Integrated inventory models with two–level credit policy and a price negotiation scenario for price–sensitive Stock–dependent demand

Article history: Received 6 September 2010 Received in revised form 18 January 2011 Accepted 20 January 2011 Available online 24 January 2011 In this research, the integrated inventory models are developed for price–sensitive stock– dependent demand and delay in payments are permissible. Two level trade credit police in the vendor–buyer and buyer–customer is considered. An easy–to–use solution algorithm is derived for the integrated models to determine the buyer’s optimal pricing and ordering strategy. A negotiation scenario is incorporated to distribute the extra profit between the vendor and buyer. A numerical example and sensitivity analysis are given to validate the proposed models. It is observed that the total joint profit of the integrated system can increase even if the price discount is offered to the buyer in proposed models. © 2011 Growing Science Ltd. All rights reserved


Introduction
Till early 70's, researchers were analyzing inventory models from the buyer's or vendor's end.However, Goyal (1976) argued that the supply chain comprising of the vendor and the buyer to determine optimal ordering strategy is mutually beneficial to achieve the minimum integrated total cost when the vendor's production is infinite.Banerjee (1986) assumed that the vendor's production rate is deterministic on the lot -for -lot basis for Goyal's (1976) model.Goyal (1988) relaxed lotfor -lot production and assumed that the vendor produces an integer multiple of the buyer's order per production run.There after, Ha and Kim (1997), Pan and Yang (2002), Chang et al. (2006), Hsiao (2008) presented the integrated vendor -buyer models with equal order sizes.These studies established that the vendor's cost as wall as the integrated total cost from the joint decision is smaller than the independent decision; however, the buyer's cost increases.Fascinated by the principal of mutual benefit, the vendor could make amends for the buyer's loss by using incentive strategies to entice the buyer to opt for joint decision.Chakravarty andMartin (1988), yang (2004), Wee and Yang (2007), suggested incentive of price discount to the buyer.
Most of the inventory models assumed that the buyer settles the account with the vendor immediately on the arrival of the product in the inventory system.In business, the practice of offering trade credit by the vendor to the buyer is prevailing.The offer of trade credit encourages the buyer to increase the order quantity and boost up the demand.Goyal (1985) developed mathematical model for the buyer's optimal order quantity when permissible delay in payments is possible.Shah (1993) and Aggarwal and Jaggi (1995) analyzed inventory policies for deteriorating items under the condition of a permissible delay in payments.Chung (1998) derived analytic results for optimum procurement units when credit period is offered by the vendor to the buyer.Chu et al. (1998) established convexities of the objective function for Aggarwal and Jaggi's (1995) model.Teng (2002) calculated interest earned on the selling price instead of the purchase cost as taken by Goyal (1985).Teng et al. (2005) modeled inventory model for deteriorating when trade credit is offered and demand is price -sensitive.For details of articles on trade credit, one can refer to review article by Shah et al. (2010).
The passing of credit period from the customers to the buyer who is getting from the vendor is termed as the two -level credit policy.Huang (2003) applied the two -level credit policy into the classical EOQ model to determine the optimal order quantity.Teng and Goyal (2007) advised the customers to settle the account with the buyer at the end of the credit period.Jaggi et al. (2008) discussed inventory model for the two -level trade credit policy, when the demand is sensitive to the credit period offered by the buyer.Teng and Chang (2009) analyzed two -level trade credit policy when production rate is finite.The above-cited articles were discussed from the buyer's point of view.Due to globalization, the researchers have started analyzing integrated vendor -buyer inventory models with allowable trade credit; such as Abad and Jaggi (2003), Yang and Wee (2006), Chen and Kang (2007), Ho et al. (2008), Ouyang et al. (2008), Huang et al. (2009), Chen and Kang (2009) etc. Barratt (2004) stated that in supply chain management trust, mutuality, information exchange, openness etc. are important for establishing a long -term cooperative relationship among the trading players.The studies by Goyal (1976), Pan and Yang (2002), Yang (2004), Ho et al. (2008) were made to support the Barratt's argument.Chen and Kang (2007) developed integrated inventory model with two -level trade credit policy.The Chung et al. (2006), Yang et al. (2007) and Chung and Wee (2008) incorporated a negotiation scheme for agreeing to vendor's offering price to the buyer to achieve a win -win strategy.Chen and Kang (2009-a) formulated integrated inventory models under the two -level trade credit policy with price -sensitive demand and negotiation scheme.In this paper, the integrated inventory models are analyzed for price -sensitive stockdependent demand with the two -level trade credit policy and a negotiation scheme.The concept of stock -dependent demand makes this contribution more practicable in the prevailing concept of super -malls.
The notations and assumptions are given in the following section.Section 3 deals with the development of three models considering the two -level credit policy with price -sensitive stockdependent demand, viz. a non -integrated model, integrated model and integrated vendor -buyer model with a negotiation scheme to make compensation for the buyer's loss.A numerical example and sensitivity analysis are given to study the derived models in section 4. The conclusions are given in section 5.

