An Analytical Investigation of the Bullwhip Effect Production and Operations Management 13(2), Pp

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

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 desired 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 replenishment 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 fluctuations 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 received 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. which the inventory returns to its desired value, ID,is defined as ⌻ .At⌻ , the order rate goes to zero, but 2.1. The Exact Solution D D items continue arriving, because there are still orders The exact solution to (5) is derived in the Appendix. in the pipeline. The inventory overshoots and it takes The critical feature is that the solution contains the a further time, ␶, the replenishment time, until the Lambert W function (Corless et al. 1996). The solution pipeline empties. After t ϭ T ϩ ␶, only the continuing is for t Յ ␶: D consumer demand remains, and so (6) applies again. ͑ ͒ ϭ Ϫ I t Io dt, (8) When the inventory falls to its desired level, the cycle repeats. The different characteristics of Figures 1 and 2 Ն ␶ and for t : are due to the orders stopping when the inventory ͑ ͒ ϭ Ϫ ϩ ͓ ␶͔ ϭ ϩ ␣ reaches its desired value. I t ID dT A exp Wt/ A a i W ϭ W͑Ϫ␶/T͒ W ϭ ␻ ϩ i⍀ (9) 2.3. Permanent Inventory Deficits The theoretical solution in the stable regime predicts ϭ ͑ Ϫ␻ ⍀͓͒ ͑⍀ ⍀ ϩ ␻ ⍀͒ Ϫ ⍀͔ a e / J cos sin K sin permanent inventory deficits. When ␶/T Ͻ ␲/2, the real ϭ Ϫ ϩ ͑ Ϫ ␶͒ part of W(z) is negative, and for large t, therefore, (9) J Io ID d T (10) becomes: ␣ ϭ ͑eϪ␻/⍀͓͒J͑␻ cos ⍀ Ϫ ⍀ sin ⍀͒ Ϫ K cos ⍀͔ I͑t 3 ϱ͒ 3 I Ϫ Td. (12) ϭ ͑ Ϫ ␶ Ϫ ␶ D K ID Io ) /T d . (11) The inventory falls to a value permanently below its Equation (9) shows that the inventory behavior is desired value by Td. Using the Final Value Theorem, dominated by the properties of the Lambert W func- John, Naim, and Towill (1994) proved analytically that tion, W(z), which can be either real or complex, and permanent deficits occur. Figure 3 shows solutions in whose real part can be either positive or negative this stable regime, which approach the predicted long- (Corless et al. 1996). Figure 1 shows several solutions term deficits (dashed lines). For small z ϾϪ1/e, W(z) and verifies that the divergence of the inventory re- ␶ is real, and the solutions no longer oscillate, which can sponse is extremely sensitive to /T, eloquently sug- be seen in Figure 3, where the curve for T ϭ 30 shows gesting a relation to the Bullwhip Effect, which we will no oscillation. pursue in Section 4. The accuracy of these solutions is explored in the Appendix. 3. Managing the Inventory 2.2. Inventory Cycles Having derived the analytical solutions, we briefly We next demonstrate the importance of the real world explore some properties that will be useful when we constraint that the orders should go to zero when the discuss the Bullwhip Effect. The time of the inventory inventory exceeds the desired value, (3b). The time at peak, referred to as t*, can be calculated by setting Warburton: An Analytical Investigation of the Bullwhip Effect Production and Operations Management 13(2), pp. 150–160, © 2004 Production and Operations Management Society 153

Figure 2 The effect of the real world constraint in which the orders go which the inventory climbs back to its desired value ␶ to zero when the inventory reaches its desired value: (3b) ( without any overshoot, i.e., I(t*) ϭ I : D .(10 ؍ ϭ Ϫ ϩ ͑ ␶͒ ID ID dT* A exp Wt*/ ). (14)

Since the argument of W(Ϫ␶/T*) depends on T*, the above equation must be solved numerically. Figure 1 includes the inventory response for the critical value of the adjustment time, T*. This provides another po- tentially valuable tool. While the manufacturer determines the delay time, the parameter, T, was introduced to allow the retailer to control the inventory response. If a surge in demand is detected and esti- mated, the retailer can adjust the orders (i.e., select T ϭ T*) so that the inventory is made up without suf- fering either deﬁcits or overshoots.

