SUPPLY CHAIN APERIODICITY, BULLWHIP AND STABILITY ANALYSIS WITH JURY S INNERS

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1 SUPPLY CHAIN APERIODICITY, BULLWHIP AND STABILITY ANALYSIS WITH JURY S INNERS STEPHEN M. DISNEY Abstract. Jury (1974) presents a novel method for the analysis of dynamical systems based on matrices of the coefficients of the systems transfer function and its Inners. Here we exploit his procedure for an analysis of a supply chain replenishment or ordering decision known as the Order-Up-To policy. We study the discrete time case and generalise the classical Order-Up-To policy by the addition of two independent proportional controllers in the policy s feedback loops. The addition of the proportional controllers is well-known to allow the Order-Up-To policy to eliminate the bullwhip problem and we quantify this herein using Jury s Inners approach. However, care has to be taken with the use of independent controllers as they can introduce stability problems. This is because the roots of the characteristic equation become complex, and they may even move out of the unit circle in the z-plane. We identify the conditions of stability using Jury s Inners approach. We also investigate further the root distribution in the characteristic equation to identify the conditions under which the Order-Up-To policy is aperiodic. An aperiodic system has only a limited number of maxima and minima in its dynamic response. Thus aperiodicity is an important characteristic of a supply chain replenishment policy as it will not induce rogue seasonality. 1. Introduction Åström, Jury and Agniel (1970) present a matrix-based procedure for evaluating complex integrals of the form (1.1) I = 1 B(z)B(z 1 ) dz 2πı A(z)A(z 1 ) z along the unit circle in a positive direction in the complex plane when A and B are polynomials with real coefficients of z. This approach was refined further by Jury (1974) and Åström (1970) and their two methodologies are surprisingly powerful and easy to use. Complex integrals of the form of 1.1 often appear in the analysis of the bullwhip effect in supply chains when control theory is exploited, Dejonckheere, Disney, Lambrecht and Towill (2003). The bullwhip effect is a common term used to describe how the long-run variance of the orders amplifies as it passes through each decision making point in a supply chain, Lee, Padmanabhan and Whang, (2004, 1997a and b), Metters (1997), Disney and Towill (2003). Thus, several echelons up a supply chain, production and distribution orders bear very little resemblance Date: July 5th, Key words and phrases. Supply chains, aperiodicity, bullwhip, stability, z-transforms, orderup-to policy. 1

2 2 STEPHEN M. DISNEY to the end consumers demand. The bullwhip effect leads to unnecessary costs in supply chains and thus the phenomena has received a lot of academic attention recently. This contribution is concerned with utilizing Jury s (1974) important work in an investigation of a common supply chain replenishment rule for bullwhip effects. We will show that the evaluation of the complex integral, I, is closely related to the issue of stability, another important supply chain characteristic that has been studied recently, see Warburton, Disney, Towill and Hodgson (2003), Riddalls and Bennett (2002) and Disney and Towill (2002). Stability is a fundamental characteristic of a supply chain ordering policy. An unstable policy will react to any demand pattern with a response that oscillates with ever increasing amplitude and is thus inherently undesirable. A stable supply chain ordering policy on the other hand will react to any finite demand signal and after a finite period of time return to steady state conditions. Obviously this is a necessary property of a practical supply chain replenishment decision. We further consider the issue of aperiodicity. This is the condition that, after a finite demand, a system will return to rest without an undue number of oscillations. This is an important criterion as it will contribute to the reduction of the tendency of a supply chain to resonate and introduce rogue seasonality or a fake business cycle, Forrester (1958). As these demand oscillations are very costly (we assume supply chain oscillations will cause capacity losses and labor scheduling and training problems), an aperiodic response by a supply chain replenishment decision is highly desirable. Aperiodicity is effectively a root distribution problem that we investigate via the Hurwitz matrix. 2. Preliminaries We are following the work of Jury (1974) and refer readers to Jury (1974) for any required proof to his approach. Let us be concerned with the exploitation of his important contribution and begin with some notation. Let z be the z-transform operator, F (z) = t=0 f(t)z t. F (z) usually manifests itself as a ratio of two polynomials in z, F (z) = B(z) A(z), where; (2.1) B(z) = b n z n + b n 1 z n b 1 z + b 0 (2.2) A(z) = a n z n + a n 1 z n a 1 z + a 0, a n > 0. Thus F (z) is the discrete time, z-domain, transfer function of the dynamic system described by f(t), a function of the discrete (time) variable t. We will use lower case letters to denote variables in the time domain and corresponding upper case letters in the frequency domain. In practice however, it is usually not necessary to start from the time domain, as a block diagram most easily defines a system. This is easily constructed from a systems description using common control engineering techniques (for example see Nise, 1995) and allows the derivation of the systems transfer function directly in the z-domain. The block diagram of a well-known supply chain ordering or replenishment algorithm is shown below in Figure 1. We refer readers to Dejonckheere, Disney, Lambrecht and Towill (2003 and 2004) for its detailed derivation where they show that the policy is a generalisation of the OUT policy. Although, here follows a brief

