Coordination in the Decentralized Assembly System with Dual Supply Modes

Abstract

This paper investigates a decentralized assembly system that consists of one assembler and two independent suppliers; wherein one supplier is perfectly reliable for the production, while the other generates yield uncertainty. Facing the random market demand, the assembler has to order the components from one supplier in advance and meanwhile requires the other supplier to deliver the components under VMI mode. We construct a Nash game between the supplier and the assembler so as to derive their equilibrium procurement/production strategies. The results show that the channel’s performance is highly undermined by the decentralization between players and also the combination of two supply modes. Compared to the centralized system, we propose an advance payment contract to perfectly coordinate the supply chain performance. The numerical examples indicate some management implications on the supply mode comparison and sensitivity analysis.

1. Introduction

Assembly systems have been widely applied in automobile, electronics, and many other manufacturing industries. One fundamental advantage of this system is to help the core assembler/manufacturer take the full advantages (e.g., low procurement cost) from his suppliers. Therefore, we have seen that many famous manufacturers (e.g., Toyota and HP) prefer to outsource their components production process to those external suppliers who locate in Asia, so as to significantly reduce their procurement cost and improve the production efficiency. Nonetheless, the accompanying challenge is how to ensure that the independent suppliers move cooperatively in the system. Difficulties arise from two aspects. First, because a single firm normally cares for its own profit, its decisions inevitably go against the channel’s overall efficiency. Second, there are multiple uncertain factors that lie in the system, for example, delivery time, customer demand, and production yield, which subsequently undermine the system’s operational performance. Therefore, to mitigate the decentralization and improve the system’s performance, scholars have made extensive researches on the optimization mechanism in the decentralized assembly systems [1–4].

Nonetheless, in this rich stream of literature most scholars assume that the suppliers have to adopt the same strategies, in which all of them are either enrolled in the ordering mode or in the VMI mode (vendor managed inventory). Ordering mode is defined as were the assembler orders the components from all his suppliers before the demand is realized. As a result, suppliers just need to follow the assembler’s instructions. For example, Yano [1] studies an assembler’s optimal order time, wherein both suppliers have stochastic supply lead times. In contrast, under VMI mode all the suppliers have to personally determine when and how to produce the components to the assembler after knowing the demand information. For this issue, Gerchak and Wang [5] investigate how to achieve the coordination with both revenue-sharing contract and buy-back contract under VMI mode.

Differently, in this paper we will combine these two supply modes into a single assembly system, wherein one supplier exerts the ordering mode, while the other supplier has to accept the VMI mode. To our knowledge, this combination is rarely discussed in the prior literature but is prevalent in practice. For example, in China’s auto industry, the assembler needs to order and pay for the key components, such as engine, from the overseas suppliers in advance. Meanwhile, the of the rest components such as windshield wipers are normally provided by the domestic suppliers under VMI mode. Also in the electronics industry, assembler always places orders in advance from key components’ provider (e.g., Intel) while requires the local suppliers to produce the other components under VMI arrangement. Given this gap between the practice and the literature, in this paper we will derive some important implications for the firms’ equilibrium strategies under the combined supply modes.

To this end, we construct a decentralized assembly system that contains two sources of uncertainty. On one hand, the market demand is stochastic so that the assembler and the supplier have to make their procurement/production decisions before the exact demand is realized. On the other hand, we assume that supplier 1’s production is perfectly reliable, while supplier 2 has a stochastic production yield rate. The assembler has to preorder the components from supplier 1 under the ordering mode but is also able to exert the VMI mode on supplier 2. Consequently, the assembler and supplier 2 have to simultaneously choose their production and procurement quantities in consideration of the other’s response. This setting helps us to solve the following questions. First, what are the firms equilibrium production and procurement strategies in this decentralized assembly system with dual supply modes? Second, compared to the centralized system how to achieve the supply chain coordination? Third, what are the differences among different supply modes, for example, ordering mode versus VMI mode versus combined mode?

Given the equilibrium strategies, our analysis has the following observations. First, double marginalization significantly undermines the channel’s efficiency, in which case both suppliers produce fewer components than that in the centralized system. In particular, the supplier with random yield produces more components than his partner’s. Second, to improve the channel efficiency, we incorporate an advance payment contract. When the sharing parameter (donated as 𝜆) falls into a rational range, the suppliers achieve perfect coordination. Moreover, we show that from the channel’s perspective, ordering mode dominates both the combined mode and VMI mode. This result coincides with the supplier’s interest but goes against the assembler’s profit. The intuition is that for any firm, the better operational mode always mitigates its inventory risk. Overall, this paper speaks to the interactions between the assembler and the suppliers under dual supply modes in a decentralized system.

Besides the literature reviewed above, another stream of the related research is random yield and random demand. Note that this issue has been extensively studied by many scholars [6–10]. However, as aforementioned the majority of these models have been developed under the condition that all the suppliers generate either production yield uncertainty or demand uncertainty, while in this paper, we assume that one supplier is perfectly reliable while the other supplier has a random yield issue. This setting is prevalent in both literature [11, 12] and practice. For example, the key component providers in assembly system (e.g., Intel) are always stable in their production capacity and can operate quite well. Therefore, they are nearly perfectly reliable. In contrast, the other component providers may suffer yield uncertainty, because they are usually in small size and unreliable in managing the production. One notable paper is Pan and So [13], in which they analyze an assemble-to-order system with these two types of supply. In their paper, assembler determines both kinds of suppliers’ production input quantity in the interest of maximizing system’s profit when facing price-dependent demand. Different from them, in our research there are multiple decision makers: the assembler decides the production quantity, while the supplier determines his production quantity.