4. The Bullwhip Effect The Bullwhip Effect is defined as the amplification of order variability along the supply chain. The retailer’s dI/dt ϭ 0 in (9). Since A and W are complex, one ϭ Ϫ orders are given by O(t) (ID I(t))/T, and we now obtains: have the exact solution for the inventory in (9). Figure ␶ ␻ Ϫ ␣⍀ 4 shows a plot of the retailer’s order rate, which AW Ϫ a exp͕Wt*/␶͖ ϭ 0 f t* ϭ tan 1 ͫ ͬ. (13) quickly grows to exceed the constant consumer de- ␶ ⍀ ␣␻ ϩ a⍀ mand rate. The orders go to zero when the inventory When a retailer detects a surge in consumer de- exceeds its desired value, and remain so for the rest of mand, the time of the peak in inventory can be fore- the inventory cycle. The impact of the zero order rate cast. The size of the peak will depend on the accuracy is noteworthy, as it affects a significant fraction of the of the estimate of d, but the analytical solutions make cycle. (Figure 4 also includes the manufacturer’s initial the computations quite straightforward. order rate to the supplier, which is discussed in Sec- Figure 2 showed that, for long replenishment de- tion 5.) lays, the inventory experiences a significant (and prob- One measure of the Bullwhip Effect is the ratio of lematic) overshoot, while Figure 3 shows that inven- the output order rate (retailer orders to manufacturer) tory deficits occur for small values of ␶/T. Therefore, to the input order rate (consumer demand). The retail there is a critical value of the adjustment time, T*, for order rate climbs to a peak, which occurs soon after ␶,

Figure 3 Stable solutions with the prediction of long-term, permanent inventory deﬁcits (dashed lines). Warburton: An Analytical Investigation of the Bullwhip Effect 154 Production and Operations Management 13(2), pp. 150–160, © 2004 Production and Operations Management Society

