Optimal Replenishment and Payment Policies for Deteriorating Inventory Model with Price-Sensitive Trapezoidal Demand Under Biddable Two-Part Trade Credit

Abstract In general, researchers analyzing inventory models under two-part trade credit considered that the buyer either pays full purchase cost during the pre-specified credit period and take the advantage of cash discount or settles the account within a longer allowable credit period at the regular purchase price. In this paper, we model decision policies when buyer may pay part of purchase cost within a permissible shorter credit period and gets a cash discount in purchase cost and then the remaining account is settled within the longer permissible credit period. The demand rate is considered to be price-sensitive trapezoidal and units in inventory are subject to constant rate of deterioration. An algorithm is proposed to compute the optimal retail price, cycle time and biddable fraction of payments to be paid which maximizes the decision maker’s profit per unit time. Numerical examples are given to illustrate the proposed problem. It is observed that the buyer is beneficial under biddable two-part trade credit scenario as compared to either not paying anything within shorter credit period or paying in full within longer permissible credit period.


Introduction
The offer of trade credit from vendor attracts a buyer to stock more goods without any investment. Zhou (2009) documented that around 70-80% of business between the players of the business is based on trade credit. In general, the vendor adopts either anet credit period in which a buyer has to settle payment in a fixed period or a two-part trade credit in which a buyer gets a discount in purchase price of an item if payment is made within an offered shorter period, otherwise the full payment is to be settled within the pre-specified longer credit period. For example, under '2/10 net 30', the widely used two-part trade credit scenario, a buyer will get 2% discount in the purchase cost of an item if the account is settled within 10-days of receipts of goods otherwise settles the full purchase cost within 30-days. This type of trade credit is termed as two-part trade credit. Two-part trade credit is denoted as ' 1 / M β net 2 M ' under which the buyer receives a cash discount of β percentage in unit purchase cost if payment is made by M 1 ; otherwise the buyer pays by M 2 at a regular purchase cost.
The buyeris prominent decision maker who bids for preferable contract terms and thereby reducinghis investments in terms of purchase cost (Klapper etal. 2012). Fabbri and Klapper (2011) advocated that when the buyer has constraint of investment, the supplier should offer flexible payment schemes to increasethe demand from the buyer resulting increase of cash in-flow. Using this concept, Zhou  The concept of trade credit was first modeled by Goyal (1985). The analysis of this promotional tool was explored by many researchers. Refer to review articles on inventory modeling and trade credit by Goyal et al. (2008) and Shah et al. (2010). Ouyang et al. (2009) determined an economic order quantity for deteriorating items with partial trade credit linked to order quantity. Huang (2003) discussed inventory model when the retailer passes some credit period to the buyer which is offered to him by the supplier. Shah and Raykundaliya (2011) determined inventory policies when demand declines under two -level trade credit schemes. Ho In this research, the optimal retail price, cycle time and payment policy for the buyer is worked out under a biddable two-part trade credit policy when demand is price-sensitive trapezoidal and units ininventory deteriorate at a constant rate. Using numerical data, it is established that the proposed concept is favorable to buyer as compared to the traditional one.
The rest of the paper is organized as follows: In section 2, notations and assumptions are listed. Section 3 is mathematical model of the proposed concept. In section 4, algorithm is proposed to determine the optimal policy. Section 5 exhibits numerical examples and sensitivity analysis. Section 6 concludes the study.

Notations
The demand ( , ) R P t is considered to be a trapezoidal type whose functional form is , ; The buyer's profit function per unit time

Assumptions
The development of mathematical model is based on following assumptions.
The buyer stocks single item. Shortages are not allowed. Lead-time is zero.
The demand rate ( , ) R P t is price-sensitive trapezoidal. , ;

Mathematical Model
The total profit per unit time of the buyer comprises of the following components.
1) Gross revenue per unit time; 2) Ordering cost per unit time;

T M and 2
M . Three cases may arise:

T M >
Using assumption (4), we compute the interest charged/earned per unit time for the buyer as follows: Case 1: Depending on values of 1 2 , M M and , T λ the possible two sub-cases are (i) The values of 1 u and 2 u will sub branch each sub-cases as discussed below.
Sub-case 3.1: Here, the buyer has sufficient amount in the account to pay λ fraction of purchase cost at time 1 and the interest paid per unit time on the unsold items by the buyer is The total profit per unit time of the buyer is given by ( ) π λ π λ π λ π λ π λ π λ Clearly, the objective function given in equation (1)

Computational Algorithm
We outline following steps to determine optimal policies for the buyer to maximize the total profit per unit time.

Optimal Replenishment and Payment Policies for Deteriorating Inventory Model with
Price-Sensitive Trapezoidal Demand Under Biddable Two-Part Trade Credit Step 1: Assign values to the inventory parameters.
Step 3: Knowing optimal policy ( ) , , , P T λ compute Q (given in Appendix A). The concavity of the profit with respect to retail price, cycle time is shown in fig. 9 and fig. 10 respectively. From fig. 8, it is observed that the buyer's profit is convex with respect to biddable payment to be made at earlier date to avail of discount in unit purchase price. The 3-D plots given in figures 11-13 suggest that buyer's profit is maximum for the proposed concept.  Next, we carry out the sensitivity analysis by changing inventory parameters given in example 1 as 20%, 10%, 10% and +20% − − + to get managerial insights for the decision maker. In Fig. 14, variations in early payment fraction λare studied. It is observed that λ has highly positive impact due to changes in scale demand and interest paid by the buyer. Though scale demand is uncontrollable, interest to be paid on unsold stock can be reduced by selling items in time λis positively sensitive to offered payment time and inventory carrying charge fraction. The ordering cost and the time point when demand starts decreasing exponentially are negatively related to λ. This suggests retailer to save in ordering cost by larger order though the inventory carrying charged fraction will hinder for larger order. So, the retailer should find out the trade-off between these two parameters. The purchase cost of an item has negative impact on λ which suggests that the decision maker should do smaller fraction of payment at earlier date for costlier items. From Fig. 15, we see that increase in purchase cost increases selling price while other inventory parameters decrease selling price. In Fig. 16, variations in cycle time are depicted. The price elasticity, larger offered delay period and the time point when demand starts decreasing increases cycle time significantly. Increase in discount rate of units in purchase price decreases cycle time. This establishes that the offer of early payment by taking advantage of discount in unit purchase price is beneficial to the retailer. In Fig. 17, profit variations are studied. The profit sharply increases with increase in discount in unit purchase price, scale demand, early payment time, the rate at which demand increases and the time in which demand increases and remains constant. Thus, the retailer is beneficial if the inventory policies are planned considering positive impact of these parameters.   M .It suggests that retailer should take advantage of time phase in which demand increases. By encouraging buyer to pay early with offer of discount in purchase price will reduce cash-out flow risk for the vendor. One can incorporate deterioration of items, finite replenishment, stochastic demand pattern etc. to get more beneficial analysis.

Appendix A: Computation of Inventory at Any Instant of Time t and Purchase
Quantity Q The inventory level in warehouse changes due to price-sensitive trapezoidal demand and deterioration rate of units in the warehouse. The rate of change of inventory at any instant of time t is governed by the differential equation with the initial condition ( ) 0 The solution of the differential equation is ;