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State space investigation of the bullwhip problem with ARMA(1,1) demand processes
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Int. J. Production Economics 104 (2006) 327–339 www.elsevier.com/locate/ijpe
State space investigation of the bullwhip problem with ARMA(1,1) demand processes Gerard Gaalmana,, Stephen M. Disneyb a
Production Systems Design Department, Faculty of Management and Organization, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands b Logistics System Dynamics Group, Cardiff Business School, Cardiff University, Aberconway Building, Colum Drive, Cardiff, CF10 3EU, UK Received 1 March 2005; accepted 1 November 2005 Available online 7 July 2006
Abstract Using state space techniques we study a ‘‘myopic’’ order-up-to policy. The policy is myopic because it is optimal at minimising local inventory holding and shortage costs. In particular we study the bullwhip effect produced by the replenishment policy reacting to a stochastic ARMA(1,1) demand processes. We reveal that bullwhip is fundamentally caused by the co-variance between the inventory level and the demand forecast. We go on to highlight the impact of a simple control engineering inspired bullwhip reduction technique, a proportional controller in the inventory feedback loop. Although it can be shown this approach is always able to remove bullwhip, we expose that it is not possible to arbitrarily ‘‘tune’’ the proportional controller, without knowing in advance the likely structure of the demand process. We conclude with an investigation of trade-offs and optimisations in the order and inventory variance (and their sum) produced by our policy. r 2006 Elsevier B.V. All rights reserved. Keywords: Order-up-to policy; Bullwhip effect; ARMA(1,1) demand process; State space representation
1. Introduction Recently, a number of papers have appeared that investigate the ‘‘bullwhip effect’’ (the variance amplification of ordering decisions in the supply chain) produced by the order-up-to (OUT) replenishment policy after the seminal papers of Chen et al. (2000) and Lee et al. (1997a). The bullwhip problem is especially important as it results in Corresponding author. Tel.: +31 50 3637196.
E-mail addresses: [email protected] (G. Gaalman), [email protected] (S.M. Disney).
unnecessary costs in supply chains (see e.g., Metters (1997) and Lee et al. (1997b)). This includes costs such as inefficient use of production, distribution and storage capacity, recruitment and training costs, increased inventory and poor customer service levels. The bullwhip problem is widespread throughout industry. Fig. 1 shows 6 months of daily data from a real supply chain. Here, ‘‘retail sales’’ represents the movement of a particular (high volume, own label) product through the supermarkets tills of a major retailer. ‘‘Production’’ represents the manufacturer’s production to satisfy this demand.
0925-5273/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2006.05.001
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Fig. 1. Bullwhip observed in the real world.
In between the retailer and the manufacturer two orders have been placed, one to replenish (the several hundred) individual supermarkets, and one to replenish the (half a dozen) distribution centres. There is a 5 to 1 increase in the variance of the (aggregate) orders placed in this real-life supply chain. The OUT policy has become a popular replenishment policy to study in bullwhip literature. This is due to its wide applicability at representing industrial ordering policies and its mathematical tractability. Indeed, a variant of the OUT policy was actually used in the supply chain of Fig. 1. Of course, the industrial strength version had a lot of ‘‘bells and whistles’’, but its fundamental structure is an OUT policy. More details on the supply chain in Fig. 1 can be found in Mason et al. (2002). Here we use it only to illustrate that real demand patterns have a complex stochastic structure and that bullwhip is present in real-supply chains. Statistical approaches are popular mechanisms to study the bullwhip problem. Lee et al. (1997a), explored different sources of the bullwhip problem in an OUT policy. They highlighted five sources: lead-times, forecasting mechanisms, gaming, batching and promotions. Chen et al. (2000) showed that the classical OUT policy with simple forecasting techniques always results in bullwhip for the class of AR demands. Control theory and transform techniques have also been applied to the bullwhip problem. The field was initiated by the Nobel Laureate, Herbert Simon (1952) who used the Laplace transform to study a continuous time-ordering policy. This was quickly replicated in discrete time by Vassian (1955), Magee
(1956) and Brown (1963). Adelson (1966) appears to be the first to explicitly study what we would now call bullwhip. Deziel and Eilon (1967) provide an insightful study of bullwhip and inventory variance of an OUT policy. SchneeweiX (1971, 1975, 1977) and Inderfurth (1977) used z transforms and Wiener filtering theory to find optimal linear production or replenishment decisions for production systems with non-linear costs. Gaalman (1976, 1978, 1979) considered the optimal linear control of a general multidimensional production-inventory system, using the so-called modern control-theoretic principles. It is often supposed that the bullwhip problem cannot be avoided. Indeed it has been shown by Dejonckheere et al. (2003) that the classical OUT policy with exponential smoothing (Brown, 1963) or moving-average forecasting, will always produce bullwhip for all possible demand patterns. However, recently, it has been shown (Gilbert, 2005; Chen and Disney, 2003) that the use of conditional expectation forecasting can allow the OUT policy to smooth demand (i.e. avoid the bullwhip problem) for some classes of stochastic-demand patterns. Modifying the classical OUT policy, by incorporating a gain in the inventory feed-back loop, can lead to ‘bullwhip’ reduction in the OUT policy. Dejonckheere et al. (2003) have shown that it is always possible to avoid the bullwhip problem with such a technique for all classes of demand processes. At first sight, this bullwhip avoidance technique will come at the cost of holding extra inventory to achieve target customer service levels. This is true for i.i.d. stochastic demands (Magee, 1956). Dejonckheere et al. (2002) investigate this scenario explicitly.
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A characteristic of recent papers is the use of slightly more realistic stochastic-demand models such as the autoregressive moving-average demand model (Box and Jenkins, 1970). Commonly, a classical control engineering approach is taken to analyse the system’s dynamic behaviour. For example, Disney et al. (2003) study the ARMA model and show there are win–win scenarios resulting from a proportional controller, 1/Ti (herein called f), placed in the inventory position feedback loop. They show it is actually possible to avoid bullwhip, reduce inventory requirements and achieve a desired fill-rate simultaneously. An expected cost approach may also be applied to the OUT policy. For example, Kim and Ryan (2003) assume the demand process is Gaussian and study the probability density function that describes the inventory levels over time; assigning costs to expected inventory holding and backlog positions. Disney and Grubbstro¨m (2004) applied the same technique to the probability density function of the orders and assigned costs to the fraction of products produced in normal working hours, and those produced in a capacitated position with subcontracting or over-time working. There the orders were generated with the OUT policy, forecasts generated with exponential smoothing and the demand was assumed to be autoregressive. Chen and Disney (2003) studied expected inventory and order related costs in the myopic OUT policy with conditional expectation forecasting and ARMA(1,1) demand. They show that the proportional controller (f ¼ 1/Ti) always results in a more economic performance than the classical OUT policy (where f ¼ 1/Ti ¼ 1). Herein we introduce a state space approach to represent the ARMA(1,1) demand model and the inventory balance equation. We derived directly, by means of matrix calculations, the variance of the states of the system (the inventory and the forecast). This results in a matrix difference equation that under stability conditions converges to stationary values. In general this stationary variance matrix equation can be solved numerically. However, in this particular case, a closed-form analytical expression for the variances of the state variables can be derived. From this the variance amplification of the ordering rule can be easily found. Analysing the ordering rule variance expression in this way highlights the essential role of the co-variance between inventory and the state of the demand forecast model on the bullwhip effect. In
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addition, given the expression of the ordering variance a polynomial equation can be derived that surrenders the value of the feedback parameter for which the variance amplification is unity. We conclude by investigating the trade-off between inventory and order variance for the ARMA(1,1) demand pattern. The structure of our paper is as follows. In Section 2 the myopic OUT policy is introduced and discussed. In Section 3, we add the proportional controller into the classical OUT policy. The role of the co-variance between the inventory and the demand forecast is indicated. In addition the need for a trade-off between inventory and ordering variance is highlighted. Section 4 presents a state space demand model and shows how ARMA(1,1) demand can be formulated. From this the conditional expectation forecast model is derived. Section 5 develops the ‘total’ system state space description of the inventory and the forecasts. Moreover the variances of the system are investigated using the matrix variance equation. Section 6 analyses the order variance for ARMA(1,1) demand In Section 7 the trade-off characteristics of inventory and ordering variance are considered. We conclude in Section 8. 2. The myopic order-up-to policy The order-up-to policy is often used to manage inventory as it minimises the long-run expected inventory related costs when they consist of piece wise linear holding and stock-out costs. In this policy the inventory is reviewed and the ordering decision is made at the beginning of the period. During the period the customer orders are received and fulfilled before or at the end of the period. The inventory balance equation is given by itþ1 ¼ it þ ot ztþ1 ,
(1)
where it is the inventory at time t, ot is the order placed to replenish the inventory and zt+1 is the demand during the period (t, t+1). The (classical) order-up-to policy is completed by the following replenishment decision ot ¼ ðit in Þ þ z^tþ1;t ,
(2)
where
in is the inventory norm value (a time invariant constant used to achieve a given customer service metric, such as availability, fill-rate or economic criteria, often solved via the ‘‘newsboy’’
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technique). We may set in to zero without loss of generality here. z^tþ1;t is the forecast of demand in the next period made at time t.
Note that the lead-time consists of a review period only and the demand forecast mechanism is based on the minimum mean squared error criterion. Thus, we are considering a ‘‘myopic’’ OUT policy (Chen and Disney, 2003). Johnson and Thomson (1975) proved that this policy is optimal at minimising inventory holding and shortage costs when demand is both stationary and non-stationary in the case where backlogs are not allowed (that is lost sales are present). In contrast, here we assume backlogs are allowed in order to preserve the linearity of our model. However this is not the only condition, the order decisions can also not be restricted. Usually it is assumed that the demand is a normally distributed variable with a constant mean and random uncorrelated error component with constant variance (demand is i.i.d.). That is, ztþ1 ¼ z¯ þ tþ1 .