3 SUPPLY CHAIN APERIODICITY, BULLWHIP AND STABILITY ANALYSIS 3 introduction. First we can see the OUT policy has three feed-forward loops; D, DWIP and TNS. D is a forecast of the demand process and is added directly to the replenishment order, O. Here the forecast is generated by exponential smoothing, with a smoothing constant of α = 1 1+T a, where T a is the average age of the data in the exponential smoothing forecast. Another feed-forward loop is used to set the Target Net Stock (TNS) position, TNS= a D. The Target Net Stock gain (a) is used to achieve a given stock availability such as the fill-rate. Disney, Farasyn, Lambrecht, Towill and van de Velde (2006) investigate the link between bullwhip and the fill-rate for the ARMA(1,1) demand process. Interestingly, bullwhip reduction may not necessarily increase inventory holding costs or reduce customer service. Finally the forecast of demand is also used to generate a dynamic Work In Progress target, DWIP by multiplying the forecast by T p, that is DWIP= T p D. T p is our estimate of the production, distribution or replenishment lead-time, T p, which we require to determine a target for the orders placed but not yet received or the WIP levels. The replenishment delay T p, is the time between placing an order and receiving it in stock. The replenishment delay in Figure 1 is represented by the delay operator z (Tp+1), where the additional unit of delay is required to ensure the correct sequence of events. That is, we receive stock and satisfy demand at the beginning of the replenishment period and observe the inventory position and placed orders at the end of the period. Thus even if our physical lead-time, (T p ) is zero, we would not observe the received stock from the suppliers or the production system until the end of the next period, hence the additional unit delay. Disney and Towill (2005) have investigated the consequences of getting the estimated delay (T p ) wrong and highlight a solution to the problem that involves calculating the WIP in a specific manner. We adopt their approach herein. Figure 1 also shows that the OUT policy has two feedback loops; the difference between the Target Net Stock (TNS) and the actual Net Stock (NS), and the difference the between Desired WIP (DWIP) and the actual WIP. Usually in inventory models these differences, or errors, are accounted for in full in the replenishment orders, thus traditionally T i = T w = 1. Herein we will consider the case of incorporating two proportional controllers into the feedback loops; a net stock proportional controller, 1 T i, and a WIP proportional controller, 1 T w. This generalisation of the OUT policy allows superior economic performance (Disney and Grubbström (2004) and Chen and Disney (2003)) and greater flexibility in shaping its dynamic response when balancing bullwhip, inventory and customer service targets, Disney, Farasyn, Lambrecht, Towill and van de Velde (2006). However the addition of the proportional controllers also introduces dynamic problems such as stability. Thus the benefits previously reported may be lost if care is not taken when designing replenishment algorithms. Furthermore, the generalisation is not new, it has a long history, for example see; Magee (1956), Deziel and Eilon (1967), Towill (1982), Bertrand (1986), Sterman (1989) to name a few. In this paper we are concerned with a single echelon of a supply chain. However, recently a number of important multi-echelon supply chain papers have appeared. Hoberg, Bradley and Thonemann (2007) investigate stability issues and bullwhip effects with a z-transform based approach for a multi-echelon supply chain under three operating policies; a base-stock, echelon stock and a replenishment rule with only an inventory feedback controller (rather than an inventory and WIP feedback

