Supply Chain Coordination for Fixed Lifetime Products with Permissible Delay in Payments

Abstract

This paper considers a two-level supply chain coordination problem for fixed lifetime products with permissible delay in payments. Two cases are discussed; that is, the retailer is required to settle the balance before the end of the ordering cycle (Case I) and after the ordering cycle (Case II). The coordination models are proposed and analyzed, respectively. The analytical methods as to how to determine the optimal policy are presented. In addition, it is indicated that the supplier's cost as well as that of the total system will be reduced no matter how much the parameters change, and the retailer will benefit from coordination, if the supplier is willing to share the cost saving with him/her in Case I. In Case II, the retailer's average cost will be reduced and the supplier will benefit from coordination only under certain conditions. Besides, the results show that, for fixed lifetime products, providing longer credit period than the retailer's order period is not commonly applicable.

1. Introduction

In recent years, the area of supply chain management has become very popular. Many researchers have devoted considerable attention to coordination issues between suppliers and retailers in a supply chain. A variety of coordination mechanisms are developed and utilized in the different settings.

Monahan [1] studied a single supplier-single buyer distribution system and indicated that the supplier could improve channel coordination and earn higher profits by offering the buyer a quantity discount. Later, Lee and Rosenblatt [2] generalized Monahan’s model by relaxing his lot-for-lot assumption for the supplier. Shin and Benton [3] studied the effectiveness of quantity discounts under different conditions. They showed that the link between quantity discounts and the performance of the chain is influenced by several other factors including the variability of demand, the relative inventory cost structures, and the buyer’s economic reorder intervals. Jaber and Osman [4] proposed a centralized model where the players in a two-level supply chain coordinated their orders through delay in payments to minimize their local costs and that of the chain. Chen and Kang [5] developed the integrated models with permissible delay in payment for determining the coefficient of negotiation and the maximum delay payment period. Sarmah et al. [6] studied a coordination mechanism through credit option such that both parties can divide the surplus equitably after satisfying their own profit targets. Emmons and Gilbert [7] studied the effect of return policies on both manufacturer and retailer’s profits. In their model, demand is uncertain and price-dependent. Giannoccaro and Pontrandolfo [8] coordinated a three-stage supply chain with a stochastic customer demand by using the revenue sharing policy. Extensive reviews of the literature on supply chain coordination are available in [9, 10].

In the supply chain coordination literatures that have been mentioned above, the researchers all ignored the constraint of items’ finite lifetime in their models. In fact, many products or goods have their own sell-by dates. Foodstuff, such as bread, milk, and beverage, all become unusable after a certain time. Photographic material, blood, and drugs are further examples of items with fixed lifetime. Moreover, the upgrade of IT products, the launch of new products, and the change of popularity make more and more products have a certain lifetime. In the past few decades, the inventory optimization of the finite lifetime product has been the focus of many researchers. Nahmias [11] classified the finite lifetime problem into two subcategories: random finite lifetime and fixed finite lifetime. The random finite lifetime products, such as fruit and vegetable, spoil over time and the value also decreases gradually. Items with the fixed finite lifetime, on the other hand, perish at the same age; that is, the value remains the same during the lifetime period and equals to zero after the expiration date. The relevant researches such as [12–17] are about the random finite lifetime products. Goyal and Giri [18] provided an excellent and detailed review of research on inventory problem of this kind of product. The researches about the fixed finite lifetime products can refer to [19–26] and their related references.

However, the above researches on fixed finite lifetime product are on single stage problem. Duan et al. [27] considered a single-vendor, single-buyer supply chain for fixed lifetime product and proposed models to analyze the benefit of coordinating supply chain through a quantity discount strategy. In addition, they proved that the quantity discount strategy can achieve system optimization and win-win outcome.

Delay in payment is one of the commonly used mechanisms in supply chain coordination practice. In real world, the supplier often utilizes this policy to promote his/her products, especially when the retailer is short of capital. In this policy, the supplier requires the retailer to increase his order size by offering the retailer a delay period. And the retailer is required to settle the balance before the end of the delay period. During the delay period, the retailer can cumulate revenues and earn interests. The policy can reduce the retailer’s capital cost tied up in stock, because it reduces the amount of capital invested in stock for the duration of the delay period, so that the retailer’s holding cost of finance is reduced. The policy can also decrease supplier’s ordering cost, but it adds an additional dimension of default risk to the supplier.

Duan et al. [28] dealt with a supply chain coordination problem through delay in payments for items with fixed lifetime, where the costs of the supplier and retailer include the ordering cost and the storage cost. The opportunity cost of capital cost is not considered. However, when the products are kept in the warehouse, a large amount of capital will be backlogged, which will incur great opportunity cost in addition to the storage cost. Furthermore, in this paper, we consider supply chain coordination issues for fixed lifetime product through delay in payment. In the delay in payment contract, the supplier will provide a credit period for the retailer. In this case, the capital cost of the retailer is a nonlinear and a piecewise function of time. So it is necessary to investigate how the optimal policy is changed when capital cost is considered. The main differences between Duan et al. [28] and this paper are as follows. Firstly, in addition to the ordering and storage cost, this paper considers the opportunity cost of capital in the model formulation, which is not considered in [28]; secondly, because the retailer’s opportunity costs of capital backlogging are dependent on the time when the balance are settled, we discuss two cases. Case I: the retailer is required to settle the balance before the end of the ordering cycle and Case II: the retailer settles the balance after the ordering cycle. Thirdly, we find that when the opportunity cost is not considered, the supplier will always benefit no matter how much the parameters change, and the cost of the retailer will not change. However, when the opportunity cost is considered and the retailer settles the balance after the ordering cycle, then the cost of the supplier may increase but the cost of the retailer may decrease. So the opportunity costs have significant influence on the optimal coordination policy.

The rest of this paper is organized as follows. In Section 2, the supply chain coordination model to minimize the supplier’s cost is formulated, and the analytically tractable solutions are derived. A numerical study is conducted in Section 3 to illustrate the performance of the proposed policy and examine the implications of the change in the value of parameters. The concluding remarks are presented in the last section.

2. Model Formulation

In this section, we formulate and analyze the decision-making models of a two-level supply chain for fixed lifetime products without and with coordination. In the absence of coordination, both of the parties make decisions to minimize their own costs. In the presence of coordination, the supplier is the decision-maker of the supply chain with the objective of minimizing his/her own cost, and the delay in payments is adopted as a coordination policy. In addition, we obtain the optimal policies for the different cases and find that the lifetime of the product is an important constraint in the optimal policies.

