An Inventory Model for Deteriorating Item with Reliability Consideration and Trade Credit

In todays global market every body want to buy products of high level quality and to achieve a high level product quality supplier have to invest in improving reliability of production process. In present article we have studies reliable production process with stock dependent unit production and holding cost. Demand is exponential function of time and infinite production process with noninstantaneous deterioration rate are considered in this paper. Whole study has been done under the effect of trade credit. The main objective of this paper is to optimize the total relevant cost for reliable production process. Numerical example and sensitivity analysis is given at the end of this paper.

Demand rate is an important factor in modeling of deteriorating inventory model.There are different types of demand functions like constant, time dependent, stock dependent, price dependent all these are of deterministic type.It is often seen that demand rate vary with time.Hence one can say that this type of demand rate is more practical then constant rate of demand.Linear trend in demand has been considered first by Resh et al. (1976) and Donaldson (1977).As the time progressed sufficient work has been done on trended demand.Giri, Chakrabarty, Chaudhuri (2000), Chang, Hung, Dye (2001), Chu and Chen (2001), Balkhi (2003), Teng and Yang (2007), Singh and Singh (2010), have studied time varying demand rate.In today's business it is seen that supplier provides a permissible delay period to their customer in settling the account, to decrease total cost and increase their profit.During this period there is no interest charged but after that period interest will be charged on unsold item.Recently Teng, Chang, Goyal (2005) have considered the effect of trade credit and developed an optimal pricing and ordering policies.Kumar, Tripathi, Singh (2008)  In this paper we have developed a production inventory model for non-instantaneous deteriorating items under consideration of reliability production process in an inflationary environment.The demand is exponential function of time with permissible delay in payment.In the next section assumptions and notations are given for mathematical model formulation which is next to it.At the end numerical illustration and sensitivity analysis is performed. The inflation rate R is difference between time discounting and inflation such that 0< R< 1.  Total cost of interest and depreciation per production cycle is inversely related to the set up cost and directly related to process reliability [1] i. e. IDP=f(C o, r) = c C o -d r e where c, d, e are all positive constants.The process reliability means only r items are of good quality and are used to satisfy the demand. During the permissible delay time M, purchaser will deposit sales revenue in interest-bearing account.There are two choices for purchaser at the end of delay period.Purchaser can pay at the end of trade period M or between M and T. The purchaser pay off for all ordered items and starts paying for the interest charges on the items in stocks when purchaser pays the amount at time M. Supplier charges high interest for unsold items when purchaser choose the payment time between M and T.

Assumption
Notations

Mathematical Model Formulation
The inventory depletion during time period [0, t d ] is due to demand only and after t d life time of an item expires and deterioration starts.Hence inventory depletion during time period [t d , T] is due to combine effect of demand and deterioration.The whole inventory function is represented by differential equations as follows Under following boundary conditions I 1 (t = 0) = rQ, I 1 (t = t d ) = I 2 (t = t d ) and I 2 (t = T) = 0. Now solving (1) and ( 2) we get The total relevant cost consists following cost parameters Now we will find Interest paid and earned by purchaser, for this there are two cases (i) T < M and (ii) M ≤ T. These two cases are graphically represented in Figure -1 & 2.

Case: 1 (T < M)
The permissible delay period M is greater than the total inventory depletion period i.e.T. Therefore there is no interest paid by purchaser to the supplier for the items.However purchaser will uses the sales revenue to earn interest at the rate of I e during time period [0, T] and interest from cash invested during period [T , M]. Hence the Present worth of interest earned is ] 0 Hence the Present worth of total relevant cost per cycle is In this case the permissible delay period M expires before the total inventory depletion period T; hence purchaser will have to pay interest charged on unsold items during (M, T).Therefore Present worth of interest paid by purchaser is Now the Present worth of interest earned during positive inventory and interest from invested cost is To minimize total relevant cost, we differentiate w. r. t to , and for optimal value necessary conditions are

Sensitivity Analysis
To check sensitivity of the model we have performed a sensitivity analysis by changing values of some important parameters like α, β, θ, t d , M, R, F, a. we have made +10%, +5%, -5%, -10% change in their original value given in numerical example.The effect of slight variations in values of parameters is given below in table 1 &2.

Conclusion
In this paper we have studied the reliability production process.Through which quality of produced amount is improved.We have developed and production inventory model for non-instantaneous deteriorating items with time dependent demand under the effect of trade credit in an inflationary environment.At the end the model is numerically illustrated and a sensitivity analysis is performed using mathematical software Mathematica7 and results are shown through graphical representation.This study is useful for the items like fruits, vegetables etc and can be extended by incorporating other inventory control parameters.
have developed a model with variable demand rate and trade credit.Many authors have focused on trade credit like Chang, Wu, Chen (2009), Singh and Jain (2009), Chen and Kang (2010), Chen and Cheng (2011), Jaggi, Goel, Mittal (2011), Singh, Kumari and Kumar (2011), Zhou, Zhong, Li (2012), etc.Before 1970's inflation is not considered by researchers.After that effect of inflation is seen in many countries.Effect of inflation has been introduced first by Buzacot (1975).After that several researchers have extended the work of Buzacot in different ways.For further review we can go through the work of Dye, Mandal, Maiti (2008), Singh Kumar and Kumari (2010), Singh and Singh (2011) etc.


Production rate is infinite with zero lead time and Infinite time horizon. Demand rate is time dependent as D(t) = α e βt where α and β > 0.  Shortages are not allowed  Deterioration rate θ is noninstantaneous as follows where 0 < θ << 1 and t d is maximum life time of an item. The unit production cost C o is order level (Q) dependent i. e. C p = a Q -b where a > 0, 0 < b < 1.  Holding cost C h per unit per unit time is unit cost dependent as follows C h = F C p where 0 < F < 1.
is the unit holding cost per unit per unit time. T is total cycle length. I e : interest earned per $ per year. I p : Interest paid by purchaser per $ in stock per year, which is charged by supplier. M: Permissible delay in payment (i.e. trade credit for purchaser to settle the account). I 1 (t) is inventory level during time period . I 2 (t) is inventory level during time period  TC 2 (Q, C 0 , r): Present worth of Total relevant cost per time unit, when M ≤ T.  TC 1 (Q, C 0 , r): Present worth of Total relevant cost per time unit, when T < M. Note: The Present worth of total relevant cost includes following costs  SC is the set-up cost. PC is present worth of purchase cost. HC is present worth of holding cost. IP: Present worth of Interest paid for unsold times at initial time or after the permissible delay M.  IE: Present worth Interest earned from sales revenue during permissible delay in payment. IDP: Cost of interest and depreciation per production cycle.

Table :
Through keen observation of Table1& 2 we found following variations: