An Integrated Supplier-Buyer Inventory Model with Conditionally Free Shipment under Permissible Delay in Payments

and Applied Analysis 3 Table 1: Comparison between the papers discussing integrated inventory model with a trade credit. Author s Demand rate Production rate Production/purchase cost Freight rate Wholesale price Abad and Jaggi 39 Retail price sensitive Constant Constant — Constant Jaber and Osman 40 Constant — — — Constant Yang and Wee 41 Retail price sensitive — Constant — Constant Sheen and Tsao 42 Retail price sensitive — Constant Quantity discounts are offered Constant Chen and Kang 43 Constant — — — Constant Su et al. 44 Customer’s credit period sensitive Constant Constant — Constant Ouyang et al. 45 Retail price sensitive Adjust with demand rate Constant Quantity discounts are offered Constant Ho et al. 46 Retail price sensitive Adjust with demand rate Constant — Constant This paper Price sensitive Adjust with demand rate Production and demand sensitive Conditionally free shipment Production cost related “—” denotes the factor is not considered in the model. Tsao 42 explored how channel coordination can be achieved using trade credit. Chen and Kang 43 developed integrated models for determining the optimal replenishment time interval and replenishment frequency. Su et al. 44 considered a seller-buyer channel in which the end demand is credit period sensitive. Recently, Ouyang et al. 45 and Ho et al. 46 considered an optimal replenishment and order policy with a credit term when the demand is price sensitive and the production is rate sensitive demand. In this study, to analyze pricing, ordering, delivery, and trade credit with comprehensive considerations of operation, marketing, and financing among channel members, a supplier-buyer inventory model is developed. We assume the supplier’s unit selling price is based on his/her unit production cost which is decided by the market demand and production rates. And the production rate is adjusted with a price-sensitive market demand. In such circumstances the unit wholesale price, reflecting the costs of the product, imposed by the seller on the buyer, does influence the end demand for the product. In addition, the supplier offers to pay freight charges if an order quantity meets or exceeds a certainminimum requirement. Furthermore, a fixed trade credit period is offered by the supplier. In this paper, we maximize the total profit of the whole supply chain i.e., treating the supply chain as a single level profit centre . An algorithm is developed to determine the optimal ordering, shipping, and pricing policy. Numerical examples with relevant data are devoted to find the optimal policies of the developed model. Sensitivity analysis for main parameters is also conducted. The major difference between our model and other related models is shown in Table 1. 4 Abstract and Applied Analysis Customers Buyer Supplier IA(Q)(h +wQ) $c(D,R) = c0D−βRγ Transportation cost per order