We adopt the same assumptions as in Duan et al. [28] which are also commonly used in the literatures, see Monahan [1] and Lee and Rosenblatt [2].(i)Demand is exogenous and constant. The main reason of this assumption is to simplify the model formulation and to derive the closed-form solution to the model.(ii)Replenishment is instantaneous and lead time is zero. Obviously, if the lead time is constant and not zero, the supplier and retailer just need to place an order a period exactly equal to the lead time in advance [2], and the same results can be obtained.(iii)Shortages are not allowed. For fixed lifetime products, due to their limited lifetime, the customers would purchase only when they are to be used up. If stock-out happens, most of them will buy the substitute products. In addition, because the demand is assumed to be constant in this paper, the shortage can be eliminated by increasing the ordering quantity.(iv)All items ordered by the supplier arrive new and fresh; that is, their age equals zero [29]. In fact, if the items’ age is 𝑙 when they arrive, we only need to replace the product lifetime 𝐿 by 𝐿=𝐿−𝑙 in the following analysis.

Notations used in this paper are as follows: 𝑖 denotes the subscript identifying a specific level in a supply chain (𝑖=𝑠,𝑟, where 𝑠= supplier and 𝑟= retailer); 𝐷 denotes the constant demand rate; 𝐴_𝑖 is the ordering cost of level 𝑖; ℎ_𝑖 is the holding cost of finance per unit per unit time, representing the cost of capital at level 𝑖 excluding the storage cost; 𝑠_𝑖 is the storage cost per unit per unit time at level 𝑖 excluding the holding cost; 𝑐_𝑖 is the procurement cost per unit for level 𝑖; 𝑘_𝑖 is the return on investment for level 𝑖; 𝑄₀ is the retailer’s EOQ, and 𝑡₀ is the order period of the retailer in the absence of any coordination; 𝐿 denotes the lifetime of product and 𝐿≥𝑡₀; 𝑇𝐶_𝑠 and 𝑇𝐶_𝑟 are the average costs of the supplier and the retailer in the absence of coordination; 𝑇𝐶_𝑠 and 𝑇𝐶_𝑟 are the average costs of the supplier and the retailer in the presence of coordination.

Decision Variables 𝜇 denotes the supplier’s order multiple in the absence of any coordination; 𝐾 denotes the retailer’s order multiple in the presence of coordination and 𝐾𝑄₀ is the retailer’s new order size; 𝑡(𝐾) denotes the credit period offered to the retailer, if he orders 𝐾𝑄₀  every interval of 𝐾𝑡₀ (interest free period); 𝜆 denotes the supplier’s order multiple in the presence of coordination.

2.1. Model Formulation for System without Coordination

In this subsection, the decision-making model for fixed lifetime products without coordination is formulated. The retailer and the supplier make decisions by minimizing their own costs.

In the absence of any coordination, the retailer’s order quantity is simply the EOQ; that is, 𝑄₀=2𝐷𝐴_𝑟ℎ_𝑟+𝑠_𝑟. (1)

Let 𝑇𝐶_𝑟 denote the retailer’s average cost per unit time and let 𝑡₀ denote the retailer’s optimal order period; then 𝑇𝐶_𝑟=2𝐷𝐴_𝑟ℎ_𝑟+𝑠_𝑟,𝑡₀=2𝐴_𝑟𝐷ℎ_𝑟+𝑠_𝑟. (2)

Since the retailer’s order quantity is fixed at 𝑄₀, the supplier is faced with a stream of demands, each with order size 𝑄₀ and at fixed intervals of 𝑡₀. Given such a stream of demands, the supplier’s economic order quantity should be some integer multiple of 𝑄₀ [2]. Let 𝜇𝑄₀ denote the supplier’s order quantity where 𝜇 is a positive integer.

The supplier’s average inventory held up per unit time is 𝜇−1𝑄₀+𝜇−2𝑄₀+⋯+𝑄₀+0𝑄₀𝜇=𝜇−1𝑄₀2. (3)

The supplier’s cost per unit time is 𝑇𝐶_𝑠𝜇=𝐷𝐴_𝑠𝜇𝑄₀+ℎ_𝑠+𝑠_𝑠𝜇−1𝑄₀2. (4)

Hence, the supplier’s problem in the absence of coordination can be formulated as follows: min𝑇𝐶_𝑠𝜇=𝐷𝐴_𝑠𝜇𝑄₀+ℎ_𝑠+𝑠_𝑠𝜇−1𝑄₀2s.t.𝜇𝑄₀𝐷≤𝐿,𝜇≥1isaninteger. (5)

By substituting 𝑄₀ into (5), we get min𝑇𝐶_𝑠𝜇=𝐴_𝑠𝜇𝐷ℎ_𝑟+𝑠_𝑟2𝐴_𝑟+ℎ_𝑠+𝑠_𝑠𝜇−1𝐷𝐴_𝑟2ℎ_𝑟+𝑠_𝑟s.t.𝜇2𝐴_𝑟𝐷ℎ_𝑟+𝑠_𝑟≤𝐿,𝜇≥1isaninteger. (6)

The first constraint in (6) is to ensure that the products are not overdue before they are sold out. In this case, the supplier will minimize his average cost by selecting the optimal order multiple 𝜇. In the next section, we will give the analytical solution to (6).

Lemma 1. If 𝐿≥𝑡₀=2𝐴_𝑟/𝐷(ℎ_𝑟+𝑠_𝑟), then the supplier’s optimal order multiple is as follows: 𝜇^∗=min𝜇^∗₁,𝜇^∗₂, (7) where 𝜇^∗₁=⌈1/4+𝐴_𝑠(ℎ_𝑟+𝑠_𝑟)/𝐴_𝑟(ℎ_𝑠+𝑠_𝑠)−1/2⌉, 𝜇^∗₂=[𝐿𝐷(ℎ_𝑟+𝑠_𝑟)/2𝐴_𝑟], ⌈𝑥⌉ denotes the least integer greater than or equal to 𝑥; [𝑥] denotes the integer part of 𝑥.

The proof of Lemma 1 can refer to Duan et al. [28]. Lemma 1 gives the supplier’s optimal order multiple of the retailer’s ordering quantity without coordination. Here, 𝐿≥𝑡₀ is to ensure that the EOQ model is to ensure that the items are not expired before they are sold out.