The remainder of this paper is organized as follows. In Section 2, we describe the model setting. The equilibrium decisions of both decentralized and centralized systems are derived in Section 3. Section 4 investigates supply chain coordination and numerical examples. Section 5 concludes the paper and discusses some future researches.

2. Model Descriptions

Consider a decentralized assembly system that consists of two suppliers and one assembler, in which all the participants are risk-neutral and in purpose of maximizing their own profits. The final product consists of two components, as we define them as the key component and the matching component. The key component is provided by supplier 1 under ordering mode, and its unit production cost is 𝑐₁ and the unit wholesale price is 𝑤₁  (𝑤₁≥𝑐₁). Differently, supplier 2 produces the matching component under VMI mode with unit production cost 𝑐₂, and the unit wholesale price for component 2 is 𝑤₂  (𝑤₂≥𝑐₂). We assume that supplier 1’s production yield is perfectly reliable, while supplier 2 has a random production yield. In particular, if supplier 2’s initial input quantity is 𝑞, finally the output quantity of the components (that meet the quality level) is 𝜃𝑞. 𝜃 is a random variable that falls into [0,1] with probability distribution function 𝑔(⋅) and cumulative distribution function 𝐺(⋅) [13–15]. On the other hand, the market demand 𝐷 is also stochastic and follows the probability distribution function 𝑓(⋅) and cumulative distribution function 𝐹(⋅) in [0,∞). Without loss of generality, we assume that the final product’s unit price is 𝑝  (𝑝≥𝑤₁+𝑤₂) and the assembly cost equals 0 (actually, if the cost 𝑐_𝑎>0, the final product’s unit price can be modified to 𝑝^′=𝑝+𝑐_𝑎).

In a single-period setting, facing a random market demand 𝐷, the assembler determines the order quantity 𝑄₁ from supplier 1 and takes the corresponding cost of over-stock and shortage. This is the interaction between supplier 1 and assembler under ordering mode. In contrast, under the VMI contract with supplier 2, supplier 2 is the only decision maker that determines the input quantity 𝑄₂ of matching component and will not receive the payment until the component is consumed. After both components’ production and delivery are finished, the assembler assembles the components together and sells them to the customer. Once the overall product quantity is lower than the market demand (min𝐸(𝑄₁,𝜃𝑄₂)<𝐷), the assembler will be penalized by the customer for shortage. The unit penalty cost is 𝛽. At the same time, if supplier 2’s production quantity 𝜃𝑄₂ is insufficient (𝜃𝑄₂<𝐷), he will also receive the punishment from the assembler. The unit penalty cost is 𝛽₂  (𝛽₂≤𝛽). Besides, we normalize the salvage value of mismatched components to zero.

In summary, in this assembly system the assembler and supplier 2 simultaneously decide the production input quantity of the key component and the matching component. Therefore, we can develop a static Nash game model between the assembler and supplier 2 to derive their equilibrium strategies. In the following section, we first set up a benchmark by investigating the centralized system. After that, we focus on the decentralized system with dual supply modes. As a matter of convenience, we use 𝑋^𝑑 to denote the condition in the decentralized system and 𝑋^𝑐 to represent the condition in the centralized system.

3. System’s Optimal Decisions

In this section, we first set up a benchmark by studying the centralized system. Afterwards, we study the Nash game between the assembler and supplier 2 in the decentralized system. Comparing the channel’s performances under these two scenraios, we finally discuss the supply chain coordination mechanism in the system.

3.1. Centralized System

In a centralized system, the assembler and the two suppliers will cooperate as one to achieve the system’s highest performance. Therefore, we first formulate a centralized system’s (denoted as 𝐵) expected profit function as follows: Π^𝑐_𝐵𝑄₁,𝑄₂=𝑝𝐸min𝑄₁,𝜃𝑄₂,𝐷−𝑐₁𝑄₁+𝑐₂𝑄₂−𝛽𝐸𝐷−min𝑄₁,𝜃𝑄₂⁺. (1)

Note that if the channel is centralized, its payoff is entirely determined by the production quantities of two components. As in (1), the first term is the final product’s expected sales revenue. The second term is the production cost of both the key and the matching components. The third term is the expected penalty when stock-out occurs. Fixing either 𝑄_𝑖  (𝑖=1,2), the maximization of Π^𝑐_𝐵(𝑄₁,𝑄₂) becomes a news-vendor problem. Thus, we can use the following lemma to characterize the property of Π^𝑐_𝐵(𝑄₁,𝑄₂). Note that throughout paper, all the proofs of lemmas and theorems are given in the appendix.

