Journal of Industrial Engineering, University of Tehran, Special Issue, 2011, PP. 95-102 95
The Impact of Bullwhip Effect in a Highly Volatile Market

Ahmad Makui *1, Seyed Jafar Sadjadi 1 and Nazli Karampour 1
Department of Industrial Engineering, Iran University of Science and Technology, Tehran, Iran
(Received 23 December 2010, Accepted 20 July 2011)

The bullwhip effect plays an important role in supply chain management especially in a highly volatile market where prices change due to many unexpected reasons brought about by different phenomenon such as global warming. Traditionally, one may expect a reduction on demand when there is a significant move on market price. However, the recent changes on global economy imply that the demand for a particular product may significantly increase as the price goes up in short time and it will come down in long run. There are many evidences to confirm this theory and as an example we could study the behaviour of price and demand for rice in September, 2008 in Iran’s economy. We present a mathematical model where demand is not only affected by price but also is influenced by the speed of price changes. Our model behaves identical the traditional demand model, where demand is only a function of price and price elasticity, when price rise is sluggish. However, in the event that there is a big shock in market price, the model has completely different attitude. The proposed model examines the bullwhip effect using the Lyapunov exponent.

Keywords: Bullwhip effect, Lyapunov exponent, Supply chain, Price Fluctuation

The Bullwhip effect is a performance index for the instability in a supply chain [1]. In a supply chain when the variation of the orders received by the supplier is greater than the variation of demand observed by the customer, we can face the bullwhip effect [2]. End users form the demand for the last level in the supply chain, but the demand for upstream levels is formed by the levels in the immediate downstream supply chain. The demand seasonality and forecast error can increase as we proceed up the supply chain. The bullwhip effect is a demand distortion and can create inefficiencies for upstream levels of supply chain [3]. Demand amplification is not a new phenomenon, and its existence has been recorded in the start of the 20th century, especially in economy. The bullwhip effect can be very costly in terms of capacity and stock-out
The most important reasons of the bullwhip effect can be listed as: Demand forecast updating, order batching, price fluctuation and rationing and shortage gaming. Demand forecast updating is the demand amplification caused by the safety stock and long lead time. Order batching causes surges in demand at a particular time period. Price discount or promotion can cause price fluctuation that modifies the buying pattern of customers and creates undesirable variations in demand. Irregular behaviours occurred in both buyers and supplier’s part causes the bullwhip effect. Also the information and material delays might be the causes of the bullwhip effect [5,6]. It is shown that by implementing Vendor Managed Inventory (VMI) system, both rationing and shortage gaming effect can be completely eliminated [7,8]. The use of VMI system also can reduce the impact of price variations or the promotion effect on the bullwhip effect. Taylor [9,10] introduced the supply variability as a possible cause of the bullwhip effect. Supply variability includes machine reliability and quality problems [11]. One of the most important reasons for bullwhip is the wrong information flow across the chain. Metters [3] studied the information distortion from the end to the beginning of the supply chain. Zhenxin [12] studied the

* Corresponding author: Tel & Fax: +98-21- 77240143 Email: [email protected]

Journal of Industrial Engineering, University of Tehran, Special Issue, 2011
effects of the information sharing among the participants in the supply chain. To reduce the negative effect of the bullwhip effect, Lee [13] categorize the topics as follows,
Reducing the information distortion from the end to the beginning of the supply chain.
Information sharing among the participants for the supply chain.
Introducing distributed controls among the supply chain, and reducing the uncertainty.
Geary [4] introduced 10 principals about bullwhip reduction: control system principle, time compression principle, information transparency principle, echelon elimination principle, synchronization principle, multiplier principle, demand forecast principle, order batching principle, price fluctuation principle and gaming principle. Lu [14] studied a nonlinear model for the bullwhip effect for order up to policy based on demand signal policy and analyse the complexity of the bullwhip effect in a supply chain. Miragliotta [15] have a complete review on bullwhip effect literature. Based on this review, some conclusions about the opinions of two schools of thoughts are extracted. The system thinking school views the bullwhip effect as an irrational reaction to a complex system and suggests teaching and training. On the other hand, the operations managers school, views the bullwhip effect as rational reaction to isolated and well perceived factors. Ozelkan [16] analysed the impact of procurement price variability in the upstream for a supply chain on the downstream retail prices. They used a game theory framework to model a serial supply chain and analysed the price variability which occurred in the upstream of the supply chain and showed that this variability could be amplified under some certain scenarios. Because of the reverse direction of price variability, compared to the direction of bullwhip effect in order variability, they named it reverse bullwhip effect in pricing (RBP). In RBP it is assumed that by augmenting the price, the demand will be diminished [16]. Ma [17] studied the behaviour of a supply chain system with a retailer and customer. In their model, a discount rate is offered by the retailer when the demand increases based on a basic level. Their analysis shows a chaotic behaviour and the bullwhip effect in the supply chain. Pan and Sinha [18] consider financial markets as complex systems in non-equilibrium steady state that one of whose most important properties is the distribution of price fluctuations. They show that the price fluctuations in the Indian stock market have a distribution that is identical to that observed for developed markets (e.g., NYSE). They represent the selforganization of price fluctuation distribution in stock market as an important component of complex systems. There are many methods to quantify the bullwhip effect [19]. Warburton [20] uses control theory and solves the fundamental differential delay equations for a retailer’s inventory reacting to a surge in demand. Bradley [21] develops model to describe the performance of supply chains based on their elasticity’s of supply and demand. The model predicts the supply chain’s ability to respond to supply interruptions, cost increases, and demand shifts, and can quantify the degree to which it is prone to the bullwhip effect. Hsieh et al [1] apply the bootstrap technique to analyze the bullwhip effect in supply chain. In this paper the method introduced by Makui and Madadi [22] using Lyapunov exponent, have been used. Alexander Mikhailovich Lyapunov [23] introduced a method to measure the rate of convergence between two orbits, one been perturbed. The quantity obtained, named Lyapunov exponent gives important information on the system’s behaviour. When Lyapunov exponent is less than zero, this means that the system is insensitive to initial conditions, a value greater than zero means that the system tend to go away from the stable attractor and has a sensitive dependence on initial conditions.