2.2. Model Formulation for System with Coordination

In this subsection, we will propose and discuss the supply chain coordination model for fixed lifetime products through delay in payments policy. In this policy, the retailer orders 𝐾𝑄₀ units products every interval of 𝐾𝑡₀ from the supplier at a unit purchasing cost 𝑐_𝑟 and the ordering cost 𝐴_𝑟. The supplier orders 𝜆𝐾𝑄₀ units every interval of 𝜆𝐾𝑡₀ and offers the credit period 𝑡(𝐾) to the retailer, where 𝐾, 𝜆, and 𝑡(𝐾) are decision variables. Meanwhile, the retailer has the opportunity to invest the unpaid balance 𝑐_𝑟𝐾𝑄₀ for a period 𝑡(𝐾) at a return rate of 𝑘_𝑟. That is, if the retailer can settle his balance at the time 𝑡(𝐾), they will have a capital gain of 𝑐_𝑟𝐾𝑄₀𝑘_𝑟𝑡(𝐾), but the supplier’s cost will be increased by 𝑐_𝑟𝐾𝑄₀𝑘_𝑠𝑡(𝐾).

Therefore, the supplier’s cost per unit time 𝑇𝐶_𝑠(𝐾,𝜆) is composed of the following:(i)the supplier’s ordering cost per unit time which is equal to  𝐷𝐴_𝑠/𝜆𝐾𝑄₀,(ii)the supplier’s holding cost of finance per unit time which is equal to ℎ_𝑠𝐾𝑄₀(𝜆−1)/2+ℎ_𝑠𝐷𝑡(𝐾),(iii)the supplier’s storage cost per unit time which is equal to 𝑠_𝑠𝐾𝑄₀(𝜆−1)/2,(iv)the supplier’s opportunity cost per unit time which is equal to 𝑐_𝑟𝐷𝑘_𝑠𝑡(𝐾).

Hence, 𝑇𝐶_𝑠(𝐾,𝜆)=𝐷𝐴_𝑠𝜆𝐾𝑄₀+ℎ_𝑠+𝑠_𝑠𝐾𝑄₀(𝜆−1)2+ℎ_𝑠𝐷𝑡(𝐾)+𝑐_𝑟𝐷𝑘_𝑠𝑡(𝐾). (8)

The retailer’s cost consists of the following:(i)the retailer’s ordering cost per cycle which is equal to 𝐴_𝑟,(ii)the retailer’s holding cost of finance per cycle which is equal to 𝐻_𝑟(𝐾)=ℎ_𝑟𝐾𝑄₀−𝐷𝑡(𝐾)²(𝐾)2𝐷,0≤𝑡≤𝐾𝑡₀0,𝑡(𝐾)≥𝐾𝑡₀, (9)(iii)the retailer’s storage cost per cycle which is equal to 𝑠_𝑟𝐾²𝑄₀𝑡₀/2,(iv)the retailer’s revenue during the credit period which is equal to 𝑐_𝑟𝐾𝑄₀𝑘_𝑟𝑡(𝐾).

Therefore, the retailer’s cost per unit time is 𝑇𝐶_𝑟(𝐾)=𝐴_𝑟+𝐻_𝑟(𝐾)+𝑠_𝑟𝐾²𝑄₀𝑡₀/2−𝑐_𝑟𝐾𝑄₀𝑘_𝑟𝑡(𝐾)𝐾𝑡₀=𝐷𝐴_𝑟𝐾𝑄₀+𝐷𝐻_𝑟(𝐾)𝐾𝑄₀+𝑠_𝑟𝐾𝑄₀2−𝑐_𝑟𝐷𝑘_𝑟𝑡(𝐾). (10)

To entice the retailer to accept this policy, the supplier should ensure that the retailer’s average cost per unit time with coordination is not greater than that without any coordination. The coordination problem can be formulated as the following mathematical programming: min𝑇𝐶_𝑠(𝐾,𝜆)s.t.𝜆𝐾𝑡₀≤𝐿𝑇𝐶_𝑟(𝐾)≤𝑇𝐶_𝑟𝜆≥1isaninteger. (11)

The first constraint of (11) is to guarantee that the supplier’s products are not overdue before they are sold out. The second constraint is to ensure that the policy proposed by the supplier is acceptable to the retailer.

Based on the value of 𝐻_𝑟(𝐾), two cases are discussed, respectively, in the following, namely, Case I: 0≤𝑡(𝐾)≤𝐾𝑡₀ and Case II: 𝑡≥𝐾𝑡₀.

Case I. When 0≤𝑡(𝐾)≤𝐾𝑡₀, 𝐻_𝑟(𝐾)=ℎ_𝑟(𝐾𝑄₀−𝐷𝑡(𝐾))²/2𝐷.
In this case, (11) is reduced to the following problem: min𝑇𝐶_𝑠(𝐾,𝜆)=𝐷𝐴_𝑠𝜆𝐾𝑄₀+ℎ_𝑠+𝑠_𝑠𝐾𝑄₀(𝜆−1)2+ℎ_𝑠𝐷𝑡(𝐾)+𝑐_𝑟𝐷𝑘_𝑠𝑡(𝐾)𝑄s.t.𝜆𝐾⁰𝐷≤𝐿𝐷𝐴_𝑟𝐾𝑄₀+ℎ_𝑟𝐾𝑄₀−𝐷𝑡(𝐾)²2𝐾𝑄₀+𝑠_𝑟𝐾𝑄₀2−𝑐_𝑟𝐷𝑘_𝑟𝑡(𝐾)≤2𝐷𝐴_𝑟ℎ_𝑟+𝑠_𝑟𝜆≥1. (12)

Theorem 2. Let 𝜇^∗ and 𝜆^∗ be the supplier’s optimal order multiple without and with coordination, and let 𝐾^∗ be the retailer’s optimal order multiple with coordination; then 𝑇𝐶_𝑠𝐾^∗,𝜆^∗≤𝑇𝐶_𝑠𝜇^∗. (13)