Introduction
Following the assumption in Harris's model 1 , most traditional inventory models assumed that the production rate is constant.However, with advanced manufacturing technologies, such as Computer-aided design/manufacturing CAD/CAM , flexible manufacturing system FMS , and computer-integrated manufacturing system CIMS , modem manufacturing industries are highly flexible, intelligent, and integrated.As stated by Schweitzer and Seidmann 2 , today it is not difficult to adjust the mechanical productivity.Over the years, a number of papers have been published dealing with economic order quantity problems under conditions of variable production rate, such as Goswami and Chaudhuri 3 , Balkhi and Benkherouf 4 , Goyal and Giri 5 , Bhunia and Maiti 6 , Kalir and Arzi 7 , Rahim and Ben-Daya 8 , Giri et al. 9 , and Öner and Bilgic ¸ 10 .Recently, Li et al. 11 developed an economic production quantity-EPQ -based model with planned backorders to evaluate the impact of the postponement strategy on a manufacturer in a supply chain.
Transportation cost is another important but often overlooked feature of real inventory systems.Transportation costs are a critical part of the total logistics costs of a commodity.Given today's intense market competition, an appropriate transportation cost function should be included into lot-sizing research and modeling with all other appropriate costs.Tersine and Toelle 12 first established an economic inventory-transport model with freight discounts.Subsequent numerous studies on transportation cost have been published, for instance, Lee 13 ,Hwang et Aggarwal 19 .Recently, Rieksts and Ventura 20 consider using both truckload TL transportation and less than truckload LTL transportation to fill order for inventory models that assume a constant demand rate that must be met without shortages.
However, the above research mainly focuses on operational aspects but neglects financial concens that firms may face when deciding the supply chain method.In real business environments, many firms are capital constrained and need to finance their operations from external capital markets.Today, payment within a specified period after delivery, usually called trade credit, is a widely observed pricing strategy for improving the profitability and cost effectiveness in sales e.g., [21][22][23] .Suppliers offer a trade credit as an incentive to increase sales and reduce stock, and the buyer can use the sale revenue to benefit without interest charged during the credit period.Goyal  The above articles have investigated the effect on inventory policy under trade credit from the perspective of the buyer or supplier only.However, decisions made by channel members are interdependent which determine the performances of other members as well as the entire channel.For example, a buyer's selling price decision will influence the customer demand and therefore the buyer's and supplier's sales volumes.As the buyer's order quantity decision based on customer demand will alter the supplier's production decision, while the supplier offers trade credit to encourage sales, the decisions on inventory and pricing of the supply chain will be altered after a while.Therefore, to improve the collaboration of supply chain partners, determining the optimal policies based on the integrated total profit function is more reasonable than using the buyer or the supplier's individual profit functions.Some scholars have noted this fact and worked on developing supply chain management decision models.Abad and Jaggi 39 developed ajoint approach to determine the optimal unit price and the length of the credit period when end demand is price sensitive under permissible delay in payments.Jaber and Osman 40 proposed a centralized model where players in a two-level supply chain coordinate their orders to minimize their local costs and that of the entire chain.Yang and Wee 41 considered an optimal replenishment policy with a credit term in a collaborative deteriorating inventory system when the demand is price sensitive and the replenishment rate is finite.Sheen and  46 considered an optimal replenishment and order policy with a credit term when the demand is price sensitive and the production is rate sensitive demand.In this study, to analyze pricing, ordering, delivery, and trade credit with comprehensive considerations of operation, marketing, and financing among channel members, a supplier-buyer inventory model is developed.We assume the supplier's unit selling price is based on his/her unit production cost which is decided by the market demand and production rates.And the production rate is adjusted with a price-sensitive market demand.In such circumstances the unit wholesale price, reflecting the costs of the product, imposed by the seller on the buyer, does influence the end demand for the product.In addition, the supplier offers to pay freight charges if an order quantity meets or exceeds a certain minimum requirement.Furthermore, a fixed trade credit period is offered by the supplier.In this paper, we maximize the total profit of the whole supply chain i.e., treating the supply chain as a single level profit centre .An algorithm is developed to determine the optimal ordering, shipping, and pricing policy.Numerical examples with relevant data are devoted to find the optimal policies of the developed model.Sensitivity analysis for main parameters is also conducted.The major difference between our model and other related models is shown in Table 1.

Mathematical Formulation
In this section, we consider an integrated inventory model with a retail price sensitive demand, where the supplier offers to pay freight charges if an order quantity exceeds or equal to a certain minimum requirement.In addition, a certain credit period is provided to the buyer.The relationship among members in this supply chain system is illustrated in Figure 1.To formulate the integrated inventory model, the supplier's total profit per unit time is discussed first.Then the buyer's total profit per unit time is discussed.

