# Effect of Ordering Policies on Complex Dynamics Behaviours in a Discrete-Type Manufacturing Industry Supply Chain System

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## Abstract

The beer game model is a typical paradigm used to study complex dynamics behaviours in production–distribution systems. The model, however, does not accord with current practical supply chain system models in discrete-type manufacturing industry, which are generally composed of retailers, distributors, manufacturers with internal supply chain, and suppliers. To describe how ordering policies influence the complex dynamics behaviour modes and operating cost in a general discrete-type manufacturing industry supply chain system, a high dimension piecewise-linear dynamics model is built for the supply chain system. Five kinds of ordering policy combination are considered. The distribution of both the largest Lyapunov exponent of effective inventory and average operating cost per cycle is obtained by simulation in a policy space. The simulation shows that for the general discrete-type manufacturing industry supply chain system, the upper chaotic corners emerge besides the lower chaotic corners in the policy space expressing the distribution of system behaviour mode, and that the ordering policies at each supply chain node as well as their combination have very significant effect on the topology of the distribution of both system behaviour mode and operating cost in the policy space. We find that chaos is not always corresponding to high cost, and the “chaos amplification” is not completely relevant to the “cost amplification”.

## Keywords

Supply chain system Ordering policy Operating cost Nonlinear dynamics behavior## 1 Introduction

The beer game model is the earliest, the most simple and complete model used to study complex nonlinear dynamics behaviours in supply chains. The beer game was derived from the product distribution game developed by Forrester [1] in the process of studying factory dynamics, which simulates the decision-making process of ordering or stocking in a chain-type production–distribution system consisting of such 4 sectors (nodes) as a retailer, a wholesaler, a distributor and a factory in total. The beer game model was early taken only as a universal case in investigating the bullwhip effect in supply chains, however, it has been considered as a classical model with nonlinear dynamics in supply chain systems since Mosekilde et al. [2] and Sterman [3] found chaos phenomenon in the game. The previous researches [2, 3, 4, 5, 6, 7, 8, 9, 10, 11] have shown that complex nonlinear dynamics behaviours, e.g., chaos/hyperchaos, can occur in the production–distribution system represented by the beer game model. The nonlinear factors come mainly from the nonlinear coupling of several oscillating feedback produced by the feedback structures among different decision-making entities, multiple inventory levels, time lag, etc., as well as the non-negativity [10, 12, 13] and piecewise-linearity [14] in ordering/dispatching decision-making. The chaos complex dynamics behaviours have been observed in multi-level series supply chain inventory systems [15, 16, 17], production-recycling system [18] and customer–supplier systems [19, 20, 21, 22] more simple than the beer game model.

There are many factors influencing complex dynamics behaviours of supply chain systems [7, 10, 11, 19, 20, 23, 24, 25, 26, 27, 28], including operating policies for ordering, production and sales, etc., as well as demand pattern, lead time, demand-information sharing, supply chain levels, etc. of which ordering policy is the most basic affecting factor. The most important effect of nonlinearity on the ordering or inventory policy of production–distribution systems is that the chaos solutions exist in significant policy parameter space to management, i.e., the chaotic corner [4, 7]. The more practical significance is that there are direct relations between the nonlinear dynamics behaviours of the system and the operating cost [7, 16]. Inside the chaotic corner area, the chaotic and hyperchaotic behaviours might intuitively be expected to generate the highest operating cost as these types of behaviour are associated with large oscillations [7]. In addition, the complex behaviour mode distribution in policy parameter space of the supply chain system is different as the managers use various ordering policies, for example, the fixed expected inventory [5, 6, 8, 12] and the variable expected inventory [7, 10].

In the above studies, the original beer game model [2, 3] was usually adopted, and the single heuristic ordering policy [12] was basically used by each node in most models of them. In this paper, based on the original beer game model, a more general supply chain system model in discrete-type manufacturing industry is considered, which consists of retailers, distributors, manufacturers and suppliers. Both the largest Lyapunov exponent of effective inventory and average operating cost at each node are obtained by simulation. The effect of different ordering policy combination is studied on the distribution of the dynamics behaviour mode (expressed by the largest Lyapunov exponent) and the corresponding average operating cost.