Proof. It is easy to verify that 𝑇𝐶_𝑠(𝐾,𝜆) is increasing in 𝑡(𝐾). 𝑇𝐶_𝑟(𝐾) is convex and decreasing in 𝑡(𝐾), for 0≤𝑡(𝐾)≤𝐾𝑡₀. As a result, the objective function of (12) is minimized when the second constraint is an equation; that is, 𝐷𝐴_𝑟𝐾𝑄₀+𝐷ℎ_𝑟𝐾𝑄₀𝐾𝑄₀−𝐷𝑡(𝐾)²+𝑠2𝐷^𝑟𝐾𝑄₀2−𝑐_𝑟𝐷𝑘_𝑟𝑡(𝐾)=2𝐷𝐴_𝑟ℎ_𝑟+𝑠_𝑟. (14)
By rearranging the terms, we have ℎ_𝑟2(𝐷𝑡(𝐾))²−ℎ_𝑟+𝑐_𝑟𝑘_𝑟𝐾𝑄₀𝐷𝑡(𝐾)+𝐴_𝑟𝐷(𝐾−1)²=0. (15)
Let 𝑡₁≤𝑡₂ be the two roots of (15); then, if (ℎ_𝑟+𝑐_𝑟𝑘_𝑟)²𝑄²₀𝐾²−2ℎ_𝑟𝐴_𝑟𝐷(𝐾−1)²≥0, 𝑡₁=ℎ_𝑟+𝑐_𝑟𝑘_𝑟𝐾𝑄₀−ℎ_𝑟+𝑐_𝑟𝑘_𝑟²𝐾²𝑄²₀−2ℎ_𝑟𝐴_𝑟𝐷(𝐾−1)²ℎ_𝑟𝐷,𝑡₂=ℎ_𝑟+𝑐_𝑟𝑘_𝑟𝐾𝑄₀+ℎ_𝑟+𝑐_𝑟𝑘_𝑟²𝐾²𝑄²₀−2ℎ_𝑟𝐴_𝑟𝐷(𝐾−1)²ℎ_𝑟𝐷. (16)
Since (ℎ_𝑟+𝑐_𝑟𝑘_𝑟)𝐾𝑄₀>0 and 𝐴_𝑟𝐷(𝐾−1)²>0, for 𝐾>1, then 𝑡₁>0 and 𝑡₂>0. In addition, by noting that 𝑡^∗(𝐾)≤𝐾𝑄₀/𝐷, we have 𝑡^∗(𝐾)=ℎ_𝑟+𝑐_𝑟𝑘_𝑟𝐾𝑄₀−ℎ_𝑟+𝑐_𝑟𝑘_𝑟²𝐾²𝑄²₀−2ℎ_𝑟𝐴_𝑟𝐷(𝐾−1)²ℎ_𝑟𝐷. (17)
It can be seen from (17) that, if 𝐾=1, then 𝑡^∗(𝐾)=0. (12) is reduced to (6), if 𝐾=1, so (13) holds. The proof of Theorem 2 is complete.

Theorem 2 indicates that the supplier’s cost as well as that of the system can be reduced in the proposed delay in payment policy, and the retailer’s cost is not changed. In practice, the supplier will usually share the saving cost with the retailer, and the sharing rate is determined by their power of balance between them.

Theorem 3. The retailer’s order size with coordination is greater than that without coordination; that is, 𝐾≥1.

Proof. Since 𝑑²𝑡^∗(𝐾)𝑑𝐾²=2ℎ_𝑟+𝑐_𝑟𝑘_𝑟²𝐾²𝑄²₀−2ℎ_𝑟𝐴_𝑟𝐷(𝐾−1)^2−3/2×ℎ_𝑟+𝑐_𝑟𝑘_𝑟²ℎ_𝑟𝐴_𝑟𝐷𝑄²₀>0, (18)
𝑡^∗(𝐾) is convex in 𝐾. In addition, it is easy to verify that 𝑑𝑡^∗(𝐾)/𝑑𝐾|_𝐾=1=0; so 𝑡^∗(𝐾) is minimized, when 𝐾=1. In practice, the more the retailer orders, the longer the delay period should be; so 𝑡^∗(𝐾) should increase in 𝐾. It is straight forward that only if 𝐾≥1, 𝑡^∗(𝐾) is increasing in 𝐾. As a result, 𝐾≥1 holds. The proof of Theorem 3 is complete.

Theorem 3 demonstrates that delay in payments policy can indeed induce the retailer to increase his/her order quantity.

Then, we will prove 𝑡^∗(𝐾)≤𝐾𝑡₀.

Proposition 4. If 𝑠_𝑟>2𝑐_𝑟𝑘_𝑟, then 0≤𝑡^∗(𝐾)≤𝐾𝑡₀ holds for 1≤𝐾≤𝐾₁, where 𝐾₁=1/(1− (ℎ_𝑟+2𝑐_𝑟𝑘_𝑟)/(ℎ_𝑟+𝑠_𝑟)).

Proof. The condition for the existence of 𝑡^∗(𝐾) is ℎ_𝑟+𝑐_𝑟𝑘_𝑟²𝑄²₀𝐾²−2ℎ_𝑟𝐴_𝑟𝐷(𝐾−1)²≥0; (19) that is ℎ_𝑟+𝑐_𝑟𝑘_𝑟²ℎ_𝑟ℎ_𝑟+𝑠_𝑟≥(𝐾−1)²𝐾². (20)
In addition, to prove that 𝑡^∗(𝐾)≤𝐾𝑄₀/𝐷 holds, that is, ℎ_𝑟+𝑐_𝑟𝑘_𝑟𝐾𝑄₀−ℎ_𝑟+𝑐_𝑟𝑘_𝑟²𝐾²𝑄²₀−2ℎ_𝑟𝐴_𝑟𝐷(𝐾−1)²ℎ_𝑟𝐷≤𝐾𝑄₀𝐷, (21)
we only need to prove that the following inequality holds: ℎ_𝑟+2𝑐_𝑟𝑘_𝑟ℎ_𝑟+𝑠_𝑟≥(𝐾−1)²𝐾². (22)
By (ℎ_𝑟+𝑐_𝑟𝑘_𝑟)²/ℎ_𝑟(ℎ_𝑟+𝑠_𝑟)≥(ℎ_𝑟+2𝑐_𝑟𝑘_𝑟)/(ℎ_𝑟+𝑠_𝑟), we know whether (ℎ_𝑟+2𝑐_𝑟𝑘_𝑟)/(ℎ_𝑟+𝑠_𝑟)≥(𝐾−1)²/𝐾²; then (ℎ_𝑟+𝑐_𝑟𝑘_𝑟)²/ℎ_𝑟(ℎ_𝑟+𝑠_𝑟)≥(𝐾−1)²/𝐾² holds. Therefore, 11≤𝐾≤1−ℎ_𝑟+2𝑐_𝑟𝑘_𝑟/ℎ_𝑟+𝑠_𝑟. (23)
Set 𝐾₁=1/(1−(ℎ_𝑟+2𝑐_𝑟𝑘_𝑟)/(ℎ_𝑟+𝑠_𝑟)). By 𝐾≥1, we have 𝐾₁≥1; that is, 𝑠_𝑟>2𝑐_𝑟𝑘_𝑟. If 𝑠_𝑟>2𝑐_𝑟𝑘_𝑟 and (23) holds, it is easy to verify that 𝑡^∗(𝐾) exists and 𝑡^∗(𝐾)≤𝐾𝑡₀. The proof of Proposition 4 is complete.

Proposition 4 shows that when the retailer’s storage cost is large, to reduce the storage cost, the retailer will not order too much items in the beginning of every ordering cycle. Accordingly, the trade credit period offered by the supplier will be less than the ordering cycle of the retailer.

