Policy Decisions for a Price Dependent Demand Rate Inventory Model with Progressive Payments Scheme

Problem statement: In this proposed research, we developed an inventor y model to formulate an optimal ordering policies for supplier who offers progressive permissible delay periods t the retailer to settle his/her account. We assumed that the annual demand rate as a decreasing functio n of price with constant rate of deterioration and ti me-varying holding cost. Shortages in inventory are allowed which is completely backlogged. Approach: The main objective of this study to frame an inventory model in real life situations. In this st udy, we introduced a new idea of trade credits, nam ely, the supplier charges the retailer progressive inter es rates if the retailer prolongs its unpaid balan ce. By offering progressive interest rates to the retailer s, a supplier, can secure competitive market advant age over the competitors and possibly improve market sh are profit. This study has two main purposes, first the mathematical model of an inventory system are e stablish under the above conditions and second demonstrate that the optimal solution not only exis ts but also feasible. We developed theoretical results to obtain the optimal replenishment interva l by examine the explicit condition. An algorithm i s given to find the flow of optimal ordering policy. Results: The results is illustrated with the help of numerical example using Mathematica software and th e optimal solution of the problem is Z (p, T 1) = 76.8586 at (p, T 1) = (0.952656, 0.128844). Conclusion: We proposed an algorithm to find the optimal ordering policy. A numerical study has been perform ed to observe the sensitivity of the effect of demand parameter changes.


INTRODUCTION
In the traditional Economic Order Quantity (EOQ) model, it is assumed that the retailer pays for the goods as soon as it is received by the system. However, in practice, the supplier offers a retailer a delay of fixed time period for setting the amount owed to him. Usually, there is no interest charge if the outstanding amount is paid within the credit period. However, if the payment is not paid in full by the end of the credit period, then interest is charged on the outstanding amount. Goyal (1985)eveloped an EOQ model under conditions of permissible delay in payments extended Goyal (1985) model by allowing shortages. Mandal and Phaujdar (1988) developed an inventory model by including interest earned from the sales revenue on the stoke remaining beyond the settlement period. Aggarwal and Jaggi (1995) extended Goyal's model for deteriorating items because the loss due to deterioration cannot be ignored. Jamal et al. (1997) generalized the model to allow for shortage and deterioration. Liao et al. (2000); Chang and Dye (2001); Teng (2002); Teng et al. (2005) and Hwang and Shinn (1997) developed the model with permissible delay in period. Chang et al. (2010) Developed an Optimal replenishment policies for non-instantaneous deteriorating items with stockdependent demand.
In the progressive trade credit period, retailer settles the outstanding amount by first credit period. Hence, the supplier does not charge any interest. Supplier charges an interest at rate Ic 1 on the un-paid balance if retailer pays after first credit period but before second period offered by supplier to retailer. If retailer settles his amount after second credit period, then supplier charges to retailer an interest at rate Ic 2 on un-paid balance (Ic 1 <Ic 2 ). By assuming progressive trade credits to the retailer supplier can secure competitive market advantage and improve market share. Goyal et al. (2007) developed an inventory model with constant demand rate and deterioration rate under progressive payment scheme. Soni and Shah (2008) developed a model for stoke-dependent demand rate under progressive payment scheme. Singh et al. (2008) extended Soni and Shah (2008) model by allowing shortages and variable holding cost. This fact attracted a number of researchers to drive inventory modals on price dependent demand rate patterns. Presented an inventory model for items havinf the demand rate is constant and variable deterioration rate under the trade credits. Some of the related works in this area are by Haley and Higgins (1973); Wee (1995); Chung and Tsai (2001); Teng (2002) and Teng et al. (2005).
In this study, we address the issues relating to progressive credit period relating to the retailer to settle his account. We developed a mathematical model when the demand rate, as a decreasing function of price and shortage which are fully backlogged with time varying holding cost. We assume that the supplier offers two progressive credit periods to the retailer to settle the account. The net profit is maximized by optimization technique. An algorithm is presented to derive the retailer's optimal solution.

Fundamental assumptions and notations:
The following assumptions are used to develop the model: • With boundary conditions, Q (0) = Q, Q (T 1 ) = 0, consequently, the solution of the above Eq. 3-5 are: And the order quantity is The cost components per unit time are as follows Eq. 6: Inventory holding cost Eq. 7: The deterioration cost in the time interval [0, T 1 ] is Eq. 8: p and T 1 are continuous variables. Hence the optimal values of p and T 1 can be obtained by setting Eq. 14, 15: And: To maximize the net profit, provided Eq. 16: Where: The optimal values of p = p 2.1 and T1 = T 2.1 are solutions of Eq. 21 and 22: And: For maximizing the total net profit, provided Eq. 23-26: And:

Z (p,T ) GR OC HC DC SC Ic IE
The optimal values of p=p 2.2 and T 1 =T 2.2 are solutions of Eq. 30 and 31: Case 3: T 1 ≥ N: Based on the total purchased cost, CQ, total money pD (p) M+IE 2 in account at M and total money pD (p) N+IE 2 at N, there are three sub cases may arise:

Sub Case 3.1: Let pD (p) M+IE 2 ≥CQ
This sub case is same as sub case 2.1; here sub case 3.1 designate decision variables and objective function.
with interest rate Ic 2 during (N, T 1 ). Therefore, total interest charged on retailer; IC 33 per unit time is Eq. 36: Interest earned per unit time is: The net profit is: If pD (p) M+IE 2 ≥CQ is true then compute T 1 = T 2.1 and p = p 2.1 from sub case 2.1or T 1 = T 3.1 and p = p 3.1 from sub case 3.1, repeat step 2 and stop.
If pD (p) M+IE 2 ≥CQ is not true but pD ( is not true, then compute T 1 = T 3.3 and p = p 3.3 from sub case 3.3, repeat step 2 and stop.
Step 4: M<T 1 <N is not true then computes T 1 = T 3.3 and p = p 3.3 from sub case 3.3, repeat step 2 and stop.

Results:
The data obtained clearly shows that individual optimal solutions are very different from each other. However, there exists a solution which ultimately provides the Maximize the total profit operating of inventory system. In the above tables, it is observed that as the value of T 1 and p are increased and then the total cost is increased. Thus, the optimal solution of the problem is Z (p, T 1 ) = 76.8586 at(p, T 1 ) = (0.952656, 0.128844).

CONCLUSION
In this study, we introduced a new idea of trade credits, namely, the supplier charges the retailer progressive interest rates if the retailer prolongs its unpaid balance. By offering progressive interest rates to the retailers, a supplier, can secure competitive market advantage over the competitors and possibly improve market share profit.
Shortages are allowed and completely backlogged in the present model. In many practical situations, stock out is unavoidable due to various uncertainties. There are many situations in which the profit of the stored item is higher than its back order cost. Consideration of shortages is economically desirable in these cases. The traditional parameters of holding cost is assumed here to be time varying. As the changes in the time value of money and in the price, index, holding cost cannot remain constant over time. It is assumed that the holding cost is linearly increasing function of time.
We developed theoretical results to obtain the optimal replenishment interval by examine the explicit condition. We proposed an algorithm to find the optimal ordering policy. A numerical study has been performed to observe the sensitivity of the effect of demand parameter changes. Further, the model can be enriched by incorporating other realistic parameters such as Weibull distribution deterioration rate, inflation rate, partial backlogging and in progressive interest charges.