# The Study of the Chaotic Behavior in Retailer's Demand Model

Authors: Junhai Ma; Yun Feng
Journal: Discrete Dynamics in Nature and Society
 (2008)

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## Abstract

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 parametersk, r  and changing initial condition, the system can be led to chaos.


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## Body

## 1. Introduction

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 [1], which increases the variability of demand in the supply chain.
Lee et al. [2] identified four uncertain causes of the bullwhip effect: demand forecast,
reorder batching, pricing fluctuation, and shortage gaming. Later, Chen et al. [3] quantify the effect in a simple two-stage supply system with statistics
method and AR(1) demand process. However, statistic method only deals with continuous
time and linear problems, the uncertain activities generate interactions among
customers, and suppliers usually lead to discrete time and nonlinear dynamic behaviors
or even chaos. Many researches focus on the uncertainty field with chaotic
theory.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 [4] study in weather forecasting system. Subsequently, many researchers have
contributed to greater interest in studying and applying chaos theory to
inventory, management, and supply chain studies. It was reported that a chaotic
demand for liquid detergent exists at the interface of a manufacturer and its
distributors [5].
Kumara et al. [6] demonstrated that in a queuing model of supply chain logistics systems, the behaviors
of the system exhibit chaos.  Wang et al. [7] explore an efficient approach for identifying chaotic phenomena in demands and
develops a production lot-sizing method for chaotic demands. Makui and Madadi [8] use the maximal Lyapunov exponent, a chaos parameter, to measure the bullwhip effect and
get some useful results on the behavior of the bullwhip effect. Wu and Zhang [9] present an investigation of interaction between customers and suppliers in a
three-tier supply chain system, and found that the attractors of the model move
with the environment and the initial states. Hwarng and Xie [10] investigate how this class of variability, chaos, may occur in a multilevel
supply chain and offer insights into how to manage relevant supply chain
factors to eliminate or reduce system chaos.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.


## 2. Demand Model

There were various
supply chain models studied previously. Most of them were based on a simple
beer distribution model [11] which includes three levels: brewery or factory, wholesaler, and retailer. In
this system, orders propagate from customers to factory. In order to
distinguish the traditional beer model, this paper focuses on the model of
retailer’s demand decision and study the complex behavior between retailer and
the whole supply chain. The
model is constructed on a customer-supplier system, including customers, a
retailer, and a manufacturer. The customers buy products from the retailer. The
demand of the customers depends on the price offered by the retailer. The
retailer forecasts the demand based on the previous sales and makes the new
order to the manufacturer accordingly. The manufacturer makes only what the
retailer is ordered. However, the manufacturer does not accept the order with
quantity in excess of its capacity. The retailer determines the sales price for
the next period according to the current stock. If the stock level is high, the
retailer offers a discount to encourage the customer to buy more. The process is
deterministic on the basis of settled rules or formulae and repeats
automatically in the next period.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 [12]. That is, (2.1)St+1=qt+1+St−Dt,where Dt is the actual demand at t period. qt+1 is the order from retailer to manufacture and
is equal to Pt+1 which is the output of manufacturer at t period.It is assumed [12] that the retailer sends a request for
quotation of the product in a quantity as forecasted. On receiving the request,
the manufacture plans the production and accepts the order with limited
production capacity in the period. If the production capacity is infinite, the Pt+1 should be equal to the expected demand D^t+1 which is the expected demand of customer in
next period. Then, (2.2)St+1=D^t+1+St−Dt. The retailer forecasts the demand by simple exponential
smoothing by Towill [13],
the expected demand at time t can be defined as follows: (2.3)D^t=ωDt−1+(1−ω)D^t−1,0<ω<1, where Dt is the actual demand of customer. ω is an exponential constant which determines
how fast expectations are updated; when ω=1, the expected demand D^t at t period is equal to the actual demand Dt−1 at t−1 period. At the ω=0, D^t=D^t−1,
the actual demand can be ignored.If the new stock levelSt+1 excesses a threshold, T,
the retailer regards the level as overstock and prepares to make a discount
offer to reduce the stock. An overstock rate vt+1 is calculated as
(2.4)vt+1=St+1−TT. The parameter T is a policy parameter for the retail to
evaluate the stock level and determine whether to offer a price discount or
not. Price discount is an effect behavior to alleviate the inventory’s pressure
and magnify the customer’s demand at short time. But the discount offer is only
valid for one period. The discount for the next period will be re-evaluated by
the overstock rate at the end of the period. The relationship between overstock
rate and discount rate is hence calculated by (2.5)rt+1=vt+1q, where q is a constant. When the rt+1
increases, the price will be decreased. As a result, the
customer’s demand will increase. The trend of demand’s increase is the result
of the retailers’ active behaviors and avoids the venture which changes from the
uncertain demand to actual inventory.Because the customers are myopic, the customer’s demandRt+1 is deterministic by the price. If there is no
discount, the customer’ demand is equal to the last period actual demand Dt,
referred to as basic demand. When a discount rate is offered, the demand
increases based on the basic demand [9]: (2.6)Rt+1=Dt(1−rt+1/a)k, where a is the upper limit of the discount, k is a parameter which determines the demand
under discount, 0<k<1.
If not considering the price fluctuation, the customer’s demand Rt+1 is equal to the retailer’s actual demand Dt+1 among the linear models of demand. But the
linear relationship cannot describe the dynamic demand model more realistic,
many researchers prefer the nonlinear function for the demand model. According
to the model of Qi et al. [14], the market’s actual demand is a downside
nonlinear function of retail price: (2.7)Dt+1=Dp−2r, where r is a coefficient of price sensitivity, D is the maximum market scale and presents the
customer’s forecast demand in market. So the maximum scale D is equal to the customer’s demand Rt+1.From (2.6) and (2.7), the actual demand
and price can be expressed as (2.8)Dt+1=Dtp−2r(1−rt+1/a)k. Form (2.4), (2.5), and (2.8), (2.9)Dt+1=(qaT)kp−2r[(qa+1)T−St+1]kDt. The model above can be expressed as a high-demand piecewise
map in the following mathematical form: (2.10)xt+1=(qaT)kp−2rxt[(qa+1)T−yt+1]k,yt+1=yt+zt+1−xt,zt+1=ωxt+(1−ω)zt.