4 4 STEPHEN M. DISNEY Figure 1. Block diagram of the generalized Order-Up-To policy controller). Gaalman and Disney (2007) have recently derived an optimal multiechelon replenishment rule for minimising the total supply inventory costs. The policy is optimal in the sense that it is the best possible linear policy and was derived using state space methods. Hosoda and Disney (2006) study a three echelon supply chain with AR(1) demand at the end customer. The evolution of the demand process as it passes through the order-up-to policy is derived and conditional expectation is used to create minimal mean squared error forecasts. 3. Transfer function and Jury s matrices of coefficients We are interesting in the dynamic (time-varying) performance of the Demand rate and the Order rates. The Demand rate is a discrete time series of the customer demand or requirements. The Order rate is a discrete time series of the replenishment orders placed on the supplier (or the production requests placed on a factory). We may manipulate the block diagram (Figure 1), to obtain following transfer function that relates the Order rate to the Demand rate. This transfer function completely describes the linear dynamic behaviour of our system. (3.1) O(z) = z1+tp (T p T i (z 1) (a + T a + T i )T w + (1 + a + T a + T i )T w z) (T a (z 1) + z)(t w + T i ((1 + T w (z 1))z Tp 1)) Jury s Inners approach exploits matrices that are a function of the order (n) of the transfer function. Thus we have to specify a value of T p here. We choose T p = 2 as the case of T p = 1 can be readily solved using time domain techniques such as those exploited by Disney and Towill (2003) and Disney, Towill and van de Velde (2004). Herein we also assume the demand pattern is a stationary independently and identically distributed (i.i.d.) stochastic process. Although this is a rather restrictive assumption we do know this type of demand process presents at least one product in Procter and Gamble s product range, (Disney, Farasyn, Lambrecht, Towill and Van de Velde (2006). The best possible forecast of an i.i.d. process (that is, the one with the minimum mean squared error) is the unconditional mean of the demand process. If demand is not i.i.d. then other types of forecasting mechanisms may be more desirable. For example for some integrated demand processes then exponential smoothing (with 0.5 < T a < ) will be optimal, Chatfield et al, (2001). If demand is of

5 SUPPLY CHAIN APERIODICITY, BULLWHIP AND STABILITY ANALYSIS 5 an ARIMA type, the Box and Jenkins (1970) approach will be helpful. In any case, the general goal is to create two forecasts; one is a forecast of the demand over the replenishment lead-time, the other is a forecast in the period after the replenishment lead-time. The variance of the inventory (net stock) levels will equal the sum of the variances of the forecast errors of demand over the lead-time and review period. This was proved, for a single echelon of a supply chain by Vassian (1955) and extended to the multi-echelon case by Hosoda and Disney (2006). Thus, it is always important to create the best possible forecasts from an inventory cost perspective. We may also set the mean demand to zero without loss of generality as we are considering a linear system. The linear system assumption also implies that all unmet demand is fully backlogged (or available from another source). To achieve the constant forecast of the unconditional mean we set T a =. The gain in the inventory feed-forward loop, a, and the Work-In-Progress (WIP) feedforward gain, T p, will simplify out of the system s transfer function when T a =. This is the optimal exponential forecasting constant for the demand process we have assumed. The transfer function then becomes (3.2) O(z) = B(z) A(z) = T w z 3 T w + T i (z 1)(1 + z + T w z 2 ). This transfer function can be expressed in the form of 2.1 and 2.2 as coefficients of z with (3.3) b 0 = b 1 = b 2 = 0, b 3 = T w and (3.4) a 0 = T w T i, a 1 = 0, a 2 = T i (1 T w ), a 3 = T i T w. An important sub-set of 3.2 occurs when T i = T w. In this case the transfer function simplifies down to z (3.5) O(z) = 1 + T i (z 1). Jury s Inners method makes use of the determinants of matrices whose elements consist of the coefficients of the transfer function. We require four matrices here. They are; (3.6) X n+1 = (3.7) Y n+1 = a n a n 1 a n 2... a 0 0 a n a n 1... a a n... a a n a a a 0... a n 2 0 a 0 a 1... a n 1 a 0 a 1 a 2... a n

