This paper focuses on the bargaining behavior of supply chain members and studies the stability of the bargaining system. There are two forms of bargaining in the process of negotiation. One is separate bargaining, and the other is that the automobile manufacturers form an alliance and bargain with the supplier collectively. We explore the influence of bargaining power and adjustment speed on the stability of the dynamic system and find that both of the factors need to be small to maintain the stability of the supply chain. After comparing the two forms of bargaining in terms of profits and stable regions, we find that the collective bargaining is a pattern with the existence of risk and benefit simultaneously. In order to control chaos in collective bargaining to lower the risk, we adopt the delay feedback control method. With the introduction of the control factor, the system tends to be stable finally.
The bargaining behavior is common in commercial operation and plays a critical role in the whole supply chain performance. Because of the fierce market competition, many manufacturers cannot raise their prices at will but rely more on bargaining to reduce costs to strive for greater profit space. The concept of common parts makes joint procurement possible, and many manufacturers become beneficiaries of collective bargaining. Automobile manufacturers are also one of them. Take the automobile industry as an example; after years of rapid development, the automobile industry has experienced a slowdown recently, which stimulates the fierce competition of the component market. In order to cut costs, most automobile manufacturers choose to purchase parts from external enterprises, while focusing on their core business. In the automobile manufacturing industry, the purchase cost accounts for a high proportion in the whole product cost, while the purchase cost of automobile parts is a large part of the purchase cost. Since that, the bargaining between automobile manufacturers and the supplier is of great significance. Formerly, manufacturers used to purchase components separately. With the prevalence of the supply chain and the win-win concept, many of them prefer to form an alliance to bargain with the supplier for the sake of a stronger bargaining power. We used the generalized Nash bargaining framework to model the bargaining process and compared separate bargaining and collective bargaining on their performance of maintaining stability in the dynamic system.
Applying nonlinear dynamics theory to an economic system can provide a better understanding of its complex practice, and many scholars had made their attempt in different fields. Fibich and Gavish [
The inherent randomness of chaos makes the trajectory of a dynamic system difficult to predict. When chaos is beneficial to the system, conditions should be created to guide the system into a specific chaotic orbit. However, when chaos is harmful to the system, it should be controlled. And in most cases, chaos is not desirable and should be controlled via different methods according to its characteristics. Many methods have been proposed by scholars, such as the method of parameter adjustment and the adaptive chaos control method. Later in the paper, we will adopt the delay feedback control method to control the chaos occurred in the bargaining process because of its good tracking ability and stability.
The paper is organized as follows. In Section
One stream of literature that is related to our work is the one which studied bargaining behavior. Feng et al. [
In recent years, a dynamical system has been widely used. Many researchers integrate nonlinear dynamics theory and complex system theory into the study of an economic system, which greatly enriches the study of long-term game complexity of an economic system. Zhang et al. [
Chaos control is significant since in many cases it may cause fluctuation in the supply chain and is not conducive to decision-making. Many methods have been proposed by scholars and widely used in chaos control. OGY is the earliest proposed method by Ott et al. [
We consider a supply chain that consisted of two manufacturers (marked as
In the case of separate bargaining, the two manufacturers bargain with the supplier separately, and the decision process is as follows: firstly, the manufacturer
We use the GNB framework to solve this case, and this framework has been widely used in the economic system [
The wholesale price is
As shown in the above formula, the wholesale price is based on manufacturers’ procurement quantity and the bargaining power of both. When the bargaining power of the manufacturer is extremely high, the wholesale price is close to zero; in other words, the manufacturer will gain the profit of the whole supply chain as more as possible. This can also be demonstrated in the second formula of profit. On the contrary, when the bargaining power of the manufacturer is rather small, the profit space will be tremendously compressed.
Static analysis is commonly used in the existing literature to analyze the bargaining problem in the supply chain. However, under the complicated environment, bargaining between both sides is a long-term and complex process, which cannot be solved by only one game. Next, we will employ the method of dynamic game to conduct multiperiod adjustment to achieve the equilibrium solution and analyze the system stability.