## 3. Chaos’s Identifying Method

A simple chaotic system has two characters: sensitivity to initial conditions and
the strange attractor with
fractal structure. For the model (2.10), this paper uses
the time series of retailer’s actual demand to reflect the system behavior. The
chaotic character can be identified through calculating
the embedding dimension of a scalar time series. If we only consider one variable and make it invariable
based on Cao [15] theorem, an equal
reconstructed space will be identified from the new point which is taken as the
new dimension on the time delay point. When embedding dimension accumulates to a
new level, the reconstructed space has all the same dynamic characters as the actual
system and also does not rely on the reconstructed detail of process.The method presented
in this paper supposes a time series. The time-delay vectors can be
reconstructed as follows:(3.1)Xn={x(n),x(n+τ),…,x(n+(m−1)τ)},n=1,2,…,N−(m−1)τ, where m is the embedding dimension and τ is the time delay. Note that Xn means the ith reconstructed vector with
embedding dimension m. Similar to the idea of the false
neighbor method, it is defined as follows:
(3.2)a(i,m)=∥XNN+1−Xn+1∥m+1∥XNN−Xn∥m,i=1,2,…,N−mτ, where
∥⋅∥
is some measurement of Euclidian distance and
is given in this paper by the maximum or the mean value of all a(i,m)’s: (3.3)E(m)=1N−mτ∑i=1N−mτa(i,m), where E(m) is dependent only on the dimension m and the lag τ.
To investigate its variation from m to m+1, let E1(m)=E(m+1)/E(m).  However, E1(m) stops changing when m is greater than some value m0;
if the time series comes from an attractor, then m0+1 is the minimum embedding dimension.Before numerical
tests, it is necessary to define another quantity which is useful to
distinguish deterministic signals from stochastic signals. Let(3.4)E*(m)=1N−mτ∑i=1N−mτ|x(i+mτ)−x(NN+mτ)|, where the meaning of (NN+mτ) is the same as above, that is, it is the
integer such that a(i,m) is the nearest neighbor of a(i,m).
Let (3.5)E2(m)=E*(m+1)E*(m).For time series data
from a random set of numbers,E1(m), in principle, will never attain a
saturation value as
m
increases. But in practical computations, it
is difficult to resolve whether the E1(m) is slowly increasing or has stopped changing
if m is sufficiently large. In fact, since
available observed data samples are limited, it may happen that the E1(m) stops changing at some
m
although the time series is random. It is
recommended calculating both E1(m) and E2(m) for determining the minimum embedding
dimension of a scalar time series, and distinguishing deterministic data from
random data.Above on the describe, we numerate the model
(2.10) with step 0.01, and record the time series data from the
numerical solution with sampling time 6 after all transients have been diminished.
The results for this time series are shown in Figure 1, where τ=1, sampling time.  One can see that the E1(m) attains its saturation value at m=2 and the E2(m)=1 at m=8, it shows that the time series is chaotic series. Therefore, m=2 should be the minimum embedding dimension for
the time series in our model. Obviously, the result does not strongly depend on
how many data points are used. Because the E1(m) attains its saturation at m=2, the models of retail’s demand have clearly
chaotic behaviors.Figure 1
The valuesE1(m) and E2(m) for the data from the retailer’s model.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
(2.10), is located in a reconstructed space like (3.1). The analysis presented here
first identifies the starting point xj (in the space)  which is closest to the initial reference
point xi.
The distance between the two points is defined as (3.6)dj(xi,xj,t)=|xj+m−1+t−xi+m−1+t|. Let (3.7)S(t)=1N∑j=1NIn(1|Uj|∑i∈Ujdj(xj,xi,t)), where Uj is the E-neighborhood included xi, xj.
When S(t) increase as the t,
the regression curve will be plotted by the method of least squares.
Calculating the slope of regression curve in the S(t)−t plot and the value of slope is the maximal
Lyapunov exponent.Figure2
shows the maximal Lyapunov exponent of retailer’s demand. In the S(t)−t plot, the slope of S(t)’s curve equals 0.05297. It clearly confirmed
that the dynamic system based on (2.10) has chaotic behavior. In the next step,
simulation of the model is necessary to prove that the chaotic behavior is
sensitive to initial conditions.The maximal Lyapunov exponent of retailer’s demand.
(a)(b)(c)