6 6 STEPHEN M. DISNEY (3.8) X n 1 = a n a n 1 a n 2... a 2 0 a n a n 1... a a n... a a n (3.9) Y n 1 = a a a 0... a n 4 0 a 0 a 1... a n 3 a 0 a 1 a 2... a n 2 Equations 3.6 and 3.7 are required for the bullwhip analyses, while 3.8 and 3.9 are required for the stability investigation. Here we can see the concept of the Inners. X n 1 is the first Inner of X n Stability via Jury s Inners The question of stability is a fundamental aspect of dynamic systems. A stable system will react to a finite input and return to steady state conditions in a finite time. An unstable system will either diverge exponentially to positive or negative infinity or oscillate with ever increasing amplitude. A critically stable system will fall into a limit cycle of constant amplitude to any finite input. Oscillations in the order rates in supply chain are costly. Thus as a first step to dynamically designing a supply chain replenishment rule, we must ensure the replenishment rule is stable. Jury shows that the necessary and sufficient conditions for stability of a system is given by; A(1) > 0, ( 1) n A( 1) > 0 and the matrices ± n 1 = X n 1 ± Y n 1 are positive innerwise. Lets take each criteria in turn. A(1) must be greater than zero. A(1) = A(z) z 1, that is A(1) is given by 2.2 with the z s replaced with 1. When the coefficients are substituted into A(1), A(1) = T w. Thus this condition is satisfied if T w > 0. ( 1) n A( 1) must be greater than zero. In alike manner as above, A( 1) = A(x) z 1, that is, Equation 2.2 with the z s replaced with -1. ( 1) n = 1 as the order, n, is 3 in our example. We substitute in the coefficients yielding ( 1) n A( 1) = 2T i T w T w. Thus for stability we require 2T i T w T w > 0 2T i T w > T w 2T i > 1 T i > 1 2. ± n 1 = X n 1 ±Y n 1 matrices are positive innerwise. A matrix is positive innerwise if its determinant is positive and all the determinants of its Inners are also positive. The ± n 1 = X n 1 ± Y n 1 matrices here are; [ ] [ (4.1) + n 1 = a3 a 2 + a 0 = a 0 a 1 + a 3 ] T i T w T w (1 T i ) T w T i T i T w

7 SUPPLY CHAIN APERIODICITY, BULLWHIP AND STABILITY ANALYSIS 7 [ ] [ (4.2) n 1 = a3 a 2 a 0 = a 0 a 3 a 1 ] T i T w 2T i T w (1 + T i ) T i T w T i T w We can see that both ± n 1 are 2x2, thus for our stability analysis here we only need to test whether the determinants of ± n 1 are positive as there are no Inners. The determinants are; (4.3) + n 1 = T w(t i (1 + T i (T w 1) + T w ) T w ) (4.4) n 1 = T it w (3 + T i ) 2T 2 i + (T i (T i 1) 1)T 2 w. Solving 4.3 for its roots yields a non-critical stability boundary of (4.5) T i nc = 1 T w ± 1 2T w + 5T 2 w 2(T w 1), which is completely dominated by the roots of 4.4, (4.6) T i c = T 2 w 3T w ± T w 1 2Tw + 5T 2 w 2(T w 2 + T 2 w). which describes the critical stability boundary. Investigation reveals that the value of T i required for stability is given by (4.7) T i ( 2<Tw< ) > T 2 w 3T w T w 1 2Tw + 5T 2 w 2(T w 2 + T 2 w) (4.8) T i (1<Tw< ) > T 2 w 3T w + T w 1 2Tw + 5T 2 w 2(T w 2 + T 2 w) (4.9) T i (0.4<Tw<1) < T w 2 3T w T w 1 2Tw + 5Tw 2 2(T w 2 + Tw) 2. In expression 4.8 as T w, the minimum T i required to maintain stability is the golden ratio, that is, T i > The first stability criterion 4.7 is relevant when T w < 2 and it appears to have been missed in a previous study of the stability the author has previously conducted, Disney and Towill (2002). Figure 2 highlights an elegant rule of thumb ; the system is always stable if T w > 1 and T i > , the golden ratio. In Figure 2 we have also scaled the axis by the reciprocal of the parameters T i and T w ( 1 T i and 1 T w ) as this allows us to view the complete parameter plane in one concise figure. Plotting the stability plane in this way enabled the previously missed stable solutions to be highlighted. Care has to be taken when interpreting the x-axis; as 1 T w 0 +, T w + and as 1 T w 0, T w. The system is unstable if 2 < T w < 0. We have verified the stability results via simulation of the systems impulse response as shown in Figure 3. Plot V and W confirms that there are indeed stable solutions in the region of 0.5 < 1 T w < 0, as predicted.

8 8 STEPHEN M. DISNEY Figure 2. The stability boundary for the generalized Order-UpTo policy when the lead-time, Tp = Bullwhip via Jury s Inners Jury (1974) shows us that bullwhip, as measured by the ratio of the stationary order and demand variance, is given by (5.1) Bullwhip = 2 σo Xn+1 + Yn+1 b =I=, 2 σd an Xn+1 + Yn+1 where [XP n+1 + Yn+1 ]b = [X Pn+1 + Yn+1 Pn] with the last row replaced by [2bn b0, 2 bi bi+n 1,..., 2 bi bi+1, 2 i=0 b2i ]. For our example this is given by [2b0 b3, 2(b0 b2 + b1 b3 ), 2(b0 b1 + b1 b2 + b2 b3 ), 2(b20 + b21 + b22 + b23 )]. After replacing the coefficients of the transfer function 3.3 and 3.4 and working through the tedious but simple algebraic process we arrive at the bullwhip expression. It is given by (5.2) Bullwhip = Tw (Ti (1 + Ti (Tw 1) + Tw ) Tw ). (2Ti 1)(Ti Tw (3 + Ti ) 2Ti2 + Tw2 ((Ti 1)Ti 1) The bullwhip produced by the generalised Order-Up-To policy with two independent controllers is shown in Figure 4 where we have plotted various contours for bullwhip. The contours of Bullwhip = x are given by, (5.3) Tw Bullwhip=x (Ti (Ti 1 3x + Ti (5 + 2Ti )x ± χ)) = 2(1 Ti (1 + Ti ) + x + Ti (Ti (2Ti 3) 1)x)