Figure 4 The “Bullwhip Effect”: the amplification of order rate from defined as the decline through the replenishment time, consumer to retailer to manufacturer. The flood and drought during which the inventory falls by the amount d␶. in retail orders are also apparent. Therefore, Ϫ overshoot I* ID BWR ϭ ϭ . (16) I undershoot d␶ This measure of the Bullwhip Effect has a different character from that in (15), because the behavior of the overshoot is different from that of the orders. For example, in the critical T* case, the overshoot was ϭ R ϭ adjusted to be zero, (I* ID) and so BWI 0. In other words, if the Bullwhip in inventory is adjusted to zero, the Bullwhip in orders may remain. If it is more important to replenish the inventory than to minimize order fluctuations, then (16) is a better measure of the impact of demand fluctuations. Bullwhip analyses typically concentrate on order variability, and while there is clearly a cost benefit to its reduction, other impacts, such as inventory replenishment, should be and so this is an appropriate time at which to compare considered. For example, customer service levels will the rates: depend on the ability to replenish the inventory. Fran- soo and Wouters (2000) cautioned that there are many ͑␶͒ Ϫ ͑␶͒ ␶ O ID I ways to measure the Bullwhip Effect, an argument BWR ϭ ϭ ϭ . (15) d Td T bolstered by the above discussion. The manufacturer will add yet another contribution to the Bullwhip The superscript, R, indicates that this amplification Effect, which is discussed below. in orders is attributable to the retailer’s ordering policy. The Bullwhip Effect is usually defined in terms of the instantaneous variability in orders, and (15) re- 5. The Manufacturer’s View flects this in its use of order rates. Since we have We now discuss the impact on the manufacturer of the analytical solutions, the relative contributions of the orders from the retailer. The equation for the change parameters are explicit. For example, (15) is indepen- in the manufacturer’s inventory is similar to that of the dent of the size of the surge, which is a direct conse- retailer: quence of defining the Bullwhip Effect as a ratio. dIM/dt ϭ RM͑t͒ Ϫ SM͑t͒. (17) Equation (15) also suggests that increasing the value of T can reduce the Bullwhip Effect. (We continue to Since both retail and manufacturing quantities oc- assume that the replenishment delay is not easily cur in the equations, the superscript, M, is included to changeable by the retailer.) However, from Section 2, distinguish the manufacturer’s quantities. RM repre- we know that T cannot be raised arbitrarily without sents the receiving rate, which can be either from an permanent inventory deficits occurring. If it is impor- upstream supplier, or from the manufacturer’sown, tant for the inventory to return to its desired value, internal production. The decline, SM, represents the then some Bullwhip Effect is inevitable, e.g., T* is the shipment rate to the retailer. We assume that, before logical choice because it returns the inventory to its the retailer starts ordering, the manufacturer’s inven- desired value without overshoot, and for ␶ ϭ 10, T* tory is at the desired value. ϭ 6.68, and BWR ϭ 1.5. The receiving rate from the upstream supplier will The definition in (15) only represents the orders in depend on the orders issued by the manufacturer and the early stages of the inventory cycle, and does not the associated replenishment delay. The same process account for the inventory overshoot. Figure 4 shows applies if the items are produced internally; the man- the retailer’s orders as a flood followed by a drought, ufacturer is then issuing internal orders. The manu- and the longer the delay time, the greater the over- facturer’s situation is frequently more complicated shoot, and the longer the drought. This suggests a than the retailer’s because orders to suppliers often different calibration of the Bullwhip Effect: measuring represent sub-component orders rather than complete the growth in inventory fluctuations, rather than order items. Typically, the sub-components have different fluctuations. For example, in Figure 2, one can com- manufacturing and shipment delays. Additionally, pare the peak in the inventory (overshoot) to the bot- manufacturers usually supply several retailers. How- tom of the decline (undershoot). The undershoot is ever, the general approach used in the retail case still Warburton: An Analytical Investigation of the Bullwhip Effect Production and Operations Management 13(2), pp. 150–160, © 2004 Production and Operations Management Society 155 applies: The manufacturer creates an ordering policy The manufacturer’s order rate to the supplier ini- for each item, and the receipts from the supplier will tially grows as a quadratic function of time. This is in be characterized by the supplier’s replenishment de- stark contrast to the retailer’s order rate, which ini- lay time for that item. tially grows linearly, and the constant consumer de- Before the supplier has fulfilled any orders, the mand rate, which started the whole order train (see receipts, RM, will be zero. Therefore, initially, we can Figure 4). The Bullwhip Effect again emerges. Further, ignore the upstream supplier and consider just the the manufacturer does not realize that he is witnessing interaction between the manufacturer and the retailer. the flood part of the order cycle from the retailer, and The manufacturer’s inventory declines as items are that a drought is to follow. shipped to the retailer, and these shipments are equal The Bullwhip Effect caused by the manufacturer can to the orders received from the retailer. There will be be measured as the ratio of supplier order rate (output a delay in shipping, which we refer to as the manu- by manufacturer) to retail order rate (input to manufacturing time, ␶M. When shipments begin, they reflect facturer). Using the fact that the orders peak close to the orders received at time t Ϫ ␶M: when the supplier items begin arriving, t ϭ ␶S ϩ ␶M:

SM͑t͒ ϭ OR͑t Ϫ ␶M͒. (18) ͑␶S͒2 M ϭ BW ␶ M . (23) The manufacturer faces the same dilemma as the 2 T retailer: What ordering policy to employ? We again Equation (23) measures the amplification in orders propose Forrester’s order rate, and so the manufactur- solely attributable to the manufacturer. However, the er’s orders are: manufacturer’s magnification of the order rate com- M Ϫ M͑ ͒ pounds that of the retailer’s. The input orders to the ID I t OM͑t͒ ϭ for IM͑t͒ Ͻ IM, and (19a) manufacturer are the orders output from the retailer, TM D which are an amplified version of the consumer de- OM͑t͒ ϭ 0 otherwise. (19b) mand. Therefore, the total Bullwhip Effect from consumer to supplier is the combination The retailer’s initial orders are given by (7). Assuming ͑␶S͒2 M the retailer’s inventory starts at the desired value (ID 3 d /2TT ϭ BWC S ϭ ϭ BWRBWM. (24) IO), and including the manufacturer’s time delay, d one obtains the equation for the manufacturer’s inventory as: Equation (24) shows the multiplicative nature of the Bullwhip Effect (Dejonckheere, Disney, Lambrecht, M ϭ Ϫ M͑ ͒ ϭ Ϫ R͑ Ϫ ␶M͒ ϭ Ϫ ͑ Ϫ ␶M͒ dI /dt S t O t d t /T, (20) and Towill 2002). which has the solution:

͑ ͒ ϭ M Ϫ ͑ Ϫ ␶M͒2 6. Discussion I t Io d t /2T. (21) This solution is valid until the shipments begin ar- 6.1. Historical Remark riving from the supplier. We will not discuss that case, We begin by briefly discussing why the results pre- but the exact retailer order expressions could be used sented here differ from Forrester’s classic treatment and the integrals performed. Equation (21) describes (1961). Forrester defined a pool of goods on order, the initial response of the manufacturer’s inventory, G(t), which increases as orders are issued, and is de- and this is also shown in Figure 4. It can be seen that, pleted by items going into inventory. Forrester ap- after a delay, the manufacturer’s order rate rises, and proximated the fulfillment delay by assuming that the even faster than the retailer’s. inventory receiving rate is a constant fraction (1/TG)of the accumulated goods on order, i.e., items flow into 5.1. Manufacturer’s Contribution to the Bullwhip inventory proportional to the backlog. TG is referred to Effect as the “exponential delay,” and is intended to play a Equation (21) shows that the impact of the retailer’s role similar to ␶ in the delay case. However, under orders is to cause the manufacturer’s inventory to Forrester’s assumption, ordering more items increases decline as a quadratic function of time. This is in con- the backlog and immediately results in increased de- trast to the linear decline in the retailer’s inventory, (6). liveries. In the delay case, orders are delivered after Using (21) for the manufacturer’s inventory, the man- the replenishment delay, and in the same sequence as ufacturer’s order rate to the supplier is: they are ordered. The equations are:

d͑t Ϫ ␶M͒2 dG͑t͒ OM͑t͒ ϭ . (22) ϭ O͑t͒ Ϫ R͑t͒ R͑t͒ ϭ G͑t͒/T (25) TMT dt G Warburton: An Analytical Investigation of the Bullwhip Effect 156 Production and Operations Management 13(2), pp. 150–160, © 2004 Production and Operations Management Society

The order rate is again given by Forrester’s expres- In distinct contrast to the Forrester case, the inventory sion, (3a), which results in an ordinary, second order diverges. It is moderately interesting that Forrester differential equation for the inventory: chose parameter values that are stable, but which gen- Љ ϩ Ј ϩ Ј ϭ erate unstable solutions when the more realistic re- I I /TG I /TG T ID /TG T. (26) plenishment delays are considered. Forrester ignored the real world constraint in which the order rate goes to zero when the inventory exceeds 6.2. Finite Time Length Demand its desired value. The solution to (26) is: We now consider a surge in demand that lasts for a finite time, T , and then falls to zero. The solution in ϭ ϩ ͑␣ ͓͒ ͑␤ ͒ ϩ ͑␤ ͔͒ S I ID exp t A sin t B cos t . (27) (9) is still valid up through TS. After TS, the demand is zero, but there are still items in the pipeline, and these Substitution into the homogeneous equation deter- continue arriving until t ϭ T ϩ ␶. If at this point the mines the constants ␣ and ␤, while the boundary S inventory exceeds its desired value, the order rate will conditions (G(0) ϭ Go and dI/dt ϭ Go/T ) determine G remain zero. Since there are no outstanding orders the constants A and B: and no demand, the inventory is constant. Some ex- 1 T amples of the inventory behavior when the demand ␣ ϭϪ ͑ ͒ ␤2 ϭ ͩ Ϫ ͪ 1/ 2TG 1 goes to zero are shown in Figure 6. While the length of TGT 4TG the surge is finite, and its length affects the shape and G ͑I Ϫ I ͒␣ ϭ o Ϫ o D ϭ Ϫ scale of the response, the end result is frequently an A ␤ ␤ B Io ID. (28) TG expensive overshoot of the inventory. The interesting inventory behavior is again determined by the Lam- In the above solution, the exponential term, ␣,is bert W function, which is therefore crucial in under- always negative, guaranteeing stable solutions in standing the details of the response. which the inventory always approaches the desired value, and which Forrester referred to as “second order negative feedback.” The significant difference 7. Conclusions from the solution to the delay equation is that, while We have demonstrated that it is possible to solve the constant ␣ is always negative in Forrester’s case, ␻ exactly the delay differential equation describing the can be both positive and negative in the delay equa- inventory. No approximations were required. The re- tion case. Forrester’s solutions are always stable, but plenishment delay emerges as responsible for much of the delay equation includes both stable and unstable the rich and complex behavior associated with the regimes. Figure 5 shows the exact solution to Forrest- inventory response. The ordering policy is rather more er’s approximate equation for values that he typically general than previous theoretical analyses because it ϭ ϭ used: TG 10 and T 5. includes the realistic condition that the order rate is Figure 5 also shows the delay equation solution for zero once the inventory reaches its desired value. the same case (the demand, d ϭ 0, and ␶ ϭ 10, T ϭ 5). While the constant demand case was considered, it