Lemma 1. The centralized system’s expected profit function, Π^𝑐_𝐵(𝑄₁,𝑄₂), is jointly concave in 𝑄₁∈[0,∞) and 𝑄₂∈[0,∞). The unique optimal input quantity of both the key and the matching components, 𝑄₁^𝑐∗ and 𝑄₂^𝑐∗, meets the F.O.Cs: 𝜕Π^𝑐_𝐵𝑄₁,𝑄₂𝜕𝑄₂=𝑝+𝛽^𝑄1/𝑄2₀^∞_𝜃𝑄2𝜃𝑓(𝐷)𝑔(𝜃)𝑑𝐷𝑑𝜃−𝑐₂=0,𝜕Π^𝑐_𝐵𝑄₁,𝑄₂𝜕𝑄₁=𝑝+𝛽¹_𝑄1/𝑄2^∞_𝑄1𝑓(𝐷)𝑔(𝜃)𝑑𝐷𝑑𝜃−𝑐₁=0. (2)(3)

From Lemma 1, we can always find the unique pair of (𝑄₁^𝑐∗,𝑄₂^𝑐∗) that maximizes Π^𝑐_𝐵(𝑄₁,𝑄₂). Moreover, we show that (2) and (3) are symmetric and irrespective to the parameter 𝜃. Given so, we can derive the relationship of 𝑄₁^𝑥∗ and 𝑄₂^𝑥∗ (𝑥 represents 𝑑 and 𝑐). The following theorem states that 𝑄₂^𝑥∗>𝑄₁^𝑥∗ always holds.

Theorem 2. In centralized system, the optimal input quantity of the matching component is strictly larger than the optimal input quantity of the key component, that is, 𝑄₂^𝑐∗>𝑄₁^𝑐∗.

The intuition behind Theorem 2 can be explained as follows. Consider that with random yield, if the input quantity of matching component is fewer than that of the key component, the output 𝜃𝑄₂ must be eternally fewer than 𝑄₁. This implies that the penalty cost of matching component’s shortage can be reduced by enlarging its input quantity, which is also beneficial to supplier 2 and the entire system. Therefore, only if 𝑄₂ exceeds 𝑄₁, they can reach an equilibrium. Now, we have derived the equilibrium decisions in the centralized system. This will be compared with the following scenario wherein the two players make their decisions independently.

3.2. Decentralized System

In the decentralized system, assembler (denoted as 𝐴) will pay supplier 1 (denoted as 𝑆₁) immediately after he finishes the key component’s delivery. Therefore, his profit function is Π^𝑑_𝑆1=(𝑤₁−𝑐₁)𝑄₁, which is nonnegative and meets the participant constraint.

As to supplier 2 (denoted as 𝑆₂), he will not receive the payment until the matching component is consumed. Besides, supplier 2 will be penalized by the assembler if his output is less than the customer’s demand. Thus, supplier 2’s expected profit function can be formulated as Π^𝑑_𝑆2=𝑤₂𝐸min𝑄₁,𝜃𝑄₂,𝐷−𝑐₂𝑄₂−𝛽₂𝐸𝐷−𝜃𝑄₂⁺. (4)

In (4), the first term is the expected revenue. The second term is the production cost, which is based on the input quantity. The third term stands for the expected penalty when the stock out of the matching component occurs. Similarly, we have the assembler’s expected profit function as follows: Π^𝑑_𝐴=𝑝−𝑤₂𝐸min𝑄₁,𝜃𝑄₂,𝐷−𝑤₁𝑄₁−𝛽𝐸𝐷−min𝑄₁,𝜃𝑄₂⁺+𝛽₂𝐸𝐷−𝜃𝑄₂⁺. (5)

Note that in the above equation, the first two terms are the final product’s expected sales revenue minus the procurement cost of the key and the matching components. The third term is the penalty penalized by the customer when the stock-out occurs. The fourth one is the compensation from supplier 2 when his yield cannot meet the customer’s demand. The following lemma demonstrates that the objective functions, Π^𝑑_𝑆2 and Π^𝑑_𝐴, are concave in supplier 2’s and supplier 1’s production (input) quantity, respectively.

Lemma 3. Let 𝑄_𝑖^𝑑∗(𝑄_𝑗) denote the optimal production input quantity of the key/matching component for a given 𝑄_𝑗  (𝑖,𝑗=1,2;𝑖≠𝑗). Then, we have the following.(i)Supplier 2’s profit function, Π^𝑑_𝑆2, is concave in 𝑄₂∈[0,∞). And the optimal input quantity 𝑄₂^𝑑∗(𝑄₁) meets the first-order condition: 𝜕Π^𝑑_𝑆2𝑄₁,𝑄₂𝜕𝑄₂=𝑤₂^𝑄1/𝑄2₀^∞_𝜃𝑄2𝜃𝑓(𝐷)𝑔(𝜃)𝑑𝐷𝑑𝜃+𝛽₂¹₀^∞_𝜃𝑄2𝜃𝑓(𝐷)𝑔(𝜃)𝑑𝐷𝑑𝜃−𝑐₂=0. (6)(ii)The assembler’s profit function, Π^𝑑_𝐴, is concave in 𝑄₁∈[0,∞). And, the optimal input quantity 𝑄₁^𝑑∗(𝑄₂) meets the first-order condition: 𝜕Π^𝑑_𝐴𝑄₁,𝑄₂𝜕𝑄₁=𝑝+𝛽−𝑤₂¹_𝑄1/𝑄2^∞_𝑄1𝑓(𝐷)𝑔(𝜃)𝑑𝐷𝑑𝜃−𝑤₁=0. (7)

With Lemma 3, we can characterize the optimal production quantity of both key and matching components through their first-order-conditions (F.O.Cs) and the static Nash game equilibrium. In particular, we have the following theorem.

