Vendor-buyer ordering policy when demand is trapezoidal

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Introduction
, Silver(1979), Xu and Wang (1991), Chung andTing (1993, 1994), Bose et al. (1995), Hariga (1995), Giri and Chaudhari (1997), Lin et al. (2000), etc. discussed optimal ordering policy when demand is linearly changing with respect to time which is superficial in the market of fashion good, Air seats, Smart phones etc. Mehta and Shah (2003Shah ( , 2004) ) assumed the demand to be exponentially time varying, which is again unrealistic for newly launched product.Shah et al. (2008) introduced the quadratic demand, which is again not observed in the market for indefinite period.In order to have an alternative demand pattern, the trapezoidal demand is considered.This type of demand increases for some time, then gets constant up to some time and afterwards decreases exponentially with time.
Most of the models available in the literature assumed that the buyer is dominant player to make the decision for procurement.This strategy may not be economical for the vendor.An integrated vendorbuyer policy should be analyzed which is beneficial to the players of the supply chain.Clark and Scarf(1960) proposed a mathematical model for vendor-buyer integration.Banerjee (1986) discussed an economic lot-size model when production is finite.Goyal (1988) extended Banerjee's model by relaxing the assumption of the lot-for-lot production.Shah et al. (2008) analyzed joint decision when demand is quadratic.
In this study, a joint vendor-buyer inventory system is analyzed when demand is trapezoidal.A negotiation factor is incorporated to share the savings.The credit period is offered to the buyer to attract the buyer for placing larger order.

Notations and Assumptions
The proposed study uses following notations and assumptions.

Notations
, and ( ) g t is exponentially decreasing in t where a denotes scale demand,  The lead time is zero and shortages are not allowed.
 The credit period is offered for settling the accounts due against purchases to attract the buyer to opt a joint decision policy.
with the boundary conditions ( ) I I  , the maximum procurement quantities for the buyer and the vendor are Hence, the buyer's total cost; b K per unit time is The vendor's inventory is the difference between the vendor-buyer combined inventory and the buyer's inventory during n-orders.This is known as the joint two-echelon inventory model.The vendor's Hence, the vendor's total cost; v K per unit time is The joint total cost K is the sum of b K and v K where b T T n  .Thus K is the function of discrete variable n and continuous variable.

Computational Procedure
There are two cases to analyze T.
Case 1: When the vendor and the buyer take decision independently.
satisfies.Here, the total cost per unit time with independent decision; NJ K is given by min Case 2: When vendor and buyer make decision jointly.
The optimum value of T and n must satisfy the following conditions simultaneously: Thus, the total joint cost is It is obvious that .
Hence, total cost savings J Sav is defined as factor equals to one, all saving goes to buyer; when it is equal to zero, all saving is in the vendor's pocket.When negotiation factor is 0.5, the total cost savings is equally distributed between the vendor and the buyer.The present value of unit after a time interval M is rM e  ,where r is discounting rate.Solving the following equation the buyer's credit period is given by

Numerical Example and Sensitivity Analysis
Consider following inventory parameters values in proper units: 40000 0.04 0.02 600 3000 10 6 0. 11 0.10 0.06] Let  = 0.5.The optimal solution is listed in Table 1 for independent and joint decisions.The buyer's cost and cycle time increases injoint decisions.The vendor gains $1314 and the buyer loses $741.This hinders the buyer to agree for joint decision.To entice the buyer to joint decision, the vendor offers the buyer a credit period of days with equal sharing of cost savings.This reduces the joint total cost PJCR by 0.08735049 %, where PJCR is defined as .The convexity of total integrated cost and independent costs are shown in Fig. 3.

Observations
 Increase in fixed demand a , decreases percentage of cost reduction and increases delay period.(See Fig. 4 and 5) 100

Observations
 Increase in linear rate of change of demand 1 b , may increase or decrease percentage of cost reduction and delay period.(SeeFig. 4 and 5)

Observations
 Increase in exponential rate of change of demand 2 b , increases percentage of cost reduction and delay period.(SeeFig. 4 and 5)

Observations
 Increase in buyer's purchase cost b C , decreases percentage of cost reduction and delay period significantly.(SeeFig. 4 and 5) of change of demand.(See Fig. 1)

For
the negotiation factor.When negotiation

Observations
Increase in Buyer's Ordering Cost b A , may increase or decreases percentage of cost reduction and delay period.(SeeFig. 4 and Fig. 5)

Observations
Increase in Buyer's inventory carrying charge fraction b I , decreases percentage of cost reduction and delay period significantly.(SeeFig. 4 and 5)

Fig. 4 .
Fig. 4. Total savings Vs. percentage of changes in affecting parameters

Table 1
Optimal solution for independent and joint decisions

Table 2
Sensitivity Analysis of Demand Rate

Table 3
Sensitivity Analysis of Linear Rate of Change of Demand

Table 4
Sensitivity Analysis of Exponential Rate of Change of Demand

Table 5
Sensitivity Analysis of Buyer's Ordering Cost

Table 6
Sensitivity Analysis of Vendor's Ordering Cost Increase in Vendor's Ordering Cost v A , may increase or decrease percentage of cost reduction and delay period.(SeeFig. 4 and 5) [Observations

Table 8
Sensitivity Analysis of Vendor's Purchase Cost

Table 9
Sensitivity Analysis of Inventory Carrying Charge Fraction of Buyer

Table 10
Sensitivity Analysis of Inventory Carrying Charge Fraction of Vendor