The Impact of Bullwhip Effect in a Highly Volatile Market

Journal of Industrial Engineering, University of Tehran, Special Issue, 0, PP. 95-0 95 The Impact of Bullwhip Effect in a Highly Volatile Maret Ahmad Maui *, Seyed Jafar Sadjadi and Nazli Karampour Department of Industrial Engineering, Iran University of Science and Technology, Tehran, Iran (Received 3 December 00, Accepted 0 July 0) Abstract The bullwhip effect plays an important role in supply chain management especially in a highly volatile maret 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 maret 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, 008 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 shoc in maret 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 Introduction The Bullwhip effect is a performance index for the instability in a supply chain []. 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 []. 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 0th century, especially in economy. The bullwhip effect can be very costly in terms of capacity and stoc-out [4]. 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 stoc 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 [9,0] introduced the supply variability as a possible cause of the bullwhip effect. Supply variability includes machine reliability and quality problems []. 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 [] studied the * Corresponding author: Tel & Fax: +98-- 774043 Email: amaui@iust.ac.ir

96 Journal of Industrial Engineering, University of Tehran, Special Issue, 0 effects of the information sharing among the participants in the supply chain. To reduce the negative effect of the bullwhip effect, Lee [3] 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. 3) Introducing distributed controls among the supply chain, and reducing the uncertainty. Geary [4] introduced 0 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 [4] 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 [5] 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 thining 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. Ozelan [6] analysed the impact of procurement price variability in the upstream for a supply chain on the downstream retail prices. They used a game theory framewor 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 [6]. Ma [7] 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 [8] consider financial marets 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 stoc maret have a distribution that is identical to that observed for developed marets (e.g., NYSE). They represent the selforganization of price fluctuation distribution in stoc maret as an important component of complex systems. There are many methods to quantify the bullwhip effect [9]. Warburton [0] uses control theory and solves the fundamental differential delay equations for a retailer's inventory reacting to a surge in demand. Bradley [] 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 [] apply the bootstrap technique to analyze the bullwhip effect in supply chain. In this paper the method introduced by Maui and Madadi [] using Lyapunov exponent, have been used. Alexander Mihailovich Lyapunov [3] 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

The Impact of Bullwhip Effect.. 97 stable attractor and has a sensitive dependence on initial conditions. Problem statement Raisudin and Bernard [4] have a study on rice price fluctuation in Bangladesh. They examine the sources and extend of rice price variability in Bangladesh and provide a framewor 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 008, the rice maret 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 lie 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, ( p v D ab P )* D, () 0 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 : The change on price in highly volatile maret price With a= 8.434, b= 0.874 and v= 0.6398. As we can observe, for 0.< 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 : The price change in sluggish price change Now consider a simple example where the initial price is P 0 = and for this price the basic demand is D 0 =00. let a=0 and b=0. and v=3. Therefore we have, P= D=00 P=. D=05 P=. D=09 P=.5 D=06 P=.6 D=0 P= D=80 P=.5 D=49.4 As we can observe, equation () is sensitive to the speed of the price variation. This phenomenon causes a secondary event that is the bullwhip effect. The important note is that the reverse pricing

98 Journal of Industrial Engineering, University of Tehran, Special Issue, 0 effect firstly taes 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 q be the order received by agent from the agent (-) and L is the ordering lead time for agent. We assume that the forecasting method is a moving average method with p=. Taylor expresses the following relationship (Taylor, 000). var( q ) var( q ) ( L ) L var( ) var( ) ( ) q D L L () The reverse pricing generates the bullwhip in price and its direction is from the end of the chain to the customer. Thus, (. p v D ab. p ). D (3) The speed of the price variation in the time interval t is as follows, p p v (4) t In the supply chain, D is the order quantity of agent. p p p t (.. ) q ab p q (5) Therefore we have, q q var( q) p v q ( ab.. p ) (6) By inference, equation (6) can be written as: p v q q q( ab.. p ) (7) p v q q3 q3( ab.. p ) (8) p v q q q( ab.. p ) (9) q D p v D ( ab.. p ) (0) 0 0 D 0 is the demand implemented by the customer. So we have:. p v q ab. p. D () 0 q a. b. p. D () q a b p D (3) ( pp) ( vv) 0 3 ( ppp3) ( v3vv3) 3.. 3. 0 Therefore we have, p j v j j j q a. b. p. D, (4) and q q a b p p j v j j j [.. p j j v j j a. b. p ]. D0 0 (5) and q D. v ab. p. D (6) 0 0 Which yields, q q q D 0 p j v j j j p v p v ab.. p ( ) [ pj vj. j. j.(. p. v a b p v ab p )] p a. b. p ( ab.. p ) (7) (8) Equation (8) can be interpreted as: var( q ). e (9) var( q ) Where the Lyapunov exponent is created by the bullwhip effect and can be set to the Lyapunov exponent created by the relation (). ( ) ln[ pj vj. j. j (. p. v a b p v ab p )] p (0) The condition for having the bullwhip effect is that 0, so: ( ) p j v j j j p v a. b. p ( ab.. p ) () Numerical Example Consider a supply chain with four stages as follows:

The Impact of Bullwhip Effect.. 99.q3 4 P 4 =4.q 3 P 3 =6.q P =8 Figure 3: A supply chain with four stages P =0 Where p,p,p 3 and p 4 are the prices in stages to 4 for the supply chain. The relation between q and q - is supposed to be as follows: q [8.434*(0.874) p * p ]* q () and the demand (D) is obtained as follows: D 8.434*(0.874) p * p (3) Where p is assumed to be a random uniform variable. The results of computations are represented in appendix. For instance the results of q 4 are plotted versus q 4 (n th value of q 4 versus the (n-) th value of q 4 ). In this case the value of will be 0.043 which is greater than zero that is the sign of being bullwhip. Conclusion By augmenting the price of a good, we expect that its demand will be diminished. But in a short interval of time, the price of some goods rise suddenly and simultaneously their demand rises with a D considerable speed. It is obvious that the reason must be in psychological behaviours of customers. This fact is not consistent with the traditional price and demand elasticity theorems. We have presented a new mathematical function where demand for a price is influenced by price and price velocity. When the price elevation speed is low or zero, the proposed function's behaviour is lie 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. The proposed model could explain the turmoil on maret price and demand fluctuations for some products such as rice in some countries lie Iran where demand and price had similar trend in short run and opposite trend for long run. The bullwhip effect has also been studied for the maret where our proposed model could explain the behaviour of the changes on price and demand. It is shown that the variation of price causes the variation in demand and in fact we are faced a positive feedbac between the price and demand. While these two factors amplify each other, the bullwhip occurs in both price and demand. References: - Hsieh, Kun-Lin Chen, Yan-Kwang.(006). "Analaysis bullwhip effect in supply chain model by using bootstrap technique. WSEAS Transactions on Information Science and Applications, Vol. 3, No., PP. 505-50. - H., Lee, Padmanabhan,V. and Whang, S.(997). The bullwhip effect in supply chains. Sloan Management Review. Vol. 38, No. 3, PP. 93-0. 3- Metters, R. ( 997). Quantifying the bullwhip effect in supply chains. Journal of Operations Management. Vol. 5, PP. 89-00. 4- Geary,S., Disney,S.M. and Towill,D.R. (006). "On bullwhip in supply chains-historical review, present practice and expected future impact. International Journal of Production Economics. Vol. 0, PP. -8. 5-Towill, D.R., Naim,M.M. and Winer,J.( 99). Industrial dynamics simulation models in the design of supply chains. International Journal of Physical Distribution & Logistics Management. Vol., No. 5, PP. 3-3.