Theorem 4. There is a unique static Nash equilibrium solution 𝐷(𝑄₁^𝑑∗,𝑄₂^𝑑∗) to the assembler’s and supplier 2’s decisions of the key and the matching components’ production quantity. The solution (𝑄₁^𝑑∗,𝑄₂^𝑑∗) meets the combination of (6) and (7).

Theorem 4 derives the unique optimal production quantity of the key component (𝑄₁^𝑑∗) and the matching component (𝑄₂^𝑑∗), which are given under the exogenous operational parameters, for example, the wholesale price 𝑤₁ and 𝑤₂. These indicate the highest payoffs that the assembler and suppliers can achieve in the decentralized system, which are certainly lower than those in the centralized system. Therefore, to identify the gap (inefficiency) between these two scenarios (decentralized versus centralized), we have the following theorem.

Theorem 5. In the decentralized system, the summation of all the participants’ optimal profit is strictly less than the centralized system’s optimal total profit, that is, Π^𝑑_𝑆1+Π^𝑑_𝑆2+Π^𝑑_{𝐴𝑄1=𝑄1𝑑∗,𝑄2=𝑄2𝑑∗}<Π^𝑐_𝐵𝑄₁^𝑐∗,𝑄₂^𝑐∗. (8)

From Theorem 5, we can see that although there exists a unique static Nash equilibrium in the decentralized system, it is necessary to introduce a proper contract to coordinate the supply chain to move like a centralized system. Therefore, we propose an advance payment contract to achieve the supply chain coordination.

3.3. Supply Chain Coordination

Among the previous literature, a number of different contract types aiming at coordinating the supply chain are discussed, for example, the revenue sharing contract, the buy-back contract, and so forth. For a detailed review, please refer to Cachon [16]. Differently, in this paper we focus on another contract type: advance payment contract. That is, at the beginning of a production period, the assembler pays (𝜆₁Π_𝐵^𝑐∗+𝑐₁𝑄₁^𝑐∗) to supplier 1 and (𝜆₂Π_𝐵^𝑐∗+𝑐₂𝑄₂^𝑐∗) to supplier 2. After receiving the customer’s order, the assembler announces the production input quantity of both the key and the matching components, 𝑄_𝑖^𝑐∗  (𝑖=1,2). Afterwards, the suppliers carry out the production according to their announcement. Finally, the assembler assembles the components into the products and sells them to the customer. Accordingly, he obtains the profit of (1−∑²_𝑖=1𝜆_𝑖)Π_𝐵^𝑐∗, where 𝜆_𝑖∈[0,1] and ∑²_𝑖=1𝜆_𝑖≤1.

Note that the major feature of the above contract is that the assembler should pay the suppliers in advance. Thus, if supplier 𝑖’s profit (𝜆_𝑖Π_𝐵^𝑐∗) is less than what he can gain in the decentralized system, the contract will fail in practice. Therefore, we have the following theorem to state that there always exists a proper pair of (𝜆₁,𝜆₂) which can successfully implement the contract.

Theorem 6. The decentralized assembly system with dual supply modes always can be coordinated through the advance payment contract. In which, the contract parameter 𝜆_𝑖 meets the following conditions: Π_𝑆^𝑑∗_𝑖Π_𝐵^𝑐∗≤𝜆_𝑖Π≤1−^𝐴𝑑∗+Π_𝑆^𝑑∗_𝑗Π_𝐵^𝑐∗,²_𝑖=1𝜆_𝑖Π≤1−^𝐴𝑑∗Π_𝐵^𝑐∗, (9) wherein Π_𝑦^𝑑∗=Π^𝑑_𝑦(𝑄₁^𝑑∗,𝑄₂^𝑑∗), 𝑦∈{𝑆₁,𝑆₂,𝐴}, and 𝑖,𝑗=1,2, 𝑖≠𝑗.

With Theorem 6, we can draw the insight that the advance payment successfully makes the suppliers operate as the assembler’s subsidiaries, so as to achieve the supply chain coordination. Besides, under this contract the assembler actually changes the business scenario with supplier 2 by deciding the input quantity of the matching component himself. This implies that when facing two kinds of supply, the VMI arrangement with the supplier who generates yield uncertainty is not beneficial for the entire supply chain. To this point, we further study the comparison of the decentralized and centralized systems and investigate the effectiveness of the advance payment contract in the following section.

4. Numerical Analysis

In this section, we conduct two numerical examples. In the first example, we make a comparison between components’ equilibrium quantities and firms’ expected payoffs under three alternative scenarios: the decentralized system with dual supply modes, the decentralized system with ordering mode [1, 17], and the centralized system. This helps us to identify the magnitude of decentralization in the assembly system with dual supply modes. Second, we examine the robustness of advance payment contract on the supply chain coordination by making the sensitivity analysis of sharing rate 𝜆_𝑖  (𝑖=1,2).

4.1. Comparison of the Alternative Scenarios

In Section 3.3, we have shown that dual supply modes is never beneficial to the entire supply chain, which is caused by the conflict between two supply modes: ordering mode and VMI mode. To better distinguish the difference between these two mode, we next introduce another scenario that the assembly system only contains ordering mode (denoted as 𝑑^′), wherein both the key and the matching components’ input quantity are selected by the assembler. Intuitively, under such a circumstance the supply chain’s performance should be better than that with dual supply modes; however, it is still worse than the centralized system since the decentralization still exists.