The rest parts of this paper are organized as follows: Section 2 describes the structure model and the operating mechanism of the supply chain system, and builds the dynamics model of the system; Section 3 describes the ordering policy and operating cost at each node; Section 4 simulates the supply chain system benchmark model; Section 5 analyses the effect of different ordering policies and their combination on the distribution of the dynamics behaviour mode and the corresponding average operating cost; Section 6 concludes.

## 2 Supply Chain System Model

### 2.1 Structure Model and Operating Mechanism of the Supply Chain System

For the convenience of analysis, it is assumed that the supply chain only supplies one type of product, and customer demand is exogenous [7, 10]. Each node of the supply chain is governed by the following rules in each period (to be referred to as a week): (1) receiving orders from downstream node, filling orders as possible according to current inventory and shipping goods, last round stock out being filled at this round; (2) checking inventory, determining next round order quantity and sending purchase orders to upstream node, and placed orders not being cancelled; (3) receiving goods from upstream node, or producing and replenishing stocks, and shipments made not being returned.

In terms of the above rules, each node of the supply chain uses the same mechanism for orders’ receiving and sending, as well as goods’ receiving and shipping. In addition, the manufacturer will make a production decision (i.e., production orders of product) based on the distributor’s order, demand prediction and inventory information, organize production and purchase raw material and parts for products from the supplier. These tasks will be respectively finished by every sector in the inner supply chain of the manufacturer, i.e., sales department, production department and purchasing department. The supplier will determine production of the raw material and parts based on the manufacturer’s orders and other relative information, and carry out production and supply.

In order to compare with the simulation of the original beer game model, in despite of differences from the structure of most beer game models, we suppose in the model of this paper that the production and shipment of goods and the transmission of orders involve time delays; and the production capacity of the manufacturer and the supplier is unlimited. The production time is two time periods, and both the transmission delay of orders and the shipment delay of the product between two successive levels of the outer supply chain are one time period, while there is no time delay in the transmission of orders and goods between two successive sectors of the inner supply chain of the manufacturer.

### 2.2 Heuristic Ordering Policy

Notations

Symbol | Description |
---|---|

| Time or period |

\(\hat{D}_{t}\) | The expected demand |

\(D_{t}\) | The actual demand i.e. the incoming orders |

\(I_{t}\) | The actual inventory level |

\(I_{t}^{*}\) | The desired inventory level |

\(AI_{t}\) | The adjustment for the inventory level |

\(SL_{t}\) | The actual supply line (orders placed but not yet received) |

\(SL_{t}^{*}\) | The desired supply line |

\(ASL_{t}\) | Adjustment for the supply line |

\(O_{t}\) | Order quantity |

\(\delta\) | The desired coverage |

\(\gamma\) | The expected (or perceived) lag between ordering and acquisition of goods |

\(\theta\) | A constant which determines how fast expectations are updated, \(0 \le \theta \le 1\) |

\(\alpha_{I}\) | Rate at which the discrepancy between actual and desired inventory levels is corrected, \(0 \le \alpha_{I} \le 1\) |

\(\alpha_{SL}\) | The rate at which the discrepancy between actual and desired supply line is corrected, \(0 \le \alpha_{SL} \le 1\) |

*t*can be defined as follows:

Usually \(0 \le \alpha_{SL} \le \alpha_{I}\), and \(0 \le \beta \le 1\) [12].

*t*is defined as below

*t*is defined as below [5, 6, 8]:

### 2.3 Dynamics Model of the Supply Chain System

Based on the above structure model, the operating mechanism and ordering policy of the supply chain system, the dynamics model is built by synthesizing several kinds of mathematics model for the beer game [5, 6, 7, 8, 10, 11, 12, 25].