00 Journal of Industrial Engineering, University of Tehran, Special Issue, 0 6- Dejoncheere,j., Disney,S.M., Lambrecht, M. R. and Towill, D. R. (003). "Measuring and avoiding the bullwhip effect: A control theoretic approach. European Journal of Operational Research. Vol. 47, No. 3, PP. 567-590. 7- Disney,S.M.,Towill,D.R.(003). The effect of vendor managed inventory (VMI) dynamics of the bullwhip effect in supply chains. International Journal of Production Economics.Vol 85,No,PP. 99-6. 8- Disney, S.M. and Towill, D. (003) "Vendor-managed inventory and bullwhip reduction in a two-level supply chain." International Journal of Operations & Production Management. Vol. 3, No. 6, PP. 65-65. 9- Taylor, D.H. (999). "Measurement and analysis of demand amplification across the supply chain." International Journal of Logistics management. Vol. 0, No., PP. 55-70. 0- Taylor, D.H. ( 000). Demand amplification: has it got us beat? International Journal of Physical Distribution & Logistics Managemen. Vol. 30, No. 6, PP. 55-533. - Pai, S.K. and bagchi, P.K. ( 007). Understanding the causes of the bullwhip effect in a supply chain. International Journal of Retail & Distribution Management. Vol. 35, No. 4, PP. 308-34. - Zhen, X.Y., Hong, Y. and Cheng, T.C. (00). "Benefits of information sharing with supply chain partnerships." Industrial Management & data System. Vol. 0, No. 4, PP. 4-0. 3- Lee, H., Padmanabhan, V. and Whang, S. ( 997). "Information distortion in a supply chain: The bullwhip effect. Management Science. Vol. 43, No. 4, PP. 546-548. 4- Lu, Y., Tong, Y. and Tang, X.(004). Study on the complexity of the bullwhip effect. Journal of Electronic Science Technology of China. Vol., No. 3, PP. 86-9. 5- Miragliotta, G.( 006) "Layers and mechanisms:a new taxonomy for the bullwhip effect." International Journal of production economics. Vol. 04, PP. 365-38. 6- Ozelan, E.C. and Caanyildirim, M. (009). Reverse bullwhip effect in pricing. European Journal of Operational Research. Vol. 9, PP. 30-3. 7- Ma, J. and Feng, Y. (008). The Study of the chaotic behavior in retailer s demand model. Discrete Dynamics in Nature and Society. [ID 7903]. 8 Pan, R. K and Sinha, S. (007) "Self organization of price fluctuation distribution in evolving marets. EPL(Europhysics Letters), Vo.l 77, No. 5. 9- Shalizi, C.R. ( 006) Methods and Techniques of Complex Systems Science: An Overview. In T.S. Deisboec & J.Y.Kresh (Eds), Complex Systems Science in Biomedicine, New Yor: Springer. PP. 33-4. 0- Warburton, R. D. H. (004) "An Analytical Investigation of the Bullwhip Effect." Production and Operations Management. Vol. 3, No., PP.50 60. - Hull, B. ( 005). The role of elasticity in supply chain performance. International Journal of Production Economics, Vol. 98, No. 3, PP. 30-34. - Maui, A. and Madadi, A. (007). The bullwhip effect and Lyapunov exponent. Applied Mathematics and Computation. Vol. 89, PP. 35-40. 3- Lyapunov, A.M. (89). General problem on motion stability, Kharov. Kharovsoye Matemachesoe Obshchestvo. Vol.. 4- Raisuddin, A. and Bernard, A.(989). Rice Price Fluctuation and an Approach to Price Stabilization in Bangladesh. Research Report 7, International Food Policy Research Institute, Banglades