(3)
The ‘‘myopic’’ OUT policy (Eqs. (1) and (2)) minimizes the inventory fluctuations around the average value in. Little attention has traditionally been given to the fluctuations in the ordering decision until the recent interest in the bullwhip problem. These order fluctuations might become large for some demand characteristics and thus it is important to constrain them so as not to induce unnecessary production on-costs. In this paper we want to analyse the effects on the inventory and the ordering decision relaxing the restrictive assumption in (3). That is, we wish to gain insight into the behaviour of the OUT policy for an autoregressive moving-average (ARMA(1,1)) demand process (Box and Jenkins, 1970). This is important as real-life demands are not always i.i.d. as implied by (3). Eliminating ot in the balance Eq. (1) using the replenishment Eq. (2) gives itþ1 ¼ in Ztþ1 ,
(4)
where ztþ1 z^tþ1;t ¼ Ztþ1 is the forecast error. If the demand forecast minimizes the variance of the forecast error then the inventory variance will be minimal. This can be easily seen by squaring Eq. (4) and taking the expected values: Sii;tþ1 ¼ SZZ ,
(5)
where Sii,t+1 and SZZ are the variances of the inventory at time t+1 and the forecast error, respectively. Note that the inventory variance is time independent, so Sii,t+1 ¼ Sii. From a control engineering viewpoint the explanation is simple; the ordering rule feeds back the inventory deviation from the inventory norm value (the target safety stock holding) completely and at the same time anticipates future demand as best as can possibly be achieved. From (4) it can be shown that ot ¼ ðZt Þ þ z^tþ1;t ,
(6)
and the variance of the ordering decision is Soo;t ¼ SZZ 2SZ^z;t þ Sz^z^;t ,
(7)
where Soo,t and Sz^z^;t are the variances of ot and z^tþ1;t respectively. SZ^z;t is the co-variance between the demand forecast z^tþ1;t and the forecast error Zt. For i.i.d. random-demand processes such as those described by Eq. (3), the co-variance will be zero and the ordering variance equals the demand variance (that is, we have a critical bullwhip effect). For other demand processes the co-variance will not be zero. We can see that if the co-variance is negative the variance of the ordering decision may even become very large. However without a further specification of the demand characteristics the precise bullwhip effect cannot be further analysed. 3. Adding a proportional controller to the myopic OUT policy The introduction of a proportional controller into the inventory feedback loop might improve the dynamic behaviour of the replenishment policy. Indeed, this would be an obvious suggestion from basic control theory and actually has a long history in the field of production and inventory control (see Simon (1952), Magee (1956), Deziel and Eilon (1967), Towill (1982), Matsuyama (1997), Dejonckheere et al (2003) and Chen and Disney (2003)). So let us consider the following: Incorporating the proportional controller f into the inventory feedback loop in the replenishment decision, we have ot ¼ f ðit in Þ þ z^tþ1;t .
(8)
Using Eq. (8) in the inventory balance Eq. (1) yields itþ1 ¼ in þ ð1 f Þðit in Þ ðZtþ1 Þ.
(9)
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From (9) it can be seen that the system will be stable within the interval 0ofo2. From Eq. (9) the inventory variance may be derived as Sii;tþ1 ¼ ð1 f Þ2 Sii;t þ SZZ
(10)
Here we assume that co-variance between the inventory and the forecast error is zero. This is reasonable, because from Eq. (9) we can see that it is a linear function of the sequence of forecast errors fZt ; Zt1 ; Zt2 ; . . .g. In ‘‘good’’ functioning demand forecasts, the forecast errors are uncorrelated. If autocorrelation exists in the forecast errors then there is some additional ‘‘information’’ that has not been taken into account. Within the stability interval, Eq. (10) will converge to a stationary linear expression from which the inventory variance can be calculated as 1 1 Sii ¼ SZZ SZZ ¼ f ð2 f Þ 1 ð1 ð1 f Þ2 Þ (11)
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We observe three terms, one related to the inventory variance, the second to the co-variance of the inventory and the demand forecast and finally the variance of the forecast error. Using Eq. (11) the influence f on the first term can be calculated. The third term is independent of f. When the demand is i.i.d. (see Eq. (3)) the co-variance Siz^ is zero. In this case Soo, the order variance is an increasing function of f in the interval 0ofo2. As Soo goes to infinity for f-2, for some f the variance will be larger than the variance of the demand (thus bullwhip exists). The critical point lies at f ¼ 1. The proportional OUT policy has been proposed as a method to control the inventory variance as well as the order variance. Thus a suitable balance between the variances has to be found. Obtaining a suitable trade-off between the variances can be formalized by the minimization of the objective function OF ¼ fSii þ aSoo g,
(14)
Fig. 2 shows the inventory variance as function of f. As expected the variance is minimal for f ¼ 1. For 0ofp1 the variance is a decreasing function of f and for 1pfo2 the variance is an increasing function of f. The variance of the ordering rule (8) becomes
where a is a positive weighting factor by which the relative importance of the variances can be chosen. The motivation behind this objective function is that we are assuming that costs are linearly related to the inventory and order variance. The necessary condition for minimum is given by
Soo;t ¼ f 2 Sii;t 2f Siz^;t þ Sz^z^;t ,
(12)
dSii dSoo þa ¼0 df df
(13)
For example when the demand is i.i.d. and a ¼ 1 the minimum of the inventory and order variance
which converges, in the stationary situation, to Soo ¼ f 2 Sii 2f Siz^ þ Sz^z^ .
Fig. 2. Inventory variance as a function of the inventory feedback gain, f.
(15)
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pffiffiffi sum is obtained at f ¼ ð 5 1Þ=2 ¼ 0:61803, the golden ratio (see Fig. 3). This case has been studied for all lead-times by Disney et al. (2005) with regard to the customer service metric, the fill-rate. If a41 then f will become smaller than 0.61803. If 0oao1 then 0.61803ofo1. For non-i.i.d. demand the co-variance term in Eq. (13) is non-zero. The effect of f on the ordering variance can only be shown if we know Siz^ as a function of f. Unfortunately the co-variance between the inventory and the demand forecast can only be studied further if a demand model is specified. Here we will consider the ARMA(1,1) demand process. In the next section we will introduce a general demand model that forms the basis of subsequent analysis. The policy is formulated as a state space model in order to derive a general expression for the co-variance. 4. A general linear state space demand and forecast model Assume the demand can be described by the sum of a constant mean term and a normally distributed random variable: ztþ1 ¼ z¯ þ z~tþ1 .
(16)
Since, as shown in Section 2, we are only interested in the fluctuation around a constant mean we may ignore the constant term and will model only the stochastic process. Let this term be
described by the linear vector-difference equation: ztþ1 ¼ Mytþ1 þ tþ1 ,
(17)
ytþ1 ¼ Dyt þ Ext ,
(18)
where yt is an m-dimensional state vector, M is a 1 m matrix, D is an m m matrix and E is an m k matrix. The variable et and the k-dimensional vector x form sequences of zero-mean uncorrelated normal-distributed random variables with positive definite variance matrices (See and Sxx) and co-variance matrix Sex. Thus the random variables are uncorrelated between periods. In contrast with the model introduced in Gaalman (1976) the matrix E is introduced to explicitly demonstrate that there need not necessarily be m random disturbances. This model is rather general in the sense that a number of well-known demand models can be described in these terms, for instance, the General Polynomial Model of Harrison (1967) and the polynomial, exponential and seasonal models of Brown–Meyer (Brown 1963). Furthermore, Box and Jenkins (1970) autoregressive integrated movingaverage processes (ARIMA(p,I,q)) can be studied with the state-space approach. State-space representations are also seen as a powerful tool in time series econometrics (e.g., see Harvey (1990)). In this paper we will assume the demand satisfies an ARMA(1,1) process as ARMA(1,1) demand is frequently used to describe short-term demand processes. A state space
Fig. 3. Inventory and order variance when demand is i.i.d.
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formulation of this process is ztþ1 ¼ ytþ1 þ tþ1 ; ytþ1 ¼ fyt þ ðf yÞt ;
(19)
where yt is the state variable, f the autoregressive parameter and y the moving-average parameter. So M ¼ 1, D ¼ f, E ¼ 1. The process stability condition requires fjo1. This is the same as Box and Jenkins’s invertibility condition. For f ¼ 1 we have an ARIMA(0,1,1) process. This process is not stationary but it is allowed in the forecast approach. Having explained the basic characteristics of the state space demand model we will now introduce the forecast model. The optimal forecast of zt+1 at time t (which is here equal to the minimum variance estimation) can be obtained by the Kalman filter (Jazwinski, 1970) as z^tþ1;t ¼ M y^ tþ1;t ,
(20)
y^ tþ1;t ¼ Dy^ t;t1 þ G t ðzt z^t;t1 Þ,
(21)
where z^tþ1;t ; y^ tþ1;t are conditional expectations at time t given all the observed realisations of demand; Gt is the so called Kalman gain that can be calculated from a matrix Riccati equation. It can be shown that the forecast errors zi z^i;i1 ¼ Zi ;
i ¼ ð1; 2; . . . ; tÞ
(22)
will form a sequence of zero-mean uncorrelated normally distributed random variables when the forecast is optimal. Intuitively the Kalman filter can be explained as follows. Assume we have derived at time t1 the best forecast of yt: yt;t1 . Without any additional information the best forecast of yt+1 is y^ tþ1;t1 ¼ Dy^ t;t1 . However at time t demand information zt ‘‘arrives’’, but only the non-forecasted part of it ðzt z^t;t1 Þ provides new information. Since the demand is also disturbed by random noise the difference is ‘weighted’ by Gt. Here we will consider only the steady-state situation. So we assume that at time t an ‘infinite’ number of demand observations from the past are available. If the forecast system is stable and detectable then the Kalman gain converges to the stationary value G. The forecast system of the ARMA(1,1) demand model satisfies z^tþ1;t ¼ y^ tþ1;t y^ tþ1;t ¼ fy^ t;t1 þ Gðzt y^ t;t1 Þ, G ¼ ðf yÞ; ðzt y^ t;t1 Þ ¼ t .