With only ordering mode, the assembler independently decides both the input quantities of key component and matching component. Therefore, each party’s expected profit functions change to Π^𝑑′_𝑆1=𝑤₁−𝑐₁𝑄₁,Π^𝑑′_𝑆2=𝑤₂𝐸𝜃𝑄₂−𝑐₂𝑄₂,Π^𝑑′_𝐴=𝑝𝐸min𝑄₁,𝜃𝑄₂,𝐷−𝑤₁𝑄₁−𝑤₂𝐸𝜃𝑄₂−𝛽𝐸𝐷−min𝑄₁,𝜃𝑄₂⁺. (10)(11)

Following the similar principle of Lemma 1, we can easily find that (11) is jointly concave in 𝑄₁ and 𝑄₂, and the optimal (𝑄^𝑑′∗₁,𝑄^𝑑′∗₂) meets the following F.O.Cs: 𝜕Π^𝑑′_𝐴𝑄₁,𝑄₂𝜕𝑄₁=𝑝+𝛽^𝑄1/𝑄2₀^∞_𝜃𝑄2𝜃𝑓(𝐷)𝑔(𝜃)𝑑𝐷𝑑𝜃−𝑤₂¹₀𝜃𝑔(𝜃)𝑑𝜃=0,𝜕Π^𝑑′_𝐴𝑄₁,𝑄₂𝜕𝑄₁=𝑝+𝛽¹_𝑄1/𝑄2^∞_𝑄1𝑓(𝐷)𝑔(𝜃)𝑑𝐷𝑑𝜃−𝑤₁=0. (12)

Due to the F.O.Cs and the profit functions’ complexity, we now make the comparison of the three alternative scenarios via numerical approaches. In the following numerical example, we conduct several comparisons under different groups of parameters.

Example 1. Assume that the customer’s demand follows the normal distribution and the production random variable of supplier 2 follows the uniform distribution. See Table 1.

Table 1 

Performance with different penalty, price, and demand.

Note that Table 1 represents the key and the matching components’ optimal input quantity, the expected profits of two suppliers and the assembler, and the expected total profit of the supply chain for three alternative scenarios. As observed from the above table, we have ∑_𝑦Π_𝑦^𝑑∗<∑_𝑦Π^𝑑′∗_𝑦<Π_𝐵^𝑐∗, which demonstrates that the decentralized systems generate incoordination, and the decentralized system only with ordering mode performs better than the one with dual supply modes. Specifically, both the key and the matching components’ optimal input quantity in the decentralized systems are less than those in the centralized system, that is, 𝑄₁^𝑑∗<𝑄^𝑑′∗₁<𝑄₁^𝑐∗ and 𝑄₂^𝑑∗<𝑄^𝑑′∗₂<𝑄₂^𝑐∗, which directly makes the performance of decentralized systems become worse than the centralized system. Besides, there exist Π^𝑑′∗_𝑆1>Π^𝑑′∗_𝑆1, Π^𝑑′∗_𝑆2>Π^𝑑′∗_𝑆2, and Π^𝑑′∗_𝑆2<Π_𝑆^𝑑∗₂. This implies that from the supplier’s perspective, they prefer to providing components with the ordering mode. In contrast, under the decentralized situation, the assembler will certainly choose the VMI mode with the supplier who generates yield uncertainty, even though this option significantly undermines the channel’s performance.

4.2. Sensitivity Analysis on 𝜆_𝑖

In the above discussion, we have shown that the advance payment contract can perfectly coordinate the decentralized supply chain by setting the appropriate 𝜆_𝑖. We next identify how the firms’ equilibrium payoffs react to the variance of 𝜆_𝑖.

Example 2. Given the following data: 𝑐₁=4, 𝑤₁=6, 𝑐₂=3, 𝑤₂=8, 𝛽₂=4, 𝑝=25, 𝛽=15; 𝐷∽𝑁(40,1.5), 𝜃∽𝑈(0,1), let 𝜆₁ take the value uniformly distributed over [(Π_𝑆^𝑑∗₁/Π_𝐵^𝑐∗),1−((Π_𝐴^𝑑∗+Π_𝑆^𝑑∗₂)/Π_𝐵^𝑐∗)] and 𝜆₂ take the value of [(1−𝜆₁−(Π_𝐴^𝑑∗/𝜋_𝐵^𝑐∗))+(Π_𝑆^𝑑∗₂/Π_𝐵^𝑐∗)]/2, which is in [(Π_𝑆^𝑑∗₂/Π_𝐵^𝑐∗),1−((Π_𝐴^𝑑∗+Π_𝑆^𝑑∗₁)/Π_𝐵^𝑐∗)]. Figure 1 shows the effectiveness of the advance payment contract.

(a)

(b)

(c)

(d)

Figure 1 

Effectiveness of the advance payment contract under different 𝜆𝑖.

In Figure 1, we can see that as long as the 𝜆_𝑖 is in its valid interval, all the suppliers and the assembler can be better off and the decentralized system can be perfectly coordinated with the advance payment contract. Therefore, it is effective and practicable.