Supplier's Total Profit per Unit Time
During the production period, the supplier manufactures in batches of size nQ, where n is an integer, and incurs a batch setup cost S V .The production cycle length is nQ/D nT .Once the first Q units are produced, the supplier delivers them to the buyer and then continuously making the delivery on average every T units of time until the supplier's inventory level falls to zero.Therefore, the setup cost per unit time is S V / nT .
The inventory holding cost contains two components: unit holding cost and opportunity cost.The unit holding cost relates to the actual ownership of the goods and includes storage and maintenance expenses, which is accounted on a per-unit-of-inventory basis.The opportunity holding cost is charged on the money value of the inventory on hand.The supplier's inventory per unit time is given by where ρ D/R.Note that the similar derivation of supplier's average inventory using a manufacturing lot size of Q units can be found in Joglekar 48 .With production cost per unit c, the holding cost rate excluding interest charges r V and the capital opportunity cost per dollar per unit time I V p , the supplier's holding cost per unit time is c r Because of offering a credit period M to the buyer, the supplier endures a capital opportunity cost vI V p DM θcI V p DM within the time gap between delivery and payment received of the product.The supplier determines unit wholesale price $v θc based on unit production cost $c, therefore the sales revenue per unit time is v − c D θ − 1 cD.In addition, to encourage order more, if the buyer's order quantity Q ≥ ψ, the supplier is required to pay the transportation cost per unit time 1 − I A Q h wQ /T .Therefore, the supplier's total profit per unit, which is the sales revenue minus set-up cost, holding cost, capital opportunity cost, and transportation cost, can be written as follows: 2.2

Buyer's Total Profit per Unit Time
For the buyer, the total sales profit per unit time is given by p − v D p − θc D and the ordering cost per unit time is S B /T .With the unit purchasing cost v, the holding cost rate r B , and the average inventory over the cycle Q/2, the buyer's holding cost excluding interest charges per unit time is vr B Q/2 θcr B DT/2.The transportation cost per unit time is As the payment is done before or after the total depletion of inventory, we have the following two possible cases: i T ≤ M, and ii T ≥ M. Case 1 T ≤ M .In this case, as the permissible payment time expires on or after the inventory is depleted completely, the buyer pays no opportunity cost for the purchase items.Through the credit period, buyer sells the products and uses the sales revenue to earn interest at a rate of I Be .Thus, the interest earned per unit time is

2.3
Case 2 T ≥ M .When buyer's permissible payment time expires on or before the inventory is depleted completely, the buyer can sell the items and earn interest with rate I Be until the end of the credit period M. Thus, the interest earned per unit time is pI Be /T M 0 Dt dt DpI Be M 2 /2T .On the other hand, the buyer still has some inventory on hand when paying the total purchasing amount to the supplier.Hence, for the items still in stock, buyer endures a capital opportunity cost at a rate of I Bp ; the opportunity cost per unit time for the items is obtained by vI Bp /T T M D T − t dt θcI Bp D T − M 2 /2T .Therefore, the total profit per unit time for the buyer, which is the sales profit plus the interest earned, minus the total relevant costs, composed of ordering cost, holding cost, opportunity cost and transportation cost, can be expressed as follows: where

Theoretical Results
Once the supplier and buyer have established a long-term strategic partnership and are contracted to commit to the relationship, they will determine the best joint policy in which to cooperate.Under this circumstance, joint total profit per unit time for the supplier and buyer is where

3.3
Note that Π 1 n, p, M Π 2 n, p, M , hence joint total profit per unit time Π n, p, T is continuous at point T M for fixed n and p.
To find the optimal solution, say n * , p * , T * , that maximizes the above-integrated total profit, the following procedures are taken.First, for fixed p and T , check the effect of n on the joint total profit per unit time Π n, p, T with the fact , T is a concave function of n.Therefore, for fixed p and T , the search for the optimal shipment number, n * , is reduced to find a local optimal solution.Case 1 T ≤ M .For fixed n and p, with ∂ 2 Π 1 n, p, T /∂T 2  −2S/T 3 < 0, Π 1 n, p, T is a concave function of T ; hence, there exists a unique value of T denoted by T 1 n, p which maximizes Π 1 n, p, T .By solving ∂Π 1 n, p, T /∂T −D c θr B ϕ pI Be /2 S/T 2 0, T 1 n, p can be obtained and is given as To ensure T 1 n, p ≤ M, substituting 3.4 into this inequality results in where Conversely, if 2S > Δ, we have which implies that Π 1 n, p, T is an increase function of T ∈ 0, M .Hence, for fixed n and p, Π 1 n, p, T has a maximum value at the boundary point T M.
From the above results, we can easyily obtain the following lemma.The proof is omitted here.Lemma 3.1.For any given n and p, a if 2S ≤ Δ, then T T 1 n, p is the optimal value which maximizes Π 1 n, p, T .
b if 2S > Δ, then T M is the optimal value which maximizes Π 1 n, p, T .Case 2 T ≥ M .The first order necessary condition with respect to T for Π 2 n, p, T in 3.3 to be maximized is then we obtain the value of T denoted by T 2 n, p as To ensure T 2 n, p ≥ M, substituting 3.9 into this inequality results in the following: where Δ is defined as above.

