Complexity

Volume 2018, Article ID 4752765, 13 pages

https://doi.org/10.1155/2018/4752765

## The Complexity Analysis in Dual-Channel Supply Chain Based on Fairness Concern and Different Business Objectives

^{1}Institute of Management Science and Engineering, Henan University, Kaifeng 475004, China^{2}School of Business, Henan University, Kaifeng 475004, China^{3}School of Management & Economics, North China University of Water Resources and Electric Power, Zhengzhou 450046, China

Correspondence should be addressed to Zhang Yu-hao; moc.361@gcdyhyz and Huang Yi-min; moc.361@625008gnauh

Received 6 December 2017; Accepted 11 April 2018; Published 20 May 2018

Academic Editor: Carlos Gershenson

Copyright © 2018 Li Qiu-xiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We study a dual-channel supply chain which consists of one dual-channel manufacturer and one traditional retailer considering fairness concern. The manufacturer takes the market share as one of its business objectives in the competition game. We are devoted to establishing a vertical Nash game model and analyzing the price evolution of the model via the method of complexity theory and take adaptive adjustment control method to control the system’s chaotic state. The results show that an excessive price adjustment speed will hurt the system’s stability as well as profit of the supply chain. A high level fair caring for retailer will push the system to fall into chaos earlier, while a higher level fair caring for manufacturer will enlarge the stability region of the system. Setting the different objectives by the manufacturer will cause drastic competition in dual-channel supply chain. The research of this paper is of great significance to the decision-makers’ price decision and supply chain operation management.

#### 1. Introduction

Nowadays, because of the Internet technology, e-ecommerce is developing rapidly in China. More and more online direct channels are established by many manufacturers and retailers. Compared with the traditional sale channels, the online direct channel not only can reduce operation cost but also provides high quality products, and this way consumers can choose the best way that provides commodity with the most convenience and the largest discount [1].

A significant amount of research has been done on dual-channel supply chain. Some scholars studied the influence of channel structure on participants. For example, Ma and Xie [2] considered one manufacturer and one retailer in a dual-channel supply chain, the manufacturer holds either asymmetric or symmetric channel power compared to the retailer. The result found that the system stability should be easier to maintain with asymmetric channel power. Lu and Liu [3] compared the single and dual-channel distribution systems considering the effect of e-commerce channel. The results showed that if the e-channel efficiency was low adequately, the e-commerce channel might dominate the distribution systems. Cai [4] investigated the influence of channel structure and channel coordination on the supplier, the retailer, and the entire supply chain in the context of two single-channel and two dual-channel supply chains. He suggested that the preference of the supplier and the retailer over channel structures with and without coordination is different.

The literatures on price strategies as well as channel structures in supply chain were studied extensively. Chen and Wang [5] studied smart phone supply channel selection in different power structures; they pointed out that the firm who has higher channel power will gain more profit. Liu and Xu [6] proved that dual-channel collaborative pricing enabled avoiding risk effectively. Gao et al. [7] established three price models on dual-channel closed-loop supply chain; the results showed that when customer’s acceptance with the Internet channel increases, the wholesale price and retail price would decrease, while the optimal recycling transfer price was not affected. Huang and Swaminathan [8] studied four prevalent pricing strategies which differed in the autonomy degree for the Internet channel. Xu et al. [9] investigated the influence of coordinating contract on a dual-channel supply chain and analyzed retailer’s price decision.

The literature above studied dual-supply chain from channel structure and price strategies aspects. However, they did not consider the influence of behavior features of decision-makers on the channel structure and price strategies.

As a matter of fact, the firms will have different business objectives for achieving maximum profit and expanding market shares at the same time, especially in the oligarchic market.

The interest in market-share analysis has always received more attention by scholars. Bell et al. [10] studied the relation between market share and marketing effort; they pointed out that market share equals marketing effort divided by total marketing effort when some assumptions were satisfied. Carpenter et al. [11] have expressed the relationship of profitability and market share. Szymanski et al. [12] performed a meta-analysis on market-profitability; the result showed that, on average, market share has a positive effect on business profitability. Dutta and King [13] studied the market share, profitability, and other business objectives simultaneously. Li and Ma [14] considered the fact that the manufacturer and retailer have different business objectives with bounded rationality and analyzed the system characteristic through the theory of bifurcations and simulation. The results showed that different business objectives and competition can expand the stability region of Nash equilibrium.

