Based on the work of domestic and foreign scholars and the application of chaotic systems theory, this paper presents an investigation simulation of retailer's demand and stock. In simulation of the interaction, the behavior of the system exhibits deterministic chaos with consideration of system constraints. By the method of space's reconstruction, the maximal Lyapunov exponent of retailer's demand model was calculated. The result shows the model is chaotic. By the results of bifurcation diagram of model parameters

A supply
chain is a complex system which involves multiple entities encompassing
activities from the raw material to the final delivery stage. A simple supply
chain system includes not only supplier, manufacturers, retailers, and
customers, but also all the flow of information and funds. So there exist
various types of uncertainties along the chain, for example, demand
uncertainty, production uncertainty, and lead time uncertainty. As a result,
the supply chains are much more dynamic. Such a dynamic and complex environment
presents a big challenge for researcher to handle uncertainty in an efficient and effective way.
One type of uncertain behaviors is bullwhip effect [

Chaos is
disorderly looking for long-term evolution occurring in a deterministic
nonlinear system. Chaos theory is concerned with chaos behavior in nonlinear
dynamical systems from a number of aspects. The origin of chaos theory dates
back to Lorenz’s [

A system of chaos is often characterized by a number of distinct features, for example, nonrandomness and nonlinearity, apparent disorder: the motion of the variables looks disorganized and irregular; strange attractor: pattern can be found in phase space; and sensitivity to initial conditions: a small change in initial conditions can have a large effect on the evolution of the system.

Chaotic behaviors can be either qualitatively identified by figure patterns methods which identify chaos or show whether a system is stable, periodic, quasiperiodic, or chaotic, such as Poincare map, phase plots, and power spectrum. Graphs and plots are visually efficient in showing trends and patterns. Another more accurate alternative is to calculate some quantifies, such as capacity dimension, correlative dimension, Kolmogorov entropy, and the maximal Lyapunov exponent. For example, provided the maximal Lyapunov exponent is a positive number, the investigated dynamic model is likely to be chaotic.

This paper is concerned with the model of retailer’s demand and calculates the system chaotic parameter which decides the chaotic system. By plotting the bifurcation diagram, the parameter of model can lead the system to the chaotic behavior.

There were various
supply chain models studied previously. Most of them were based on a simple
beer distribution model [

It is assumed
that the amount of the stock increase is the difference of the amount received,
which is ordered in the previous period, and the amount of sales to customers
as given by Steman [

It is assumed [

If the new stock level

Because the customers are myopic, the customer’s demand

From (

A simple chaotic system has two characters: sensitivity to initial conditions and
the strange attractor with
fractal structure. For the model (

The method presented
in this paper supposes a time series. The time-delay vectors can be
reconstructed as follows:

Before numerical
tests, it is necessary to define another quantity which is useful to
distinguish deterministic signals from stochastic signals. Let

For time series data
from a random set of numbers,

Above on the describe, we numerate the model
(

The values

After getting the embedding dimension of time
series, it is necessary to calculate the maximal Lyapunov exponent for the
chaotic model. It is assumed that the retailer’s demand, which is the origin of
(

Figure

The maximal Lyapunov exponent of retailer’s demand.

The initial data, as
show in Table

The parameters and initial data.

Item | Value |
---|---|

Initial retailer’s demand, | 500 |

Initial stock, | 1000 |

Initial expected demand | 0 |

Discount threshold, | 600 |

Ratio of overstock, | 1.2 |

Discount, | 70% |

Exponential constant, | 0.82 |

1 |

Because the
flow of demand or stock cam reflects the operational complexity of system, the
investigation of the system behavior in the paper will concentrate on the stock
and demand held by retailer. All the bifurcation diagrams in the following,
wherever it is not specified, were generated with a series time of 10000
iterations. Figure

The bifurcation diagram of actual stock and expected demand.

The above results show that the retailer’s
demand model based on (

The
bifurcation diagram of retail’s demand (

When price decreases,
it is normal that the customer’s demand will increase. If the amount of stock
is enough to satisfy the demand, which is called no stockout, the demand is positive like the
diagram at

The above results show that

The bifurcation diagram of retail’s stock (

Whether the
stockout happened or not, according to Figures

It is well known [

Bifurcation diagram of 5% and 10% increase of initial stock.

The impact of other initial data was also investigated. Figure

Bifurcation diagram of 5% and 10% increase of initial demand.

The above results show that the system state and the behavior depend not only on the model’s parameters, but also on the initial state. As a state—at any time—can be an initial state for future evolution, the behavior of such a system is thus sensitive to disturbances. A disturbance to the system states could eventually lead the system into equilibrium or chaos.

Despite the
model of retailer’s demand shows the chaotic behavior, the mechanism of
stabilizing the chaotic process in the actual supply chain is still under
discussion. In the dynamic system, Huang and Zhang [

This paper has
presented a model of retailer’s demand. In simulation of the interaction, the
behavior of the system exhibits deterministic chaos with consideration to system
constraints. By the method of space-reconstructed, this paper calculates the minimum
embedding dimension and the maximal Lyapunov exponent of retailer’s demand
model. The result shows that the model is chaotic. In the last, simulations
have demonstrated that the parameters

As a result of simulation, the model exhibits different behaviors as the initial condition varies. The sensitivity of the chaotic behavior to initial conditions makes the system easy to be disturbed in a changing environment. In reality, the system keeps disturbed all the time by various factors and the disturbance could be amplified in this way. Thus, the later state of the system remains unpredictable.

The simulation model discussed in this paper can inflect that the real supply chain is much more complex and its behavior is hence much more complicated. The managers should smoothen the uncertain behavior in supply chain. As a result, the behavior of the system can be much less chaotic than simulated.