Notations

Symbol | Description |
---|---|

I | The actual inventory level of goods |

B | Backlog of orders |

IS | Incoming shipments |

OS | Outgoing shipments |

SL | The actual supply line |

IO | Incoming orders |

OP | Orders placed |

ED | The expected demand |

PR | Production rate |

PD1, PD2 | Production delay of one period and two periods |

COR | The exogenous customer order rate |

Every variables are prescribed by a letter indicating the respective sector, i.e., using the first letter of the state variable denoting the node or sector in the supply chain, *C* stands for customer, *R* for retailer, *D* for distributor, *M*, *W*, *P* for sale department, production department and purchasing department of manufacturer, respectively, and *S* for supplier.

The dynamics behaviours of the supply chain system shown in Figure 2 can be described by a set of high-dimensional piecewise-linear and nonlinear difference equations.

- 1.
For the sale department of the manufacturer

The actual inventory of products$$MI_{t} = \text{max} \left\{ {0, \, MI_{t - 1} + MIS_{t - 1} - MB_{t - 1} - MIO_{t - 1} } \right\}.$$(8)The order backlog of products$$MB_{t} = \text{max} \left\{ {0, \, MB_{t - 1} + MIO_{t - 1} - MI_{t - 1} - MIS_{t - 1} } \right\}.$$(9)The incoming shipments of products (i.e., the production of products)$$\begin{aligned} MIS_{t} & = WOS_{t} = WPD_{2,t} , \hfill \\ WPD_{2,t} & = WPD_{1,t - 1} ,\hfill \\ WPD_{1,t} & = WPR_{t - 1} . \hfill \\ \end{aligned}$$(10)The outgoing shipments of products$$MOS_{t} = \text{min} \left\{ {MI_{t - 1} + MIS_{t - 1} , \, MB_{t - 1} + MIO_{t - 1} } \right\}.$$(11)The incoming orders of products$$MIO_{t} = DOP_{t - 1} .$$(12)The expected demand of products$$MED_{t} = \theta \cdot MIO_{t - 1} + (1 - \theta )MED_{t - 1} .$$(13)The actual supply line of products$$MSL_{t} = WPD_{1,t} + WPD_{2,t} .$$(14)The orders of products placed$$\begin{aligned} MOP_{t} & = \text{max} \{ 0,MED_{t} + \alpha_{I} (\delta MED_{t} - MI_{t} ) \hfill \\ & \quad + \alpha_{SL} (\gamma MED_{t} - MSL_{t} )\} , \hfill \\ \end{aligned}$$(15a)$$MOP_{t} = \text{max} \left\{ {0,MED_{t} + \alpha_{I} (Q - MI_{t} + MB_{t} - \beta MSL_{t} )} \right\}.$$(15b) - 2.
For the production department of the manufacturer

The actual inventory of work in process (WIP, i.e., unfinished products)$$WI_{t} = MSL_{t} .$$(16)The order backlog of WIP$$WB_{t} = \text{max} \left\{ {0, \, WB_{t - 1} + WIO_{t - 1} - WI_{t - 1} - WIS_{t - 1} } \right\}.$$(17)The volume of WIP put into production (outgoing shipments of raw material and parts)$$WIS_{t} = POS_{t} = WPR_{t} .$$(18)The quantity of finished WIP (i.e., the production of products)$$WOS_{t} = \text{min} \left\{ {WI_{t - 1} + WIS_{t - 1} , \, WB_{t - 1} + WIO_{t - 1} } \right\}.$$(19)The incoming orders of WIP (i.e., the orders of products)$$WIO_{t} = MOP_{t} .$$(20)The expected demand of WIP (i.e., the expected demand of products)$$WED_{t} = MED_{t} .$$(21)The actual supply line of WIP$$WSL_{t} = WIS_{t} + PIO_{t} + PB_{t} + POS_{t} .$$(22)The orders of WIP placed (i.e., the orders of products)$$WOP_{t} = MOP_{t} .$$(23) - 3.
For the purchasing department of manufacturer

