Complexity

Volume 2017 (2017), Article ID 1382689, 12 pages

https://doi.org/10.1155/2017/1382689

## Complexity Dynamic Character Analysis of Retailers Based on the Share of Stochastic Demand and Service

College of Management and Economics, Tianjin University, Tianjin 300072, China

Correspondence should be addressed to Weiya Di; moc.621@216041dyiew and Hao Ren; moc.621@82016102nerh

Received 27 March 2017; Revised 12 May 2017; Accepted 22 May 2017; Published 24 September 2017

Academic Editor: Sajad Jafari

Copyright © 2017 Junhai Ma 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

Apart from the price fluctuation, the retailers’ service level becomes another key factor that affects the market demand. This paper depicts a modified price and demand game model based on the stochastic demand and the retailer’s service level which influences the market demand decided by customers’ preference, while the market demand is stochastic in this model. We explore how the price adjustment speed affects the stability of the supply chain system with respect to service level and stochastic demand. The dynamic behavior of the system is researched by simulation and the stability domain and the bifurcation phenomenon are shown clearly. The largest Lyapunov exponent and the chaotic attractor are also given to confirm the chaotic characteristic of the system. The simulation results indicate that relatively small price adjustment speed may maintain the system at stable state. With the price adjustment speed gradually increasing, the price system gets unstable and finally becomes chaotic. This chaotic phenomenon will perturb the product market and this phenomenon should be controlled to keep the system stay in the stable region. So the chaos control is done and the chaos can be controlled completely. The conclusion makes significant contribution to the system referring to the price fluctuation based on the service level and stochastic demand.

#### 1. Introduction

Bertrand duopoly model is frequently used when the competitive behaviors are discussed in proper research. Hence, the dynamic behaviors of the whole supply chain system have already been probed. However, the retail service level plays an important role in influencing the retail price which indirectly changes the dynamic behaviors of the whole supply chain system that contains one supplier and two retailers.

The two retailers are duopoly in the product market. Xiao and Yang [1] established a price-service competition model with the circumstance whereby the demand is uncertainty in two competing supply chains considering the risk sensitivity. The outcome showed the higher service investment efficiency of one retailer may lead to lower optimal price and service level of his rival. Yan and Pei [2] focused on the retail service in a dual-channel supply chain, while the result suggested that the improved retail services enhanced the supply chain performance effectively in a dual-channel supply chain. Hall et al. [3] engaged in the dynamic retail price and ordering decisions in the category management setting, and this research found brand’s own price and the cross-price effect had an effect on the brand-by-brand market. Ghosh and Mitra [4] examined the different and opposite results to the ones obtained from a similar comparison in the standard setting through comparing the Bertrand and Cournot model in mixed markets with public and private enterprises considering maximum welfare and maximum profit separately.

C. H. Tremblay and V. J. Tremblay [5] introduced a Cournot-Bertrand duopoly model in a Cournot-type firm and a Bertrand-type firm based on the degree of product differentiation. The Cournot-Bertrand equilibrium was calculated and the analysis of the equilibrium was also conducted. Sieke et al. [6] investigated the service level based supply contracts including flat penalty contract and unit penalty contract in a coordinated supply chain. The optimal values of the contract parameters were decided and service level measures were connected to traditional service level measures. Ma and Zhang [7] analyzed the property of three oligarchs which have different rationalities in insurance market based on the price game model and variable feedback control method. Naimzada and Tramontana [8] studied the liner demand and cost functions with product differentiation, and a dynamic model was established to probe into the stability properties in the mixed Cournot-Bertrand duopoly model. Elsadany [9] developed a triopoly game to indicate the characteristic of a dynamic Cournot game through three bounded rational players. Jha and Shanker [10] introduced the circumstance where a supplier produced products and supplied them to a set of buyers. The lead time can be reduced with an added cash cost and a service level constraint was also included in the research. Wang and Ma [11] centered on the limited information in a Cournot-Bertrand mixed duopoly game model. The bounded rational principles were used to help the two firms make management decisions. He discussed the Nash equilibrium point and the local stability of the game. Mahmoodi and Eshghi [12] added the price competition and the stochastic demand into the duopoly supply chains; the effect of price and the uncertainty demand on the system were discussed.