Fairness concern is another important behavior factor that enterprises pay attention to. The studies about fairness concern showed that it has significant influence on price strategies and efficiency of supply chain. Some scholars studied fairness concerns in the single-channel. Cui et al. [15] showed that when fairness is concerned by supply chain members, the manufacturer tends to make simple wholesale price contract to coordinate the supply chain rather than to elaborate one. Pavlov and Katok [16] extended the research on channel coordination by developing a new model based on fairness and bounded rationality to explain rejections. Katok et al. [17] studied the performance of wholesale price when the supply chain partners’ fairness concerns were private information. They found that wholesale pricing could coordinate the supply chain when fairness concerns were strong enough. Du et al. [18] studied the influence of retailer’s fairness concerns on the coordination of supply chain and found that the fairness of retailer cannot change the coordination condition of the supply chain. Some scholars also analyzed fairness in dual-supply chain. Zhang and Ma [19] established two different pricing policies in a dual-supply chain with a fair caring retailer and showed that the higher adjustment speed of the direct price was disadvantageous to manufacturer but good for the retailer. Q.-H. Li and B. Li [20] considered the fact that the retailer had fairness concerns in the dual-channel supply chain with the services. The result pointed out that channel efficiency would decrease with the increase of the retailer’s fairness level. Wang et al. [21] discussed the coordination of a dual-channel supply chain considering a fair caring retailer and developed a Stackelberg game between manufacture and retailer and pointed out that the retailer’s fairness concern is beneficial to improve its own utility, but it damages the profits of manufacturer. Zhang and Wang [22] examined the influence of a firm’s horizontal and vertical fairness concerns on the three-party supply chain coordination in a duopoly supply chain game model. They found that a firm’s horizontal fairness concern would generate direct influence on its own traditional channel wholesale price and demand, and a firm’s vertical fairness concern, under equilibrium, will generate direct influence on its online channel equilibrium price and demand.

Some researchers center on the dynamic characteristics of model in dual-supply chain. Rand [23] found that with the adjustment process of Cournot, the Nash equilibrium might loss stability and chaotic dynamics occurs. Puu [24] pointed out that strange attractors could appear in the duopoly model and briefly analyzed three oligopoly competition conditions. Moreover, he reconsidered whether the equilibrium remains stable with an increasing number of firms in [25]. Sun and Ma [26] analyzed bifurcation and cyclic attractors in Cournot duopoly game model. Li and Ma [27] established a Bertrand model based on nonlinear demand functions in Chinese cold rolled steel market. They found that the system’s parameters could affect the stability of Nash equilibrium point and made the system go into chaos. Ma and Wang [28] considered a dual-channel retailer who made delay decision, and showed that adopting price delay strategy in traditional channel would make the system more stable, while adopting price delay decision in network channel would make the system less stable. Li and Ma [29] considered a closed-loop supply chain (CLSC) with product recovery and found that improving the retailer’s competitive position made the CLSC system get into chaos easier. Huang et al. [30] considered the fact that the customer had risk aversion behavior for probabilistic product and analyzed the influences of parameters on the system stability nonlinear science. Zheng et al. [31] developed three dynamic game models to investigate the influences of parameters on the stability of risk-averse complementary product manufacturer. Li et al. [32] analyzed the dynamics of dual-channel energy supply chain model with heterogeneous retailers and showed that the adjustment speed of price has obvious impact on the complexity of competition. Matouk et al. [33] established a nonlinear quadropoly game based on Cournot model, fully heterogeneous, and analyzed the stability of discrete-time dynamical system and discussed the Neimark-Sacker bifurcation. Wu and Van Gorder [34] studied dynamic multiple-player multiple-level discrete-time leader-followers games in the vein of Cournot or Stackelberg games under incomplete information and complete information. The results found that under perfect information players tended to quickly converge upon their respective equilibrium values, while incomplete information could result in loss of regularity and the emergence of periodic or chaotic dynamics. The papers mentioned above provided many methods and perspectives with analyzing the complex characteristic of the supply chain.

However, we can hardly find few papers that simultaneously considered fairness concerns and different business objectives of the manufacturer and retailer in dual-channel supply chain as well as investigating the stability and complexity features of the system with different parameters values combination.

In this paper, we considered a fair caring manufacturer concern about its market share and a fair caring retailer in the dual-cannel supply chain. Then, a vertical Nash game model was developed; meanwhile the complex dynamic behaviors of the system were explored under different parameters conditions by chaos theory, such as chaos, chaotic attractors, 2D period diagram, and sensitivity dependence to initial conditions.

The paper is organized as follows. In Section 2, the model assumptions and notations were described. The model analysis of the system is shown in Section 3. In Section 4, numerical simulations are presented. Chaos control of the system is adopted in Section 5. Conclusions are drawn in Section 6.

#### 2. Model Assumptions and Model Construction

The model we have made is based on the following assumptions:

(a) The linear demand functions were taken to describe consumer demand.

(b) is wholesale price from manufacturer to retailer for per unit goods; the marginal cost of the product is and .

(c) The traditional retailer only considers his profits maximization; meanwhile the manufacturer not only tries to maximize his own profits but also expands the whole market share as much as possible.