The actual inventory of raw material and parts$$PI_{t} = \text{max} \left\{ {0, \, PI_{t - 1} + PIS_{t - 1} - PB_{t - 1} - PIO_{t - 1} } \right\}.$$(24)The order backlog of raw material and parts$$PB_{t} = \text{max} \left\{ {0, \, PB_{t - 1} + PIO_{t - 1} - PI_{t - 1} - PIS_{t - 1} } \right\}.$$(25)The incoming shipments of raw material and parts$$PIS_{t} = SOS_{t - 1} .$$(26)The outgoing shipments of raw material and parts$$POS_{t} = \text{min} \left\{ {PI_{t - 1} + PIS_{t - 1} , \, PB_{t - 1} + PIO_{t - 1} } \right\}.$$(27)The incoming orders of raw material and parts (i.e., the orders of products)$$PIO_{t} = WOP_{t} .$$(28)The expected demand of raw material and parts (i.e., the expected demand of products)$$PED_{t} = MED_{t} .$$(29)The actual supply line of raw material and parts$$PSL_{t} = PIS_{t} + SIO_{t} + SB_{t} + SOS_{t} .$$(30)The orders of raw material and parts placed (i.e., the orders of products)where subscript$$POP_{t} = MOP_{t} ,$$(31)*t*denotes the time period.

## 3 Ordering Policy and Operating Cost of Each Node

### 3.1 Ordering Policy of Each Node and Their Combination

Ordering behaviour is basic one of each node in supply chain. Different ordering policies and combination will influence not only operating state of each node, but also dynamics behaviour of the whole supply chain system. In this paper, the following nodes’ ordering policies and their combinations are taken into consideration.

Order policy 1, i.e., all supply chain nodes use the same heuristic ordering policy as presented in Section 2.2, keep the ordering adjusting coefficients \(\alpha_{I}\) and \(\alpha_{SL}\) of these nodes consistent with each other, and determine the order quantity \(O_{t}\) by Eq. (7), e.g., Eq. (15b).

Order policy 2, i.e., all supply chain nodes use the same heuristic ordering policy, keep the ordering adjusting coefficients \(\alpha_{I}\), \(\alpha_{SL}\), the desired coverage time \(\delta\) and \(\gamma\) of these nodes consistent with each other (\(\delta\) = 3, \(\gamma\) = 2), and determine the order quantity \(O_{t}\) by Eq. (6), e.g., Eq. (15a).

Order policy 3 allows different ordering policies for different supply chain nodes, assuming that the manufacturer uses order policy 1, and the other nodes adopt an improved (\(t, \, R, \, S\)) ordering policy or replenishing policy. Similar to the heuristic ordering policy, (\(t, \, R, \, S\)) replenishing policy is also a sort of feedback based ordering policy, i.e., checking inventory level at each period, placing order if the stock level is lower than ordering point \(R_{P}\) (in this study \(R_{P}\) = 20), otherwise, no ordering. Let \(S\) be the maximum stock capacity (in this study \(S\) = 500), if the stock level \(I_{t} < R_{P}\), the order quantity \(O_{t} = S - I_{t}\). The improved (\(t, \, R, \, S\)) replenishing policy considers the supply line \(SL_{t}\), in which the actual order quantity \(O_{t} = S - I_{t} - SL_{t}\).

Order policy 4 allows different ordering policies for different supply chain nodes, assuming that the supplier and the manufacturer use order policy 1 and order policy 2, respectively, the other nodes adopt the improved (\(t, \, R, \, S\)) replenishing policy.

Order policy 5 allows different ordering policies for different supply chain nodes, assuming that the manufacturer uses the improved (\(t, \, R, \, S\)) replenishing policy, and the other nodes adopt order policy 1.

There would be quite a few ordering policies and their combination in actual supply chains. In this paper, we focus on the effect of the above ordering policies and combination on the dynamics behaviour modes and operation cost.

### 3.2 Operation Cost of Nodes

There is a relation between the dynamics behaviour modes of order or inventory and the operating cost of a supply chain, where chaotic/hyperchaotic behaviour modes would bring intuitively high cost, since this kind of behaviour modes usually result in large oscillation amplitude [7]. Therefore, it is very meaningful to study the relation between behaviour modes and operating cost in the parameter space \((\alpha_{I} ,\alpha_{SL} )\) for the ordering and inventory management decision of a supply chain.