Ahmed et al. [13] pointed out that the gradient adjustment mechanism of price was applied in a dynamic Bertrand duopoly in which the goods were differentiated. Kawabata and Takarada [14] probed into the effect of free trade agreements on the welfare in Bertrand and Cournot competition model under differentiated products. Brianzoni et al. [15] studied the mathematical properties and dynamics of a Bertrand duopoly with horizontal differentiated products and nonlinear costs, and the new evidence has been offered. Liu and Wang [16] investigated the quality control game that discussed the different risk attitudes affecting quality control game of supply chain especially in logistics service supply chain, instead of considering one member’s risk attitude and ignoring the combination of two members’ risk attitudes. Fang and Shou [17] managed supply uncertainty under supply chain Cournot competition between two supply chains consisting of a retailer and an exclusive supplier, respectively.

Esmaeilzadeh and Taleizadeh [18] conducted the optimal pricing decisions for two complementary products in a two-echelon supply chain under different market powers with game theory approaches. Ma and Guo [19] examined the impacts of information on the dynamical price game in two Bertrand game models, whereby the player obtains information of his rival before making decisions. Q.-H. Li and B. Li [20] designed the dual-channel supply chain; thus channel competition became inevitable. Hence, value-added services provided by retailers were considered. Equilibrium problems regarding retail services and fairness concerns were analyzed. Protopappa-Sieke et al. [21] developed the optimal two-period inventory allocation policies under multiple service level contracts in view of the fact that optimal inventory allocation had a significant impact on profits in the retail industry. How a manufacturer responds to a service level contract was analyzed as considering to minimize the expected costs.

The structure of the paper is as follows. Firstly, the basic Bertrand duopoly model considering the service level and stochastic demand is established in Section 2. Secondly, the analysis of the model mentioned in the above section has been conducted in Section 3. Then, we simulate the stability domain and the dynamical behaviors, and the related discussions have been made in Section 4. Lastly, the conclusion will be given in Section 5.

#### 2. Demand Model

Since oligopoly game model has already been researched in previous literature, while the oligopoly situation has been discussed, we consider that the other retailer comes into the monopoly market. Due to the entrance of the other retailer, exclusive monopoly was broken. What is more, the marketing competition is more vehement. Retailer’s demand is influenced by the other retailer’s price. In addition to retail price, we also consider that the market demand is associated with the service level of themselves and the competitors. Meanwhile, we assume the products that two retailers sell are homogeneous. Hence, the demand function of the two retailers is shown as follows:where () shows the ratio of customers buying the product, which represents the service discrepancy level of the products. () represents the price sensitive coefficient, which expresses the degree of market demand influenced by the price of their products and competition’s product price. Due to the difference of the price and service level between two products, customers will make their own choice according to the price and service level. We must be clear that the higher the service level of the products is, the more the consumers will choose to buy the products in condition of the same price level facing the homogeneous product.

In the expression above, we find that the market demand is affected by both retail price and the service level. The higher service level will bring about the larger cost. The cost of service level is shown in

In (2), means the service level of two retailers. is the influence coefficient, which is used to signify the influence degree of the service level to cost.

While the product market is complex and stochastic, the final demand of the customers is uncertain. We set the demand as stochastic variable , whereby the stochastic demand is the sum of the fixed demand and the potential stochastic part with market uncertainty , that is, . And the expectation and variance of are 0 and , respectively. We assume that retailers can predict the stochastic demand of their products. Retailers can help to make decisions with the market information forecast, when facing the uncertain factors in the product market. The forecast value of the stochastic demand is , and is composed with stochastic variable and the error term , that is, . The expectation and variance of are 0 and , separately. We have to point out that and are independent. Just like the former research that in Li [22], we know

The profit function of the manufacturer is depicted in (4). In this equation, represents the wholesale prices that manufacturer sells the products to retailer:

The retailers’ operating goal is to maximize profits. The retailer’s profit function is shown as follows:

When retailers and manufacturer in the market share the information, the expected profits decision models of manufacturer and retailers are shown as follows, respectively:

We can deduce the marginal profits of two retailers through calculating the first-order partial derivative of (6):

For the sake of simplicity, we make () in (7).