Next, we will determine the supplier and the retailer’s optimal ordering quantities 𝐾^∗ and 𝜆^∗.

By substituting (17) into (12), we have min𝑇𝐶_𝑠()=𝐴𝐾,𝜆^𝑠𝐷𝜆𝐾𝑄₀+ℎ_𝑠+𝑠_𝑠(𝜆−1)2+𝑐_𝑟𝑘_𝑠+ℎ_𝑠ℎ_𝑟+𝑐_𝑟𝑘_𝑟ℎ_𝑟𝐾𝑄₀−𝑐_𝑟𝑘_𝑠+ℎ_𝑠ℎ_𝑟×ℎ_𝑟+𝑐_𝑟𝑘_𝑟²𝐾²𝑄²₀−2ℎ_𝑟𝐴_𝑟𝐷(𝐾−1)^21/2𝑄s.t.𝜆𝐾⁰𝐷≤𝐿,𝜆≥1. (24)

Since 𝜕²𝑇𝐶_𝑠(𝐾,𝜆)𝜕𝐾²=2𝐴_𝑠𝐷𝜆𝑄₀𝐾³+2𝑐_𝑟𝑘_𝑠+ℎ_𝑠ℎ_𝑟+𝑐_𝑟𝑘_𝑟²𝐴_𝑟𝐷𝑄²₀×ℎ_𝑟+𝑐_𝑟𝑘_𝑟²𝑄²₀𝐾²−2ℎ_𝑟𝐴_𝑟𝐷(𝐾−1)^2−3/2>0, (25) then 𝑇𝐶_𝑠(𝐾,𝜆) is convex in 𝐾, for 1≤𝐾≤𝐾₁. Let 𝐾^∗₁ and 𝜆^∗₁ be the solutions of min𝑇𝐶_𝑠(𝐾,𝜆), then, for given 𝜆, 𝐾^∗₁(𝜆) satisfies 𝜕𝑇𝐶_𝑠(𝐾,𝜆)/𝜕𝐾=0.

Since 𝜆 is an integer and 𝜆≤𝐿/𝑡₀, we can get 𝐾^∗₁ and 𝜆^∗₁ by the enumeration method.

Therefore, we have the following results:(a)if 1≤𝐾^∗₁<𝐾₁, then 𝐾^∗=𝐾^∗₁ and 𝜆^∗=𝜆^∗₁;(b)if 𝐾^∗₁≥𝐾₁, then 𝐾^∗=𝐾₁.

If 𝐾^∗₁≥𝐾₁, then 𝐾^∗=𝐾₁. By substituting 𝐾^∗=𝐾₁ into (24), we get min𝑇𝐶_𝑠(𝜆)=𝐴_𝑠𝐷𝜆𝑄₀1−ℎ_𝑟+2𝑐_𝑟𝑘_𝑟ℎ_𝑟+𝑠_𝑟+ℎ_𝑠+𝑠_𝑠2()+𝑐𝜆−1^𝑟𝑘_𝑠+ℎ_𝑠ℎ_𝑟+𝑐_𝑟𝑘_𝑟ℎ_𝑟×𝑄₀1−ℎ_𝑟+2𝑐_𝑟𝑘_𝑟/ℎ_𝑟+𝑠_𝑟−𝑐_𝑟𝑘_𝑠+ℎ_𝑠ℎ_𝑟ℎ_𝑟+𝑐_𝑟𝑘_𝑟𝑄₀1−ℎ_𝑟+2𝑐_𝑟𝑘_𝑟/ℎ_𝑟+𝑠_𝑟²−2ℎ_𝑟𝐴_𝑟𝐷ℎ_𝑟+2𝑐_𝑟𝑘_𝑟ℎ_𝑟+𝑠_𝑟−ℎ_𝑟+2𝑐_𝑟𝑘_𝑟^21/2𝜆s.t.1−ℎ_𝑟+2𝑐_𝑟𝑘_𝑟/ℎ_𝑟+𝑠_𝑟𝑄₀𝐷≤𝐿,𝜆≥1. (26)

It is easy to verify that 𝑇𝐶_𝑠(𝜆) is convex in 𝜆. Similar to the Proof of Lemma 1, we can get that if 𝐿≥2𝐴_𝑟/√𝐷(ℎ_𝑟+𝑠_𝑟−ℎ_𝑟+2𝑐_𝑟𝑘_𝑟),𝜆^∗=min{𝜆^∗₂,𝜆^∗₃}, where 𝜆^∗₂=14+𝐴_𝑠𝐴_𝑟1−ℎ_𝑟+2𝑐_𝑟𝑘_𝑟ℎ_𝑟+𝑠_𝑟²−12,𝜆^∗₃=𝐿√𝐷ℎ_𝑟+𝑠_𝑟−ℎ_𝑟+2𝑐_𝑟𝑘_𝑟2𝐴_𝑟, (27)

⌈𝑥⌉ is the least integer greater than or equal to 𝑥; [𝑥] denotes the integer part of 𝑥.

