Modeling and analyzing information delays in supply chains using transfer functions

Author's Accepted Manuscript

Modeling and Analyzing Information Delays in Supply Chains Using Transfer Functions Kai Hoberg, Ulrich W. Thonemann

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S0925-5273(14)00182-0 http://dx.doi.org/10.1016/j.ijpe.2014.05.019 PROECO5784

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Int. J. Production Economics

Received date: 6 November 2012 Accepted date: 24 May 2014 Cite this article as: Kai Hoberg, Ulrich W. Thonemann, Modeling and Analyzing Information Delays in Supply Chains Using Transfer Functions, Int. J. Production Economics, http://dx.doi.org/10.1016/j.ijpe.2014.05.019 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Modeling and Analyzing Information Delays in Supply Chains Using Transfer Functions

May 31, 2014

Abstract Advanced inventory policies require timely system-wide information on inventories and customer demand to accurately control the entire supply chain. However, the presence of unsynchronized processes, processing lags or inadequate communication structures hinder the widespread availability of real-time information. Therefore, inventory systems often have to deal with obsolete data which can seriously harm the overall supply chain performance. In this paper, we apply transfer function methods to analyze the effect of information delays on the performance of supply chains. We expose the common echelonstock policy to information delays and determine to what extent a delay in inventory information and point-of-sale data deteriorates the inventory policies’ performance. We compare the performance of this policy with the performance of an installation-stock policy that is independent of information delays since it only requires local information. We find that this simple policy should be preferred in certain settings compared to relying on a complex policy with outdated system-wide information. We derive an echelon-stock policy that compensates for information delays and show that its performance improves significantly in their presence. We note potential applications of the approach in service parts supply chains, the hardwood supply chain, and in fast moving consumer goods settings. (Keywords: Supply Chain Management; Control Theory; Multi-Echelon Inventory Control; Information Delays)

1

Introduction

In the past decade, research in supply chain management has highlighted the value of information for managing inventories. In numerous settings, many authors (see among many others Gavirneni et al. (1999), Lee et al. (2000) and Croson and Donohue (2006)) have demonstrated that sharing information between the partners in the supply chain can significantly cut operational costs, thus increasing competitiveness and 1

profitability of the supply chain. Advances in information technology play a crucial role as an enabler of this development (Cachon and Fisher 2000). However, investments into IT systems are costly and adaptation is slow. Therefore, the availability of real-time information among all partners in the supply chain is often still limited. In an inventory management context the availability of timely information is particularly important since more advanced inventory policies require information across the entire supply chain. A classic inventory policy that utilizes inventory information from downstream stages of the supply chain is the echelon-stock policy (Clark and Scarf 1960). Axs¨ ater and Rosling (1993) and Chen (1998) have shown that the echelonstock policy outperforms another classic inventory policy that utilizes local information, i.e., the installationstock policy. However, timely information must be applied since using delayed data may create potential problems down the supply chain. As Chen (1999) points out, both, the material and the information flows in the supply chain are subject to delays. However, most research on supply chain management focuses on delays in the material flow, e.g., lead times or shipping delays. Delays in the information flow that are also common in practice are less frequent considered. Among the few articles that address the problem are Chen (1999), Angulo et al. (2004) and Bensoussan et al. (2006). Information delays can arise due to an inadequate communication structure, unsynchronized processes, the lack of real-time systems or due to processing delays. In this paper, we analyze the effect of a static, known and supply-chain inherent information delay on the echelon-stock policy. To assess this effect, we compare the performance of the supply chain in response to three inventory policies: the installation-stock policy that is not affected by an information delay, the echelon-stock policy that is affected by an information delay and a compensated echelon-stock policy that we design to account for the information delay. The installation-stock policy is a classic inventory policy that bases the order decision on installation stock, which is the amount of inventory on hand at an echelon plus the amount of inventory that is on order from the preceding echelon. Since all required information is locally available, there is no need for information sharing and an information delay is not considered. In contrast to the installation-stock policy, an echelon-stock policy bases order decisions on echelon stock, which is the sum of installation stock at an echelon and the installation stocks of all downstream echelons. Therefore, information on downstream inventories must be shared to calculate the echelon stock. Further, end-customer demand is shared with the upstream echelons. Thus, two flows of information are affected by information delays. Figure 1 summarizes the material and information flows for the installation-stock policy and the echelon-stock policy. We assume fixed information delays that are known to the inventory

2

Installation-Stock Policy Installation Stock 1

Installation Stock 2 Echelon 2

Echelon 1

Supplier

Customer

Echelon-Stock Policy Echelon Stock 2 Echelon 2

Echelon Stock 1

Echelon 1

Customer

Supplier

Information Flow

Material Flow

Figure 1: Information and material flows in installation-stock and echelon-stock policies. manager, such that the echelon-stock policy can be adjusted to counter the effect. Therefore, we introduce a compensated echelon-stock policy that considers the delay and estimates a reference echelon stock based on latest shipping and demand information locally available at the echelon. To analyze the effect of the different inventory policies we apply linear control theory. The contributions of this paper are threefold. First, we model a supply chain that operates under an echelon-stock policy which is subject to information delays. We derive transfer functions that fully describe the behavior of orders and inventories under any customer demand. Second, we show that a supply chain that operates under an echelon-stock policy with information delays can perform worse than a supply chain that operates under the classic installation-stock policy. Third, we show how to adjust the echelon-stock

3

policy in order to incorporate information delays. We prove the superiority of the compensated echelon-stock policy over the installation-stock policy We also show several properties of the inventory policies. The remainder of this paper is organized as follows. In Section 2, we provide an overview on the relevant literature from two streams of research: literature addressing information delays in inventory management and literature applying linear control theory in a supply chain context. In Section 3, we model the echelonstock policy that is subject to delays using transfer functions. In Section 4, we analyze the performance of the echelon-stock policy and compare it with the performance of the installation-stock policy. In Section 5, we assume that the length of the static delays is known and model an adjusted echelon-stock policy that is considering these delays. We show that the supply chain benefits significantly from the adjustment. In Section 6, we conclude.

2

Literature Review

Linear control theory has been applied widely in engineering for many years. For an introduction to linear control theory we refer the reader to Franklin et al. (1997) and Nise (2008). Simon (1952) and Vassian (1955) were the first authors who applied linear control theory in an inventory management context. Simon considered a continuous-time representation for the steady-state analysis of an inventory system whereas Vassian used a discrete-time representation to design an order rule as applied in this paper. Meyer and Groover (1972) and Burns and Sivazlian (1978) were the first to apply linear control theory to the analysis of multi-echelon problems. Towill (1982) presents a new inventory control rule called the IOBPCS-policy (Inventory and Order Based Production Control System) which inspired a new stream of research. John et al. (1994) extends the policy and introduces the APIOBPCS (Automatic Pipeline, Inventory and Order Based Production Control System) inventory policy that bases orders on a demand forecast, the error between target and actual inventory and the error between target and actual work in progress. In recent years linear control theory has been widely used for studying the bullwhip effect that was coined by Lee et al. (1997a, 1997b). In this context, Dejonckheere et al. (2003) investigate the order amplification induced by order-up-to policies and the APIOPBCS policy. Dejonckheere et al. (2004) extend this work by analyzing the effect of information sharing to reduce order variability. In addition to order variability, Hosoda and Disney (2006b) analyze a three echelon supply chain that applies an order-up-to policy with the minimum mean square error forecasting and show that the number of echelons of the supply chain has no impact upon the bullwhip effect, while the variance amplification is determined by the accumulated lead-time from the customer and the local replenishment lead-time. Hosoda and Disney (2006a) show that the local optimization of the previous inventory policy for each stage of a supply chain does not yield globally optimal solutions considering the 4

dynamics of the whole supply chain. They find that in order to reduce total cost, the bullwhip effect must be considered and the ordering policy should be adjusted accordingly. Boute et al. (2007) review a setting with integrated production and inventory planning. Disney et al. (2008) review different coordination strategies for sharing details of replenishment rules, lead times and demand patterns in a supply chain with a supplierretailer setting. Results show that the retailer can dampen her order variability in order to improve supply chain performance even in a situation with non-cooperative suppliers. Gaalman and Disney (2009) review a proportional order up to policy for ARMA(2,2) demand in situation with arbitrary lead-times and identify a full-state-feedback policy with improved performance and reduced variability. Disney et al. (2008) analyze the value of coordination in a two echelon supply chain including information sharing. Potter et al. (2009) examine the relationship between production strategy and bullwhip where constraints exist and conclude that a certain order amplification is acceptable in order to minimize total supply chain costs. In a recent paper, Li et al. (2014) apply a new dampend trend forecasting method and review its stability. In additon, they identify regions for the bullwhip generating and dampening behaviour. In a different stream of research, delays in the information flows in the supply chain are addressed. Chen (1999) analyzes information lead times in a multi-echelon inventory model that occur due to order processing at an echelon. The results show that information delays play exactly the same role as the production/transportation lead times in defining the optimal replenishment strategies but have different cost affects. Watson and Zeng (2005) analyze a similar setting with information delays and show that sharing real-time sales data across the chain can eliminate the delay and mitigate its costs. In a series of papers Bensoussan, Cakanyildirim and Sethi study dynamic information delays as a Markov process: Bensoussan et al. (2006) consider constant and random delays of inventory level information and show the optimality of base-stock policies and (s, S)-type policies. Bensoussan et al. (2007) review dynamic information delays in a standard multi-period stochastic inventory problem with backorders. They show that optimal base stock levels depend on the latest delay observation and the age of this observation. Bensoussan et al. (2009) develop a framework to describe information flows in an inventory system with information delays and show that information can substitute inventory and vice versa. Marion and Sipahi (2010) study (among different production and transportation delays) a communication delay and introduce a visual decision-making tableau that is used to review the stability of the inventory system. Two recent papers by Hosoda and Disney (2012a, 2012b) consider information delays in a related situation. Hosoda and Disney (2012a) analyze the impact of a time delay on a serial two-stage supply chain where demand information is shared. Demand follows a first-order autoregressive process and an orderup-to (OUT) policy is applied for replenishment. Analysis shows that reducing time delays has a positive

5

impact on the downstream echelon while the impact on the upstream echelon depends on the forecasting approach applied at the downstream stage. Hosoda and Disney (2012b) model a supply chain that consists of a single manufacturer that is in a consignment stock situation with a customer. Internal processes at the manufacturer create an information delay at the production planning. To deal with the delay a controlling dynamics strategy (CDS) is introduced and the performance of the strategy is compared with an approach that eliminates time lags (TES). The analysis highlights that the TES strategy is particularly beneficial for the reduction of inventory cost, whereas the CDS strategy can reduce production cost significantly. The research closest related to this research in terms of methodology are the papers by Hoberg et al. (2007a, 2007b) and Hoberg and Thonemann (2013). Hoberg et al. (2007a) compare the effect of three inventory policies (inventory on-hand, installation-stock and echelon-stock policies) on stationary supply chain performance, i.e. on stability, order variability and inventory variability. Hoberg et al. (2007b) extend this research by also considering the non-stationary performance using the ITAE metric (integral of time and absolute error). ITAE measures a system’s transient reponse to a shock (Dorf and Bishop 2012) and was first introduced by Graham and Lathrop (1953). Hoberg et al. (2007b) further analyze the trade-offs between stationary and non-stationary performance for classic inventory policies. Hoberg and Thonemann (2013) introduce another base-stock policy that interchangeably applies demand and order smoothing and analyze different performance measures. However, the papers analyzing echelon stock policies do not consider possible delays in the information flows. In addition, no other papers have explictly addressed the idea to handle information delays in an echelon stock policy setting by adjusting the policy.

3

Modeling Information Delays in the Supply Chain

In this section, we model a two-stage inventory system that is subject to information delays. First, we summarize the basic framework for applying linear control theory to supply chain. Next, we describe the model for representing the echelon policy subject to information delays. Finally, we use transfer functions to describe the inventory system. These transfer functions comprehensively describe the inventory system and are applied in the subsequent analysis.

3.1

Modeling Inventory Policies with Linear Control Theory

When applying linear control theory, inventory systems are represented in the frequency domain. The frequency domain is an alternative representation to the time domain that is applied since different important system properties can be more easily identified. In the case of a continuous-time representation the Laplace-

6

transform is used and in the case of discrete-time-representation the z-transform is applicable (Franklin et al. 1997). The fundamental difference between a continuous-time and a discrete-time control system is that the latter system operates on samples rather than on the continuous signal (Brogan 1993). Here, we apply the discrete-time-representation with the z-transform since it more accurately describes the shipping delays and information delays that are relevant for this research. The z-transform Z{·} is defined1 as Z{x(kT )} = X(z) =

