PRODUCTION AND OPERATIONS MANAGEMENT POMS Vol. 13, No. 2, Summer 2004, pp. 150–160 issn 1059-1478 ͉ 04 ͉ 1302 ͉ 150$1.25 © 2004 Production and Operations Management Society An Analytical Investigation of the Bullwhip Effect Roger D. H. Warburton University of Massachusetts, Dartmouth, North Dartmouth, Massachusetts 02747, USA [email protected] he Bullwhip Effect is problematic: order variability increases as orders propagate along the supply Tchain. The fundamental differential delay equations for a retailer’s inventory reacting to a surge in demand are solved exactly. Much of the rich and complex inventory behavior is determined by the replenishment delay. The analytical solutions agree with numerical integrations and previous control theory results. Managerially useful ordering strategies are proposed. Exact expressions are derived for the retailer’s orders to the manufacturer, and the Bullwhip Effect arises naturally. The approach is quite general and applicable to a wide variety of supply chain problems. Key words: Bullwhip Effect; supply chain management; ordering policy Submissions and Acceptance: Received March 2002; revisions received November 2002 and May 2003; accepted October 2003 by Seungjin Wang. 1. Introduction Parker (2000) suggest the amplification of demand vol- Lee, Padmanabhan, and Whang (1997a) and Lee, So, and atility is particularly large in distribution and component Tang (2000) popularized the term “Bullwhip Effect,” parts supply chains, e.g., machine tools. Johnson and where a retailer’s orders to their suppliers tend to have a Whang (2002) survey emerging research on the impact larger variance than the consumer demand that trig- of e-business on supply chains. gered the orders. This demand distortion propagates Much earlier, however, Forrester (1961) had defined a upstream with amplification occurring at each echelon. simplified form for the equations describing the relation Lee, Padmanabhan, and Whang (1997b) identified four between inventory and orders. In this paper, it is dem- major causes of the Bullwhip Effect: (1) users interpret- onstrated that the fundamental differential delay equa- ing orders (the demand); (2) order batching; (3) promo- tions describing an inventory reacting to a surge in con- tions, which artificially stimulate demand; and (4) sup- sumer demand can be solved exactly. Forrester ply shortages, which also lead to artificial demands. The pioneered the simulation approach and established the Bullwhip Effect has been documented as a significant importance of integrating information flow with mate- problem in an experimental, managerial context (Ster- rial flow. Burbidge (1961) emphasized the now well- man 1989), as well as in a wide variety of companies and accepted principles of cycle time reduction and order industries (Buzzell, Quelch, and Salmon 1990; Kelly synchronization, and later coined his Law of Industrial 1995; Holmstrom 1997; Metters 1997). Many proposed Dynamics (Burbidge 1984): “If demand is transmitted strategies for mitigating the Bullwhip Effect have a his- along a series of inventories using stock control ordering, tory of successful application (Clark 1994; Gill and then the demand variation will increase with each trans- Abend 1997; Hammond 1993; Towill 1997). fer.” Simulation has since been employed extensively to Fine (2000) discusses the Bullwhip Effect as one of two analyze supply chains (Berry and Towill 1995; Disney laws that govern supply chain dynamics, focusing on the and Towill 2002a, 2002c). strategic issues that arise. Anderson and Morrice (2000) analyzed the Bullwhip Effect in service industries, which 1.1. Related Theoretical Analyses cannot hold inventory, and in which backlogs can only Kahn (1987) showed that a serially correlated demand be managed by adjusting capacity. Anderson, Fine, and results in the Bullwhip Effect. Lee, Padmanabhan, and 150 Warburton: An Analytical Investigation of the Bullwhip Effect Production and Operations Management 13(2), pp. 150–160, © 2004 Production and Operations Management Society 151 Whang (1997a) used the same demand assumption in As the consumer demand depletes the retailer’s in- which orders, Dt, depend on the orders in the previous ventory, replenishment orders are issued to the man- time interval, DtϪ1, as: ufacturer to bring the inventory back toward the de- sired value. A typical ordering policy is to order ϭ ␳ ϩ ϩ Dt DtϪ1 d ut, (1) proportional to the inventory deficit: where d and ␳ are constants such that d Ͼ 0 and Ϫ1 Ϫ ͑ ͒ Ͻ ␳ Ͻ ID I t 1, and ut is normally distributed with zero mean ͑ ͒ ϭ ͑ ͒ Ͻ O t for I t ID and (3a) and variance, ␴2. (Negative demands are unlikely T when ␴ ϽϽ d.) A cost minimization approach showed that distortion in demand arises when retailers opti- O͑t͒ ϭ 0 otherwise. (3b) mize orders, and amplification increases as the replen- ishment lead-time increases. Various demand distri- Forrester (1961) originally proposed the policy in (3a) butions and numerical experiments have been and referred to the quantity, T,asthe“adjustment employed to study the Bullwhip Effect. Bourland, time.” This policy has been extensively studied, and Powell, and Pyke (1996) examined the case in which both simulation and control theory analyses suggest it the review period of the manufacturer is not synchro- is misguided practice to attempt to recover the entire ϭ nized with the retailer, while Gavirneni, Kapuscinski, deficit in one time period, i.e., setting T 1 (Disney, and Tayur (1999) considered a manufacturer with lim- Naim, and Towill 2000). ited capacity. Equation (3b) adds the realistic constraint in which Disney and Towill (2002a) provide a useful compi- retailers and manufacturers stop ordering when their lation of the control theory literature applicable to the inventory exceeds its desired value. Using (3a) in that Bullwhip Effect. Dejonckheere, Disney, Lambrecht, situation would represent negative production orders, and Towill (2002) used z-transforms to investigate i.e., returning items. Including (3b) is somewhat more bullwhip performance of order-up-to models. Partic- realistic than previous approaches, all of which as- ularly relevant is that John, Naim, and Towill (1994) sume that excess inventory can be returned at no cost used the Final Value Theorem to prove that, for step (Kahn 1987; Lee, Padmanabhan, and Whang 1997a, function shocks to the inventory, a long-term inven- 1997b; Disney and Towill 2002c). We shall see that tory deficit can occur; and they verified the prediction including (3b) significantly impacts inventory behav- through simulation. ior. Due to manufacturing and shipping times, there is a delay in replenishment. The retailer’s receipts from 2. The Retailer’s Supply Chain the manufacturer are the retailer’s orders, just delayed by the replenishment time, ␶, which is assumed to be Retailers attempt to minimize their inventory while constant: maintaining sufficient on hand to guard against fluc- tuations in demand. The challenge is to formalize the R͑t͒ ϭ O͑t Ϫ ␶͒. (4) ordering process with simple, robust policies that ac- complish optimal inventory replenishment. The in- If the manufacturer carries inventory, the replenish- ventory, I(t), is depleted by the demand rate, D(t), and ment time will be much shorter than if goods are made increased by the receiving rate, R(t), so the inventory to order. However, in either case, the manufacturer, balance equation is: and not the retailer, determines the replenishment dI time. The adjustment time, T, allows the retailer to ϭ R͑t͒ Ϫ D͑t͒. (2) dt tune the order rate, and since T can be adjusted more easily and quickly than ␶, it is reasonable to consider ␶ ϭ Initially (t 0), the inventory has the value, Io, to be a constant. which may be different from its desired value, ID.We The “inventory position” is usually considered to be consider the impact of a step function surge in de- the deficit minus the unfulfilled orders, or work in mand rate beginning at t ϭ 0, and assume the surge, d, process (WIP), which was not included in the above to be constant and permanent. In Section 6, we discuss equations. However, we present evidence in the Ap- the impact of relaxing this assumption. The response pendix that many of the critical features of the above to a deterministic step input is important because the equations should also occur in models that include responses are easily interpreted, and it is a useful WIP terms. For example, the inventory deficit predic- measure of a system’s ability to cope with sudden tions of this model are corroborated by results from changes. The step function surge in demand is also a control theory (John, Naim, and Towill 1994). Also, common feature in control theory analyses (John, several of the model’s theoretical implications (e.g., Naim, and Towill 1994; Disney and Towill 2002a, stability) depend directly on the replenishment delay, 2002b). and any model or simulation that correctly treats the Warburton: An Analytical Investigation of the Bullwhip Effect 152 Production and Operations Management 13(2), pp. 150–160, © 2004 Production and Operations Management Society replenishment delay should inherit the stability prop- Figure 1 Four exact solutions of the inventory equation. T* is the critical erties discussed here. Therefore, even without the WIP adjustment rate, which brings the inventory exactly back to its .(10 ؍ desired value (␶ term, the above ordering policy will turn out to be of considerable theoretical, as well as practical, interest. Substituting the ordering policy in (2) and the time delay in (4) gives: ͑ Ϫ ␶͒ dI I t ID ϩ ϭ Ϫ d. (5) dt T T Because of the replenishment delay, no items are re- ceived for t Յ ␶, and (5) becomes: dI ϭ Ϫd f I͑t͒ ϭ I Ϫ dt. (6) dt o Since the inventory is less than its desired value, the order rate, (3a), is for t Յ ␶: ͑ ͒ ϭ ͑ Ϫ ͒ ϩ O t ID Io /T td/T. (7) The order rate has two contributions: a term at- tempting to fill the initial inventory deficit and another reacting to the demand.