5. Conclusion

In this paper, we investigate the equilibrium production and procurement strategies in a decentralized assembly system consisting of a single assembler and two suppliers. In particular, the assembler orders the key component from supplier 1 who is perfectly reliable, while supplier 2 provides the matching component under VMI mode with yield uncertainty. We derive the components’ optimal production input quantity in a static Nash game model and also set up a benchmark case by identifying the centralized system. Given the comparison between these two scenarios, we propose an advance payment contract to achieve supply chain coordination. We also make two numerical examples and find that (1) the less components’ optimal production input quantity makes the decentralized systems perform worse than the centralized system, (2) the assembler prefers VMI mode in practice so as to maximize his own expected profit, and (3) advance payment contract is effective regardless the sharing rate.

One extension of our model is to consider a more general assembly system with one assembler and 𝑁 (≥3) suppliers, where the decisions of those suppliers with yield uncertainty can affect each other. Besides, it can be extended to other random yield models. For example, in semiconductor industry, the output of chips 𝑌(𝑞) is a nonlinear function of the input of silicon wafers 𝑞, and other related factors. These extensions certainly have the potential to be better explored in the future.

Appendix

Proof of Lemma 1. Equation (1) can be rewritten as Π^𝑐_𝐵(𝑄₁,𝑄₂)=𝑝𝐶−𝛽𝐷−(𝑐₁𝑄₁+𝑐₂𝑄₂), wherein 𝐶=¹_𝑄1/𝑄2^∞_𝑄1𝑄₁𝑓(𝐷)𝑔(𝜃)+𝑑𝐷𝑑𝜃_{𝑄1/𝑄20∞𝜃𝑄2}𝜃𝑄₂𝑓(𝐷)𝑔(𝜃)+𝑑𝐷𝑑𝜃_{𝑄101𝐷/𝑄2}𝐷𝑔(𝜃)𝑓(𝐷)𝑑𝜃𝑑𝐷,𝐷=¹_𝑄1/𝑄2^∞_𝑄1𝐷−𝑄₁𝑓(𝐷)𝑔(𝜃)+𝑑𝐷𝑑𝜃_{𝑄1/𝑄20∞𝜃𝑄2}𝐷−𝜃𝑄₂𝑓(𝐷)𝑔(𝜃)𝑑𝐷𝑑𝜃. (A.1)
Taking first- and second-order derivatives with respect to 𝑄₁ and 𝑄₂ in (1) separately, we get 𝜕Π^𝑐_𝐵𝑄₁,𝑄₂𝜕𝑄₂=𝑝+𝛽^𝑄1/𝑄2₀^∞_𝜃𝑄2𝜃𝑓(𝐷)𝑔(𝜃)𝑑𝐷𝑑𝜃−𝑐₂,𝜕Π^𝑐_𝐵𝑄₁,𝑄₂𝜕𝑄₁=𝑝+𝛽¹_𝑄1/𝑄2^∞_𝑄1𝑓(𝐷)𝑔(𝜃)𝑑𝐷𝑑𝜃−𝑐₁,𝜕²Π^𝑐_𝐵𝑄₁,𝑄₂𝜕𝑄²₂=𝑄𝑝+𝛽𝐸+²¹𝑄³₂𝐹,𝜕²Π^𝑐_𝐵𝑄₁,𝑄₂𝜕𝑄²₁1=−𝑝+𝛽𝐺+𝑄₂𝐹,𝜕²Π^𝑐_𝐵𝑄₁,𝑄₂𝜕𝑄₁𝜕𝑄₂=𝑄𝑝+𝛽¹𝑄²₂𝐹. (A.2)
Here, 𝐸=∫^𝑄1/𝑄2₀𝜃²𝑓(𝜃𝑄₂)𝑔(𝜃)𝑑𝜃>0, 𝐹=∫^∞_𝑄1𝑓(𝐷)𝑔(𝑄₁/𝑄₂)𝑑𝐷>0, and 𝐺=∫¹_𝑄1/𝑄2𝑓(𝑄₁)𝑔(𝜃)𝑑𝜃>0. Then, we derive the Hessen Matrix as follows: 𝜕𝐻=²Π^𝑐_𝐵𝑄₁,𝑄₂𝜕𝑄²₁𝜕²Π^𝑐_𝐵𝑄₁,𝑄₂𝜕𝑄₁𝜕𝑄₂𝜕²Π^𝑐_𝐵𝑄₁,𝑄₂𝜕𝑄²₂𝜕²Π^𝑐_𝐵𝑄₁,𝑄₂𝜕𝑄₂𝜕𝑄₁=−1𝑝+𝛽𝐺+𝑄₂𝐹𝑄₁𝑄²₂𝐹𝑄₁𝑄²₂𝑄𝐹−𝐸+²¹𝑄³₂𝐹. (A.3)
In (A.3), the value of the first-order determinant is −(𝑝+𝛽)(𝐺+(1/𝑄₂)𝐹)<0 and the value of the second-order determinant is (𝑝+𝛽)(𝐸𝐺+(𝑄²₁/𝑄³₂)𝐹𝐺+(1/𝑄₂)𝐸𝐹)>0. As such, the Hessen Matrix is negative definite. Therefore, Π^𝑐_𝐵(𝑄₁,𝑄₂) is joint concave in 𝑄₁∈[0,∞) and 𝑄₂∈[0,∞). And the optimal production input quantity (𝑄₁^𝑐∗,𝑄₂^𝑐∗) meets the F.O.Cs, which consists of (𝜕Π^𝑐_𝐵(𝑄₁,𝑄₂))/𝜕𝑄₂=0 and (𝜕Π^𝑐_𝐵(𝑄₁,𝑄₂))/𝜕𝑄₁=0.