∞ 

x(kT ) · z −kT ,

(1)

k=0

where z is a complex variable and x(kT ) is the value of a time series x(·) in period k. T refers to the sampling time. For simplicity, we can set T = 1 and it could refer to a day or a week. The z-transform can be related to the Laplace transform by substituting z = ejw . The region of convergence of the z-transform is a complex topic to which we refer to Franklin et al. (1997) and Oppenheim et al. (1996). However, we can generally assume that for the signals applied in this research the convergence is confirmed. Using linear control theory, different operations (e.g. summations, time delays) can easily be represented using z-transform building blocks. Among the most important building blocks are the z-transform t of the time-discrete summation Z{ τ =0 x(τ ) dτ } = z/(z − 1) · X(z), the z-transform of a time delay of T periods Z{x(t−T )} = z −T X(z), and the z-transform of a single-parameter, exponentially-smoothed forecast with smoothing factor α applied to the time series x(t), and αz/(z − 1 + α) · X(z). Note, that we use capital letters to represent variables in the frequency domain and lower case to represent them in the time domain. In the general inventory system, we model the sequence of events as follows: At the beginning of each period the demand forecast is computed. Accordingly, the target inventory is calculated. Next, the inventory error is computed as the difference between the target inventory and current inventory level. Then, the order is placed. Finally, the placed orders arrive and demand is filled to the extent possible. Excess demand is backordered at each echelon. The order quantity can be negative, that is, excess stock can be returned to the supplier. Both assumptions are required to keep the model linear and it has been frequently used in the inventory-control literature (see Chen et al. 2000 and Lee et al. 1997a). In addition, multiple relations across parameters cannot be embedded. Note that these assumptions are commonly applied in the large body of supply chain management literature applying linear control theory. 1 Systems

are commonly assumed to be causal, i.e., the system cannot respond to stimuli (an input) before it has occured.

Therefore, the lower summation limit given above can be set to 0 rather than −∞. In this case, we refer to the unilateral z-transform, rather than to the bilateral z-transform.

7

3.2

The Echelon-Stock Policy with Information Delays

In order to minimize system-wide costs in a multi-stage supply chain it is beneficial to base the inventory policy on system-wide information (Simchi-Levi et al. 2007). The echelon stock can be understood as the system-wide stock that is relevant for each stage. It is defined as the sum of the installation stock at that stage and all downstream stages. The concept was first introduced by Clark and Scarf (1960). We apply an echelon-stock policy in a base-stock context. Base-stock policies are used in situations where fixed ordering costs are relatively small compared to other costs, such as inventory-holding costs. Base-stock policies or order-up-to-policies are applied to bring the relevant stock up to the target level. In the echelon-stock policy, the target echelon-stock is based on the lead times, the demand forecast and the desired safety stock. To calculate the safety stock, we apply the cover time concept that requires the policy to keep safety inventory for a given time of demand to hedge against demand uncertainty (Segerstedt 1995). The cover time concept is widely applied in practice, e.g., inventory planners at Gillette set the target inventory as 35 days demand (Duffy 2004) and Dell chooses an inventory target of 10 days of demand (Kapuscinski et al. 2004). Here, we model an echelon-stock policy in a serial two-echelon supply chain. We focus on the twoechelon situation, although the analysis can be extended to more than two echelons. However, this extension does not allow us to receive more general insights while the analysis is largely complicated. In a classic inventory system orders are the only information that is shared with the supplier. In the considered echelonstock policy, in addition to order data, inventory information and point-of-sale data is shared from the downstream stage (echelon 1) with the upstream stage (echelon 2) as shown in Figure 2. However, both information flows are vulnerable to possible delays. As pointed out by Watson and Zheng (2005), modern information technologies (e.g. POS, ERP, intranet, and Internet) have been made available in many supply chains. Nevertheless, delays in the information flows are still frequent, e.g. due to delays in the data entry, asynchronized processes, transmission delays or bureaucratic review processes (Bensoussan et al. 2009). To keep the model concise we assume a common static delay for the point-of-sale and the inventory data, i.e. the length of all delays is fixed over time and known to all parties. In the supply chain model, we use several parameters in line with Hoberg et al. (2007a,b): the lead time Ln at echelon n describes the shipping delay between placing an order and arrival of the goods. The cover time factor Cn is the multiple of the current demand forecast that should be held at echelon n to hedge against demand uncertainty. The parameter αn is the exponential smoothing factor used at echelon n to calculate the demand forecast with single exponential smoothing. While α1 and α2 will be set as 0 < α1 , α2 ≤ 1 in most practical settings (Nahmias 2009), we enable 0 < α1 , α2 < 2 as indicated in the stability analysis in Section 3.4. The sequence of events in the time-domain representation is as follows.. First, echelon 1 and 2 forecast end-customer demand using

8

Echelon 1

Echelon 2

Supplier

Customer Order Data

DELAY Data on Inventory Position DELAY Point-Of-Sale Data Information Flow

Material Flow

Figure 2: Effects of delays on information flows in echelon-stock policy.