Notations and assumptions
The proposed models are derived using the following notations and assumptions.

Notations
, , TPV n T P i i bi bi : total profit per unit time for the vendor (also denoted as TPV i ) in model i

( )
, , TP n T P i i bi bi : total profit per unit time which is sum of TPB i and TPV i (also denoted asTP i PTPG : percentage total profit gain GMR : gross margin ratio Assumptions 1.The supply chain under consideration consists of a single vendor and single buyer for a single product.2. Demand rate is price -sensitive stock -dependent and production rate is greater than the demand rate.3. Shortages are not allowed.4. The lead -time is zero. 5.The two -level credit policy is implemented in which the vendor offers the buyer a credit period and the buyer also gives a credit period to the customers.The customers settle the account with the buyer when the credit period offered by the buyer is due.

Mathematical models
Here, the rate of change of inventory is due to price -sensitive stock -dependent demand.The differential equation governing the inventory status at any instant of time t is given by with ( ) Then the solution of the differential equation is and optimum procurement quantity is In this section, we develop three models.Firstly, we compute the vendor's profit model, which will be applied to all the three proposed models.
Vendor's profit model The buyer's order ( ) n Q i i is produced n i times by the vendor.The different cost components of vendor's profit are as follows: 1. Set -up cost for the vendor 4. Opportunity interest loss per unit time 1 Revenue per unit time for the vendor ( ) Hence, the total profit per unit time for the vendor is ( )

Model 1: Non -integrated vendor -buyer model
The buyer's cost components are as follows: Interest earned and interest charged are computed on the basis of the lengths of 1 b T , M and N as follows.Various scenarios could be possible due to offer of two -level trade credit.Under the two -level trade credit policy, the customer is allowed to settle for account when the credit period offered by the vendor is due.Therefore, the customer can pay off the buyer during Here, the customers settle the accounts with the buyer when the credit period offered by the buyer is due.The buyer's interest charges are zero; and 10.Interest earned per unit time during [ N, M ] ( ) In this situation, the buyer does not have purchase cost to be paid against the procured units, and hence, he borrows the total purchase cost from the bank.Hence, the interest earned is zero; and 11.Interest charged per unit time ( ) 12. Revenue per unit time for the buyer is ( ) Based on the above scenarios, the buyer's total profit per unit time is as follows: Case 1. M N ≥ Scenario1: Scenario2: Case 2. M N ≤ ( ) Solution procedure 1: The optimal solution of 1 n (number of shipments) which maximizes the vendor's total profit per unit time can be computed, and hence total profit of the system (which is sum of the buyer's total profit per unit time and the vendor's total profit per unit time) can be obtained.

Model 2: Vendor -buyer integrated model
Here, the vendor and buyer determine optimal policy which maximizes the total profit of both the players, collectively.For the attainment of the goal when the end -user's demand is price -sensitive and stock -dependent, the total profit per unit time for the integrated vendor -buyer model is as follows.Here, it is assumed that the buyer's price is given.Similar to model 1, the total profit per unit time is formulated depending on lengths of 2 b T , M and N as follows.