9 SUPPLY CHAIN APERIODICITY, BULLWHIP AND STABILITY ANALYSIS 9 Figure 3. Verification of the stability results where χ = (Ti 1)2 2(1 + Ti (Ti 4 + 6Ti2 ))x + (1 + Ti (9Ti 2))(x 2Ti x)2. Here the bullwhip boundary, that is when Bullwhip = 1, simplifies down to, (5.4) Tw Bullwhip=1 p 2Ti 3Ti2 Ti3 ± Ti2 5 14Ti + 9Ti2 =. 2(1 Ti 2Ti2 + Ti3 ) 5.1. Integration of bullwhip and stability analysis. Jury (1974) notes that Xn+1 + Yn+1 = 2A(1)( 1)n A( 1) n 1 which we will notice are the conditions for stability. Calculating the denominator of 5.1 in this way has advantages as n 1 is of one less order than Xn+1 + Yn+1 and thus it is easier to obtain the determinant. Furthermore we already know of n 1, A(1) and A( 1) from the stability analysis. The coefficients of 3.3 mean that Xn+1 + Yn+1 b is also easy to calculate in our system. An opportunity now exists to further explore Jury s stability test here. Obviously, when we have a critically stable system, the bullwhip generated by the system is infinite. Infinite bullwhip is observed when the denominator of 5.1 becomes zero. So setting an Xn+1 + Yn+1 = 2an A(1)( 1)n A( 1) n 1 = 0 and solving for Ti yields the following stability conditions;

10 10 STEPHEN M. DISNEY Figure 4. Bullwhip produced by the generalised OUT policy with independent controllers (5.5) Ti = 0, Ti = p Tw (Tw 3 ± 1 2Tw + 5Tw2 1, Ti = 2 2(Tw 1 + Tw2 ) These are the same conditions as found earlier, but notice that we did not generate the obsolete criterion (Equation 4.5) or the rather misleading and unnecessary (but correct) criterion Tw > 0 that was given in the first step of Jury s approach. We may also observe that when Ti or Tw then the numerator of 5.1 become infinite, giving us the last boundary of Figure 3, (Ti < ). 6. Aperiodicity via Jury s Inners An aperiodic system has a time domain response with only a finite number of maxima or minima in its time domain response (less than n, the order of the system), Jury (1974) and Meerov and Jury (1998). Thus an aperiodic supply chain ordering policy will avoid costly long-term oscillations in the ordering patterns generated. This will obviously contribute to a reduction of any system induced cycles such as the well-known business cycle. An aperiodic condition occurs when all the roots of the characteristic equation are distinct and lie on the real axis in the interval [0, 1) in the z-plane, Jury (1974). Lets first consider the case of Equation 3.5, the generalised OUT policy when Ta = and Tw = Ti. The characteristic equation is 1 + Ti z Ti that clearly has a single, simple root at z = TiT 1. Thus the generalised OUT policy is aperiodic i when Tw = Ti and Ti = [1, ). The case of Equation 3.2, the generalised OUT policy with independent controllers, is more difficult to study as there are three roots to the characteristic