Because of the lack of market information, the manufacturers are not always rational. For a better practical fit, we assume that the manufacturer adopts the adjustment rule called the gradient adjustment mechanism where they decide their procurement quantity based on the counterpart and margin profit in the last period. The application of this mechanism can also be found in other literatures [
Under this rule, the manufacturers will increase their procurement quantity in period
With the gradient adjustment mechanism, the manufacturers will stop adjusting their procurement quantity when
Equilibrium points’ stability is decided by the characteristic roots of the Jacobi matrix. When the absolute values of the characteristic roots are less than one, the equilibrium point tends to be stable, or not, otherwise. And, the Jacobi matrix can be shown as follows:
Equilibrium solutions
More detailed proof can be found in Appendix. Lacking of stability implies that the solution cannot return to a fixed position in a certain period, and in this bargaining problem in the supply chain, it means the quantity will not turn to a fixed value after iterations during multiple periods. At these three boundary equilibrium points, at least one of the manufacturers decides not to place order, which signifies abandoning the next bargaining cycle. This will do harm to his/her interests or even force him/her to withdraw from the market in the long term. Therefore, the supply chain is not sustainable and unpredictable, which brings more difficulty to decision-making.
In terms of the Nash equilibrium point, since the characteristic roots of its Jacobi matrix are difficult to compute, we can use the Jury criterion [
The conditions for the stability of the supply chain can be described as follows:
Proposition
To better explain the influence of parameters in Proposition
In Figures
The stable region of the separate bargaining system.
The parameter basin with respect to
The bifurcation diagram of the model with respect to
LLE diagram of the model with respect to
The stable region of both bargaining systems.
Besides separate bargaining, manufacturers can also form an alliance to negotiate with the supplier collectively. Leader-based collective bargaining is one of the most popular forms of collective bargaining in practice. In this case, manufacturers decide their procurement quantity
The profit functions of manufacturers and the supplier can be described as follows:
Based on the GNB framework, the bargaining problem between the leader manufacturer and the supplier can be formulated as
The outcomes of the wholesale price is
Both functions of the wholesale price and the profit have the same form as the counterpart under separate bargaining. But, considering the enhanced bargaining power, it is actually different. Compared with the separate bargaining case, with the stronger bargaining power, the wholesale price is more close to zero and the manufacturer will gain more share of the whole profit in the supply chain. This provides manufacturers with a stand point to form an alliance willingly.
Likewise, under the gradient adjustment mechanism, the system can be formulated as
By solving the equation
Equilibrium solutions
At least one of the quantity decisions is set to zero in these three boundary equilibrium points, which is the lack of economic significance and is not sustainable. It means that the manufacturer chooses to abandon the next bargaining period, which may cause damage to him/her and may even be forced out of the market in the long run. Therefore, boundary equilibrium points cannot maintain stability and is not sustainable in reality. Since the absolute values of the characteristic roots are less than one, it means that the three boundary equilibrium points are lacking stability and will approach to chaos after a few iterations. Under these circumstances, the supply chain is in a shambles, and the economic activities in the supply chain are unpredictable, which add more difficulty for decision-making.
In terms of the Nash equilibrium point, the conditions under the Jury criterion for the stability of the supply chain can be described as follows:
Proposition
Just as stated in Proposition
The stable region of the collective bargaining system.
The bifurcation diagram of the model with respect to
LLE diagram of the model with respect to
The formation of chaotic attractors.
In practice, risk and profit are two key factors emphasized by enterprises. Considering that, we will present the comparisons on these two factors between separate bargaining and collective bargaining in the following.
Total profit of the alliance under collective bargaining is greater than that under separate bargaining at the Nash equilibrium point, if
According to Lemma
Figure
The total profit of manufacturers.
Stability of the supply chain under separate bargaining is better than that under collective bargaining.