Model 3: Integrated vendor -buyer model with a negotiation scheme
In negotiation scheme, the vendor offers a price discount to the buyer to compensate for the loss.The total profit per unit time is computed as follows: Case 2. M N < 33 TP = ( ) For fixed 3 n , To make amends for the buyer's loss, the price scheme is carried out by computing the difference of total profits of models 1 and 3 for buyer and vendor both.Define, vendors extra profit as For 0 δ = , the total profit increment will be for the buyer.
For 1 δ = the extra profit will be distributed equally between two players.The vendor will be beneficial for higher value of δ .
Step 2: Calculate the maximum profit of  1, the solution is given for N = 0.0 and 0.05.Clearly, N = 0.0 is one -level trade credit policy which is the special case of the developed model.For N = 0.0, the total profit per unit time for the integrated vendor -buyer model increases by 11.03 % ( 2PTPG ) compared to non -integrated model (Model 1).The buyer looses $3629 and vendor gains $ 5651 in the integrated system.Hence, the buyer will be reluctant to opt for joint decision.To reduce the buyer's loss, the vendor offers the price discount to the buyer.Table 1 depicts that 3 PTPG increases to 11.99 % i.e. increases by 0.96 % when negotiation scenario is implemented.In Fig. 4, the changes in EPB and EPV are plotted with respect to discounted purchase cost bd C per unit item.It indicates that both players will be benefited when bd C = $ 3.85.For N = 0.05, i.e. offering trade credit to the customers lowers total profit and PTPG for the vendor and buyer.However, the buyer's optimal selling price b P * for each model increases very slightly compared to that when N = 0.0.This proves that the increase of the buyer's selling price cannot counter act the decrease of profit due to offer of credit period, N = 0.05 to the customers.In Table 2, the sensitivity analysis of the optimal solutions with respect to the negotiation factor; δ is carried out for N = 0.05 and it is observed the that increase in the negotiation factor δ increases the buyer's purchase cost per unit and selling price per unit slightly.It is observed that the gross margin ratio (GMR) has a negative change with increase in values of δ which is in favor of the buyer.On the other hand, for δ greater than 1, vendor is benefited the most.From Table 3, it is observed that gross margin ratio decreases for scale demand, stock -dependent parameter, mark -up, production utility ratio; δ , buyer's ordering cost, vendor's ordering cost, interest charged by the vendor to the buyer on the unsold stock whereas it is insensitive to interest earn by the buyer on the generated revenue.(See Fig. 5 and 6).The percentage gain in total profit of the integrated system increases significantly when there is an increase on stock -dependent parameter and interest charged by the vendor to the buyer.

Conclusions
This paper analyzes the impact of stock -dependent demand and the credit period offered by the buyer to the customer, called as the two -level trade credit policy, in the integrated models.The recursive solution procedure is established to determine the optimal solutions.It is suggested that the buyer can be encouraged for taking joint decision by incorporating offer of price discount in the unit price.The developed model is illustrated by a numerical example, and also sensitivity analysis is performed with respect to the negotiation factor to observe the changes in the buyer's gross margin ratio and percentage in total profit gain.For M = 0.1 and N = 0.05, for equal increase inδ , the buyer's total profit decreases and vendor's total profit increases and the gross margin reverses.
This study favors the mutual benefits and information sharing between the players of supply chain.In future research, the integrated models could be developed to study the other promotional scheme.
Appendix A.
cost incurred by the vendor in dealing with each order

Fig. 1 .Fig. 2 .Fig
Fig. 1.Interest earned and charged under M ≥ N and T N M 2,3 for suitable model parametric values with the help of mathematical software.Then optimum value of 1 In this section, we exhibit working of the proposed models by a numerical example.Consider, the parametric values as follows: α = 100000, β = 3.5, η = 1.5, b A = 100, b C = 5, b I = 0.10, v I = 0.16, v A = 1200, F = 100, e I = 0.09, o I = 0.09, c I = 0.12, v C = 2.5, ρ = 0.8 and δ = 1.In Table

Table 3
Sensitivity a