11 SUPPLY CHAIN APERIODICITY, BULLWHIP AND STABILITY ANALYSIS 11 equation, of which two may be complex. In order to determine whether the roots are distinct, real and in the interval of [0, 1) in the z-plane we may apply the linear fractional transformation, 6.1, to the characteristic equation to map the segment [0, 1) in the z-plane onto the negative real axis in the s-plane. (6.1) z = s s 1 After this transformation is applied to Equation 3.2 we may use a continuous time stability test to determine the aperiodicity condition has now reduced to the requirement that the all the roots of A(s) are distinct and lie on the left-hand side of s-plane. (6.2) A(s) = (s 1) 3 T w + T i (1 + s(s(2 + T w ) 3)) Collecting together terms surrenders the following coefficients of s; (6.3) a 0 = T i T w, a 1 = 3(T w T i ), a 2 = T i (2 + T w ) 3T w, a 3 = T w. Jury (1974) and Meerov and Jury (1998) show us that the aperiodicity constraints are that a i > 0, i = 0,.., n and that the Hurwitz matrix of A(s) must be positive innerwise. Lets first consider the signs of the coefficients a i, i = 0..n; (6.4) a 0 > 0 T i > T w, a 1 > 0 T w > T i, a 2 > 0 T i > 3T w 2 + T w, a 3 > 0 T w > 0 We can show that the third order representation of the matched controller case, that is when T i = T w, contains two roots at the origin. From the requirement that the coefficients, a i, must be positive, we require that T i > T w and T w > T i simultaneously. This does not surprise us, as the double root is not distinct, contravening Meerov and Jury s conditions for aperiodicity and re-enforcing the need to simplify first. The conditions for a positive a 2 and a 3 ensure that when T i = [1, ) the distinct root of A(z) lies on the real axis of the z-plane between [0, 1). When T w T i the double root at the origin will become either a pair of roots on the real line; one positive, one negative, or they will become complex. We may determine when the roots of A(z) become complex by considering when the Hurwitz matrix of A(s) as given by 6.5 is positive innerwise, (in fact Meerov and Jury (1998) assert that, in their experience, only the determinant of 5 is critical and needs to be considered and not it s inners; we concur herein). (6.5) 5 = a 3 a 2 a 1 a a 3 a 2 a 1 a a 3 2a 2 a 1 0 3a 3 2a 2 a 1 0 3a 3 2a 2 a Substituting in the values of a 0..a n (the a i for A(s), not (A(z)) yields.

12 12 STEPHEN M. DISNEY Figure 5. Aperiodicity of the generalised OUT policy (6.6) 5 = Tw Tw Ti (2 + Tw ) 3Tw 3(Tw Ti ) Ti T w Tw Ti (2 + Tw ) 3Tw 3(Tw Ti ) 0 3Tw 2(Ti (2 + Tw ) 3Tw ) 3Tw 2(Ti (2 + Tw ) 3Tw ) 3(Tw Ti ) 2(Ti (2 + Tw ) 3Tw ) 3(Tw Ti ) 0 0 Ti Tw 3(Tw Ti ) 0 0 and its determinant is, (6.7) 5 = 8Ti2 Tw 6Ti Tw2 + 8Ti2 Tw2 12Ti Tw3 + 2Ti Tw3. Solving 5 = 0 for Ti yields the following roots; (6.8) Ti = 0, Ti = 0, Ti = Tw, Ti = 27Tw3. (2 + Tw )2 (4Tw 1) Together with the positive innerwise requirement, 6.8, shows that the congruent 3 27Tw roots of A(z) are real if Tw > Ti > (2+Tw )2 (4T ; complex otherwise. So although, w 1) the OUT policy is not aperiodic when Ti 6= Tw as there is always one root with a neg3 27Tw ative real part, the system is weakly aperiodic when Tw > Ti > (2+Tw )2 (4T. w 1) We use the term weakly aperiodic to denote the case when the congruent roots of A(z) lie between (-1,1) on the real line and the simple root lies between [0,1) on the real line. We have collected these aperiodicity results graphically in Figure 5. We have verified our findings on the distribution of real roots of A(z) by plotting them in the complex z-plane for a selection of settings of parameters Ti and Tw in Figure 6. Here we can see that plots; E,G and H are aperiodic, B,C,D,I and

13 SUPPLY CHAIN APERIODICITY, BULLWHIP AND STABILITY ANALYSIS 13 Figure 6. Verification of the poles of A(z) Figure 7. Simulation of impulse responses J are weakly aperiodic and that plots A,F,K,L,M,N and O are non-aperiodic as predicted. For insights into the implications for the time domain solution of aperiodic systems we have further plotted the impulse response for the 15 sets of parameters of Figure 6 in Figure 7. From Figure 7 we can see that weakly aperiodic systems do not oscillate around zero. Strictly aperiodic systems approach zero exponentially directly from the demand impulse. Weakly periodic systems may initially diverge but soon return to exponential decay. Non-aperiodic systems possess a greater number of longer lasting oscillations and presumably these oscillations are costly in real supply chains.