Figure 5 Comparison of stable Forrester and unstable Lambert W-based solutions. Warburton: An Analytical Investigation of the Bullwhip Effect Production and Operations Management 13(2), pp. 150–160, © 2004 Production and Operations Management Society 157

Figure 6 The inventory response for surges in demand that last for a We calculated the replenishment rate for orders is- .(10 ؍ finite time, T (␶ S sued by the retailer to the manufacturer, and solved the equation for the manufacturer’s inventory reacting to the retailer’s orders. The Bullwhip Effect emerged naturally. The manufacturer’s situation is complicated by shipments to multiple retailers and orders to many suppliers. In principle, however, the manufacturer can also create a critical ordering policy to reduce inventory overshoot, but must also communicate the size of the consumer demand to the supplier. This reinforces the suggestion that the sharing of retail sales information is a major strategy for countering the Bullwhip Effect (Lee, Padmanabhan, and Whang 2000; Towill 1996). The additional Bullwhip Effect attributable to the manufacturer was calculated. Bullwhip analyses are typically based on inventory cost minimization, or profit maximization (Kahn 1987; Lee, Padmanabhan, and Whang 1997b). Here, we took a rather different approach. A form for the consumer is not true that the approach only applies to this demand was proposed, and solving the resulting assumption. The demand term only contributes to the equations determined the inventory behavior over inhomogeneous equation, and a different form for the time. It was then possible to adjust the parameters to demand would result in a new inhomogeneous solu- achieve a desired inventory response. It emerged that tion, but the homogeneous Lambert W-based solution one must trade the reduction in the Bullwhip Effect would remain unchanged. Thus, the process is com- against competing processes, such as permanent in- pletely general, and applicable to any form for the ventory deficits or inventory excesses. demand. Using simulation, it is difficult to understand a sys- More important, however, is that, for any demand tem’s underlying structure and behavior, and so op- for which the inhomogeneous solution can be found, portunities for performance enhancements may be the full solution will always contain the same homo- missed (Towill 1989). The model described here can geneous solution and, therefore, inherit its properties help determine the interesting regions in the parame- (i.e., the Lambert W function). This was confirmed in ter state space, and aid direct experimentation with Figure 6, where even though the demand was turned simulation. We conclude that these analytical case off, the inventory response was still dominated by the studies are extremely useful in precisely defining and Lambert W function. Also, this model’s prediction of quantifying the inventory behavior, and they provide permanent inventory deficits reproduces behavior tools to analyze important management issues, such found in more complex models previously analyzed as the Bullwhip Effect. The solutions clarify the con- with control theory. tribution of the various parameters, which helps in Comparison with direct numerical integration (see assessing and measuring the relative impact of pro- Appendix) established that the theoretical solutions posed remedies. All of the presented solutions are provide an excellent representation of the inventory exact and deterministic. Somewhat surprisingly, the behavior in both the stable and unstable regimes. The interesting scenarios are mathematically quite tracta- theory predicts a critical stability point, which is of ble, and so the general approach introduced here considerable practical value, since a diverging inven- should prove useful in a wide variety of supply chain tory is expensive. Numerical integrations verified that problems. the critical stability point is an inherent feature of the Acknowledgments differential equation, and its value is accurately pre- This work was supported in part by Griffin Manufacturing dicted by the theory. The ordering policy was param- Co., Inc. and a grant from the National Textile Center. eterized so as to allow the retailer to calculate a critical Steven Warner and Jonathan Hodgson contributed excellent order adjustment rate that returns the inventory back suggestions. The author would like to thank the referees, to its desired value exponentially fast, while not gen- whose questions resulted in new insights. erating an overshoot. The availability of analytical solutions makes such computations quite straightfor- Appendix ward. Confidence is thus established in both the accu- A.1. The Exact Solution racy and applicability of the Lambert W function- In this section, we derive the exact solution to (5) for t Ͼ ␶. based analytical solutions. As is usual, we begin with the homogeneous equation: Warburton: An Analytical Investigation of the Bullwhip Effect 158 Production and Operations Management 13(2), pp. 150–160, © 2004 Production and Operations Management Society

