Optional Inventory Ordering Policies of Multiple Non-Instantaneous Deteriorating Items Under Trade Credit and Price Discount

When ordering multiple items, retailers tend to be used the collaborative procurement, so conducive to the formation of scale in quantities utility. Therefore, this paper established the inventory model of the retailer’ in a multiple non-instantaneous deterioration items supply chain, with two kinds of incentives-credits and price discounts. Then we use the theorems to proof the existence of the optimal solution of inventory, and compare the profits under different scenarios. Finally, we use numerical examples to validate the model, we found that the optimal profits of retailer under trade credit is better than that of price discount in the multiple non-instantaneous deterioration items supply chain.


INTRODUCTION
No-matter how big the company, they all face up two tough problems: the shortage of money and the inventory ordering policies of multiple non-instantaneous deterioration items with uncertainty demand. Especially, the extension of the concept of multiple non-instantaneous deterioration items is involving most of our daily products, with the development of society and technology. One of the most important branches of operation is the inventory model of multiple noninstantaneous deterioration items, and has been systematically summarized. But, in practice, retailers always order multiple non-instantaneous deterioration items, and the single item inventory model is not so precisely for the multiple non-instantaneous deterioration items. Therefore, to tackle the inventory ordering policies of multiple non -instantaneous deterioration items is aroused highly attention in the field of scholars. Bo se et al. (1995) built up the EOQ model for deterioration goods with liner demand, allowing shortage and backlogging, and the influence of time value, firstly. And they acquire the inventory policies in difference scenarios. Moon et al. (2005) described a special kind of deterioration items, which the price goes up with time, for example, wine, poultry. And they also build up the inventory model with considering time value; Yang and Chang. (2013) formulated the optimal replenishment model by considering trade credit, time value, two warehouse and allowing shortage and backlogging; Tsao (2010) considered multi -echelon multi-item channels subject to supplier's credit period and retailer's promotional effort. They analyse two trade allowances, the promotion costs sharing and the cash discount, finding that coordination mechanism is better; Wen and Da (2006) discuss single ordering, partly ordering and coordination ordering from the perspective of retailer's, and they got the optimal ordering policies; Wang and Jiang (2007) compare four different circumstance of the combination of decentralized decision and cooperative decision, finding that the cooperative circumstance is better; Mo (2011) cons idered a model with demand depend on inventory and resisted space. They got the result by line search algorithm; Zhang et al. (2012) provide a completed literature review of the joint inventory policies of multi -deterioration items, and find out the inventory model of multiple deterioration items will be hot in the future. Above all, they just consider the goods is deteriorating, when the goods arrive at the store, but, in practice, some kinds of goods have a certain life long time. So the existing models is not so precisely to tackle the inventory issues of noninstantaneous deterioration items. Wu (2006) provide the concept of non -instantaneous deterioration items, and build the related models to solve it; Chang et al. Above all, deterioration items inventory research can be divided into two categories: deteriorating immediately and noninstantaneous deterioration. Even there are some literatures have been studied the multi -deterioration items, but the research about non-instantaneous deterioration items with trade credit is rarely. We build up the inventory ordering model under price discount and trade credit to maximize the retailer's profit, and compare the two strategies, get that the profit under trade credit is better than that of under price discount in multiple non -instantaneous deterioration items supply chain.
1. PROBLEM DESCRIPTION, ASSUMPTIONS AND NOTATIONS

Problem description
In this paper, we consider the retailer how to make inventory policies in a multiple non -instantaneous deterioration items supply chain. In order opening, the retailer will order all the non -instantaneous deterioration items together, and at the end of ordering period, all products are already completely sold. Items have no effect to others. Meanwhile, suppliers will provide the trade credit or price discount to retailer, to stimulate the order quantity. Modelling the problem based on the process shown in Fig. 1: (3)Deterioration rate i  depends on inventory, and the beginning time of deterioration is smaller than ordering period max( )< di tT . (4) Bad items can't be repaired or replaced in the order cycle.
(5) Capital rate of inventory opportunity cost is greater than the current interest rates, > pe II .

Notations
S : the ordering cost per cycle, has nothing to do with the kinds of items per order . First of all, the supplier provide price discount to retailer, retailers make his inventory policies by maximizing the profit. When , inventory system under the price discount is shown in fig.2: Fig. 2 Retailer's inventory levels under the price discount The inventory goes like this: during the time interval [0, tdi], the inventory level is decreasing only owing to demand. The inventory level is dropping to zero due to demand and deterioration during the time interval [tdi, T].
Using the differential equation representing the inventory status is given by: According to the boundary conditions, yields: We have the retailer's total profit function:  fig.3. 219 According to Fig.3, we can get the retailer's total profit function, when MT  .
In the order of Eq.

Results with price discount
We discuss the optimal ordering quantity and ordering period by maximizing the profit of retailer. As i p has the same identity and no effect to each other. We use p to alternative i p for simple. First, we discuss the () p,T not only exist and the only, for any R. Then, we discuss the R not only exist and the only, for any p and T. Taking the partial derivative of 1 TP with respective to p and T, we have: ( - Taking the second partial derivative of 1 TP with respective to p and T, and we have the Hessian matrix:


, retailer's profit is inversely proportional to the price discount rate.
Let Eq. (8) and Eq.(9) to be zero, we obtain the optimal value of 1* i p and T in price discount: According to Theorem1, we get:

Results under trade credit
First of all, we should to determine the () i pT , of item i is not only exist but the only for any M . For any i p and T , the M is not only exist but the only.

Scenario 1 if <
MT , there is an optimal solution of the profits. Taking the partial derivative of 2 TP with respective to p and T, we have: Taking the second partial derivative of 2 TP with respective to p and T, and we have the Hessian matrix:

Theorem 3 if <
MT , retailer always get the optimum values for any M. Scenario 2 if MT  , there is an optimal solution of the profits. Taking the partial derivative of 3 TP with respective to p and T, we have:  (15) Taking the second partial derivative of 3 TP with respective to p and T, and we have the Hessian matrix: According to table 1, the price of the two non-instantaneous deterioration items decrease with the price discount rate; the ordering quantity of the two non-instantaneous deterioration items increase with the price discount rate. Under the coordination order cycle, due to different deterioration rate and deterioration time will lead to different retailer's strategy of different items order quantity and price. It means if the deterioration time is shorter, the retailer will order more item s and the price will be lower. The profit and order period are increasing with the discount rate.