The Impact of Bullwhip Effect.. 0 Appendix: The computations of a numerical example of a supply chain with four stages: p D q q q3 q4.4 33.499877 59.87 604605 5797849 78336567 8.634 96.37339 43065.93 506559 6909648 499463074 9.63667 3.9446 4695.56 559498 3634938 633846963.8388 45.74889 007.9 79506 5050743 34949983.57638 45.3586 53793.37 637307 70940404 87969384.078 3.8606 50953 59936 56634366 774073947 6.763404 55.6794 34036.7 400347 74956 850663 7.359987 69.5577889 3793.8 437488 8733368 95008959 7.33590 68.99737 37070.84 436036 867453 9076943.66803 38.554495 538.77 655038 63563708 897548 7.79673 79.30737 39335.3 46674 989634 369570590 5.7958 50.904448 55037.4 6473634 770649 9684 0.95969 3.5360998 50789.5 5973955 55809599 76837456 7.47405 7.7994 37757.6 44443 9073339 34639074 3.5904 49.586594 54748.75 643968 7575377 9063389 6.348 4.545743 3044.5 36553 5636555 080903407.73868 39.655804 546.83 670806 6438930 866434 8.048343 84.80583 40538.6 476848 048059 4466764 7.95533 8.8394 4005.8 477345 000056 396398603.6058 37.9973763 506.58 640665 694855 8770887.753 46.40648 5404.8 6353353 705575 88067947 7.746074 78.975087 390.95 460035 96989645 3675954 0.8773 30.04693 5046.45 593556 5463606 756993958 0.4344 5.096 49403.8 580996 4883556 703437 7.90805 8.7094646 39859.39 4688358 00759300 3878895.08986 3.9489874 5099.8 600409 57370609 77963480 9.54489.4778843 46608.68 5486 3475338 684059 9.80608 6.46937 4748.6 558500 3955095 653447 8.7609 98.90393 43496.54 5669 907857 544563 0.8899 30.6658866 50598.36 59550 5484855 767644 9.965 08.498743 4578.5 537754 3069943 5980737 5.0659 5.699679 55.7 64940 78086995 937674 0.869 9.94405 50433.5 5933 5407888 755986634 6.390795 45.6666395 3953. 3758403 6093786 538707.60 33.70797 565.67 60999 58096 78496068 5.67 5.6998 5595.07 64979 78000356 977375 0.7486.807553 48655.9 57307 4506468 694094487 7.6754 65.0004 3698.87 457799 83398 6036673 8.735 87.388778 405.08 4834879 0703346 43905 5.646499 5.70835 7573.68 34383 388800 96005676 7.00356 6.076394 3533.54 455899 77958976 300306 0.5009 6.04433 49597.77 583380 49808493 7688773 9.484903.57445 46398 5457445 336930 65478484.9096 47.00495 5485.87 6373474 79734 886635534 8.850844 00.608559 44003.58 575808 63309 530098 6.40888 49.507 54653.48 648475 75757 909660 5.9503 50.4993 54948.96 646330 76760753 9304565 9.909448 8.04607 47830. 565893 4090530 66534467 0.05359 0.0775 4885. 567944 439758 688750 7.66739 76.55470 3878.65 4555358 950648 34844835 5.584 5.374046 550.88 64876 7757635 9884490.57853 39.647663 869.00 0374 43778899 30636804.85649 40.4966 5678.7 69697 653679 834594 9.405057 0.9653 463.3 54396 358405 605566668 5.43568 9.5066909 64.7 3083440 3035388 9740349 p D q q q3 q4 0.6633 8.0844093 5003.09 5884896 5996044 74009938 9.4047 0.73578 4603. 5477 30749 605478 4.64598 5.68635 5508.55 6493763 780689 94879 6.53486 49.3667807 3764.77 385387 650593 4079878 9.6703 4.474797 47046.7 5533749 36959644 63806564 6.434 49.3775 54650.3 6480 755655 9080646.445 43.084674 533.5 6794 68568889 856575466 5.389 5.586359 5586.6 64984 77957749 94796 7.9983 8.06389 39936.58 4697438 048 390505987 3.097 47.95656 54383.75 6396749 7393969 8935584

0 Journal of Industrial Engineering, University of Tehran, Special Issue, 0 7.3644 68.548394 3697.9 434864 8638 87574 0.9950 3.9578 50874.6 5983994 5639467 77344068.6445 38.3450966 58.85 649637 633349 8037665 9.30646 08.5857945 45754.9 538805 3045357 593087963 9.9597 8.6647999 47965.83 564857 458893 67006704 7.5534 73.3583004 3807.49 447886 95366 3403569 6.5048 38.679555 3040.44 35786 53873 0597438 5.36695 7.996085 5866.59 30449 3089 9006890 0.585 6.368037 49655.59 58406 50099733 7890030 9.0657 03.6973408 4468.6 555676 50534 5557507 4.45808 5.559064 5579.87 649039 7793798 94458.837 0.07636957 0.33 59984.4 3744 76958946.9 5.99457 33.6483 90.3 3435778 47809 07037375.8487 46.646369 54099.04 636360 7479970 883668 9.75474 5.739354 4734. 5566377 38356773 64773754 4.7985 5.349587 5535.48 648569 7770093 99698769 5.000053 07.533066 3588. 774503 880648 890953.79959 46.490563 54069.73 635984 733385 8859936.5735 34.6847549 5479.93 605595 5988349 794059 8.73797 89.4636704 4560.33 488847 093643 4470446 0.8653 9.97596 50434. 59384 5400934 75600769.7657 46.30907 5409.84 6355 7347 880305 3.546 49.67509 54768. 644970 75850367 906907 8.304578 90.0863849 4696.93 4904494 00449 45797484.954 40.937679 5850.48 6640 669383 8404037 0.7389 9.007886 5035.3 5908798 530954 749085 6.9076 59.060069 3489.06 40397 75735408 483874 7.99787 83.7399496 40304.79 4740747 0300654 4033660.4389 44.4968 5357.89 63056 698489 8655804.77 4.89383 5380.65 667000 6835807 8557745 7.90749 8.80487 3988.5 4690958 00870650 388587940 0.06346 0.6079 4835.79 568300 4335539 68580 9.06997 04.538668 44865.66 57708 597439 56589 6.3779 45.768 3844.76 3745658 60397 08766066