According to the above assumptions, the demand functions of the manufacturer and retailer are introduced as follows:where is the potential market size, means the customer’s loyalty to the traditional retailer channel, and expresses the number of customer preferring the traditional retailer channel, while represents the number of customer preferring the online channel of manufacturer. In addition, , denote the price elasticity of , , means the cross-price sensitivity between two channels, and satisfies , .

Suppose that the price decision of the manufacturer and retailer occurs within the discrete-time period , the price of the manufacturer and retailer in the period is , and the profit functions of the manufacturer and retailer are

Zhang and Ma [19] gave the retailer’s utility function:where , represent the profits of the manufacturer and retailer and denotes fairness concern coefficient. They assumed that the fairness feeling depends on the comparison of the absolute profit of the manufacturer and retailer.

Pavlov and Katok [16] gave the retailer’s fairness utility function as follows:where

, denote the monetary payoff of the manufacturer and retailer, respectively, and , represent the sensitivity coefficient of difference in payoff between and . Retailer’s fairness feelings depended on the comparison between part profit of the manufacturer and retailer’ profit.

Inspired by the above literatures, the utility function of the manufacturer and retailer can be constructed as follows:where , represent the fairness concern coefficient of the manufacturer and retailer. denotes the relative profit coefficient and denotes the balance coefficient between the profits and the market share of manufacturer. is the manufacturer’s profit from the traditional sale channel; the retailer cares about the absolute profit of himself comparing with . means the manufacturer only cares about the part profits of himself comparing with retailer’s profit in traditional sale channel. On the one hand, only if the retailer’s utility will decrease; on the other hand only if , the utility of manufacturer will decrease.

denotes the whole market share of manufacturer which can be represented by the ratio of sales revenue of manufacturer to total sales revenue in the market; namely, .

Let , ; thus, . Then taking partial derivative of with respect to yieldsBecause manufacturer’s sales revenue has the same change trends with its market share, we can use sale revenue substitute for market share [10, 15].

Substituting (1) and (2) into (6) and (7), the marginal utility of retailer can be obtained by taking the first-order partial derivatives of utility function ():Similarly, we obtain the marginal utility of manufacturer as follows:By solving , we can get the best decisions of the manufacturer and retailer: where , .

As a matter of fact, when players are making price decisions, they cannot get the complete information of the whole market and adjust their price strategy with bounded rationality. The manufacturer and retailer make their next period price decision based on the marginal utility of the current period. When the current marginal utility is positive, the price will be raised in the next period; otherwise, the price will be reduced in next period. The price adjustment process is as follows: where represents the price adjustment speed and denote the marginal utility of the manufacturer or retailer.

Then, a discrete dynamic dual-channel Nash game model can be described:

#### 3. Model Stability Analysis

We can get four equilibrium solutions by making :

The Jacobian matrix of system (13) is given bywhere

The stability of equilibrium points is determined by the eigenvalues of Jacobian matrix evaluated at the corresponding equilibrium points. We substitute , , , and into Jacobian matrix (15) and get the following proposition.

Proposition 1. *, , and are boundary equilibrium points and unstable points; is the only Nash equilibrium solution.*

*Proof. *For equilibrium point , the Jacobian matrix of system (13) is equal towhich gives two eigenvalues:Obviously, , imply that is unstable.

For , we calculate the eigenvalues of its Jacobian matrix: We can see that , which indicates that is an unstable point. More precisely, is a saddle point for . We can prove is also an unstable point in the same way; is the unique Nash equilibrium solution of the system.

Next, we only consider the stability of the Nash equilibrium point, the corresponding characteristic polynomial of system (13) can be written as follows:where and represent the trace and determinant of the Jacobian matrix , respectively.

The necessary and sufficient condition of asymptotic stability is that all the eigenvalues are inside the unit circle in complex plane, so the stability of system should satisfy the Jury’s conditions known as

#### 4. Numerical Simulations

In this section, numerical simulations are used to show the dynamic behaviors and features of system (13). We choose some values for parameters based on actual competition; values for parameters are as follows: , , , , , , and .

##### 4.1. The Influence of , , and on the Stability Region of Nash Equilibrium

Only if the initial values are in the stability region of Nash equilibrium, the prices of players will be fixed at Nash equilibrium point after a series of iterations.

Figure 1(a) shows the stability regions of the Nash equilibrium point with different values of when and are fixed at 0.2. We can see that the stability region is the biggest with = 0.2, when = 0.6, 0.9; the stable region becomes small gradually. Figure 1(b) shows the stability regions of the Nash equilibrium point with different values of when = 0.6, ; the stability region becomes smaller with increasing and the scope of is decreasing at the same time; the system stability is more sensitive to than ; in this case, the retailer should be more cautious than manufacturer in adjusting the price strategy. Similarly, Figure 1(c) shows that the stability region of Nash equilibrium is expanded with increasing.