Case II. When 𝑡(𝐾)≥𝐾𝑡₀, 𝐻_𝑟(𝐾)=0.
In this case, (11) is reduced to the following problem: min𝑇𝐶_𝑠(𝐾,𝜆)=𝐷𝐴_𝑠𝜆𝐾𝑄₀+ℎ_𝑠+𝑠_𝑠𝐾𝑄₀(𝜆−1)2+ℎ_𝑠𝐷𝑡(𝐾)+𝑐_𝑟𝑘_𝑠(𝐾)𝑄𝐷𝑡s.t.𝜆𝐾⁰𝐷≤𝐿𝐷𝐴_𝑟𝐾𝑄₀+𝑠_𝑟𝐾𝑄₀2−𝑐_𝑟𝑘_𝑟𝐷𝑡(𝐾)≤2𝐷𝐴_𝑟ℎ_𝑟+𝑠_𝑟𝜆≥1. (28)
By the second constraint of (28), we have 𝑡(𝐾)≥𝐴_𝑟𝐾𝑄₀𝑐_𝑟𝑘_𝑟+𝑠_𝑟𝐾𝑄₀2𝐷𝑐_𝑟𝑘_𝑟−𝑄₀ℎ_𝑟+𝑠_𝑟𝐷𝑐_𝑟𝑘_𝑟. (29)
Set 𝑡₃=𝐴_𝑟/𝐾𝑄₀𝑐_𝑟𝑘_𝑟+𝑠_𝑟𝐾𝑄₀/2𝐷𝑐_𝑟𝑘_𝑟−𝑄₀(ℎ_𝑟+𝑠_𝑟)/𝐷𝑐_𝑟𝑘_𝑟. In this case, 𝑡(𝐾)≥𝐾𝑡₀ holds. So, 𝑡(𝐾)≥max𝐾𝑡₀,𝑡₃. (30)
Let 𝐹(𝐾)=𝑡₃−𝐾𝑡₀; that is 𝐹(𝐾)=𝐴_𝑟𝐾𝑄₀𝑐_𝑟𝑘_𝑟+𝑠_𝑟𝑡₀𝐾2𝑐_𝑟𝑘_𝑟−𝑡₀ℎ_𝑟+𝑠_𝑟𝑐_𝑟𝑘_𝑟𝑄−𝐾⁰𝐷. (31)
It is easy to verify that 𝐹(𝐾) is convex in 𝐾 and 𝐹(𝐾=1)<0. Let 𝐾₂ and 𝐾₃(𝐾₂>𝐾₃) be the solutions of 𝐹(𝐾)=0; then 𝐾₂=2𝐷𝐴_𝑟1−ℎ_𝑟+2𝑐_𝑟𝑘_𝑟/ℎ_𝑟+𝑠_𝑟𝐾>1,₃=2𝐷𝐴_𝑟1+ℎ_𝑟+2𝑐_𝑟𝑘_𝑟/ℎ_𝑟+𝑠_𝑟<1. (32)
Therefore, if 1≤𝐾≤𝐾₂, then 𝑡₃≤𝐾𝑡₀, 𝑡^∗(𝐾)=𝐾𝑡₀; if 𝐾>𝐾₂, then 𝑡₃>𝐾𝑡₀, 𝑡^∗(𝐾)=𝑡₃.
The following analysis is conducted based on the following two situations. (i) If 1≤𝐾≤𝐾₂, 𝑡^∗(𝐾)=𝐾𝑡₀.
It is straight forward that, in this case, the retailer’s cost will reduce no matter how much the parameters change. By substituting 𝑡^∗(𝐾) into (28), we get the following problem: min𝑇𝐶_𝑠(𝐾,𝜆)=𝐷𝐴_𝑠𝜆𝐾𝑄₀+ℎ_𝑠+𝑠_𝑠2(𝜆−1)+ℎ_𝑠+𝑐_𝑟𝑘_𝑠𝐾𝑄₀𝑄s.t.𝜆𝐾⁰𝐷≤𝐿,𝜆≥1. (33)

Theorem 5. If ℎ_𝑠+𝑐_𝑟𝑘_𝑠−(ℎ_𝑠+𝑠_𝑠)/2<0 and 𝐾^∗ℎ≥−^𝑠+𝑠_𝑠ℎ_𝑠−𝑠_𝑠+2𝑐_𝑟𝑘_𝑠, (34) then 𝑇𝐶_𝑠𝐾^∗,𝜆^∗≤𝑇𝐶_𝑠𝜇^∗. (35)

Proof. Let 𝜌=𝜆𝐾; (33) is equivalent to the following problem: min𝑇𝐶_𝑠𝜌=𝐷𝐴_𝑠𝜌𝑄₀+ℎ_𝑠+𝑠_𝑠2𝜌−1𝑄₀+ℎ_𝑠+𝑐_𝑟𝑘_𝑠−ℎ_𝑠+𝑠_𝑠2𝐾𝑄₀+ℎ_𝑠+𝑠_𝑠2𝑄₀s.t.𝜌𝑡₀≤𝐿,𝜌≥1. (36)
It is easy to verify that if ℎ_𝑠+𝑐_𝑟𝑘_𝑠−(ℎ_𝑠+𝑠_𝑠)/2<0 and 𝐾^∗≥−(ℎ_𝑠+𝑠_𝑠)/(ℎ_𝑠−𝑠_𝑠+2𝑐_𝑟𝑘_𝑠), then 𝑇𝐶_𝑠(𝐾^∗,𝜆^∗)≤𝑇𝐶_𝑠(𝜇^∗). The proof of Theorem 5 is complete.

Theorem 5 indicates that when the retailer is required to settle the balance after the ordering cycle, the retailer’s average cost will be reduced, but the supplier will benefit from coordination only under certain conditions.

Next, we will determine the supplier and the retailer’s optimal ordering quantities.

It is easy to verify that 𝑇𝐶_𝑠(𝐾,𝜆) is convex in 𝐾 for 1≤𝐾≤𝐾₂. Let 𝐾^∗₂ be the solution of 𝜕𝑇𝐶_𝑠(𝐾,𝜆)/𝜕𝐾=0.(a)If 𝐾^∗₂≥𝐾₂, then 𝐾^∗=𝐾₂.

By substituting 𝐾^∗=𝐾₂ into (33), we have min𝑇𝐶_𝑠(𝜆)=𝐷𝐴_𝑠1−ℎ_𝑟+2𝑐_𝑟𝑘_𝑟/ℎ_𝑟+𝑠_𝑟𝜆𝑄₀2𝐷𝐴_𝑟+ℎ_𝑠+𝑠_𝑠2(𝜆−1)+ℎ_𝑠+𝑐_𝑟𝑘_𝑠×𝑄₀2𝐷𝐴_𝑟1−ℎ_𝑟+2𝑐_𝑟𝑘_𝑟/ℎ_𝑟+𝑠_𝑟𝜆s.t.2𝐷𝐴_𝑟1−ℎ_𝑟+2𝑐_𝑟𝑘_𝑟/ℎ_𝑟+𝑠_𝑟𝑄₀𝐷≤𝐿,𝜆≥1. (37)

We can easily verify that 𝑇𝐶_𝑠(𝜆) is convex in 𝜆. Similar to the Proof of Lemma 1, we can get the following result: if 𝐿≥2𝐴_𝑟/(ℎ_𝑟+𝑠_𝑟−ℎ_𝑟+2𝑐_𝑟𝑘_𝑟), 𝜆^∗=min{𝜆^∗₄,𝜆^∗₅}, where 𝜆^∗₄=14+𝐴_𝑠2𝐷𝐴²_𝑟1−ℎ_𝑟+2𝑐_𝑟𝑘_𝑟ℎ_𝑟+𝑠_𝑟²−12,𝜆^∗₅=𝐿ℎ_𝑟+𝑠_𝑟−ℎ_𝑟+2𝑐_𝑟𝑘_𝑟2𝐴_𝑟, (38)

⌈𝑥⌉ is the least integer greater than or equal to 𝑥; [𝑥] denotes the integer part of 𝑥.(b) If 1≤𝐾^∗₂<𝐾₂, then 𝐾^∗=𝐾^∗₂.