## 4. Simulation Results and Discussion

The initial data, as
show in Table1, for running the model described above were chosen arbitrarily.
But the value of price must be equal to 1, it easy to find the relationship between
the actual stock and expected demand.Table 1
The parameters and initial data.
ItemValueInitial retailer’s demand,D0500Initial stock,S01000Initial expected demandD^00Discount threshold,T600Ratio of overstock,q1.2Discount,a70%Exponential constant,ω0.82p−2r1Because 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.  Figure3 shows the chaotic
behavior of retail’s actual and expected demand, which is based on (2.10). It is
clear in the bifurcation diagram that the dynamic system does not attain an
equilibrium state in the simulated interval between 0.1 and 1 of parameter k.
Aperiodic behavior happens in two intervals (0.16, 0.19) and (0.404, 1) and the
stock and demand appear to be periodically oscillatory in the intervals from
0.1 to 1.0.Figure 3
The
bifurcation diagram of actual stock and expected demand.


### 4.1. Impact of The Price Fluctuation

The above results show that the retailer’s
demand model based on (2.10) under the given conditions, including initial states
as described previously, will go into chaos in certain ranges of k.
So it is necessary to investigate the impact of the price fluctuation on the
system. Based on formula (2.7), Dt+1=Dp−2r,
the retailer’s price is decided by the parameter r which is the exponential forms. The results of
simulation of exponential parameter r and retailer’s demand are shown in Figure 4.
It is clearly chaotic behavior in retailer’s demand as the r increases. When r varies from 0.48 to 1.2, the retailer’s demand
shows the chaotic
behavior. At r=0.48, the period doubling bifurcation which appeared
and decreased the initial r to 0.14 will bring the system shifts from oscillation
to equilibrium.Figure 4
The
bifurcation diagram of retail’s demand (k = 0.6, p = 10).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 atr∈(0,0.48) in Figure 4. When r>0.48, the demand is negative, it obviously means that
the stockout happened. As a result, the system of (2.10) shows chaotic behavior. So
the stockout is the uncertain behavior and can lead the system to the chaos. Please
refer to the Xu et al.’s article [16] for the further discussion about the coefficient of the price
sensitivity r.The above results show thatr>0.48 would lead the system into chaos and stockout.
It is necessary to discuss the equilibrium point when the stockout does not happen.
Figure 5 shows the retail’s stock bifurcation as the r changes and the system of (2.10) shows chaotic behavior.  When the coefficient 0.41>r>0.48,
the retail’s stock into oscillation and  r>0.48,
the system from oscillation to chaos.Figure 5
The bifurcation diagram of retail’s stock (k = 0.6, p = 10).Whether the
stockout happened or not, according to Figures4 and 5, r>0.48 can lead the system of (2.10) to chaos. Because coefficient
of price sensitivity r is decided by (4.1)Dt+1=Dp−2r,r>0.48, 2r>1, p2r≥p1,
the price is not a fluctuation
at 2r=1.
As an increase of r from 0.48, the price fluctuation gets worsened,
the stockout and chaos happened. The system of (2.10) shows the chaotic behavior.


### 4.2. Impact of The Initial Condition

It is well known [17] that attractors of a high-order map can be sensitive to initial conditions. Any
change of initial stock or demand can be regarded as a disturbance of system
and can lead to chaos. Figure 6 shows the bifurcation diagrams when the initial
stock increases by 5% and 10%, respectively. In the case of initial stock
increase by 5%, the major aperiodic interval moves to 0.68. In the case of
initial stock increase by 10%, the bifurcation diagram shows much less chaotic
behavior. It should be noted that the change of initial stock also caused
variance of the bifurcation diagram sharps.Bifurcation diagram of 5% and 10% increase of initial stock.
(a)(b)The impact of other initial data was also investigated. Figure7 shows the bifurcation
diagrams when the initial demand increases by 5% and 10%, respectively. It is clear
that the change of initial demand also leads the system to chaos.Bifurcation diagram of 5% and 10% increase of initial demand.
(a)(b)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.