##### 2.1. Optimal Decision

If there is only one retailer in the market, the retailer regards profit maximization as the goal to give the optimal decision in the circumstance of discrete game. When there are two retailers in the market, the optimal decision of retailer 1 is influenced by retailer 2. Meanwhile, retailer 2 optimal decision is also affected by retailer 1 decision. Hence, we assume the information in the market can be shared between two retailers; that is, the optimal decisions of two retailers are known to each other. When the marginal profits reduce to zero, we have the optimal decision for retailers. The retail price and the service level are the decision variables.

For retailer 1, we calculate the first-order derivative of its profit. When the marginal profit gets to zero, the optimal retail price and optimal service level will be derived in equation. The detailed process is as follows:

Analogously, the optimal decision of retailer 2 will be conducted. Equation (10) has the same meaning as (8). The optimal decision of retailer 2 is expressed in (11):

When two retailers make their own optimal decision, we may further find the second derivatives of the retail prices and service level are less than zero. Therefore, the optimal decision of the system will be derived, which means the system has the equilibrium point: . The calculation will be expressed as follows:

Through jointing (9) and (11), the optimal solution of the system will be solved:

In (13), we have , , , .

##### 2.2. The Complex Dynamics of the System

Due to the complexity of the product market, the competition between enterprises in the market is increasingly fierce. Retailers do not know the perfect information of the market, while the market is complex and customers’ demands are uncertain. In addition, the more the transactions are, the greater the uncertainty will be, the more incomplete the information will be. Hence, the market demand cannot be forecasted accurately. The bounded rationality expectations solve this problem faultlessly. The retail price of next period will be decided according to the retail price in this period and the marginal profits, which makes up the discrete dynamic system of the product market.

In the equation above, () is the price adjustment speed of retailer , and the price adjustment speed is positive, which is set to make sure that retail price of next period is practically significant. When the marginal profit takes positive value, the retail price of next period will increase. However, when the marginal profit takes negative value, the retail price of next period will decrease.

The system (14) is nonlinear; the system is running under constant interference. When the system is disturbed, whether it can maintain a predetermined trajectory, the stability of the system becomes the primary question to study. What is different from the linear system is that the nonlinear system has multiple equilibriums. We obtain four equilibriums of system (14): , , , .

We may find that , , are boundary points, and is within the system. We guess that is the unique Nash equilibrium. In order to verify the guess and analyze the stability of the system, we calculate the Jacobian matrix of system (14):

In the equation above,

is the unique Nash equilibrium; the best profit of the system can be obtained when the best decision is in . We all know that when the margin profit is more than zero, the retailers will adjust their retail prices, until the margin profit goes up to zero. At this time, the system reaches to equilibrium state. Retailers will not change their own retail price any more, and the equilibrium state will be maintained by these two game players. Therefore, among all the equilibrium points, only the Nash equilibrium has practical significance. Then, this part will explore the local stability of the Nash equilibrium by using the Jacobian matrix and the July criterion.

The characteristic polynomial of the Jacobian matrix takes the following form:

In order to guarantee that the Nash equilibrium is local stability, () must satisfy the following limitations:

Due to the fact that limitations are so complex, the process to solve inequality (18) is very complicated. If the Nash equilibrium satisfies inequality (18), we may insure that Nash equilibrium is locally stable. We will probe into the dynamic characteristic of the system through numerical simulation in next section.

#### 3. Dynamic Characteristics of the System

In consideration of the complexity of system (14), the numerical simulation method will be employed in this paper as this method has been widely used in most of the literatures. The stable region, bifurcation, and chaos will be investigated; meanwhile, the largest Lyapunov exponent, entropy, chaotic attractor, and the time domain response will be discussed to verify the dynamic characteristics of this system. At last, the sensitivity of initial value will be analyzed. We take the parameter values as follows: , , , , , , , , , , , , , , .

##### 3.1. The Stable Region

The decision variables of the system are retail prices of two retailers. As retail prices obtain different values, the system will change to either the stable range or the unstable range. Figure 1 depicts the stable region of the system with blue range and the unstable of the system with white region. When employed values are in the blue range, the system will tend to the equilibrium state after iteration period by period. Then the system will stay at the Nash equilibrium state unless some factors outside break the equilibrium state.