9

exponential smoothing. We denote the forecasts in period t at echelon n by ft,n . Due to the information delay of τ periods for transmitting point-of-sale data to echelon 2 the demand forecast for echelon 2 is based on the demand dt−τ , i.e., the customer demand observed in period t − τ . Next, both echelons compute n their target echelon-stock positions test,n = ft,n · i=1 (Li + Ci ) by extending the demand forecasts over the lead times and inventory cover times of the downstream stages of the supply chain. Then, the echelon-stock n position est,n = i=1 ist,i of echelon n is the sum of the installation-stock position of the echelon and all downstream installations. However, due to the information delay the echelon-stock that is applied at echelon 2 is est,2 = ist−τ ,1 + ist,2 . Next, both echelons compute their order quantities ot,n = Δest,n + ft,n , where Δest,n = test,n − est,n is the error between the target echelon-stock test,n and the actual echelon-stock est,n . Accordingly, echelon 2 bases the order on delayed data. Then, echelon 1 receives the order that was placed L1 periods earlier and echelon 2 receives the order that was placed L2 periods earlier. Next, demand occurs at echelon 1 and is filled, and echelon 1 places its orders with echelon 2. Finally, information on customer demand, inventory, and orders from echelon 1 is shared with the upstream echelon 2. Figure 3 shows the block diagram of the echelon-stock policy in the z-domain. The basic block diagram resembles mostly the one introduced by Hoberg et al. (2007a) who first modeled the echelon-stock policy with linear control theory. In contrast to this contribution we incorporate information delays into the signal flow using time shifts. Using this block diagram representation we derive the transfer functions in the following section that are used throughout the analysis.

3.3

Transfer Functions

In linear control theory, the properties of an inventory system are represented by their transfer functions. A transfer function G (z) represents the system output Y (z) in relation to its input X(z) in the frequency domain and is defined as G(z) = Y (z)/X(z). We denote the transfer function of the orders On (z) at echelon n in response to the customer demand D(z) by GOn (z) and the transfer function of the inventory level In (z) at echelon n in response to the customer demand D(z) by GI (z). The transfer functions are derived from the set of basic system equations that we can identify from the block diagram in Figure 3. Note that the time-domain representation is given in the appendix. The set of basic equations at echelon 1 for the

10

Figure 3: Block diagram of echelon-stock policy with information delay τ .

11

echelon-stock policy in the z-domain is as follows:  z  O1 (z)z −L1 − D(z) z−1  z  O1 (z) − O1 (z)z −L1 z−1

I1 (z)

=

P I1

=

ES1 (z)

=

I1 (z) + P I1 (z)

(4)

F1 (z)

=

α1 z D(z) z − 1 + α1

(5)

T ES1 (z)

=

(L1 + C1 ) · F1 (z)

(6)

ΔES1 (z)

=

T ES1 (z) − ES1 (z)

(7)

O1 (z)

=

[ΔS1 (z) + F1 (z)]z −1 .

(8)

(2) (3)

Note that these basic equations at echelon 1 resemble the ones of the installation-stock policy as presented by Hoberg et al. (2007a) (with the echelon stock ES1 resembling the installation stock IS1 ). The set of basic equations at echelon 2 is as follows:  z  O2 (z)z −L2 − O1 (z) z−1  z  O2 (z) − O2 (z)z −L2 z−1

I2 (z)

=

(9)

P I2

=

IS2 (z)

=

I2 (z) + P I2 (z)

(11)

ES2 (z)

=

IS2 (z) + IS1 (z) · z −τ

(12)

F2 (z)

=

α2 z D(z) · z −τ z − 1 + α2

(13)

T ES2 (z)

=

(L1 + L2 + C1 + C2 ) · F2 (z)

(14)

ΔES2 (z)

=

T ES2 (z) − ES2 (z)

(15)

O2 (z)

=

[ΔES2 (z) + F2 (z)]z −1 .