3.10
Note that when 2S ≥ Δ holds, then holds, which implies that T 2 n, p in 3.9 is well defined.Besides, we can show that Hence, T 2 n, p in 3.9 is a unique value which maximizes Π 2 n, p, T .Conversely, if 2S < Δ, we have

3.13
Thus, Π 2 n, p, T is a strictly decreasing function of T ∈ M, ∞ , which implies that Π 2 n, p, T has a maximum value at the boundary point T M for fixed n and p.
From the above results, we can easily obtain the following lemma.The proof is omitted here.

Lemma 3.2. For any given n and p,
Combining Lemmas 3.1 and 3.2, we obtain the following result.Theorem 3.3.For any given n and p, Proof.It immediately follows from the facts that Π 1 n, p, M Π 2 n, p, M , Lemmas 3.1 and 3.2.

Next, let
3.14 and taking the derivative of f p with respect to p, it gets Therefore f p is a strictly increasing function of p. Furthermore, we have lim p → 0 f p −∞ and lim p → ∞ f p 2S > 0. Hence, a unique value p such that f p 0 exists, that is, Thus, we have 2S ≤ Δ iff p ≤ p.

3.17
Based on the above arguments and Theorem 3.3, we can easily obtain the following lemma.The proof is omitted here.Lemma 3.4.For any given n and p, From Lemma 3.4, when n and p are given, we can get the maximum joint total profit per unit time as follows:

3.20
Now, to obtain the optimal retail price p which maximizes Π n, p for fixed n, by taking the first-order partial derivative of Π i n, p, T i n, p , i 1, 2 in 3.19 and 3.20 with respect to p and by setting the result to be zero, we have
Therefore, we obtain the following result.Summarizing the above arguments, an efficient algorithm for obtaining the optimal solution n * , p * , T * is depicted.Algorithm 3.6.
Step 3. Find p 1 which satisfies p 1 ≤ p, ∂Π Step 5. Find Max i 1,2 {Π i n, p i , T i n, p i }.Set Π n n, p n , T n Max i 1,2 {Π i n, p i , T i n, p i }, then p n , T n is the optimal solution for this given n.
Step 6. Set n n 1. Repeat Steps 2 to 5 to find Π n n, p n , T n .
Once the optimal solution n * , p * , T * is obtained, the optimal order quantity Q * D p * T * follows.