Problem statement
Raisudin and Bernard [24] have a study on rice price fluctuation in Bangladesh. They examine the sources and extend of rice price variability in Bangladesh and provide a framework for the implementation of an effective and simple mechanism to limit the variability in future. In their report the main causes of variation are shown to be the interactions between demand, supply and import. In the September 2008, the rice market in Iran faced a new phenomenon. As suggested in the literature, by augmenting the price of a good, its demand will be diminished. But in a short interval of time, the price of different types of rice rise suddenly and simultaneously the demand of rice rises with a considerable speed. This phenomenon is not consistent with the traditional price and demand elasticity theorems and it becomes our main incentive for this research. Our studies show that the demand for essential and critical goods is not only dependent on the price, but also it has a considerable correlation to the elevation speed of price. It is obvious that the reason can be found in psychological behaviours of customers. In this situation, a new mathematical function is developed to simulate the relation between demands, price and the speed of price elevation. When the price elevation speed is low or zero, the proposed function’s behaviour is like the classical iso-elastic function and when the price rises, the demand will be diminished. But when the speed is high enough, the behaviour of the proposed function becomes different from classical one. When the price rises, the demand also rises.
Consider demand (D) for a product which is an exponential function of price (p), velocity (v) of price with the following function,
Where a and b are assumed to be nonnegative arbitrary numbers. The following is an example graph which shows the behaviour of the function,

Figure 1: The change on price in highly volatile market price

With a= 8.4341, b= 0.874 and v= 0.6398. As we can observe, for 0.1< p < 5, the demand is increasing, but for p>5, the demand begin to diminishes. Now assume v= 0.05 meaning a low speed then the function has different behaviour which is shown in the following graph,

Figure 2: The price change in sluggish price change

Now consider a simple example where the initial price is P0=1 and for this price the basic demand is D0=100. let a=10 and b=0.1 and v=3. Therefore we have,
P=1 → D=100
P=1.1 → D=105
P=1.2 → D=109
P=1.5 → D=106
P=1.6 → D=102
P=2 → D=80
P=2.5 → D=49.4
As we can observe, equation (1) is sensitive to the speed of the price variation.
This phenomenon causes a secondary event that is the bullwhip effect. The
D ab P (p v)*D0 , (1) important note is that the reverse pricing
Journal of Industrial Engineering, University of Tehran, Special Issue, 2011
effect firstly takes place and then, the bullwhip effect occurs in the supply chain. In the next section we will analyse these effects mathematically.

Mathematical Analysis
Assume a supply chain with N agents. Let qk be the order received by agent k from the agent (k-1) and Lk is the ordering lead time for agent k. We assume that the forecasting method is a moving average method with p=1. Taylor expresses the following relationship (Taylor, 2000).

var(q ) var(q)(L )2  L
q a b 2. (p p1 2).p (v v1 2).D (12)
q a b 3. (p p p1 23).p (v v v3 23).D (13)

Therefore we have,
qa bk  j1pj p  j1vj D
k  .. k. 0, (14)

q qk  k1 [a bk.  j1 pj .pk j1vj k1k1 (15) a bk1  j1 pj p  j1vj D
 .. k1]. 0


var(qk1) var(Dk)1  (Lk1)2  L1k q D ab p D1  0.1. 1 1. 0 (16)
Which yields,
The reverse pricing generates the bullwhip
in price and its direction is from the end of qk qk1
the chain to the customer. Thus, 

The speed of the price variation in the time
interval ∆t is as follows, ak1 b j11 pj pk1 j11vj abpk .pkvk 1)
..( .

ab. p1 .p1v1
Dk  ( .ab p Dp. v). k1 (3) 1  D0 (17) q

v 

k tpk1 (4) p11v1 [a b(k2). j2 pj .pk1j1vj .( .ab ppk . kvk 1)] (18) p
In the supply chain, Dk is the order quantity of agent k. Equation (18) can be interpreted as:
pk pkvar(qk)

qk  ( .ab ppk . k t 1 )qk1 (5) var(q1) e.k (19)

Therefore we have, Where the Lyapunov exponent is qk qk1  var(qk) created by the bullwhip effect and can be pk .pkvk 1) (6) set to the Lyapunov exponent created by the relation (1). qk1( .ab
By inference, equation (6) can be written 

kp11v1 ln[a( 2)k .bkj1kj11vj ( .ab ppk. kvk 1)] (20) as:
qk1 qk2 q ab pk2( . pk1. k1vk1 1) (7)
qk2 qk3 q abk3( . pk2vk2 1) (8) The condition for having the bullwhip effect is that  0, so:
… q q q ab p2p2v2a(k2).bkj12pj . kj11vj ab ppk . kvk  1)1 (21)
 11( .. 2 1) (9) pk1( .
q D D ab p1  00( . p1. 1v1 1) (10)
Numerical Example

D0 is the demand implemented by the Consider a supply chain with four customer. So we have: stages as follows:
q ab p D1  . p1. 1v1. 0 (11)

.q3 .q2 .q1 D






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