dI I͑t Ϫ ␶͒ A ϭ a ϩ i␣ W ϭ ␻ ϩ i⍀. (A.8) ϩ ϭ 0. (A.1) dt T Matching the solution in (6) to that of (A.7) at t ϭ ␶, gives: The complexity of this equation is associated with the a cos ⍀ Ϫ ␣ sin ⍀ ϭ eϪ␻͑I Ϫ I ϩ d͑T Ϫ ␶͒͒. (A.9) replenishment delay. The equation is referred to as the lo- o D gistics equation, and is a member of the class of delay Differentiating (A.7) at t ϭ ␶, gives the required condition on differential equations (Bellman and Cooke 1963). We pro- the derivative: pose a solution of the form I ϭ A exp(st), to obtain: dI Aest͕s ϩ eϪs␶/T͖ ϭ 0. (A.2) ͯ ϭ R͑␶͒ Ϫ d ϭ O͑0͒ Ϫ d dt tϭ␶ The term in braces can be rearranged into the following Ϫ ID Io AW ϭ Ϫ d ϭ eW. (A.10) suggestive form: T ␶ ␶ s␶ ϭ Ϫ␶ s e /T. (A.3) This provides a second relation between a and ␣, which ␻ ⍀ Equation (A.3) can be solved exactly in terms of the Lambert can thus be determined in terms of and . The complete, W function, which is defined as: exact solution to (5) is thus determined, and is given in (9). A number of combinations of solutions from (A.5) were W͑ z͒eW͑ z͒ ϭ z. (A.4) tried. Despite adding complexity, including more Lambert W functions, doesn’t significantly improve the match to the A review of the history, theory, and applications of the numerical solution. Considering the constant, A, as complex, Lambert W function may be found in Corless et al. (1996). provides two parameters, and guarantees the continuity of There are an infinite number of complex values of the Lam- the solution and its derivative at t ϭ ␶. From a practical bert W function, denoted as W(k, z). By the linearity of (A.1), perspective, once the inventory climbs back to ID, (3b) ap- any combination of the W(k, z) can be used in the solution, plies, and the solution changes. In practice, therefore, the and we could consider a solution of the form: ϭ Lambert W function is only required until I ID. We conclude that the k ϭ 0 term with a complex constant provides I͑t͒ ϭ ͸ c exp͓W͑k, Ϫ␶/T͒t/␶͔. (A.5) k a practical representation of the inventory behavior in both k the stable and unstable regimes. This is an infinite formula and we are interested in practical, easy-to-implement solutions. It turns out that using A.2. The Accuracy of the Theoretical Solution just the first term (k ϭ 0) results in a sufficiently accurate The above discussion suggests that only the first term in the representation of the inventory response. Therefore, our series in (A.5) is required, and we now examine the accuracy “practical” solution is: of that assumption. Figure 7 compares the analytical solutions with numerical integrations of the same differential I͑t͒ ϭ A exp͑Wt/␶͒ with W ϭ W͑0, Ϫ␶/T͒. (A.6) equation. The difference between the theoretical and numerical solutions in the unstable regime is less than 3% at the The inventory behavior is extremely sensitive to the prop- first peak. In the stable regime, there is very little difference erties of W(z), because it occurs in the exponential. Fortu- between the theoretical and numerical solutions. The theo- nately, the Lambert W function is readily available in effi- retical solution also predicts the period of oscillation to cient and accurate implementations, such as in MAPLE better than 0.4%. Therefore, in practical situations where (2002), where it is defined as Lambert W(k, z). W(z) only there is likely to be noise in the data, the one-term Lambert enters (A.6) as a parameter; there is no time dependence in W function provides an easy-to-compute, accurate represen- the W(z) term. tation of the inventory response. We now proceed to solve the inhomogeneous equation. The inhomogeneous term is a constant, so we propose the A.3. The Impact of the Lambert W Function constant, K, as a solution. Substitution in (5) determines K as The properties of the solutions are determined by the char- ϭ Ϫ K ID dT. Adding this to the homogeneous solution acteristics of the Lambert W function. For large ␶/T, the real gives: part of the Lambert function, ␻, is positive and the solutions ͑ ͒ ϭ Ϫ ϩ ͓ ␶͔ diverge (Figure 1). However, for small values of ␶/T ϭ ⑀, I t ID dT A exp Wt/ . (A.7) W(Ϫ⑀) ϭϪ⑀, which is real, and results in stable, decaying The constant, A, is determined by the imposition of solutions. The issue is to determine the critical stability boundary conditions. The inventory must be continuous, point. There are two interesting values for the argument of and so (A.7) must match the solution in (6) at t ϭ ␶. In order W(z), Ϫ1/e and Ϫ␲/2. to guarantee appropriate behavior at t ϭ ␶, it is also neces- When ␶/T Ͼ ␲/2, the real part of W(z) is positive, and the sary to specify the slope of the inventory there, and (2) solution in (9) applies. The oscillation is due to the imagi- provides the required condition. Two conditions on the nary part of the Lambert function, ⍀.As␶/T increases, ⍀ inventory require two integration constants. Noting that W increases, resulting in a decrease in the period, which can be is complex suggests that A can be also. Treating A as a clearly seen in Figure 1. complex constant results in a solution that turns out to The real part of W(z) ϭ 0atϪ␲/2. Therefore, the transition provide an excellent representation of the inventory over the from unstable to stable solutions occurs when ␶/T ϭ ␲/2. A entire range of ␶/T. Therefore, we consider A and W as: detailed numerical integration of the delay equation was Warburton: An Analytical Investigation of the Bullwhip Effect Production and Operations Management 13(2), pp. 150–160, © 2004 Production and Operations Management Society 159