ð23Þ
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Eq. (23) shows how the new demand information is contained in the forecast. The forecast error is weighted by the demand parameters and used to update the current demand forecast. For the ARMA(0,1,1) model, f ¼ 1, and we get the single exponential smoothing model of Brown (1963). Brown argues possible useful values of the smoothing parameter. For instance use a smoothing parameter 0.1 for slowly changing means values and 0.3 for very fast variations. Here the (optimal) value of G is linked to the two demand process parameters, f and y. To summarize this section we have shown how a demand model may be formulated as a state space model and we have introduced a minimum-variance forecast system. Also we have shown how the ARMA(1,1) demand process can be formulated as a state space model. We will not go into the calculation of the Kalman gain G by means of the matrix Riccati equation. We simply assume G is known. In the next section we will integrate this forecast system into the inventory (state) model. 5. The state space system and variance equations In order to derive expressions for the variances and co-variances of the relevant variables in our system we need to integrate the forecast system of the ARMA(1,1) process and the inventory system. To do this we will define Xt as a two-dimensional state variable of the joined system, consisting of the inventory state variable it and y^ tþ1;t the state variable of the forecast system at time t (note that since we calculate this forecast at time t it is part of Xt and not y^ t;t1 ). The total system is now be described by the difference equation X tþ1 ¼ AX t þ But þ CZtþ1 ,
(24)
with A¼
1
0
0
D
;
1 B¼ ; 0
C¼
1 G
,
(25)
where D ¼ f, G ¼ (fy) and ut ¼ ot z^tþ1;t . Since we are only interested in the characteristics of the variances we may ignore the mean values. It is not possible to influence the demand part of this system by ut. Thus the system is not completely controllable. Furthermore, if the eigenvalue of D, then the system is not stationary. So from here on we assume that fo1. This implies that the exponential smoothing model is not allowed. From a practical point of view this is not a serious
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problem since, in practice, an infinite set of demand data is neither available nor relevant for forecasting. Thus, the exact determination of demand parameters is not possible. So, one might choose f near unity. For example, instead of f ¼ 1 we might choose f ¼ 0:95. By this, the ARIMA(0,1,1) model becomes an ARMA(1,1) model. By introducing the inventory feedback rule ut ¼ fit , we are studying the OUT policy given by Eq. (8) and the system becomes X tþ1 ¼ ðA BF ÞX t þ CZtþ1 ,
(26)
where F ¼ ðf ; 0Þ;
A BF ¼
1f
0
0
f
! and Zt ¼ t .
As mentioned in Section 4 and Eq. (23), Zi (where i ¼ ðt; t 1; t 2; . . .Þ) forms a sequence of zeromean uncorrelated normally distributed random variables. This property makes it easy to derive a matrix variance differential equation from (26). Let the variance matrix-of the state vector at time t be ( ) ^ tþ1 t Þ0 i i i ð y t t t SXX ;t ¼ E X t X 0t ¼ E . ðy^ tþ1 t Þit ðy^ tþ1 t Þðy^ tþ1 t Þ0 (27) Then by multiplying Eq. (26) by the transpose of X tþ1 and taking expectations we get the variance equation: SXX ; tþ1 ¼ ðA BF ÞSXX ; t ðA BF Þ0 þ CSZZ C 0 . (28) EfZ2tþ1 g
Note that SZZ ¼ is the variance of the forecast error. Under the condition 0of o2 and fo1; the matrix variance equation converges to the stationary situation, expressed by SXX ðA BF ÞSXX ðA BF Þ0 þ CSZZ C 0 .
(29)
This equation can, in general, be solved by a number of numerical techniques. In this particular case the structure of A and B allows further analytical investigation. Eq. (27) shows that three components are relevant to us here (Sii ¼ Efi2t g; Syi^ ¼ Efðy^ tþ1; t Þit g; Sy^y^ ¼ Efðy^ tþ1; t Þðy^ tþ1; t Þ0 g). They are: Sii ¼ ð1 f Þ2 Sii þ SZZ ,
(30)
Syi^ ¼ ð1 f ÞDSyi^ GSZZ ,
(31)
Sy^y^ ¼ DSy^y^ D0 þ GG 0 SZZ .
(32)
Eq. (30) corresponds with (10), the inventory variance whose closed-form solution was shown in (11). As expected the inventory variance is not influenced by the demand but only by the (minimum) variance of the forecast error. The variance of the forecast, Sy^y^ is not influenced by the feedback factor f, see (32). The co-variance between the inventory and the forecasted demand state is dependent on the feedback factor (f), the demand model (D) and the Kalman gain (G). Eq. (31) can be written as Syi^ ¼ ð1 ð1 f ÞfÞ1 ðf yÞSZZ (33) Since f o1; ð1 ð1 f ÞfÞ cannot be zero in the stability interval 0of o2. In the next section we will analyse the effects of the ARMA(1,1) demand model on the variance of the ordering policy. Here, Eq. (33) will play an important role. 6. Analysis of the order variance for ARMA(1,1) demand The variance of the ordering policy was given in Eq. (13). Since M ¼ 1 we have Siz^ ¼ Sz^i ¼ Syi^ and Sz^z^ ¼ Sy^y^ . Thus the ordering variance becomes Soo ¼ f 2 Sii 2f Syi^ þ Sy^y^ .
(34)
The first and third components are positive, but the sign of the second component depends on the sign of the co-variance. Thus we will first explore the characteristic of the co-variance. From (33) we can observe that the co-variance has an asymptote for f ¼ ð1 fÞ=f. So for 1ofo0 the asymptote is42 and for 0ofo1 the asymptote is o0. Thus the co-variance is bounded on the stability interval, but might be large if ð1 ð1 f ÞfÞ is near zero. That is, when f is near 71 (regardless of the value of f). Depending on the demand model parameters the co-variance can be positive or negative, increasing or decreasing, function of f in the stability interval. For ðf yÞ40 the co-variance is negative, for ðf yÞo0, positive. The co-variance is zero when f ¼ y, which corresponds with the i.i.d. demand case. With this insight into the characteristics of co-variance we may now consider its influence on the ordering variance. Substituting (30) and (33) into (34) gives Soo ¼
f ðf yÞ þ 2f SZZ þ Sy^y^ . ð2 f Þ 1 fð1 f Þ (35)
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The ordering variance has an asymptote to þ1 for f ! 2. At f ¼ 0, Soo ¼ Sy^y^ . For f ¼1 the ordering variance is Soo ¼ 1 þ 2ðf yÞ SZZ þ Sy^y^ ¼ Szz þ 2ðf yÞSZZ . From this we can observe that for ðf yÞ40 the bullwhip effect exists. Since the third term in Eq. (34) does not depend on f we only need to consider the first two terms. We introduce H(f) as the part of Soo dependent on f:
variance is increasing (region 4 of Fig. 4) and there is a bullwhip effect in 1of o2. To find the location of the minimum we will consider the derivative of H(f):
f ðf yÞ þ 2f Hðf Þ ¼ ð2 f Þ 1 fð1 f Þ f ð1 þ 3f 4y þ f ð2y fÞÞ . ¼ ð2 f Þð1 ð1 f ÞfÞ
The denominator is always positive and the numerator consists of the sum of two quadratic terms and is positive for ðf yÞ40, confirming the earlier results. If ðf yÞo0 then the numerator can be reduced to
ð36Þ
If ðf yÞ40, the co-variance is negative and so H(f) is an increasing function in the stability interval (region 5 of Fig. 4). In addition the bullwhip effect occurs in the interval 0of o1. If ðf yÞo0, the covariance is positive and H(f) might have a (absolute) minimum. The denominator of H(f) is positive in the stability interval. H(f) has two zeros, one at f ¼ 0 and the other one for f ¼ ð1 þ 3f 4yÞ= ð2y fÞ. For ð1 þ 3f 4yÞo0 the second zero lies in the stability interval and we have a minimum. So, if ðf yÞo0 and ð1 þ 3f 4yÞ40, the ordering
dH 2½ð1 ð1 f ÞfÞ2 ðy fÞð1 fÞð2 f Þ2 ¼ . df ð2 f Þ2 ð1 ð1 f ÞfÞ2 (37)
½1 ð1 f ÞfÞ bð2 f Þ½1 ð1 f Þf þ bð2 f Þ, (38) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where b ¼ ðy fÞð1 fÞ. The necessary condition for an optimum gives two solutions: f1 ¼
2b þ f 1 2b þ f 1 ; f2 ¼ , bþf b þ f
(39)
where f1 gives the minimum and f2 the maximum value. The maximum value always lies outside the stability region. However the minimum does not
Fig. 4. The behaviour of the order variance in the ARMA(1,1) plane.
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necessarily lie within the stability interval. The necessary and sufficient conditions for the minimum being in the stability interval is that ðf yÞo0 and ð1 þ 3f 4yÞo0. In this case f ¼ f 1 . For b41 the minimum lies in the interval 1of o2 (region 2 in Fig. 4) and for bo1 the minimum lies in the interval 1of o1 (region 3 of Fig. 4). Fig. 4 maps these insights into the ARMA plane. Here the y ¼ f line corresponds to the i.i.d. demand patterns studied in Section 3. Typical plots in regions 2–5 are shown in Fig. 5 for verification. The critical bullwhip effect condition occurs when Soo ¼ Szz . This is equivalent to Soo Szz ¼ ðHðf Þ 1ÞSZZ ¼ 0.
(40)
The numerator of (40) is important here. It is a second degree polynomial. The critical bullwhip
condition occurs when yf 2 þ ð1 2yÞf þ ðf 1Þ ¼ 0.