By 𝜕𝑇𝐶_𝑠(𝐾,𝜆)/𝜕𝐾=0, we have 𝐾^∗₂=ℎ_𝑟+𝑠_𝑟𝐴_𝑠2𝐴_𝑟𝜆ℎ_𝑠+𝑠_𝑠/2(𝜆−1)+ℎ_𝑠+𝑐_𝑟𝑘_𝑠. (39)

Substituting (39) into (33), we have min𝑇𝐶_𝑠(𝜆)=2𝐴_𝑠𝐷ℎ_𝑠+𝑠_𝑠/2(𝜆−1)+ℎ_𝑠+𝑐_𝑟𝑘_𝑠𝜆𝐴s.t.^𝑠𝜆𝐷ℎ_𝑠+𝑠_𝑠/2(𝜆−1)+ℎ_𝑠+𝑐_𝑟𝑘_𝑠≤𝐿²,𝜆≥1. (40)

Since √𝑥 is strictly increasing in 𝑥, so (40) is equivalent to the following problem: min𝑇𝐶_𝑠(𝜆)=𝐴_𝑠𝐷ℎ_𝑠+𝑠_𝑠/2(𝜆−1)+ℎ_𝑠+𝑐_𝑟𝑘_𝑠𝜆𝐿s.t.²𝐷ℎ_𝑠+𝑠_𝑠2−𝐴_𝑠𝜆+𝐿²𝐷ℎ_𝑠+𝑐_𝑟𝑘_𝑠−ℎ_𝑠+𝑠_𝑠2≥0𝜆≥1. (41)

Since ℎ_𝑠+𝑐_𝑟𝑘_𝑠−(ℎ_𝑠+𝑠_𝑠)/2<0, then 𝑇𝐶_𝑠(𝜆) is increasing in 𝜆. If ℎ_𝑠+𝑐_𝑟𝑘_𝑠<(ℎ_𝑠+𝑠_𝑠)/2 and 𝐿²𝐷(ℎ_𝑠+𝑠_𝑠)/2−𝐴_𝑠>0, then the first constraint of (41) holds for 𝜆≥⌈−[ℎ_𝑠+𝑐_𝑟𝑘_𝑠− (ℎ_𝑠+𝑠_𝑠)/2]/((ℎ_𝑠+𝑠_𝑠)/2−𝐴_𝑠/𝐿²𝐷)⌉. Due to the fact that 𝑇𝐶_𝑠(𝜆) is increasing in 𝜆 and ⌈−[ℎ_𝑠+𝑐_𝑟𝑘_𝑠− (ℎ_𝑠+𝑠_𝑠)/2]/((ℎ_𝑠+𝑠_𝑠)/2−𝐴_𝑠/𝐿²𝐷)⌉≥1, it is straight forward that 𝜆^∗=⌈−[ℎ_𝑠+𝑐_𝑟𝑘_𝑠− (ℎ_𝑠+𝑠_𝑠)/2]/((ℎ_𝑠+𝑠_𝑠)/2−𝐴_𝑠/𝐿²𝐷)⌉. (ii) If 𝐾>𝐾₂, 𝑡^∗(𝐾)=𝑡₃=𝐴_𝑟𝐾𝑄₀𝑐_𝑟𝑘_𝑟+𝑠_𝑟𝑡₀𝐾2𝑐_𝑟𝑘_𝑟−𝑡₀ℎ_𝑟+𝑠_𝑟𝑐_𝑟𝑘_𝑟. (42)

By substituting (42) into (28), the problem is reduced to min𝑇𝐶_𝑠()=ℎ𝐾,𝜆^𝑠+𝑠_𝑠2()+ℎ𝜆−1^𝑠+𝑐_𝑟𝑘_𝑠𝑠_𝑟2𝑐_𝑟𝑘_𝑟𝐾𝑄₀+𝐴_𝑠𝐷𝜆+ℎ_𝑠+𝑐_𝑟𝑘_𝑠𝐴_𝑟𝐷𝑐_𝑟𝑘_𝑟1𝐾𝑄₀−ℎ_𝑠+𝑐_𝑟𝑘_𝑠2𝐷𝐴_𝑟ℎ_𝑟+𝑠_𝑟𝑐_𝑟𝑘_𝑟𝑄s.t.𝜆𝐾⁰𝐷≤𝐿,𝜆≥1. (43)

According to the constraints in (43), we get 1≤𝜆≤𝐿𝐷/𝐾𝑄₀. In order to guarantee that there exits the feasible solution for the problem (43), the lifetime needs to satisfy 𝐿≥𝐾₂(𝑄₀/𝐷); that is 𝐿≥2𝐷𝐴_𝑟ℎ_𝑟+𝑠_𝑟−ℎ_𝑟+2𝑐_𝑟𝑘_𝑟. (44)

It is easy to verify that the right side of (44) is usually very large, so (44) is not easy to be satisfied for fixed lifetime products. Hence, for fixed lifetime products, providing longer credit period than the retailer’s order period is not commonly applicable.

Remark 6. By (12) and (28), it is straightforward that for items with infinite lifetime, it is possible that 𝑇𝐶_𝑠(𝐾,𝜆) is minimized when both 𝑡(𝐾) and 𝐾 are very large, which is meaningless in practice. So, an upper bound 𝑡 is necessary to be set on delay period 𝑡(𝐾) to avoid very large delay period. But for items with finite lifetime, the first constraints of (12) and (28) ensure that 𝐾 will not be very large. Particularly, in Case I (0≤𝑡(𝐾)≤𝐾𝑡₀), by (17), 𝑡^∗(𝐾) will not be very large. In Case II (𝑡(𝐾)≥𝐾𝑡₀), we have 𝑡^∗(𝐾)=𝐾𝑡₀, for 1≤𝐾≤𝐾₂. And we have proved that 𝐾>𝐾₂ is not applicable for fixed lifetime products. Hence, the upper bound on 𝑡 is not necessary any more for fixed lifetime products.

3. Numerical Example

In this section, numerical examples are presented to evaluate the benefit of the proposed policy in Section 2. Moreover, we will examine the implications of the change in the value of parameters.

Let 𝐿=0.2 year and 𝐷=150000 units per year. SSP_𝛼(%) and RSP_𝛼(%) denote the savings in percentage of the supplier and the retailer when the supplier shares 𝛼 percentage of his/her cost saving with the retailer and let 𝛼=0.5. TSP(%) denotes the saving in percentage of the supply chain.