Figure 7 Comparison of theoretical and numerical solutions, showing differences of less than 3% (at the peak) in the unstable regime, and small .(10 ؍ differences in the stable regime (␶

performed on either side of the critical stability value (␶ References ϭ f ϭ Ϯ 10.0 T 6.36 0.1). One slowly diverged and the other Anderson Jr., E. G., D. J. Morrice. 2000. A simulation game for slowly converged, validating that the critical stability point service-oriented supply chain management: Does information is in fact an inherent feature of the differential equation. The sharing help managers with service capacity decisions? Produc- ability to predict this critical stability point is very valuable, tion and Operations Management 9(1) 40–55. because it divides stable from unstable inventory behavior. Anderson Jr., E. G., C. H. Fine, G. G. Parker. 2000. Upstream volatility in the supply chain: The machine tool industry as a A.4. Analysis of Assumptions case study. Production and Operations Management 9(3) 239–261. While we considered the constant demand case, it is not true Bellman, R. E., K. L. Cooke. 1963. Differential-difference equations. that the solution only applies to this assumption. Since (5) is Academic Press, New York. linear, its solution is always the sum of homogeneous and Berry, D., D. R. Towill. 1995. Reduce costs: Use a more intelligent inhomogeneous components. A different demand would production and inventory policy. BPICS Control Journal 1(7) result in a new inhomogeneous component, but the homo- 26–30. geneous Lambert W-based component would remain un- Bourland, K., S. Powell, D. Pyke. 1996. Exploring timely demand changed. Therefore, the process is completely general and information to reduce inventories. European Journal of Opera- applicable to any form for the demand. tions Research 92 239–253. The analytical solutions reproduced the inventory def- Burbidge, J. L. 1961. The “new approach” to production. Production icit prediction of control theory, and we can hypothesize Engineer 40 3–19. as to why the simplified model (without a WIP term) Burbidge, J. L. 1984. Automated production control with a simula- includes such behavior found in more complex models. tion capability. Proceedings of IFIP Conference WG 5–7, Copen- The critical stability point and other interesting character- hagen, 1–14. istics are properties of the exponential Lambert W func- Buzzell, R. D., J. A. Quelch, W. J. Salmon. 1990. The costly bargain tion, which arises from the solution to the homogeneous of trade promotion. Harvard Business Review 68(March April) 141–148. equation. Therefore, any system of equations that includes the replenishment delay (a homogeneous term) Clark, T. 1994. Campbell Soup: A leader in continuous replenishment innovations. Harvard Business School Case, Boston, Massa- should inherit these characteristics. In contrast, the de- chusetts. mand leads to inhomogeneous terms, e.g., d is on the Corless, R. M., G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, D. E. Knuth. right-hand side of (5). As long as the ordering policy 1996. On the Lambert W function. Advances in Computational includes a time delay in fulfillment, adding WIP terms Mathematics 5 329–359. should merely add inhomogeneous terms to the homoge- Dejonckheere, J., S. M. Disney, M. R. Lambrecht, D. R. Towill. 2002. neous Lambert W solutions. We tentatively conclude Transfer function analysis of forecasting induced bullwhip in that it is reasonable to expect more complex models to supply chains. International Journal of Production Economics 78 inherit the properties described here. In which case, de- 133–144. spite its apparent simplicity, the present model has sig- Disney, S. M., M. M. Naim, D. R. Towill. 2000. Genetic algorithm nificant predictive power, and is of considerable theoret- optimization of a class of inventory control systems. Interna- ical interest. Further research is underway to confirm this tional Journal of Production Economics 68 259–278. conjecture. Disney, S. M., D. R. Towill. 2002a. Inventory drift and instability in Warburton: An Analytical Investigation of the Bullwhip Effect 160 Production and Operations Management 13(2), pp. 150–160, © 2004 Production and Operations Management Society