(41)
This gives two bullwhip conditions. However only one lies in the stability region 0of o2. It is f
þ
¼
2y 1 þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 4yðy fÞ . 2y
(42)
This result corresponds to the bullwhip boundary given in Chen and Disney (2003), thus supporting our findings. In Fig. 5 the bullwhip conditions are plotted for the four typical variance situations, showing where the maximum f lies to avoid bullwhip. In this section we have shown that the ordering variance can have four different patterns in the stability interval (Fig. 4). With a suitable choice of the feedback parameter the bullwhip effect can be
Fig. 5. Variances for different ARMA(1,1) demands.
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avoided. The critical bullwhip point can lie in 0of o1 or in 1of o2. In the next section we will analyse the trade-off between inventory variance and ordering variance. 7. The trade-off of inventory and ordering variance Finally we will analyse the trade-off conditions for this demand model. As we have analytical expressions for the order and inventory variance, the derivatives can be calculated and used in the following objective function. dSii dSoo þa df df d 1 f 2f ðf yÞ þa þ ¼ . df f ð2 f Þ ð2 f Þ 1 fð1 f Þ ð43Þ The numerator yields the necessary condition for an optimum (minimum) and can be written as
af 2 þ f 1 ð1 ð1 f ÞfÞ2 aðy fÞð1 fÞð f ð2 f ÞÞ2 ¼ 0.
(44) We see that this condition is a polynomial of the 4th degree, consisting of the sum of a 4th degree term and a 2nd degree term. In the i.i.d. demand case (that is, when y ¼ f) the necessary condition is given by the first term and results in ðaf 2 þ f 1Þ ¼ aðf f o1 Þðf f o2 Þ ¼ 0, pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 þ 4a , ð45Þ f o1;2 ¼ 2a where 0of o1 o1 and f o2 o0. For i.i.d. demand the optimal value fT for which the trade-off between the variances is found lies, for all a, in the interval 0of o1. For ARMA(1,1) demand we are able to distinguish a number of possibilities. Since a40, for f in the interval f o2 of of o1 , the first term of (44) is negative. The second term of (44) is positive for f4y and negative for foy. Combining these insights we get, for a given a:
if f4y then the trade-off will be found in the interval 1=ð1 þ 2aÞof of o1 . From Section 6 we know that the ordering variance is an increasing function over the stability interval and larger than the ordering variance of the i.i.d. demand case. This can be dampened by increasing a, resulting in a lower fT. For a given a the minimum value of fT occurs when y ¼ 1 and f ¼ 2að1 þ 2aÞ=ð1 þ 8aÞ.
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if foy the trade-off lies in the interval f o1 of o2. This situation might be combined with the insights obtained from analysing Hðf Þ. If 1 þ 3f 4y40 the order variance is still increasing and the trade-off will be found in the region f o1 of o1. If 1 þ 3f 4yo0 and bo1 the tradeoff will be in the region maxðf o1 ; f Þof o1. Otherwise if b41 then 1of of . However f in the interval of 1of o2 is usually not advised because it over compensates for inventory deviations. When b ¼ 1, f T ¼ f ¼ 1. Here the derivative of both the inventory variance and the order variance is zero.
In Fig. 6 for the case of a ¼ 1, the fT has been plotted, showing the minimum of the sum of order and inventory variances. We have also plotted in Fig. 6 the contour plot in the ARMA plane to illustrate the impact of the demand parameters on the optimum value of f, fT. From Eq. (44) the closed form expression of fT can be obtained. Unfortunately, this expression is rather large and unmanageable, hence this graphical approach. Fig. 6 also highlights 15 real-life demand patterns that have been identified by Disney et al (2003). We can see that one demand pattern is i.i.d. and thus our discussion in Section 3 is relevant here. Two demand patterns lie in region 3 of Fig. 4 and 12 demand patterns lie in region 5 of Fig. 4, so our investigations in Section 7 are relevant here. Happily the paradoxical trade-off in region 2 of Fig. 4 does not seem to be relevant in the practical situation of these 15 real-life demand patterns. Of course, an important question has now arisen; do real-life demand patterns fall in regions 2 and 3 of Fig. 4? We intuitively suspect that only demand patterns in regions 4 and 5 are common in many real-life supply chains. 8. Conclusions The classical order-up-to policy minimizes inventory fluctuations (variance) but does not attempt to control ordering fluctuations. This may lead to the so-called bullwhip problem where the ordering variance is amplified as it moves through a production/distribution/replenishment decision. The use of a proportional controller has often been proposed as a solution to the bullwhip problems caused by the classical order-up-to policy. By a suitable choice of the feedback parameter value the
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Fig. 6. The order and inventory variance trade-off in the ARMA(1,1) plane for a ¼ 1.
proportional controller strives for a good balance between the ordering and inventory variances. However, relatively less is known about the behaviour of the proportional policy for arbitrary demand characteristics. In this paper the performance of the proportional policy has been analysed for the ARMA(1,1) demand model. The crucial role of the co-variance between the inventory and the demand forecast on the bullwhip effect has been highlighted. Analytical expressions illustrating the influence of the demand model parameters are derived. From this expression bullwhip generating and order smoothing components can be uncovered. In general this leads to a paradoxical situation. If the co-variance is negative, the ordering variance increases (compared with the inventory-related component or the random-demand case) and bullwhip arises in the control interval 0ofo1, but at the same time a good balance between inventory and ordering variance can be found in this interval. If the co-variance is positive the ordering variance
decreases with respect to the i.i.d. random-demand case and could even have a minimum in the interval 1ofo2. In the latter case, though the ordering variance is relatively low, no balance can be found between inventory and ordering variance in the control interval 0ofo1. However, a balance can be found in the region of 1ofo2, but this is not an attractive situation since this leads to an overreaction to inventory deviations. Our overall conclusion is that the functioning of the proportional controller depends heavily on the demand model and its parameter values and as such requires considerable care when used in a real-life situation. Moreover, a reasonably functioning robust controller suitable for a broad class of demand models with their parameter values is not immediately obvious. However, for the real-life demand patterns pictured in Fig. 6, and for equal weight of the inventory and ordering variance (a ¼ 1) the golden ratio controller (fT ¼ 0.61803) seems to be reasonably close to the optimal
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solution. Indeed two-thirds of the demand patterns have an optimal feedback controller within 20% of the golden ratio. Future research will analyse the effects of ordering delays when using this general demand model. Further work will be devoted to finding better-performing ordering rules. References Adelson, R.M., 1966. The dynamic behaviour of linear forecasting and scheduling rules. Operational Research Quarterly 17 (4), 447–462. Box, G.E., Jenkins, G.M., 1970. Time Series Analysis Forecasting and Control. Holden-Day, San Francisco. Brown, R.G., 1963. Smoothing, Forecasting and Prediction of Discrete Time Series. Prentice-Hall, Englewood Cliffs, NJ. Chen, Y.F., Disney, S.M., 2003. The order-up-to policy ‘‘sweet spot’’—using proportional controllers to eliminate the bullwhip problem. In: Proceedings of EUROMA POMS Conference, 16–18 June 2003, Como Lake, Italy,vol. 2, pp. 551–560. Chen, Y.F., Drezner, Z., Ryan, J.K., Simchi-Levi, D., 2000. Quantifying the bullwhip effect in a simple supply chain: the impact of forecasting, lead-times and information. Management Science 46, 436–443. Dejonckheere, J., Disney, S.M., Farasyn, I., Janssen, F., Lambrecht, M., Towill, D.R., Van de Velde, W., 2002. Production and inventory control: the variability trade-off. In: Proceedings of the Ninth EUROMA Conference, 2–4 June, Copenhagen, Denmark. Dejonckheere, J., Disney, S.M., Lambrecht, M.R., Towill, D.R., 2003. Measuring and avoiding the bullwhip effect: a control theoretic approach. European Journal of Operational Research 147 (3), 567–590. Deziel, D.P., Eilon, S., 1967. A linear production-inventory control rule. The Production Engineer 43, 93–104. Disney, S.M., Grubbstro¨m, R.W., 2004. The economic consequences of a production and inventory control policy. International Journal of Production Research 42 (17), 3419–3431. Disney, S.M., Farasyn, I., Lambrecht, M., Towill, D.R., Van de Velde, W., 2003. Creating win–win scenarios from the bullwhip problem—by design not accident. In: Proceedings of EUROMA POMS Conference, 16–18 June 2003, Como Lake, Italy, vol. 2, pp. 561–570. Disney, S.M., Farasyn, I., Lambrecht, M., Towill, D.R., Van de Velde, W., 2005. Smooth is smart: bullwhip, inventory, fillrates and the golden ratio. Tijdschrift voor Economie en Management L, 167–186. Gaalman, G.J., 1976. Optimal linear control of linear production-inventory systems. International Journal of Systems Science 7, 1005–1015. Gaalman, G.J., 1978. On the optimal time-invariant control of linear Gaussian non-quadratic problems. International Journal of Control 28, 635–648.