*T*

_{r}are calculated by Eqs. (32) and (33), respectively,

## 4 Simulation Analysis for the Supply Chain Benchmark Model

### 4.1 Benchmark Model

- i.
The customer demand mode is a step-function pattern, i.e., the demand is four units per period in the first four periods, and is increased to eight units per period from the fifth period till the end of the simulation;

- ii.
Ordering policy 1 is used;

- iii.
Order and shipments in each node equal to 4 units per week. All inventories are initialized at 12 units and all initial order backlog are zero. Moreover, for each node,

*θ*= 0.25, \(Q = 17\) units of goods, \(C_{H}\) = 0.5 and \(C_{B}\) = 1.0 units of money.

### 4.2 Simulation Design

Provided that the simulation run length *T* is 2000 time periods, \(T_{r}\) = 500, ordering parameters \((\alpha_{I} ,\alpha_{SL} )\) for each node are the same and constant. In the range of \(0 \le \alpha_{I} \le 1\) and \(0 \le \alpha_{SL} \le 1\) (increment 0.02), the actual inventory \(I_{i,t}\) and the order backlog \(B_{i,t}\), as well as the corresponding stock holding cost \(C_{Hi} I_{i,t}\) and out-of-stock cost \(C_{Hi} B_{i,t}\) are obtained by simulation, thus the effective inventory of each node \(EI_{i,t}\) is calculated, where \(EI_{i,t}\) is defined as the difference value of \(I_{i,t}\) and \(B_{i,t}\), i.e., \(EI_{i,t} = I_{i,t} - B_{i,t}\) [4, 5, 6, 7, 8]. According to the largest Lyapunov exponent LLE corresponding to \(EI_{i,t}\), we can depict quantitatively the nonlinear behaviour modes of the supply chain system [7, 10, 11], i.e., if the LLE is positive, the system is chaotic; otherwise the system is stable, periodic or quasi-periodic, where the LLE is calculated by a method of small data sets [29].

### 4.3 Distribution of Behaviour Modes and Operating Cost at Each Node in the Benchmark Model

- i.
From the downstream node (retailer) successively to the upstream node (supplier), the blue area (\(\lambda_{i\text{max} }\) ≤ 0, corresponding to stable or periodic behaviour mode) is gradually reduced, while the non blue area (\(\lambda_{i\text{max} }\) > 0, corresponding to chaotic/hyperchaotic behaviour mode) is inchmeal increased, and the color of the area is in a gradual transition from mainly green (smaller LLE value means weaker chaotic degree) to dominated yellow with saffron yellow (bigger LLE value means stronger chaotic degree), which indicates that along the supply chain from the downstream node to the upper-stream node, more and more complex the behaviours of the system appear, more and more prominent the chaotic range and degree become, i.e., there exists so-called “chaos amplification” phenomenon [10, 11];

- ii.
There exists so-called “chaotic corner” [7, 10] in the distributions of behaviour mode at each node, i.e., the region at the lower right corner of these figures (hereinafter referred to as “lower chaotic corner”), where the solutions with different properties are complicatedly mixed, and from the downstream node to the upperstream node, the “chaotic corner” region gradually diffuses into the upper right corner of these figures, which means that there exists so-called “upper chaotic corner”, and is not mentioned in Refs. [7, 10, 11];

- iii.
Although each node belongs to a part of the supply chain system, and there exists a relation of information and goods flow between the adjacent nodes, the states and behaviours at each node seem to have relative independence, which means that for the same \((\alpha_{I} ,\alpha_{SL} )\) values, not only is the magnitude of the LLE values of each node different, but also there may exist difference between the properties of the corresponding behaviour modes, i.e., it is stable for some node, but periodic or chaotic for other nodes, which can be proved by the effective inventory time series of each node and the research result of Hwarng et al. [10, 11].