To focus on the common factor, i.e., adjustment speed, we fixed bargaining power as
In collective bargaining, manufacturers form the alliance designate manufacturer 1 as a leader to negotiate with the supplier. Since the leader pools the quantity of both manufacturers, a slight fluctuation will bring great changes to the system. Therefore, the adjustment speed should be controlled to a smaller extent. Compared with separate bargaining, the adjustment speed of the follower not only influences his/her own decision but also has indirect impact on the alliance. With the double influence, the collective bargaining system will be easier to be chaotic if the adjustment speed increased. Meanwhile, according to Proposition
We depict the bifurcation diagrams of the profit under separate bargaining in Figure
The transformation of relationship focus between manufacturers, from competition to cooperation, can explain the change of adjustment speed status. In separate bargaining, competition is more emphasized, and the position of the manufacturer determines the influence of his/her adjustment speed. Since we assumed that manufacturer 1 is more powerful than manufacturer 2, his/her adjustment speed has more influence on the system. When two manufacturers form an alliance in collective bargaining, competition is weakened and cooperation plays a more critical role in the bargaining system. Since manufacturer 1 is the representative of the alliance to negotiate with the supplier, the adjustment speed of manufacturer 2 not only influences his/her own decision directly but also has an impact on the alliance. Because the influence of
Propositions
The stable region of both bargaining systems.
The bifurcation diagram of the profit under separate bargaining.
The bifurcation diagram of the profit under collective bargaining.
Chaos is inherently random, nonlinear, and sensitive to an initial value. Sometimes, chaos benefits firms [
As one of the methods of chaos control, the delay feedback control method does not change the structure of the controlled system and has good tracking ability and stability. Its main idea is to feedback partial information of the output signal of the system, instead of the external input, to the control system with delay time. The control system can be described as follows:
At the Nash equilibrium point
From the former numerical analysis, we know that the system is chaotic when
The stability is achieved when the Jury criterion is satisfied. Therefore, the control system is stable around the Nash equilibrium point when
As shown in Figures
The quantity fluctuations of manufacturer 1.
The quantity fluctuations of manufacturer 2.
The bifurcation diagram of the controlling factors.
This paper analyzed the system of two forms of bargaining: the separate one and the collective one. The boundary equilibrium points of both are unstable. In fact, neither of the manufacturers wants to keep his quantity at zero because that would cost him market share. Therefore, the motivation to change will damage the stability of the bargaining system. The Nash equilibrium point is sensitive to the value of adjustment speed and bargaining power of manufacturers. The result demonstrated that the bargaining power and adjustment speed should be small at the same time to ensure the stability of the system. The adjustment speed reflects the sensitivity of manufacturers to the profits of the last period. When the adjustment speed is large, it will magnify the fluctuation in the system and cause chaos. Meanwhile, the bargaining power should be small to reduce the probability of negotiation failure. Comparing these two forms of bargaining, we found that the collective bargaining brings about more profits and risks at the same time. However, chaos will cause damage to the gross profit of manufacturers, which reduces the profit advantage of collective bargaining. By introducing the controlling factors, the delay feedback method can control the chaos effectively. It means that manufacturers can make their decisions not only based on the information from the last period but also from previous periods to improve decision effectiveness. This result can help manufacturers make decisions accurately and is beneficial for maintaining the stability of the economic system.
The four equilibrium points is the solution of
And, can take
The Jacobi matrix of
The characteristic roots of
One of the characteristic roots is
The Jacobi matrix of
The trace and determinant of
Solving three inequalities:
The proof of Lemma
In terms of collective bargaining, the gradient adjustment mechanism of manufacturer 1 is the same as that in separate bargaining, while the formula of manufacturer 2 is adjusted as
When
The Jacobi matrix of
The absolute values of both characteristic roots are more than 1, and that means
The Jacobi matrix of
One of the characteristic roots is
Similarly, the instability of
The Jacobi matrix of
The trace and determinant of
According to the Jury criterion, the following inequalities are solved:
And, Proposition
The sum of two manufacturers’ profits under separate bargaining is
The gross profit of the alliance of these two under collective bargaining is
The comparison between
Since
Suppose
No data were used to support this study.
The authors declare that they have no conflicts of interest.
This research was supported by the Humanities and Social Science Foundation in the Hubei Provincial Education Department and the Research Center of Enterprise Decision Support, Key Research Institute of Humanities and Social Sciences in Universities of Hubei Province.