [T, M], the retailer deposits only the total revenue into an account that earns I e /year. Hence, interest earned per time unit is

The optimal values of t 1 and T which minimize total average cost per unit per time, can be obtained by solving following equations 0 ) , (

To illustrate the inventory model we consider the following hypothetical values: A 0 =150,β=0.95,C 0 =2,e=2.718,t=0.5,k=0.1,α=0.5,h=0.2,s=7,D=10,C=12,I C =0.15,I e =0.11 Based on above data and using the software Mathematica5.1, we calculate the optimal value of M=50 where time t 1 =2.0089, optimal values of cycle length T=4.4277, optimal values of total average cost TC 1 =240.395 Illustration-ll: (M>t 1 ) To illustrate the inventory model we consider the following hypothetical values: A 0 =150,β=0.95,C 0 =2,e=2.718,t=0.5,k=0.1,α=0.5,h=0.2,s=7,D=10,C=12,I C =0.15,I e =0.11 Based on above data and using the software Mathematica5.1, we calculate the optimal value at M=35 where time t 1 =1.6162, optimal values of cycle length T=3.699907, optimal values of total average cost TC 2 =222.786.

Sensitivity analysis

The sensitivity analysis is carried to explore the effect of changes in model parameters and total average cost of the optimal policies, by varying each parameter (-20%,-10%, 0%, 10%,20%) at a time for the model under study for section 3.The results are presented in Table-1,2.The relationship between the parameters ,optimal cycle length T and optimal total average cost TC 1, TC 2 is shown in figure

Results and Discussions

The two numerical investigations for case-l ( 1 t M  ) and case-ll (M>t 1 ) are considered to study the effect of changes of the effect of changes of the system parameters A 0 , C 0 , α and β on the total average cost. Whenever one parameter is changing by some percentage all other parameters are kept at their original values. Investigation has been done for positive and negative changes of these four parameters. The results obtained are discussed in Table-l (case-1) and Table-ll(case-ll). A sensitivity analysis has also been conducted to reflect the effects of varied system performance. It is evident that the parameter β is comparatively less sensitive then other parameters in Table-l. Where as in table-ll parameter α is more sensitive for positive changes then negative changes. Further it can examine various policy scenarios that are associated with different levels of economic inputs.

Conclusion

In this paper, we have developed an inventory model for Pareto deterioration with parabolic time varying holding cost and inflation under trade credit. Shortages are allowed. It is assumed that the supplier offers credit period to the retailer who has the reserve money to make the payments at the time of receiving items from supplier. Finally different costs have been illustrated through the numerical example and inferences of the theoretical results.

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