Proof of Lemma 3. Equation (4) can be rewritten as Π^𝑑_𝑆2=𝑤₂𝐴−𝛽₂𝐵−𝑐₂𝑄₂, wherein 𝐴=¹_𝑄1/𝑄2^∞_𝑄1𝑄₁𝑓(𝐷)𝑔(𝜃)+𝑑𝐷𝑑𝜃_{𝑄1/𝑄20∞𝜃𝑄2}𝜃𝑄₂𝑓(𝐷)𝑔(𝜃)+𝑑𝐷𝑑𝜃_{𝑄101𝐷/𝑄2}𝐷𝑔(𝜃)𝑓(𝐷)𝑑𝜃𝑑𝐷,𝐵=¹₀^∞_𝜃𝑄2𝐷−𝜃𝑄₂𝑓(𝐷)𝑔(𝜃)𝑑𝐷𝑑𝜃. (A.4)
Fix 𝑄₁, taking first- and second-order derivatives with respect to 𝑄₂ in (4), we get 𝜕Π^𝑑_𝑆2𝜕𝑄₂=𝑤₂^𝑄1/𝑄2₀^∞_𝜃𝑄2𝜃𝑓(𝐷)𝑔(𝜃)𝑑𝐷𝑑𝜃+𝛽₂¹₀^∞_𝜃𝑄2𝜃𝑓(𝐷)𝑔(𝜃)𝑑𝐷𝑑𝜃−𝑐₂,𝜕²Π^𝑑_𝑆2𝜕𝑄²₂=−𝑤₂^𝑄1/𝑄2₀𝜃²𝑓𝜃𝑄₂𝑔(𝜃)+𝑑𝜃_∞𝑄1𝑄²₁𝑄³₂𝑓(𝐷)𝑔𝑄₁𝑄₂𝑑𝐷−𝛽₂¹₀𝜃²𝑓𝜃𝑄₂𝑔(𝜃)𝑑𝜃. (A.5)
It is easy to find that 𝜕²Π^𝑑_𝑆2/𝜕𝑄²₂<0. Therefore, Π^𝑑_𝑆2 is concave in 𝑄₂. Besides, we observe that 𝜕Π^𝑑_𝑆2/𝜕𝑄₂|_𝑄2=0=𝑤₂+𝛽₂𝜃−𝑐₂>0 and 𝜕Π^𝑑_𝑆2/𝜕𝑄₂|_𝑄2→∞=−𝑐₂<0. As such, there must exist a 𝑄₂∈[0,∞) that meets the first-order condition 𝜕Π^𝑑_𝑆2/𝜕𝑄₂=0.
Similarly, fix 𝑄₂, taking first- and second-order derivatives with respect to 𝑄₁ in (5), we get 𝜕Π^𝑑_𝐴𝜕𝑄₁=𝑝+𝛽−𝑤₂¹_𝑄1/𝑄2^∞_𝑄1𝑓(𝐷)𝑔(𝜃)𝑑𝐷𝑑𝜃−𝑤₁,𝜕²Π^𝑑_𝐴𝜕𝑄²₁=−𝑝+𝛽−𝑤₂×¹_𝑄1/𝑄2𝑓𝑄₁𝑔(𝜃)+1𝑑𝜃𝑄₂^∞_𝑄1𝑓(𝐷)𝑔𝑄₁𝑄₂𝑑𝐷. (A.6)
We can easily derive 𝜕²Π^𝑑_𝐴/𝜕𝑄²₁<0, 𝜕Π^𝑑_𝐴/𝜕𝑄₁|_𝑄1=0=𝑝+𝛽−𝑤₂−𝑤₁>0, and 𝜕Π^𝑑_𝐴/𝜕𝑄₁|_𝑄1→∞=−𝑤₁<0. Therefore, we can claim that Π^𝑑_𝐴 is concave in 𝑄₁ and the optimal 𝑄₁^𝑑∗(𝑄₂) meets the first-order condition 𝜕Π^𝑑_𝐴/𝜕𝑄₁=0.

Proof of Theorem 2. Simplifying (7) and (3), we haveing 1−𝐹𝑄₁^𝑑∗×𝑄1−𝐺^1𝑑∗𝑄₂^𝑑∗=𝑤₁𝑝+𝛽−𝑤₂𝑄,1−𝐹₁^𝑐∗×𝑄1−𝐺^1𝑐∗𝑄₂^𝑐∗=𝑐₁.𝑝+𝛽 (A.7)(A.8)
Consider the value of 𝐺(𝑄₁^𝑥∗/𝑄₂^𝑥∗). If 𝑄₁^𝑑∗≥𝑄₂^𝑑∗ or 𝑄₁^𝑐∗≥𝑄₂^𝑐∗, 𝐺(𝑄₁^𝑥∗/𝑄₂^𝑥∗)=1. As such, the left parts of (A.7) and (A.8) both equal 0. Meanwhile, the right part of (A.7) and (A.8) both are larger than 0. Thus, both (A.7) and (A.8) cannot be balanced unless 𝑄₁^𝑥∗<𝑄₂^𝑥∗.