Numerical Examples and Discussion
Example 4.1.In order to verify the proposed model, a numerical case will be used to demonstrate our model.We consider a company produces a product for an industrial client.It has experienced that the demand rate is price-relative.This item is produced with the market demand.To increase sales, a credit term "net 30" i.e., M 30 days is offered by the supplier, I V p 0.04/$/year, I Be 0.09/$/year, and I Bp 0.10/$/year.In addition, per shipment from the company to the client is assessed a fixed cost h $200/shipment and a variable cost w $0.5/unit.In addition, all shipments on orders equal to or over ψ 900 units are shipped free of charge.Summary of other parameters used is as follows: a 100000, δ 1.5, ρ 0.95, c 0 4.75, θ 2, γ 1.5, β 1.52, S V $1000/setup, S B $200/order, r V 0.05, and r B 0.1.
After running the basic set of parameters, the computational results are that the buyer charges his/her customers a retail price p * $12.8258/unit, the optimal replenishment cycle length is T * T 2 0.4170 per year and the demand rate is D p * a p * −δ 2177 units/year.The optimal lot size Q * D p * T * 908 units per order which is over the minimum boundary of free deliveries, so the shipping cost h wQ * $654 per shipment will be paid by the supplier.Under this condition, the buyer's annual total profit is $7653.Per production run, the supplier spends c c 0 D −β R γ $4.3990/unit to produce the items at rate R D/ρ 2, 292 units/year and sells them at a wholesale price v θc $8.7979/unit to the buyer.As there are 16 deliveries from the supplier to the buyer per production run, the supplier's annual total profit is $7490, and the maximumjoint total annual profit Π * n * , p * , T * Π * 16, 12.8258, 0.4170 $15143.When p and T are decision variables, the surface generated by the total profit function Π 16, p, T over a wide range of values of p and T is shown in Figure 2. The graph is drawn using MATHEMATICA 4.0.Furthermore, we show the numerical results with values of n 1, 2, . . ., 200.The numerical results indicate that there is a unique integer n which maximizes the value of Π n ≡ Π n n, p n , T n , as shown in Figure 3. Consequently, the solution obtained through this Algorithm is the optimal solution.Example 4.2.Using the same data as in Example 4.1, we investigate some values of M to analyze the effects of credit period on performance.Consider M ∈ {0, 10, 30, 45, 60, 75, 90}, the optimal solutions obtained through this Algorithm are presented in Table 2. To illustrate the relationship between credit terms and profit progress, we also demonstrate profit gain comparing with no trade credit, i.e., M 0 in percentage in the last three columns of Table 2.We define the profit gain profit with trade credit − profit without trade credit /profit without trade credit × 100%.Table 2 shows the optimal retail price p * and replenishment cycle length T * decreases when the credit period M is increasing.However, the optimal order quantities Q * increase first then drop.Observe from Table 2 that as the credit period M increases, the profit gains in percentage are positive for the supplier and entire supply chain system, but are not always positive for the buyer.These results reveal that a larger credit period may motivate the buyer to order a smaller quantity and shorten the replenishment cycle length in order to take advantages of the trade credit more frequently.Yet, if the buyer orders less than ψ 900 units , shipping costs will be charged.The result indicates that when a firm faces a trade-off between trade credit and free-freight order, receiving a trade credit is not always a good idea.3.
Table 3 indicates that the lower the supplier's capacity utilization, that is, the greater inefficient production in the supply chain, the higher production cost.The increasing cost in production leads to rising retail price which in turn reduces market demand and drop profits of the supply chain system.Therefore, while the value of ρ decreases, the expected profits per unit time of the buyer and the entire supply chain decrease.On the other hand, the supplier's expected profits per unit time increase first then drop.This results from free shipping offered by the supplier with an order amount over 900 units as ρ 0.95.The managerial implication of the result is that if the supplier can obtain the demand information of final customers through the buyer, then he/she may employ the information to adjust his/her production to meet this demand and optimize the entire supply chain.
Example 4.4.Following assumptions 5 in Section 2, the supplier's unit production cost is directly related to the production rate R and inversely related to demand rate D, which is given by c D, R c 0 D −β R γ .To understand the effect of various values of γ and β on the channel performance, using the same parameter values as in Example 4.1, we apply the Algorithm to obtain the optimal solutions.The results are shown in Tables 4 and 5.
From Table 4, we can see that for a certain value γ 1.5 , increasing β leads to a higher order size as well profit gains for the buyer and entire channel.However, with the increase in β, the supplier's profit increases first then drops.The results indicate that, while unit production cost is sensitive to market demand rate, the supplier will make a mass production to reduce unit production cost.A decrease retail price resulting from lower unit production cost yields an increase in market demand increase.Then the buyer's profit gain increases with an upward market demand.On the other hand, though the supplier can reduce unit production cost by producing in large batches, however, the holding costs also increase with large production quantity.Consequently, a larger production lot size is not always more beneficial for the supplier.Furthermore, it can be noted from Table 5 that the optimal order sizes decline with the increase in γ for a specific β 1.52 .In addition, profits for the buyer and entire channel decrease but increase first then drop for the supplier.Since the unit production cost increases with γ, therefore, higher unit wholesale price and retail price are required.As expected, the increasing retail price significantly reduces market demand significantly resulting in lower total profit for the buyer.On the other hand, first the supplier's profit goes up with the increasing wholesale price, however, as the demand level falls below its desired target this leads to a much lower annual profit for the supplier.
Example 4.5.The purpose of this example is to evaluate the relative performances for various values of the problem parameters.The study was conducted for different values of h, w, and r B /r V .With the exception of the selected parameters, the values of other parameters have been kept the same as in Example 4.1.The optimal policy maximizing the channel's profit for the various problem parameters is reported in Table 6.
From Table 6, it is observed that as the value of r B /r V increases, that is, as the relative carrying cost rate excluding interest charge for the buyer increases, the buyer will order a smaller lot size within a shorter inventory cycle, so more replenishments for each production run higher value of n are required.While the value of w increases i.e., the relative unit shipping cost increases , the buyer will order a smaller lot size within a longer inventory cycle in order to save cost while selling the items to customers at a higher retail price.Also, the entire channel's expected total profit reduces as the ratio r B /r V and w increase.Finally, Table 6 shows with the increase in h the replenishment cycle and lot size both increase first then decrease.And the number of shipments from the supplier to the buyer and channel profit decrease first then increase.