Case I (example). In Case I, we prove that the retailer’s cost with coordination is the same as that without coordination, but the supplier’s cost is reduced. If the supplier would like to share the saving cost (𝑇𝐶_𝑠−𝑇𝐶_𝑠) with the retailer, the retailer’s cost can also be reduced.
Let 𝐴_𝑠=$70 per order, 𝐴_𝑟=$50 per order, 𝑐_𝑟=$10 per unit, ℎ_𝑠=$4 per unit per year, ℎ_𝑟=$7 per unit per year, 𝑠_𝑠=$12 per unit per year, 𝑠_𝑟=$20 per unit per year, 𝑘_𝑠=$0.1 per dollar per year, and 𝑘_𝑟=$0.1 per dollar per year.
When the values of 𝐴_𝑠, ℎ_𝑠, and 𝑘_𝑠 change, the corresponding computing results are as specified in Tables 1–3.
As illustrated in Table 1, as 𝐴_𝑠 increases, 𝐾^∗, 𝑡^∗, and the supplier’s cost without and with coordination all increase. In fact, when the supplier’s ordering cost 𝐴_𝑠 is high, the supplier will reduce his/her ordering cost by increasing the order quantity per cycle, that is, increasing 𝜇^∗ in the absence of coordination or 𝐾^∗ in the presence of coordination. In order to entice the retailer to order more, the supplier will provide a longer credit period 𝑡^∗. Moreover, as 𝐴_𝑠 increases, the supplier and the retailer’s savings in percentage decrease or increase, but they are both positive. The total cost saving of the supply chain is also positive; that is, the proposed policy is effective to the system.
By Table 2, as ℎ_𝑠 increases, 𝐾^∗ and 𝑡^∗ both decrease, but the supplier’s cost without and with coordination both increase. Actually, if the supplier’s capital cost is high, then he/she will provide a shorter delay period to reduce his/her capital cost. The shorter credit period offered by the supplier will cause the less order multiple of the retailer. In addition, as ℎ_𝑠 increases, the supplier and the retailer’s savings in percentage both decrease. Hence, if the supplier’s holding cost is high, the supplier and the retailer cannot gain much from coordination.
From Table 3, as 𝑘_𝑠 increases, the supplier’s cost with coordination increases, but 𝐾^∗, 𝑡^∗, and the supplier and the retailer’s savings in percentage decrease. If 𝑘_𝑠 is large, it is profitable for the supplier to invest, and accordingly the credit period offered to the retailer will be shortened. In this case, the supplier and the retailer cannot gain much from coordination.

Table 1 

Sample computing results for different values of 𝐴_𝑠.

Table 2 

Sample computing results for different values of ℎ_𝑠.

Table 3 

Sample computing results for different values of 𝑘_𝑠.

Case II (example). In Case II, we prove that if 1≤𝐾≤𝐾₂, the retailer’s cost with coordination is less than that without coordination no matter how much the parameters change, but the supplier will benefit only if (34) holds. In Tables 4 and 5, “R of (34)” denotes the right hand side of (34). (1)Let 𝐴_𝑠=$200 per order, 𝑐_𝑟=$5 per unit, ℎ_𝑠=$6 per unit per year, ℎ_𝑟=$8 per unit per year, 𝑠_𝑠=$12 per unit per year, 𝑠_𝑟=$20 per unit per year, 𝑘_𝑠=$0.1 per dollar per year, and 𝑘_𝑟=$0.1 per dollar per year. The different values of 𝐴_𝑟 and the computing results are as specified in Table 4.
As shown in Table 4, when 𝐾^∗ is less than “R of (34),” the supplier’s cost saving in percentage is negative, but the retailer’s is positive. It is consistent with Theorem 5; that is, if (34) does not hold, the supplier will not benefit from coordination. (2)Let 𝐴_𝑠=$200 per order, 𝐴_𝑟=$80 per order, 𝑐_𝑟=$10 per unit, 𝑠_𝑠=$12 per unit per year, 𝑠_𝑟=$20 per unit per year, ℎ_𝑠=$4 per unit per year, 𝑘_𝑠=$0.1 per dollar per year, and 𝑘_𝑟=$0.1 per dollar per year. The different values of ℎ_𝑟 and the computing results are as specified in Table 5.
In Table 5, 𝐾^∗ is greater than “R of (34).” That is, (34) holds. In this case, we observe that the supplier and the retailer’s cost savings in percentage are both positive and increase as ℎ_𝑟 increases. Hence, if the retailer’s holding cost is high, both parties can gain much from coordination.
In Case II, the retailer’s cost decreases, but the supplier’s cost possibly decreases or increases, which depends on the value of the parameters and 𝐾^∗. As proved in Theorem 5, if (34) holds, then the supplier’s cost can be reduced by coordination as shown in Table 5, else as illustrated in Table 4, the supplier’s average cost increases.

Table 4 

Sample computing results for different values of 𝐴_𝑟.

Table 5 

Sample computing results for different values of ℎ_𝑟.

4. Conclusions

In this paper, supply chain coordination for fixed lifetime products through delay in payments is considered and modeled. The analytical method as to how to determine the optimal strategy is proposed, and analytically tractable solutions are obtained. It is indicated that if the retailer is required to settle the balance with the supplier before the end of the ordering cycle, then the supplier can gain with coordination. The supplier’s total cost as well as that of the total system can be reduced no matter how much the parameters change. Thus, the supplier can not only compensate the retailer for his increased inventory cost by offering the retailer an order size dependent delay period, but also provide the retailer with an additional saving of 𝛼(𝑇𝐶_𝑠(𝜇^∗)−𝑇𝐶_𝑠(𝐾^∗,𝜆^∗)). Here, 𝛼 is determined through negotiations between the supplier and the retailer and is generally dependent on the existing balance of power between them. Furthermore, it is indicated that, in this case, the retailer’s order size is larger at cooperation against noncooperation (𝐾>1). As proved in Theorem 5, if the retailer is permitted to settle the balance after the ordering cycle, the supplier can gain only under certain conditions. In this case, the credit period is equal to the ordering cycle.

In the proposed model, demand is assumed to be a deterministic constant. In fact, the demand is influenced by many factors, such as price [30], inventory level [31], advertising strategy [32], and other uncertainties or noises from market; so the varying or uncertain demand functions are more practical. However, the analysis of the models in these cases would be more complicated and we need to analyze the characteristics of noises first [33, 34]. Perhaps it is difficult to obtain the analytical solutions, but one could try to obtain the optimal numerical solutions. Anyway, further works could extend the models in these ways.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors gratefully acknowledge the supports from the National Natural Science Foundation of China (71371139 and 71002020), the Shanghai Pujiang Program (12PJC069), the “Shu Guang" project supported by the Shanghai Municipal Education Commission and the Shanghai Education Development Foundation (13SG24), and the Fundamental Research Funds for the Central Universities.