Proof of Theorem 4. Considering the profit curves of supplier 2 and the assembler in (4) and (5), we let 𝑁₁𝑄₁,𝑄₂=𝜕Π^𝑑_𝐴𝑄₁,𝑄₂𝜕𝑄₁,𝑁₂𝑄₁,𝑄₂=𝜕Π^𝑑_𝑆2𝑄₁,𝑄₂𝜕𝑄₂. (A.9)
Taking first-order derivative with respect to 𝑄₁ in 𝑁₁(𝑄₁,𝑄₂) and 𝑁₂(𝑄₁,𝑄₂), we have 𝜕𝑁₁𝑄₁,𝑄₂𝜕𝑄₁=𝜕²Π^𝑑_𝐴𝑄₁,𝑄₂𝜕𝑄²₁=−𝑝+𝛽−𝑤₂¹_𝑄1/𝑄2𝑓𝑄₁𝑔(𝜃)+1𝑑𝜃𝑄₂^∞_𝑄1𝑓(𝐷)𝑔𝑄₁𝑄₂𝑑𝐷<0,𝜕𝑁₂𝑄₁,𝑄₂𝜕𝑄₁=𝜕²Π^𝑑_𝑆2𝑄₁,𝑄₂𝜕𝑄₂𝜕𝑄₁=𝑤₂^∞_𝑄1𝑄₁𝑄²₂𝑓(𝐷)𝑔𝑄₁𝑄₂𝑑𝐷>0. (A.10)
Then, we get (𝜕𝑁₁(𝑄₁,𝑄₂)/𝑄₁)−(𝜕𝑁₂(𝑄₁,𝑄₂)/𝑄₁)<0, for all (𝑄₁,𝑄₂). As such, the players’ profit functions, Π_𝑆2(𝑄₁,𝑄₂) and Π_𝐴(𝑄₁,𝑄₂), will meet once at most [3, Proposition  2.1]. Also, referring to Theorem  2.4 in Friedman [18], there is always a Nash equilibrium existing for concave-payoff-functions game like ours. Therefore, the static Nash equilibrium of both component’s production input quantity is unique and the solution, (𝑄₁^𝑑∗,𝑄₂^𝑑∗), meets the combination of (6) and (7).

Proof of Theorem 5. Sum Π^𝑑_𝑆1, (4), and (5), we get the decentralized system’s total profit function: Π^𝑑_𝑆1+Π^𝑑_𝑆2+Π^𝑑_𝐴=𝑝𝐸min𝑄₁,𝜃𝑄₂,𝐷−𝑐₁𝑄₁+𝑐₂𝑄₂−𝛽𝐸𝐷−min𝑄₁,𝜃𝑄₂⁺. (A.11)
Notice that the formulation is the same as the one of the centralized system. As such, we can compare their optimal performance through the components’ optimal production input quantity. Considering (A.7) and (A.8), we have 𝑤₁𝑝+𝛽−𝑤₂−𝑐₁=𝑤𝑝+𝛽𝑝+𝛽₁−𝑐₁+𝑐₁𝑤₂𝑝+𝛽𝑝+𝛽−𝑤₂>0. (A.12)
As a result, we can derive that (𝑄₁^𝑑∗,𝑄₂^𝑑∗)≠(𝑄₁^𝑐∗,𝑄₂^𝑐∗). Otherwise, if 𝑄₁^𝑑∗=𝑄₁^𝑐∗ and 𝑄₂^𝑑∗=𝑄₂^𝑐∗, (A.7) and (A.8) cannot hold simultaneously. Then, plug (𝑄₁^𝑑∗,𝑄₂^𝑑∗) and (𝑄₁^𝑐∗,𝑄₂^𝑐∗) into the system’s profit function separately; we can easily get Π^𝑑_𝑆1+Π^𝑑_𝑆2+Π^𝑑_{𝐴𝑄1=𝑄1𝑑∗,𝑄2=𝑄2𝑑∗}<Π^𝑐_𝐵𝑄₁^𝑐∗,𝑄₂^𝑐∗, (A.13) since (𝑄₁^𝑐∗,𝑄₂^𝑐∗) is the unique optimal decision.

Proof of Theorem 6. Under the advance payment contract, each party’s expected profit (denoted as Π_𝑦^co) can be formulated as follows: Π_𝑆^co₁=𝜆₁Π_𝐵^𝑐∗,Π_𝑆^co₂=𝜆₂Π_𝐵^𝑐∗,Π_𝐴^co=1−²_𝑖=1𝜆_𝑖Π_𝐵^𝑐∗. (A.14)
To ensure the contract’s success, one must satisfy the following constraints: Π_𝑦^co≥Π_𝑦^𝑑∗,∀𝑦∈𝑆₁,𝑆₂,𝐴. (A.15)
Substituting 𝜆_𝑖  (𝑖=1,2) into (A.15) and combining with Π_𝐵^𝑐∗>∑_𝑥Π_𝑥^𝑑∗, we can easily derive that 𝜆_𝑖 meets equation (9).

Acknowledgment

This work was supported by the National Natural Science Foundation of China (nos. 71231007, 71102174, and 71372019).