order-up-to replenishment policies. Proceedings of the 12th Inter- Johnson, M. E., S. Whang. 2002. E-business and supply chain man- national Symposium on Inventories, August, Budapest, Hungary. agement: An overview and framework. Production and Opera- Disney, S. M., D. R. Towill. 2002b. A discrete transfer function tions Management 11(4) 413–423. model to determine the dynamic stability of a vendor managed Kahn, J. 1987. Inventories and the volatility of production. American inventory supply chain. International Journal of Production Re- Economic Review 77 667–679. search 40(1) 179–204. Kelly, K. 1995. Burned by busy signals: Why Motorola ramped up Disney, S. M., D. R. Towill. 2002c. A procedure for the optimization production way past demand. Business Week 6 36. of the dynamic response of a vendor managed inventory sys- Lee, H. L., V. Padmanabhan, S. Whang. 1997a. Information distor- tem. Computers and Industrial Engineering 43 27–58. tion in the supply chain: The Bullwhip Effect. Management Fine, C. 2000. Clockspeed-based strategies for supply chain design. Science 43(4) 546–558. Production and Operations Management 9(3) 213–221. Lee, H. L., V. Padmanabhan, S. Whang. 1997b. The bullwhip effect Forrester, J. W., 1961. Industrial Dynamics, MIT Press, Cambridge, in supply chains. Sloan Management Review 38(3) 93–102. Massachusetts. Lee, H. L., K. C. So, C. S. Tang. 2000. The value of information Fransoo, M., J. F. Wouters. 2000. Measuring the bullwhip effect in sharing in a two-level supply chain. Management Science 46(5) the supply chain. Supply Chain Management 5(2) 78. 626–643. Gavirneni, S., R. Kapuscinski, S. Tayur. 1999. Value of information MAPLE. 2002. Computer Program, Version 8. Waterloo Maple, Inc., in capacitated supply chains. Management Science 45(1) 16–24. Ontario, Canada. Gill, P., J. Abend. 1997. Wal-Mart: The supply chain heavyweight Metters, R. 1997. Quantifying the bullwhip effect in supply chains. champ. Supply Chain Management Review 1(1) 8–16. Journal of Operations Management, Columbia, May. Hammond, J. 1993. Quick response in retail/manufacturing chan- Sterman, J. D. 1989. Modeling managerial behavior: Misperceptions nels in Globalization, technology and competition: The fusion of of feedback in a dynamic decision-making experiment. Man- computers and telecommunication in the 1990’s, Bradley et al. (ed.), Harvard Business School Press, Boston, Massachusetts, 185– agement Science 35(3) 321–339. 214. Towill, D. 1989. The dynamic analysis approach to manufacturing Holmstrom, J. 1997. Product range management: a case study of systems design. Journal of Advanced Management Engineering 1 supply chain operations in the European grocery industry. 131–140. Supply Chain Management 2(3) 107–115. Towill, D. 1996. Time compression and the supply chain: A guided John, S., M. M. Naim, D. R. Towill. 1994. Dynamic analysis of a WIP tour. Supply Chain Management 1(1) 15–27. compensated decision support system. International Journal of Towill, D. 1997. FORRIDGE: Principles of good practice in material Manufacturing Design 1(4) 283–297. ﬂow. Production Planning and Control 8(7) 622–632.