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Gaalman, G.J., 1979. On the optimal time-variant control of linear production inventory systems. Engineering Costs and Production Economics 4, 147–163. Gilbert, K., 2005. An ARIMA supply chain model. Management Science 51, 305–310. Harrison, P.J., 1967. Exponetial smoothing and short term salesforecasting. Management Science 13, 831–842. Harvey, A.C., 1990. The Econometric Analysis of Time Series. Philip Allan, New York. Inderfurth, K., 1977. Zur Gute linearer Entscheidungsregeln in Produktions-Lagerhaltungsmodellen. Westdeutschen, Opladen. Jazwinski, A., 1970. Stochastic Processes and Linear Filtering Theory. Academic Press, New York. Johnson, G., Thomson, H., 1975. Optimality of myopic inventory policies for certain dependent demand processes. Management Science 21, 1303–1307. Kim, H.-K., Ryan, J.K., 2003. The cost impact of using simple forecasting techniques in a supply chain. Navel Research Logistics 50, 388–411. Lee, H.L., Padmanabhan, P., Whang, S., 1997a. Information distortion in a supply chain: the bullwhip effect. Management Science 43, 543–558. Lee, H.L., Padmanabhan, P., Whang, S., 1997b. The bullwhip effect in supply chains. Sloan Management Review Spring, 93–102. Magee, J.F., 1956. Guides to Inventory Control Part II. Harvard Business Review Mar–April, 106-116. Matsuyama, K., 1997. Maintaining optimal inventory level by feedbacks. International Journal of Production Economics 53, 57–69. Metters, R., 1997. Quantifying the bullwhip effect in supply chains. Journal of Operations Management 15, 89–100. Mason, R., Potter, A., Lalwani, C., 2002. Trends in ICT and their impact on grocery supply chains. In: Proceedings of the Seventh International Symposium of Logistics, 14–17 July, Melbourne, Australia, 255-262. SchneeweiX, C., 1971. Smoothing production by inventory—an application of the Wiener filtering theory. Management Science 17, 472–483. SchneeweiX, C., 1975. Dynamic certainty equivalents in production smoothing theory. International Journal of Systems Science 6 (4), 353–365. SchneeweiX, C., 1977. Inventory-production Theory. Springer, Berlin. Simon, H.A., 1952. On the application of servomechanism theory to the study of production control. Econometrica 20, 247–268. Towill, D.R., 1982. Dynamic analysis of an inventory and order based production control system. International Journal of Production Research 20, 369–383. Vassian, H.J., 1955. Application of discrete variable servo theory to inventory control. Journal of the Operations Research Society of America 3 (3), 272–282.
Int. J. Production Economics 104 (2006) 327–339 www.elsevier.com/locate/ijpe
State space investigation of the bullwhip problem with ARMA(1,1) demand processes Gerard Gaalmana,, Stephen M. Disneyb a
Production Systems Design Department, Faculty of Management and Organization, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands b Logistics System Dynamics Group, Cardiff Business School, Cardiff University, Aberconway Building, Colum Drive, Cardiff, CF10 3EU, UK Received 1 March 2005; accepted 1 November 2005 Available online 7 July 2006
Abstract Using state space techniques we study a ‘‘myopic’’ order-up-to policy. The policy is myopic because it is optimal at minimising local inventory holding and shortage costs. In particular we study the bullwhip effect produced by the replenishment policy reacting to a stochastic ARMA(1,1) demand processes. We reveal that bullwhip is fundamentally caused by the co-variance between the inventory level and the demand forecast. We go on to highlight the impact of a simple control engineering inspired bullwhip reduction technique, a proportional controller in the inventory feedback loop. Although it can be shown this approach is always able to remove bullwhip, we expose that it is not possible to arbitrarily ‘‘tune’’ the proportional controller, without knowing in advance the likely structure of the demand process. We conclude with an investigation of trade-offs and optimisations in the order and inventory variance (and their sum) produced by our policy. r 2006 Elsevier B.V. All rights reserved. Keywords: Order-up-to policy; Bullwhip effect; ARMA(1,1) demand process; State space representation
1. Introduction Recently, a number of papers have appeared that investigate the ‘‘bullwhip effect’’ (the variance amplification of ordering decisions in the supply chain) produced by the order-up-to (OUT) replenishment policy after the seminal papers of Chen et al. (2000) and Lee et al. (1997a). The bullwhip problem is especially important as it results in Corresponding author. Tel.: +31 50 3637196.
E-mail addresses: [email protected] (G. Gaalman), [email protected] (S.M. Disney).
unnecessary costs in supply chains (see e.g., Metters (1997) and Lee et al. (1997b)). This includes costs such as inefficient use of production, distribution and storage capacity, recruitment and training costs, increased inventory and poor customer service levels. The bullwhip problem is widespread throughout industry. Fig. 1 shows 6 months of daily data from a real supply chain. Here, ‘‘retail sales’’ represents the movement of a particular (high volume, own label) product through the supermarkets tills of a major retailer. ‘‘Production’’ represents the manufacturer’s production to satisfy this demand.
0925-5273/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2006.05.001
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Fig. 1. Bullwhip observed in the real world.
In between the retailer and the manufacturer two orders have been placed, one to replenish (the several hundred) individual supermarkets, and one to replenish the (half a dozen) distribution centres. There is a 5 to 1 increase in the variance of the (aggregate) orders placed in this real-life supply chain. The OUT policy has become a popular replenishment policy to study in bullwhip literature. This is due to its wide applicability at representing industrial ordering policies and its mathematical tractability. Indeed, a variant of the OUT policy was actually used in the supply chain of Fig. 1. Of course, the industrial strength version had a lot of ‘‘bells and whistles’’, but its fundamental structure is an OUT policy. More details on the supply chain in Fig. 1 can be found in Mason et al. (2002). Here we use it only to illustrate that real demand patterns have a complex stochastic structure and that bullwhip is present in real-supply chains. Statistical approaches are popular mechanisms to study the bullwhip problem. Lee et al. (1997a), explored different sources of the bullwhip problem in an OUT policy. They highlighted five sources: lead-times, forecasting mechanisms, gaming, batching and promotions. Chen et al. (2000) showed that the classical OUT policy with simple forecasting techniques always results in bullwhip for the class of AR demands. Control theory and transform techniques have also been applied to the bullwhip problem. The field was initiated by the Nobel Laureate, Herbert Simon (1952) who used the Laplace transform to study a continuous time-ordering policy. This was quickly replicated in discrete time by Vassian (1955), Magee
(1956) and Brown (1963). Adelson (1966) appears to be the first to explicitly study what we would now call bullwhip. Deziel and Eilon (1967) provide an insightful study of bullwhip and inventory variance of an OUT policy. SchneeweiX (1971, 1975, 1977) and Inderfurth (1977) used z transforms and Wiener filtering theory to find optimal linear production or replenishment decisions for production systems with non-linear costs. Gaalman (1976, 1978, 1979) considered the optimal linear control of a general multidimensional production-inventory system, using the so-called modern control-theoretic principles. It is often supposed that the bullwhip problem cannot be avoided. Indeed it has been shown by Dejonckheere et al. (2003) that the classical OUT policy with exponential smoothing (Brown, 1963) or moving-average forecasting, will always produce bullwhip for all possible demand patterns. However, recently, it has been shown (Gilbert, 2005; Chen and Disney, 2003) that the use of conditional expectation forecasting can allow the OUT policy to smooth demand (i.e. avoid the bullwhip problem) for some classes of stochastic-demand patterns. Modifying the classical OUT policy, by incorporating a gain in the inventory feed-back loop, can lead to ‘bullwhip’ reduction in the OUT policy. Dejonckheere et al. (2003) have shown that it is always possible to avoid the bullwhip problem with such a technique for all classes of demand processes. At first sight, this bullwhip avoidance technique will come at the cost of holding extra inventory to achieve target customer service levels. This is true for i.i.d. stochastic demands (Magee, 1956). Dejonckheere et al. (2002) investigate this scenario explicitly.
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A characteristic of recent papers is the use of slightly more realistic stochastic-demand models such as the autoregressive moving-average demand model (Box and Jenkins, 1970). Commonly, a classical control engineering approach is taken to analyse the system’s dynamic behaviour. For example, Disney et al. (2003) study the ARMA model and show there are win–win scenarios resulting from a proportional controller, 1/Ti (herein called f), placed in the inventory position feedback loop. They show it is actually possible to avoid bullwhip, reduce inventory requirements and achieve a desired fill-rate simultaneously. An expected cost approach may also be applied to the OUT policy. For example, Kim and Ryan (2003) assume the demand process is Gaussian and study the probability density function that describes the inventory levels over time; assigning costs to expected inventory holding and backlog positions. Disney and Grubbstro¨m (2004) applied the same technique to the probability density function of the orders and assigned costs to the fraction of products produced in normal working hours, and those produced in a capacitated position with subcontracting or over-time working. There the orders were generated with the OUT policy, forecasts generated with exponential smoothing and the demand was assumed to be autoregressive. Chen and Disney (2003) studied expected inventory and order related costs in the myopic OUT policy with conditional expectation forecasting and ARMA(1,1) demand. They show that the proportional controller (f ¼ 1/Ti) always results in a more economic performance than the classical OUT policy (where f ¼ 1/Ti ¼ 1). Herein we introduce a state space approach to represent the ARMA(1,1) demand model and the inventory balance equation. We derived directly, by means of matrix calculations, the variance of the states of the system (the inventory and the forecast). This results in a matrix difference equation that under stability conditions converges to stationary values. In general this stationary variance matrix equation can be solved numerically. However, in this particular case, a closed-form analytical expression for the variances of the state variables can be derived. From this the variance amplification of the ordering rule can be easily found. Analysing the ordering rule variance expression in this way highlights the essential role of the co-variance between inventory and the state of the demand forecast model on the bullwhip effect. In
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addition, given the expression of the ordering variance a polynomial equation can be derived that surrenders the value of the feedback parameter for which the variance amplification is unity. We conclude by investigating the trade-off between inventory and order variance for the ARMA(1,1) demand pattern. The structure of our paper is as follows. In Section 2 the myopic OUT policy is introduced and discussed. In Section 3, we add the proportional controller into the classical OUT policy. The role of the co-variance between the inventory and the demand forecast is indicated. In addition the need for a trade-off between inventory and ordering variance is highlighted. Section 4 presents a state space demand model and shows how ARMA(1,1) demand can be formulated. From this the conditional expectation forecast model is derived. Section 5 develops the ‘total’ system state space description of the inventory and the forecasts. Moreover the variances of the system are investigated using the matrix variance equation. Section 6 analyses the order variance for ARMA(1,1) demand In Section 7 the trade-off characteristics of inventory and ordering variance are considered. We conclude in Section 8. 2. The myopic order-up-to policy The order-up-to policy is often used to manage inventory as it minimises the long-run expected inventory related costs when they consist of piece wise linear holding and stock-out costs. In this policy the inventory is reviewed and the ordering decision is made at the beginning of the period. During the period the customer orders are received and fulfilled before or at the end of the period. The inventory balance equation is given by itþ1 ¼ it þ ot ztþ1 ,
(1)
where it is the inventory at time t, ot is the order placed to replenish the inventory and zt+1 is the demand during the period (t, t+1). The (classical) order-up-to policy is completed by the following replenishment decision ot ¼ ðit in Þ þ z^tþ1;t ,
(2)
where
in is the inventory norm value (a time invariant constant used to achieve a given customer service metric, such as availability, fill-rate or economic criteria, often solved via the ‘‘newsboy’’
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technique). We may set in to zero without loss of generality here. z^tþ1;t is the forecast of demand in the next period made at time t.