Conclusion
In this paper, we consider a single-supplier single/buyer supply chain problem where the production rate of the supplier is assumed to be linearly related to the market demand rate, while demand is sensitive to retail price.The wholesale price imposed by the supplier on the buyer is based on his/her unit production cost which is determined by the market demand rate and production rate.The supplier produces one product in batches and periodically delivers the product at a fixed lot size to the buyer.In addition, to encourage the retailer to order more, the supplier offers a trade credit and a quantity-dependent free freight.By analyzing the total channel profit function, we then developed a solution algorithm to determine the optimal retail price, replenishment cycle length and the number of shipments per production cycle from the supplier to the buyer.Numerical examples are presented to illustrate this model.Comprehensive sensitivity analyses for the effects of the parameters on the optimal solutions are also offered.
The following observations could be made from the numerical examples.First, when the buyer faces a trade-off between ordering a smaller quantity to take advantages of the trade credit more frequently against ordering a larger quantity to take advantages of free shipping, he/she must carefully weigh the pros of each.Second, it is found that if the supplier can acquire "real time" market demand rate from the buyer and adjust production rate to match, this not only helps to reduce production cost for the supplier but also increases the profit gain for the entire supply chain.Finally, the result indicates that a larger production lot size is not always more economical for the supplier.It is because though the supplier can achieve economies of scale by producing in large batches, however, his/her holding and transportation costs also increase with a large order quantity.
As for future research, our model can be extended to more general supply chain networks, for example, multiechelon or assembly supply chains.Also, it is interesting to consider deteriorating items into the proposed model.

Notation and Assumptions
The following notations are adopted throughout this paper: R: Supplier's production rate.T : Buyer's replenishment cycle length decision variable .
Q: Buyer's order quantity per order decision variable .
h: Fixed shipping cost per delivery.
w: Unit shipping cost.
TV P : The supplier's expected total profit per unit time.
TBP : The buyer's expected total profit per unit time.
Π: The channel's expected total profit per unit time.
In addition, the following assumptions are made in deriving the model.
1 There is single supplier and single buyer for a single product.
2 Shortages are not permitted.