14 14 STEPHEN M. DISNEY Figure 8. Simulation to i.i.d. random demand 7. Simulation to a normal, independently and identically distributed demand pattern In order to gain an understanding of what our stability, bullwhip and aperiodic results might mean to a supply chain, Figure 8 shows six responses of the generalized OUT policy to an i.i.d. random demand. Plot A shows the demand pattern and the response produced by the classical OUT policy. They are the same because when T w = T i = 1 the OUT policy simply passes on orders. This is the response that would have been obtained if a standard, text book like approach was used to design a supply chain replenishment rule. Plot B shows a typical aperiodic response in the generalised OUT policy. Compared to the nearby non-aperiodic response (Plot C) we can see that there is a reduction in the fluctuations later in time. Clearly the amplitudes of the cylical demands in periods has been attenuated. In Plot D we have set T w = to effectively remove the WIP feedback loop. The plot shows a critically stable system. Here the order rate is stuck in an orbit around the mean demand with a large amplitude. Obviously this is a response to avoid in a real supply chain. Plot E highlights a weakly aperiodic response in the newly discovered stable region, that is when T w < 2. Clearly this is a very damped response. The final Plot (F) demonstrates that the generalised OUT policy is also capable of level scheduling. In this policy T i = T w = This has the effect of removing both the inventory and the WIP feedback loops. Care must be taken with this level scheduling policy as there is no control of the inventory levels. In fact the inventory levels are simply an accumulation of the random demands and is thus a random walk. It is also possible to minimise the inventory and order variances in a trade-off as was shown in Disney, Towill and van de Velde (2004) for the case of T w = T i. Indeed, for the case where costs are equally related to inventory and order variance, the optimal T w = T i = , the golden ratio for all lead-times. This was illustrated in plot B.

15 SUPPLY CHAIN APERIODICITY, BULLWHIP AND STABILITY ANALYSIS Conclusions Using Jury s Inners approach we have studied the dynamics of a generalized OUT policy, a replenishment rule often used in supply chain to control material flow. To use the Inner s approach we needed to specify a lead-time, we chose leadtime T p = 2. Having to specify the lead-time appears to be the only weakness of Jury s Inners approach. There appears to be no limit to the order of the transfer function when using Jury s Inners. In other (time-based) methods, there is often a limit to the order of the transfer function that can be considered as inversion of a transfer function often requires us to find its roots; not an easy task and impossible in radicals for orders greater than 5. With the Inners approach we may avoid this issue altogether as we only require the determinants of matrices of the coefficients of the transfer function. This can always be done analytically, but the expressions may become large for very high order transfer functions. These high order transfer functions are generated when long lead-times or complex, multi-echelon supply chain models are present. Neither-the-less, computers are very good at dealing with matrix operations numerically if large, awkward determinants exist. A new stability test has been developed that is more direct than the traditional matrix based procedures as it simply involves solving 2a n A(1)( 1) n A( 1) n 1 = 0. This was achieved by integrating the bullwhip and stability analysis of Jury (1974). It appears not to generate redundant stability criteria. The stability analysis together with appropriate scaling of the parameter plane (by using 1/T i and 1/T w ), identified new areas of stable responses that were previously overlooked. We have also investigated the roots distribution of the transfer function in order to identify when aperiodic responses are generated by the OUT policy. The classical OUT policy (with T w = T i = 1) was found to be aperiodic. This means in a practical sense that the system does not oscillate unduly and has a damped response. An aperiodic response is also generated when T w = T i in the generalized OUT policy, thus confirming the importance of the so called Deziel and Eilon (1967) settings. We also found weakly aperiodic responses in the parameter plane. Weakly aperiodic systems have only real roots, but one or more of them may be negative and they may have small short-term oscillations in the time domain, although they quickly dissipate and the long-term behaviour is damped. Aperiodicity is important in supply chains, as it will contribute to a reduction in rogue seasonality. The Inners approach is a very powerful tool for the supply chain analyst. Here we have used it to generate closed form bullwhip expressions for a system that is too complex to be solved using other techniques. For example, the squared impulse response method of Disney and Towill (2003) is unable to solve the system considered herein. The Inners approach is also very efficient to use for both stability and bullwhip questions as many of the steps are similar in each analysis and we may gain some economies of scale. It is hoped this analytical approach will provide solutions to many more supply chain problems. In summary, the unique contributions of this paper are; the derivation of the bullwhip expression for the T p = 2 case of the OUT policy with independent feedback controllers, the identification of stable responses with negative values of the WIP feedback controller (T w ), a more direct stability test based on Jury s approach, and the identification of two new supply chain performance characteristics - aperiodic responses and weakly aperiodic responses.