Note that the lead-time consists of a review period only and the demand forecast mechanism is based on the minimum mean squared error criterion. Thus, we are considering a ‘‘myopic’’ OUT policy (Chen and Disney, 2003). Johnson and Thomson (1975) proved that this policy is optimal at minimising inventory holding and shortage costs when demand is both stationary and non-stationary in the case where backlogs are not allowed (that is lost sales are present). In contrast, here we assume backlogs are allowed in order to preserve the linearity of our model. However this is not the only condition, the order decisions can also not be restricted. Usually it is assumed that the demand is a normally distributed variable with a constant mean and random uncorrelated error component with constant variance (demand is i.i.d.). That is, ztþ1 ¼ z¯ þ tþ1 .
(3)
The ‘‘myopic’’ OUT policy (Eqs. (1) and (2)) minimizes the inventory fluctuations around the average value in. Little attention has traditionally been given to the fluctuations in the ordering decision until the recent interest in the bullwhip problem. These order fluctuations might become large for some demand characteristics and thus it is important to constrain them so as not to induce unnecessary production on-costs. In this paper we want to analyse the effects on the inventory and the ordering decision relaxing the restrictive assumption in (3). That is, we wish to gain insight into the behaviour of the OUT policy for an autoregressive moving-average (ARMA(1,1)) demand process (Box and Jenkins, 1970). This is important as real-life demands are not always i.i.d. as implied by (3). Eliminating ot in the balance Eq. (1) using the replenishment Eq. (2) gives itþ1 ¼ in Ztþ1 ,
(4)
where ztþ1 z^tþ1;t ¼ Ztþ1 is the forecast error. If the demand forecast minimizes the variance of the forecast error then the inventory variance will be minimal. This can be easily seen by squaring Eq. (4) and taking the expected values: Sii;tþ1 ¼ SZZ ,
(5)
where Sii,t+1 and SZZ are the variances of the inventory at time t+1 and the forecast error, respectively. Note that the inventory variance is time independent, so Sii,t+1 ¼ Sii. From a control engineering viewpoint the explanation is simple; the ordering rule feeds back the inventory deviation from the inventory norm value (the target safety stock holding) completely and at the same time anticipates future demand as best as can possibly be achieved. From (4) it can be shown that ot ¼ ðZt Þ þ z^tþ1;t ,
(6)
and the variance of the ordering decision is Soo;t ¼ SZZ 2SZ^z;t þ Sz^z^;t ,
(7)
where Soo,t and Sz^z^;t are the variances of ot and z^tþ1;t respectively. SZ^z;t is the co-variance between the demand forecast z^tþ1;t and the forecast error Zt. For i.i.d. random-demand processes such as those described by Eq. (3), the co-variance will be zero and the ordering variance equals the demand variance (that is, we have a critical bullwhip effect). For other demand processes the co-variance will not be zero. We can see that if the co-variance is negative the variance of the ordering decision may even become very large. However without a further specification of the demand characteristics the precise bullwhip effect cannot be further analysed. 3. Adding a proportional controller to the myopic OUT policy The introduction of a proportional controller into the inventory feedback loop might improve the dynamic behaviour of the replenishment policy. Indeed, this would be an obvious suggestion from basic control theory and actually has a long history in the field of production and inventory control (see Simon (1952), Magee (1956), Deziel and Eilon (1967), Towill (1982), Matsuyama (1997), Dejonckheere et al (2003) and Chen and Disney (2003)). So let us consider the following: Incorporating the proportional controller f into the inventory feedback loop in the replenishment decision, we have ot ¼ f ðit in Þ þ z^tþ1;t .
(8)
Using Eq. (8) in the inventory balance Eq. (1) yields itþ1 ¼ in þ ð1 f Þðit in Þ ðZtþ1 Þ.
(9)
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From (9) it can be seen that the system will be stable within the interval 0ofo2. From Eq. (9) the inventory variance may be derived as Sii;tþ1 ¼ ð1 f Þ2 Sii;t þ SZZ
(10)
Here we assume that co-variance between the inventory and the forecast error is zero. This is reasonable, because from Eq. (9) we can see that it is a linear function of the sequence of forecast errors fZt ; Zt1 ; Zt2 ; . . .g. In ‘‘good’’ functioning demand forecasts, the forecast errors are uncorrelated. If autocorrelation exists in the forecast errors then there is some additional ‘‘information’’ that has not been taken into account. Within the stability interval, Eq. (10) will converge to a stationary linear expression from which the inventory variance can be calculated as 1 1 Sii ¼ SZZ SZZ ¼ f ð2 f Þ 1 ð1 ð1 f Þ2 Þ (11)
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We observe three terms, one related to the inventory variance, the second to the co-variance of the inventory and the demand forecast and finally the variance of the forecast error. Using Eq. (11) the influence f on the first term can be calculated. The third term is independent of f. When the demand is i.i.d. (see Eq. (3)) the co-variance Siz^ is zero. In this case Soo, the order variance is an increasing function of f in the interval 0ofo2. As Soo goes to infinity for f-2, for some f the variance will be larger than the variance of the demand (thus bullwhip exists). The critical point lies at f ¼ 1. The proportional OUT policy has been proposed as a method to control the inventory variance as well as the order variance. Thus a suitable balance between the variances has to be found. Obtaining a suitable trade-off between the variances can be formalized by the minimization of the objective function OF ¼ fSii þ aSoo g,
(14)
Fig. 2 shows the inventory variance as function of f. As expected the variance is minimal for f ¼ 1. For 0ofp1 the variance is a decreasing function of f and for 1pfo2 the variance is an increasing function of f. The variance of the ordering rule (8) becomes
where a is a positive weighting factor by which the relative importance of the variances can be chosen. The motivation behind this objective function is that we are assuming that costs are linearly related to the inventory and order variance. The necessary condition for minimum is given by
Soo;t ¼ f 2 Sii;t 2f Siz^;t þ Sz^z^;t ,
(12)
dSii dSoo þa ¼0 df df
(13)
For example when the demand is i.i.d. and a ¼ 1 the minimum of the inventory and order variance
which converges, in the stationary situation, to Soo ¼ f 2 Sii 2f Siz^ þ Sz^z^ .
Fig. 2. Inventory variance as a function of the inventory feedback gain, f.
(15)
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pffiffiffi sum is obtained at f ¼ ð 5 1Þ=2 ¼ 0:61803, the golden ratio (see Fig. 3). This case has been studied for all lead-times by Disney et al. (2005) with regard to the customer service metric, the fill-rate. If a41 then f will become smaller than 0.61803. If 0oao1 then 0.61803ofo1. For non-i.i.d. demand the co-variance term in Eq. (13) is non-zero. The effect of f on the ordering variance can only be shown if we know Siz^ as a function of f. Unfortunately the co-variance between the inventory and the demand forecast can only be studied further if a demand model is specified. Here we will consider the ARMA(1,1) demand process. In the next section we will introduce a general demand model that forms the basis of subsequent analysis. The policy is formulated as a state space model in order to derive a general expression for the co-variance. 4. A general linear state space demand and forecast model Assume the demand can be described by the sum of a constant mean term and a normally distributed random variable: ztþ1 ¼ z¯ þ z~tþ1 .
(16)
Since, as shown in Section 2, we are only interested in the fluctuation around a constant mean we may ignore the constant term and will model only the stochastic process. Let this term be
described by the linear vector-difference equation: ztþ1 ¼ Mytþ1 þ tþ1 ,
(17)
ytþ1 ¼ Dyt þ Ext ,
(18)
where yt is an m-dimensional state vector, M is a 1 m matrix, D is an m m matrix and E is an m k matrix. The variable et and the k-dimensional vector x form sequences of zero-mean uncorrelated normal-distributed random variables with positive definite variance matrices (See and Sxx) and co-variance matrix Sex. Thus the random variables are uncorrelated between periods. In contrast with the model introduced in Gaalman (1976) the matrix E is introduced to explicitly demonstrate that there need not necessarily be m random disturbances. This model is rather general in the sense that a number of well-known demand models can be described in these terms, for instance, the General Polynomial Model of Harrison (1967) and the polynomial, exponential and seasonal models of Brown–Meyer (Brown 1963). Furthermore, Box and Jenkins (1970) autoregressive integrated movingaverage processes (ARIMA(p,I,q)) can be studied with the state-space approach. State-space representations are also seen as a powerful tool in time series econometrics (e.g., see Harvey (1990)). In this paper we will assume the demand satisfies an ARMA(1,1) process as ARMA(1,1) demand is frequently used to describe short-term demand processes. A state space
Fig. 3. Inventory and order variance when demand is i.i.d.