18
Abstract and Applied Analysis 3 The market demand rate for the product is assumed sensitive to the buyer's selling price p and is given by D p ap −δ , where a > 0 is a scaling factor, and δ > 0 is a price-elasticity coefficient.For notational simplicity, D p and D will be used interchangeably in this paper.4 The supplier's capacity utilization, ρ, is the ratio of the demand rate, D, to the production rate, R, which is given as less than 1, that is, ρ D/R and ρ < 1.
5 The supplier's unit production cost is directly related to the production rate R and inversely related to demand rate D, which is given by c D, R c 0 D −β R γ , where c 0 , β and γ are nonnegative real numbers and satisfy 0 < β − γ < 1.Similar assumption has been considered in Cheng 47 .For notational simplicity, c D, R and c will be used interchangeably in this paper.
6 Each unit is produced for $c and sold $v to the buyer, where v θc, θ > 1. Afterward, each unit is sold by the buyer on the market for $p > v .
7 The buyer's replenishment cycle length is T and order quantity is Q DT per order.
8 The supplier manufactures, at rate R, in batches of sizes of nQ and incurs a batch set up cost S V .Each batch is dispatched to the buyer in n equal size shipments.
9 Per shipment from the supplier to the buyer is assessed a fixed cost h that includes insurance on consignment invoice value, trucking costs, and a variable cost w for the unit shipping.In addition, free shipping is offered when the amount ordered reaches the minimum amount ψ.That is the buyer's transportation cost per order is I A Q h wQ , where 10 During the credit period, the buyer sells the items and uses the sales revenue to earn interest at a rate of I Be .At the end of the permissible delay period, the buyer pays the purchasing cost to the supplier and incurs an opportunity cost at a rate of I Bp for the items in stock.
24 is the first one who developed an EOQ model under the condition of permissible delay in payments.Numerous other studies on trade credit have since been published including Aggarwal and Jaggi 25 , Jamal et al. 26 , Chang and Dye 27 , Teng 28 , Arcelus et al. 29 , Biskup et al. 30 , Chang et al. 31 , Huang 32 , Chang 33 , Chung et al. 34 , Ouyang et al. 35 , Teng et al. 36 , Chung and Liao 37 , and Sarmah et al. 38 .

Figure 1 :
Figure 1: The supply chain system.

Figure 2 :
Figure 2: Concavity of the total profit function Π n n, p, T for fixed n 16.

Example 4 . 3 .
Here we examine the issue of how sensitive the performances of the supply chain are to the supplier's capacity utilization parameter ρ.Using the same data as in Example 4.1 except the value of ρ belongs to the set {0.95, 0.75, 0.50, 0.25}.The results are reported in Table S V : Supplier's setup cost per setup.S B : Buyer's ordering cost per order.
r V : Supplier's holding cost rate, excluding interest charges.
r B : Buyer's holding cost rate, excluding interest charges.c: Supplier's unit production cost.v: Supplier's unit wholesale price.p: Buyer's unit retail price decision variable .D: Market demand rate for the product.M: Buyer's credit period offered by the supplier per order.I V p : Supplier's capital opportunity cost per dollar per unit time.I Bp : Buyer's capital opportunity cost per dollar per unit time.I Be : Buyer's interest earned per dollar per unit time.n: Number of shipments from supplier to buyer per production run, a positive integer decision variable .

Table 1 :
Comparison between the papers discussing integrated inventory model with a trade credit.
"-" denotes the factor is not considered in the model.
Theorem 3.5.For any given n, a if there exists a value p 1 which satisfies the corresponding ∂Π 1 n, p, T 1 n, p /∂p 0 in 3.21 , ∂ 2 Π 1 n, p, T 1 n, p /∂p 2 < 0 and p 1 ≤ p, then p, T p 1 , T 1 n, p 1 is the optimal solution such that Π 1 n, p 1 , T 1 n, p 1 has a maximum value, b if there exists a value p 2 which satisfies the corresponding ∂Π 2 n, p, T 2 n, p /∂p 0 in 3.22 , ∂ 2 Π 2 n, p, T 2 n, p /∂p 2 < 0, and p 2 > p, then p, T p 2 , T 2 n, p 2 is the optimal solution such that Π 2 n, p 2 , T 2 n, p 2 has a maximum value.Proof.It immediately follows from Lemma 3.4.

Table 2 :
Influence of optimal solution with different M values.

Table 3 :
Influence of optimal solution with different ρ values.

Table 4 :
Computation results for various values of β γ 1.5 .

Table 5 :
Computation results for various values of γ β 1.52 .

Table 6 :
The results of sensitivity analysis.