### 4.3. Adaptive Adjustment

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 [18] advanced the theory of the adaptive adjustment which is the effective method to smooth the
chaotic behavior. In Huang’s discussion [19],
conventional adaptive expectations as a mechanism of stabilizing an unstable
economic process are reexamined through a generalization to an adaptive
adjustment framework. The generic structures of equilibrium that can be
stabilized through adaptive adjustment mechanisms are identified theoretically
and numerically. The adaptive adjustment schemes can be applied to a broader
class of discrete economic processes, such as in cobweb dynamics, price
adjustment, tariff game, population control model.


## 4.1. Impact of The Price Fluctuation

The above results show that the retailer’s
demand model based on (2.10) under the given conditions, including initial states
as described previously, will go into chaos in certain ranges of k.
So it is necessary to investigate the impact of the price fluctuation on the
system. Based on formula (2.7), Dt+1=Dp−2r,
the retailer’s price is decided by the parameter r which is the exponential forms. The results of
simulation of exponential parameter r and retailer’s demand are shown in Figure 4.
It is clearly chaotic behavior in retailer’s demand as the r increases. When r varies from 0.48 to 1.2, the retailer’s demand
shows the chaotic
behavior. At r=0.48, the period doubling bifurcation which appeared
and decreased the initial r to 0.14 will bring the system shifts from oscillation
to equilibrium.Figure 4
The
bifurcation diagram of retail’s demand (k = 0.6, p = 10).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 atr∈(0,0.48) in Figure 4. When r>0.48, the demand is negative, it obviously means that
the stockout happened. As a result, the system of (2.10) shows chaotic behavior. So
the stockout is the uncertain behavior and can lead the system to the chaos. Please
refer to the Xu et al.’s article [16] for the further discussion about the coefficient of the price
sensitivity r.The above results show thatr>0.48 would lead the system into chaos and stockout.
It is necessary to discuss the equilibrium point when the stockout does not happen.
Figure 5 shows the retail’s stock bifurcation as the r changes and the system of (2.10) shows chaotic behavior.  When the coefficient 0.41>r>0.48,
the retail’s stock into oscillation and  r>0.48,
the system from oscillation to chaos.Figure 5
The bifurcation diagram of retail’s stock (k = 0.6, p = 10).Whether the
stockout happened or not, according to Figures4 and 5, r>0.48 can lead the system of (2.10) to chaos. Because coefficient
of price sensitivity r is decided by (4.1)Dt+1=Dp−2r,r>0.48, 2r>1, p2r≥p1,
the price is not a fluctuation
at 2r=1.
As an increase of r from 0.48, the price fluctuation gets worsened,
the stockout and chaos happened. The system of (2.10) shows the chaotic behavior.


## 4.2. Impact of The Initial Condition

It is well known [17] that attractors of a high-order map can be sensitive to initial conditions. Any
change of initial stock or demand can be regarded as a disturbance of system
and can lead to chaos. Figure 6 shows the bifurcation diagrams when the initial
stock increases by 5% and 10%, respectively. In the case of initial stock
increase by 5%, the major aperiodic interval moves to 0.68. In the case of
initial stock increase by 10%, the bifurcation diagram shows much less chaotic
behavior. It should be noted that the change of initial stock also caused
variance of the bifurcation diagram sharps.Bifurcation diagram of 5% and 10% increase of initial stock.
(a)(b)The impact of other initial data was also investigated. Figure7 shows the bifurcation
diagrams when the initial demand increases by 5% and 10%, respectively. It is clear
that the change of initial demand also leads the system to chaos.Bifurcation diagram of 5% and 10% increase of initial demand.
(a)(b)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.


## 4.3. Adaptive Adjustment

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 [18] advanced the theory of the adaptive adjustment which is the effective method to smooth the
chaotic behavior. In Huang’s discussion [19],
conventional adaptive expectations as a mechanism of stabilizing an unstable
economic process are reexamined through a generalization to an adaptive
adjustment framework. The generic structures of equilibrium that can be
stabilized through adaptive adjustment mechanisms are identified theoretically
and numerically. The adaptive adjustment schemes can be applied to a broader
class of discrete economic processes, such as in cobweb dynamics, price
adjustment, tariff game, population control model.


## 5. Conclusion

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 parametersk and r can lead the system to the chaotic behavior.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.



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*Source: 792031-2008-12-04.xml*