16 16 STEPHEN M. DISNEY 9. Biblography Åström, K.J Introduction to stochastic control theory. Richard Bellman, University of Southern California. Åström, K.J., Jury, E.I., Agnail, R.G A numerical method for the evaluation of complex integrals. IEEE Trans on Automatic Control AC Aug Bertrand, J.W.M Balancing production level variations and inventory variations in complex production systems. International Journal of Production Research Box, G.E. and Jenkins, G.M Time series analysis forecasting and control. Holden-Day, San Francisco. Chatfield, C., Koehler, A.B., Ord, J.K. and Snyder, R.D A new look at models for exponential smoothing. The Statistician Chen, Y.F., Disney, S.M The order-up-to policy sweet spot - Using proportional controllers to eliminate the bullwhip problem. EUROMA POMS Conference, Como Lake, Italy, June Dejonckheere, J., Disney, S.M., Lambrecht, M.R., Towill, D.R Measuring the Bullwhip Effect: A control theoretic approach to analyse forecasting induced Bullwhip in order-up-to policies. European Journal of Operations Research Dejonckheere, J., Disney, S.M., Lambrecht, M.R., Towill, D.R The impact of information enrichment on the Bullwhip effect in supply chains: A control engineering perspective. European Journal of Operational Research, Deziel, D.P., Eilon, S A linear production - inventory control rule. Production Engineer Disney, S.M., Farasyn, I., Lambrecht, M., Towill, D.R., Van de Velde, W Taming the bullwhip effect whilst watching customer service in a single echelon of a supply chain. European Journal of Operational Research Disney, S.M., Grubbström, R.W The economic consequences of a production and inventory control policy. International Journal of Production Research Disney, S.M., Towill, D.R A discrete linear control theory model to determine the dynamic stability of Vendor Managed Inventory supply chains. International Journal of Production Research Disney, S.M., Towill, D.R On the bullwhip and inventory variance produced by an ordering policy. OMEGA: The International Journal of Management Science Disney, S.M., Towill, D.R Eliminating inventory drift in supply chains. International Journal of Production Economics Disney, S.M., Towill, D.R., van de Velde, W Variance amplification and the golden ratio in production and inventory control. International Journal of Production Economics Forrester, J Industrial Dynamics, A major breakthrough for decision makers. Harvard Business Review, July-August Gaalman, G. and Disney, S. M. (2007) On the echelon-order-up-to policy: A stochastic optimal control approach, Manufacturing and Service Operations Management Conference, Beijing, June

17 SUPPLY CHAIN APERIODICITY, BULLWHIP AND STABILITY ANALYSIS 17 Hoberg, K., Bradley, J.R. and Thonemann, U.W Analyzing the effect of the inventory policy on order and inventory variability with linear control theory, European Journal of Operational Research 176 (3) Hosoda, T. and Disney, S.M., (2006), On variance amplification in a threeechelon supply chain with minimum mean squared error forecasting, OMEGA: The International Journal of Management Science Jury, E.I., 1974, Inners and the stability of dynamic systems, John Wiley and Sons, New York. Lee, H.L., Padmanabhan, V., Whang, S. 1997a. The bullwhip effect in supply chains. Sloan Management Review Spring Lee, H.L., Padmanabhan, V., Whang, S. 1997b. Information distortion in a supply chain: the Bullwhip Effect. Management Science Lee, H.L., Padmanabhan, V., Whang, S Comments on Information Distortion in a Supply Chain: The Bullwhip Effect. The Bullwhip Effects: A reflection, Management Science Magee, J.F Guides to inventory policy-ii. Problems of uncertainty. Harvard Business Review, March-April, Meerov, M.V., Jury, E.I On periodicity robustness. International Journal of Control Metters, R Quantifying the bullwhip effect in supply chains. Journal of Operations Management Nise, N.S Control systems engineering. The Benjamin Cummings Publishing Company, Inc., California. Riddalls, C.E., Bennett, S The stability of supply chains. International Journal of Production Research Sterman, J Modelling managerial behaviour: Misperceptions of feedback in a dynamic decision making experiment. Management Science Towill, D.R Dynamic analysis of an inventory and order based production control system. International Journal of Production Research Warburton, R.D.H., Disney, S.M., Towill, D.R., Hodgson, J.P.E Further insights into The stability of supply chains. International Journal of Production Research Vassian H.J Application of discrete variable servo theory to inventory control. Journal of the Operations Research Society of America Cardiff Business School, Cardiff University, Aberconway Building, Colum Drive, Cardiff, CF10 3EU, UK. address: DisneySM@Cardiff.ac.uk. Tel: +44(0) Fax: +44(0)