- i.
Corresponding to the “lower chaotic corner” in the distribution \(\lambda_{i\text{max} } (\alpha_{I} ,\alpha_{SL} )\), there exist high cost areas at the lower right corner of the distribution \(C_{i} (\alpha_{I} ,\alpha_{SL} )\) of each node (hereinafter referred to as “lower loss corner”), where the operation cost and loss risk of node is higher.

- ii.
The distribution \(C_{i} (\alpha_{I} ,\alpha_{SL} )\) in the “lower loss corner” of each node shows more obvious hierarchical structure, where along the direction of the diagonal line \(\alpha_{I} = \alpha_{SL}\) in the parameter space \((\alpha_{I} ,\alpha_{SL} ),\) an approximately parallel banded region appears, and the corresponding cost value decreases successively from the vertex at the lower right corner in the distribution \(C_{i} (\alpha_{I} ,\alpha_{SL} )\) along the direction approximately perpendicular to the diagonal line \(\alpha_{I} = \alpha_{SL}\).

- iii.
From the downstream node (retailer) successively to the upstream node (supplier), the dark blue area (lower cost region) is gradually diminishing, while the non dark blue area (higher cost region) is inchmeal expanding, or the cost value in the “lower loss corner” is gently increasing, i.e., there exists so-called bullwhip effect or “cost amplification” phenomenon, which is similar to the “chaos amplification” phenomenon in the distribution \(\lambda_{i\text{max} } (\alpha_{I} ,\alpha_{SL} )\) [10].

- iv.
In the distribution \(\lambda_{i\text{max} } (\alpha_{I} ,\alpha_{SL} )\) and \(C_{i} (\alpha_{I} ,\alpha_{SL} )\) of each node, there exist all the corresponding “lower chaotic corner” and “lower loss corner”, respectively, while only for the upstream manufacturer and supplier, there exists the “upper chaotic corner” besides the “lower chaotic corner”, but there does not exist the corresponding “upper loss corner” at the upper right corner in the distribution \(C_{i} (\alpha_{I} ,\alpha_{SL} )\), which means that chaos is unnecessarily corresponding to high cost, or rather, the “chaos amplification” does not completely respond to the bullwhip effect or “cost amplification”, and it is not mentioned in Refs. [7, 10, 11].

## 5 Effect of Ordering Policies on the Distribution of Behaviour Mode and Operation Cost

- i.
Not only does ordering policy 2 make the non chaotic connected region embedded in the “lower chaotic corner” obviously increasing, but also the “upper chaotic corner” and the dark blue lowest cost area in the banded region approximately parallel to \(\alpha_{I} = \alpha_{SL}\) almost disappear;

- ii.
Ordering policy 3 causes the distribution \(\lambda_{M\text{max} } (\alpha_{I} ,\alpha_{SL} )\) and \(C_{M} (\alpha_{I} ,\alpha_{SL} )\) to fundamentally change, in the distribution \(\lambda_{M\text{max} } (\alpha_{I} ,\alpha_{SL} )\), the “lower chaotic corner” connects with the “upper chaotic corner” into a piece of region approximately parallel to \(\alpha_{SL}\) axis, while the change of topological structure similar with the distribution \(\lambda_{M\text{max} } (\alpha_{I} ,\alpha_{SL} )\) also occurs in the corresponding distribution \(C_{M} (\alpha_{I} ,\alpha_{SL} )\), where the cost values in the entire region become distinctly bigger, and the degressive direction of cost turns from the direction approximately vertical to the diagonal line \(\alpha_{I} = \alpha_{SL}\) into \(\alpha_{SL}\) axis;

- iii.
Not only does ordering policy 4 make the “lower chaotic corner” clearly minish, but also the “upper chaotic corner” vanishes, it is different from other cases that the “loss corner” usually with the largest cost does not appear at the lower right corner of the distribution \(C_{M} (\alpha_{I} ,\alpha_{SL} )\), but transfers to the upper right corner corresponding to the “upper chaotic corner”, which results in the corresponding relation between the “chaotic corner” and the “loss corner” into disappearance;

- iv.
Not only does ordering policy 5 make the area of the “lower chaotic corner” extend slightly, but also the “upper chaotic corner” vanishes, though the distribution \(C_{M} (\alpha_{I} ,\alpha_{SL} )\) still has a normal topological structure, the cost values in the entire region become remarkably bigger.