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formulation of this process is ztþ1 ¼ ytþ1 þ tþ1 ; ytþ1 ¼ fyt þ ðf yÞt ;
(19)
where yt is the state variable, f the autoregressive parameter and y the moving-average parameter. So M ¼ 1, D ¼ f, E ¼ 1. The process stability condition requires fjo1. This is the same as Box and Jenkins’s invertibility condition. For f ¼ 1 we have an ARIMA(0,1,1) process. This process is not stationary but it is allowed in the forecast approach. Having explained the basic characteristics of the state space demand model we will now introduce the forecast model. The optimal forecast of zt+1 at time t (which is here equal to the minimum variance estimation) can be obtained by the Kalman filter (Jazwinski, 1970) as z^tþ1;t ¼ M y^ tþ1;t ,
(20)
y^ tþ1;t ¼ Dy^ t;t1 þ G t ðzt z^t;t1 Þ,
(21)
where z^tþ1;t ; y^ tþ1;t are conditional expectations at time t given all the observed realisations of demand; Gt is the so called Kalman gain that can be calculated from a matrix Riccati equation. It can be shown that the forecast errors zi z^i;i1 ¼ Zi ;
i ¼ ð1; 2; . . . ; tÞ
(22)
will form a sequence of zero-mean uncorrelated normally distributed random variables when the forecast is optimal. Intuitively the Kalman filter can be explained as follows. Assume we have derived at time t1 the best forecast of yt: yt;t1 . Without any additional information the best forecast of yt+1 is y^ tþ1;t1 ¼ Dy^ t;t1 . However at time t demand information zt ‘‘arrives’’, but only the non-forecasted part of it ðzt z^t;t1 Þ provides new information. Since the demand is also disturbed by random noise the difference is ‘weighted’ by Gt. Here we will consider only the steady-state situation. So we assume that at time t an ‘infinite’ number of demand observations from the past are available. If the forecast system is stable and detectable then the Kalman gain converges to the stationary value G. The forecast system of the ARMA(1,1) demand model satisfies z^tþ1;t ¼ y^ tþ1;t y^ tþ1;t ¼ fy^ t;t1 þ Gðzt y^ t;t1 Þ, G ¼ ðf yÞ; ðzt y^ t;t1 Þ ¼ t .
ð23Þ
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Eq. (23) shows how the new demand information is contained in the forecast. The forecast error is weighted by the demand parameters and used to update the current demand forecast. For the ARMA(0,1,1) model, f ¼ 1, and we get the single exponential smoothing model of Brown (1963). Brown argues possible useful values of the smoothing parameter. For instance use a smoothing parameter 0.1 for slowly changing means values and 0.3 for very fast variations. Here the (optimal) value of G is linked to the two demand process parameters, f and y. To summarize this section we have shown how a demand model may be formulated as a state space model and we have introduced a minimum-variance forecast system. Also we have shown how the ARMA(1,1) demand process can be formulated as a state space model. We will not go into the calculation of the Kalman gain G by means of the matrix Riccati equation. We simply assume G is known. In the next section we will integrate this forecast system into the inventory (state) model. 5. The state space system and variance equations In order to derive expressions for the variances and co-variances of the relevant variables in our system we need to integrate the forecast system of the ARMA(1,1) process and the inventory system. To do this we will define Xt as a two-dimensional state variable of the joined system, consisting of the inventory state variable it and y^ tþ1;t the state variable of the forecast system at time t (note that since we calculate this forecast at time t it is part of Xt and not y^ t;t1 ). The total system is now be described by the difference equation X tþ1 ¼ AX t þ But þ CZtþ1 ,
(24)
with A¼
1
0
0
D
;
1 B¼ ; 0
C¼
1 G
,
(25)
where D ¼ f, G ¼ (fy) and ut ¼ ot z^tþ1;t . Since we are only interested in the characteristics of the variances we may ignore the mean values. It is not possible to influence the demand part of this system by ut. Thus the system is not completely controllable. Furthermore, if the eigenvalue of D, then the system is not stationary. So from here on we assume that fo1. This implies that the exponential smoothing model is not allowed. From a practical point of view this is not a serious
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problem since, in practice, an infinite set of demand data is neither available nor relevant for forecasting. Thus, the exact determination of demand parameters is not possible. So, one might choose f near unity. For example, instead of f ¼ 1 we might choose f ¼ 0:95. By this, the ARIMA(0,1,1) model becomes an ARMA(1,1) model. By introducing the inventory feedback rule ut ¼ fit , we are studying the OUT policy given by Eq. (8) and the system becomes X tþ1 ¼ ðA BF ÞX t þ CZtþ1 ,
(26)
where F ¼ ðf ; 0Þ;
A BF ¼
1f
0
0
f
! and Zt ¼ t .
As mentioned in Section 4 and Eq. (23), Zi (where i ¼ ðt; t 1; t 2; . . .Þ) forms a sequence of zeromean uncorrelated normally distributed random variables. This property makes it easy to derive a matrix variance differential equation from (26). Let the variance matrix-of the state vector at time t be ( ) ^ tþ1 t Þ0 i i i ð y t t t SXX ;t ¼ E X t X 0t ¼ E . ðy^ tþ1 t Þit ðy^ tþ1 t Þðy^ tþ1 t Þ0 (27) Then by multiplying Eq. (26) by the transpose of X tþ1 and taking expectations we get the variance equation: SXX ; tþ1 ¼ ðA BF ÞSXX ; t ðA BF Þ0 þ CSZZ C 0 . (28) EfZ2tþ1 g
Note that SZZ ¼ is the variance of the forecast error. Under the condition 0of o2 and fo1; the matrix variance equation converges to the stationary situation, expressed by SXX ðA BF ÞSXX ðA BF Þ0 þ CSZZ C 0 .
(29)
This equation can, in general, be solved by a number of numerical techniques. In this particular case the structure of A and B allows further analytical investigation. Eq. (27) shows that three components are relevant to us here (Sii ¼ Efi2t g; Syi^ ¼ Efðy^ tþ1; t Þit g; Sy^y^ ¼ Efðy^ tþ1; t Þðy^ tþ1; t Þ0 g). They are: Sii ¼ ð1 f Þ2 Sii þ SZZ ,
(30)
Syi^ ¼ ð1 f ÞDSyi^ GSZZ ,
(31)
Sy^y^ ¼ DSy^y^ D0 þ GG 0 SZZ .
(32)
Eq. (30) corresponds with (10), the inventory variance whose closed-form solution was shown in (11). As expected the inventory variance is not influenced by the demand but only by the (minimum) variance of the forecast error. The variance of the forecast, Sy^y^ is not influenced by the feedback factor f, see (32). The co-variance between the inventory and the forecasted demand state is dependent on the feedback factor (f), the demand model (D) and the Kalman gain (G). Eq. (31) can be written as Syi^ ¼ ð1 ð1 f ÞfÞ1 ðf yÞSZZ (33) Since f o1; ð1 ð1 f ÞfÞ cannot be zero in the stability interval 0of o2. In the next section we will analyse the effects of the ARMA(1,1) demand model on the variance of the ordering policy. Here, Eq. (33) will play an important role. 6. Analysis of the order variance for ARMA(1,1) demand The variance of the ordering policy was given in Eq. (13). Since M ¼ 1 we have Siz^ ¼ Sz^i ¼ Syi^ and Sz^z^ ¼ Sy^y^ . Thus the ordering variance becomes Soo ¼ f 2 Sii 2f Syi^ þ Sy^y^ .
(34)
The first and third components are positive, but the sign of the second component depends on the sign of the co-variance. Thus we will first explore the characteristic of the co-variance. From (33) we can observe that the co-variance has an asymptote for f ¼ ð1 fÞ=f. So for 1ofo0 the asymptote is42 and for 0ofo1 the asymptote is o0. Thus the co-variance is bounded on the stability interval, but might be large if ð1 ð1 f ÞfÞ is near zero. That is, when f is near 71 (regardless of the value of f). Depending on the demand model parameters the co-variance can be positive or negative, increasing or decreasing, function of f in the stability interval. For ðf yÞ40 the co-variance is negative, for ðf yÞo0, positive. The co-variance is zero when f ¼ y, which corresponds with the i.i.d. demand case. With this insight into the characteristics of co-variance we may now consider its influence on the ordering variance. Substituting (30) and (33) into (34) gives Soo ¼
f ðf yÞ þ 2f SZZ þ Sy^y^ . ð2 f Þ 1 fð1 f Þ (35)
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The ordering variance has an asymptote to þ1 for f ! 2. At f ¼ 0, Soo ¼ Sy^y^ . For f ¼1 the ordering variance is Soo ¼ 1 þ 2ðf yÞ SZZ þ Sy^y^ ¼ Szz þ 2ðf yÞSZZ . From this we can observe that for ðf yÞ40 the bullwhip effect exists. Since the third term in Eq. (34) does not depend on f we only need to consider the first two terms. We introduce H(f) as the part of Soo dependent on f:
variance is increasing (region 4 of Fig. 4) and there is a bullwhip effect in 1of o2. To find the location of the minimum we will consider the derivative of H(f):
f ðf yÞ þ 2f Hðf Þ ¼ ð2 f Þ 1 fð1 f Þ f ð1 þ 3f 4y þ f ð2y fÞÞ . ¼ ð2 f Þð1 ð1 f ÞfÞ
The denominator is always positive and the numerator consists of the sum of two quadratic terms and is positive for ðf yÞ40, confirming the earlier results. If ðf yÞo0 then the numerator can be reduced to
ð36Þ
If ðf yÞ40, the co-variance is negative and so H(f) is an increasing function in the stability interval (region 5 of Fig. 4). In addition the bullwhip effect occurs in the interval 0of o1. If ðf yÞo0, the covariance is positive and H(f) might have a (absolute) minimum. The denominator of H(f) is positive in the stability interval. H(f) has two zeros, one at f ¼ 0 and the other one for f ¼ ð1 þ 3f 4yÞ= ð2y fÞ. For ð1 þ 3f 4yÞo0 the second zero lies in the stability interval and we have a minimum. So, if ðf yÞo0 and ð1 þ 3f 4yÞ40, the ordering
dH 2½ð1 ð1 f ÞfÞ2 ðy fÞð1 fÞð2 f Þ2 ¼ . df ð2 f Þ2 ð1 ð1 f ÞfÞ2 (37)
½1 ð1 f ÞfÞ bð2 f Þ½1 ð1 f Þf þ bð2 f Þ, (38) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where b ¼ ðy fÞð1 fÞ. The necessary condition for an optimum gives two solutions: f1 ¼
2b þ f 1 2b þ f 1 ; f2 ¼ , bþf b þ f
(39)
where f1 gives the minimum and f2 the maximum value. The maximum value always lies outside the stability region. However the minimum does not
Fig. 4. The behaviour of the order variance in the ARMA(1,1) plane.