## 6 Discussions and Conclusions

Ordering is the most basic behaviour in the operation of a supply chain system, while ordering policy is one of the most primary factors influencing the operating states of the system. In this paper, we have emphatically studied the effect of several ordering policies on the dynamics behaviour modes and operating cost of the supply chain system in discrete-type manufacturing industry.

The study shows that when the same ordering policy (i.e., ordering policy 1 in Section 3.1) with the original beer game model is adopted, the supply chain system model (i.e., the benchmark model in Section 4.1) in this paper has the similar distribution characteristics of dynamics behaviour modes and operating cost in the ordering parameter space with the original beer game model, i.e., in the behaviour mode distribution \(\lambda_{i\text{max} } (\alpha_{I} ,\alpha_{SL} )\) so that there exist the “lower chaotic corner” [7, 10] and the “chaos amplification” phenomenon [10], while in the operating cost distribution \(C_{i} (\alpha_{I} ,\alpha_{SL} )\) there exist the “lower loss corner” corresponding to the “lower chaotic corner” and the “cost amplification” phenomenon similar to the “chaos amplification”. The difference, however, is that there exists still the “upper chaotic corner” in the behaviour mode distribution \(\lambda_{i\text{max} } (\alpha_{I} ,\alpha_{SL} )\) of the upper stream nodes, while there does not exist the relevant “upper loss corner” in the cost distribution \(C_{i} (\alpha_{I} ,\alpha_{SL} )\), which means that chaos is not always corresponding to high cost, or say, the “chaos amplification” is not completely relevant to the “cost amplification”.

The research also indicates that using different ordering policies in the benchmark model of the supply chain system may lead to distinct variation in the topological structure of the distribution \(\lambda_{i\text{max} } (\alpha_{I} ,\alpha_{SL} )\) and \(C_{i} (\alpha_{I} ,\alpha_{SL} )\), which may make not only the relevant relation between the “lower chaotic corner” and the “lower loss corner” no longer exist, but also the “upper chaotic corner” disappear, and the “upper loss corner” appear, i.e., the relevant relation between the “chaotic corner” and the “loss corner” does not always exist.

In the view of reducing the cost of nodes or supply chain, we can select proper decision parameters in the policy space \((\alpha_{I} ,\alpha_{SL} )\) to avoid the “chaotic corner” and the “loss corner”. Since the one-to-one corresponding relation between the “chaotic corner” and the “loss corner” does not always exist, thus each node can not select the ordering parameters simply according to the relevant relation between the “chaotic corner” and the “loss corner”. Adopting learning policy [24] or optimizing policy [30] is more effective method to select the ordering parameters.

## Notes

### Authors’ Contributions

W-PF was in charge of the whole trial; WW wrote the manuscript; WW assisted with sampling and laboratory analyses. Both authors read and approved the final manuscript.

### Authors’ Information

Wen Wang, born in 1966, is currently a professor at *Xi’an University of Technology*, *China*. She received her PhD degree from *Xi’an University of Technology*, *China*, in 2010. Her research interests include supply chain dynamics.

Wei-Ping Fu, born in 1957, is currently a professor and supervisor for PhD candidate at *Xi’an University of Technology*, *China*. He received his PhD degree from *Xi’an Jiaotong University*, *China*, in 1996. His research interests include nonlinear dynamics, electromechanical system dynamics and control, intelligent robot and vehicle control.

### Competing Interests

The authors declare that they have no competing interests.

### Funding

Supported by National Natural Science Foundation of China (Grant No. 11072192), and Shaanxi Provincial Industrial Technology Research Projects of China (Grant No. 2015GY118).

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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