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necessarily lie within the stability interval. The necessary and sufficient conditions for the minimum being in the stability interval is that ðf yÞo0 and ð1 þ 3f 4yÞo0. In this case f ¼ f 1 . For b41 the minimum lies in the interval 1of o2 (region 2 in Fig. 4) and for bo1 the minimum lies in the interval 1of o1 (region 3 of Fig. 4). Fig. 4 maps these insights into the ARMA plane. Here the y ¼ f line corresponds to the i.i.d. demand patterns studied in Section 3. Typical plots in regions 2–5 are shown in Fig. 5 for verification. The critical bullwhip effect condition occurs when Soo ¼ Szz . This is equivalent to Soo Szz ¼ ðHðf Þ 1ÞSZZ ¼ 0.
(40)
The numerator of (40) is important here. It is a second degree polynomial. The critical bullwhip
condition occurs when yf 2 þ ð1 2yÞf þ ðf 1Þ ¼ 0.
(41)
This gives two bullwhip conditions. However only one lies in the stability region 0of o2. It is f
þ
¼
2y 1 þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 4yðy fÞ . 2y
(42)
This result corresponds to the bullwhip boundary given in Chen and Disney (2003), thus supporting our findings. In Fig. 5 the bullwhip conditions are plotted for the four typical variance situations, showing where the maximum f lies to avoid bullwhip. In this section we have shown that the ordering variance can have four different patterns in the stability interval (Fig. 4). With a suitable choice of the feedback parameter the bullwhip effect can be
Fig. 5. Variances for different ARMA(1,1) demands.
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avoided. The critical bullwhip point can lie in 0of o1 or in 1of o2. In the next section we will analyse the trade-off between inventory variance and ordering variance. 7. The trade-off of inventory and ordering variance Finally we will analyse the trade-off conditions for this demand model. As we have analytical expressions for the order and inventory variance, the derivatives can be calculated and used in the following objective function. dSii dSoo þa df df d 1 f 2f ðf yÞ þa þ ¼ . df f ð2 f Þ ð2 f Þ 1 fð1 f Þ ð43Þ The numerator yields the necessary condition for an optimum (minimum) and can be written as
af 2 þ f 1 ð1 ð1 f ÞfÞ2 aðy fÞð1 fÞð f ð2 f ÞÞ2 ¼ 0.
(44) We see that this condition is a polynomial of the 4th degree, consisting of the sum of a 4th degree term and a 2nd degree term. In the i.i.d. demand case (that is, when y ¼ f) the necessary condition is given by the first term and results in ðaf 2 þ f 1Þ ¼ aðf f o1 Þðf f o2 Þ ¼ 0, pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 þ 4a , ð45Þ f o1;2 ¼ 2a where 0of o1 o1 and f o2 o0. For i.i.d. demand the optimal value fT for which the trade-off between the variances is found lies, for all a, in the interval 0of o1. For ARMA(1,1) demand we are able to distinguish a number of possibilities. Since a40, for f in the interval f o2 of of o1 , the first term of (44) is negative. The second term of (44) is positive for f4y and negative for foy. Combining these insights we get, for a given a:
if f4y then the trade-off will be found in the interval 1=ð1 þ 2aÞof of o1 . From Section 6 we know that the ordering variance is an increasing function over the stability interval and larger than the ordering variance of the i.i.d. demand case. This can be dampened by increasing a, resulting in a lower fT. For a given a the minimum value of fT occurs when y ¼ 1 and f ¼ 2að1 þ 2aÞ=ð1 þ 8aÞ.
337
if foy the trade-off lies in the interval f o1 of o2. This situation might be combined with the insights obtained from analysing Hðf Þ. If 1 þ 3f 4y40 the order variance is still increasing and the trade-off will be found in the region f o1 of o1. If 1 þ 3f 4yo0 and bo1 the tradeoff will be in the region maxðf o1 ; f Þof o1. Otherwise if b41 then 1of of . However f in the interval of 1of o2 is usually not advised because it over compensates for inventory deviations. When b ¼ 1, f T ¼ f ¼ 1. Here the derivative of both the inventory variance and the order variance is zero.
In Fig. 6 for the case of a ¼ 1, the fT has been plotted, showing the minimum of the sum of order and inventory variances. We have also plotted in Fig. 6 the contour plot in the ARMA plane to illustrate the impact of the demand parameters on the optimum value of f, fT. From Eq. (44) the closed form expression of fT can be obtained. Unfortunately, this expression is rather large and unmanageable, hence this graphical approach. Fig. 6 also highlights 15 real-life demand patterns that have been identified by Disney et al (2003). We can see that one demand pattern is i.i.d. and thus our discussion in Section 3 is relevant here. Two demand patterns lie in region 3 of Fig. 4 and 12 demand patterns lie in region 5 of Fig. 4, so our investigations in Section 7 are relevant here. Happily the paradoxical trade-off in region 2 of Fig. 4 does not seem to be relevant in the practical situation of these 15 real-life demand patterns. Of course, an important question has now arisen; do real-life demand patterns fall in regions 2 and 3 of Fig. 4? We intuitively suspect that only demand patterns in regions 4 and 5 are common in many real-life supply chains. 8. Conclusions The classical order-up-to policy minimizes inventory fluctuations (variance) but does not attempt to control ordering fluctuations. This may lead to the so-called bullwhip problem where the ordering variance is amplified as it moves through a production/distribution/replenishment decision. The use of a proportional controller has often been proposed as a solution to the bullwhip problems caused by the classical order-up-to policy. By a suitable choice of the feedback parameter value the
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Fig. 6. The order and inventory variance trade-off in the ARMA(1,1) plane for a ¼ 1.
proportional controller strives for a good balance between the ordering and inventory variances. However, relatively less is known about the behaviour of the proportional policy for arbitrary demand characteristics. In this paper the performance of the proportional policy has been analysed for the ARMA(1,1) demand model. The crucial role of the co-variance between the inventory and the demand forecast on the bullwhip effect has been highlighted. Analytical expressions illustrating the influence of the demand model parameters are derived. From this expression bullwhip generating and order smoothing components can be uncovered. In general this leads to a paradoxical situation. If the co-variance is negative, the ordering variance increases (compared with the inventory-related component or the random-demand case) and bullwhip arises in the control interval 0ofo1, but at the same time a good balance between inventory and ordering variance can be found in this interval. If the co-variance is positive the ordering variance
decreases with respect to the i.i.d. random-demand case and could even have a minimum in the interval 1ofo2. In the latter case, though the ordering variance is relatively low, no balance can be found between inventory and ordering variance in the control interval 0ofo1. However, a balance can be found in the region of 1ofo2, but this is not an attractive situation since this leads to an overreaction to inventory deviations. Our overall conclusion is that the functioning of the proportional controller depends heavily on the demand model and its parameter values and as such requires considerable care when used in a real-life situation. Moreover, a reasonably functioning robust controller suitable for a broad class of demand models with their parameter values is not immediately obvious. However, for the real-life demand patterns pictured in Fig. 6, and for equal weight of the inventory and ordering variance (a ¼ 1) the golden ratio controller (fT ¼ 0.61803) seems to be reasonably close to the optimal
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solution. Indeed two-thirds of the demand patterns have an optimal feedback controller within 20% of the golden ratio. Future research will analyse the effects of ordering delays when using this general demand model. Further work will be devoted to finding better-performing ordering rules. References Adelson, R.M., 1966. The dynamic behaviour of linear forecasting and scheduling rules. Operational Research Quarterly 17 (4), 447–462. Box, G.E., Jenkins, G.M., 1970. Time Series Analysis Forecasting and Control. Holden-Day, San Francisco. Brown, R.G., 1963. Smoothing, Forecasting and Prediction of Discrete Time Series. Prentice-Hall, Englewood Cliffs, NJ. Chen, Y.F., Disney, S.M., 2003. The order-up-to policy ‘‘sweet spot’’—using proportional controllers to eliminate the bullwhip problem. In: Proceedings of EUROMA POMS Conference, 16–18 June 2003, Como Lake, Italy,vol. 2, pp. 551–560. Chen, Y.F., Drezner, Z., Ryan, J.K., Simchi-Levi, D., 2000. Quantifying the bullwhip effect in a simple supply chain: the impact of forecasting, lead-times and information. Management Science 46, 436–443. Dejonckheere, J., Disney, S.M., Farasyn, I., Janssen, F., Lambrecht, M., Towill, D.R., Van de Velde, W., 2002. Production and inventory control: the variability trade-off. In: Proceedings of the Ninth EUROMA Conference, 2–4 June, Copenhagen, Denmark. Dejonckheere, J., Disney, S.M., Lambrecht, M.R., Towill, D.R., 2003. Measuring and avoiding the bullwhip effect: a control theoretic approach. European Journal of Operational Research 147 (3), 567–590. Deziel, D.P., Eilon, S., 1967. A linear production-inventory control rule. The Production Engineer 43, 93–104. Disney, S.M., Grubbstro¨m, R.W., 2004. The economic consequences of a production and inventory control policy. International Journal of Production Research 42 (17), 3419–3431. Disney, S.M., Farasyn, I., Lambrecht, M., Towill, D.R., Van de Velde, W., 2003. Creating win–win scenarios from the bullwhip problem—by design not accident. In: Proceedings of EUROMA POMS Conference, 16–18 June 2003, Como Lake, Italy, vol. 2, pp. 561–570. Disney, S.M., Farasyn, I., Lambrecht, M., Towill, D.R., Van de Velde, W., 2005. Smooth is smart: bullwhip, inventory, fillrates and the golden ratio. Tijdschrift voor Economie en Management L, 167–186. Gaalman, G.J., 1976. Optimal linear control of linear production-inventory systems. International Journal of Systems Science 7, 1005–1015. Gaalman, G.J., 1978. On the optimal time-invariant control of linear Gaussian non-quadratic problems. International Journal of Control 28, 635–648.
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