(16)

(10)

Equation (12) incorporates the delay in the inventory data from echelon 1 while Equation (13) incorporates the delay in the point-of-sale data. To receive the transfer functions for the orders and the inventory in response to the customer demand we use relatively simple algebraic operations. Equations and (2)-(16) are substituted into each other until we receive the relevant ratios On (z)/D(z) and In (z)/D(z). The resulting

12

transfer functions are as follows: GES O1

GES I1

GES O2

GES I2

=

[(T1 + 1)α1 + 1]z − (T1 )α1 − 1 O1 (z) = D(z) (z − 1 + α1 )z

=

z I1 (z) =− D(z)

=

=

L1 +1

+

L1 i=0

α1 z i − (T1 + 1)α1 − 1

(z − 1 + α1 )z L1

[α1 + 1 + α1 T1 ] · z τ +2 [(α1 + 1 + α1 T1 )(α2 − 1) − 1 − α1 T1 ] · z τ +1 −(1 + α1 T1 )(α2 − 1)z T + [α2 (T2 + T1 + 1) + 1] · z 3 +[α1 T1 (α2 − 1) + α2 T2 (α1 − 2) + α1 α2 − α2 (2T1 + 1) − 3] · z 2 +[3 + α1 T1 (2 − 2α2 ) − α1 α2 T2 α1 + α2 (T1 + T2 − α1 − 1)] · z 1 +α2 − 1 + α1 T1 (α2 − 1) O2 (z) = D(z) (z − 1 + α1 )(z − 1 + α2 )z τ +2

I2 (z) = D(z)

[(α1 T1 + 1)(z − 1) + α1 z(z − 1 + α2 )] · z τ +L2 (α1 + 1 + α1 T1 ) · z τ +3 [(α1 + 1 + α1 T1 )(α2 − 1) − 1 − α1 T1 ] · z τ +2 −(1 + α1 T1 )(α2 − 1)z T +1 + [α2 (T2 + T1 + 1) + 1] · z 4 + [α1 T1 (α2 − 1) + α2 T2 (α1 − 2) + α1 α2 − α2 (2T1 + 1) − 3] · z 3 +[3 + α1 T1 (2 − 2α2 ) − α1 α2 T2 α1 + α2 (T1 + T2 − α1 − 1)] · z +α2 − 1 + α1 T1 (α2 − 1) (z − 1)(z − 1 + α1 )(z − 1 + α2 )z τ +2+L2

(17)

(18)

(19)

(20)

Note, that we use T1 = C1 + L1 and T2 = C2 + L2 to simplify notation. If we set the information delay to zero, the transfer functions resemble the ones used by Hoberg et al. (2007a) for the echelon-stock policy that we use as a benchmark as well as the installation-stock policy. Based on Equations (19) and (20) we find that information delay τ and the physical delays L1 and L2 have different effects in our model and are not completely interchangeable. The transfer functions of the installation-stock policy at echelon 2 are as follows (transfer functions at echelon 1 are identical). GIS O2 (z)

= =

GIS I2 (z)

= =

O2 (z) D(z) [(L1 + C1 + 1)α1 + 1]z − (C1 + L1 )α1 − 1 (z − 1 + α1 )z [(L2 + C2 + 1)α2 + 1]z − (L2 + C2 )α2 − 1 × (z − 1 + α2 )z I2 (z) D(z) (α1 (L1 + C1 ) + α1 + 1)z − (C1 + L1 )α1 − 1 − (z − 1 + α1 )z L2 L2 +1 z + α2 z i − (L2 + C2 + 1)α2 − 1 i=0 × (z − 1 + α2 )z L2

13

(21)

(22)

(23)

3.4

Stability of Transfer Functions

A key property of a dynamic system is its stability (Dorf et al. 2010). To evaluate the stability of a system, its natural response is analyzed. The natural response refers to the system output in response to an impulse system input. A system is stable if the natural response of the system approaches zero as time approaches infinity. A system is marginally stable if the natural response neither decays nor grows but remains constant or oscillates with constant amplitude. A system is unstable if the natural response grows to infinity as time approaches infinity. Unstable policies create fluctuating order and inventory behaviour that cannot be controlled without intervention. Hoberg et al. (2007a) have shown that the commonly applied inventory-onhand policy, in line with the behaviour frequently observed in the classic beer game (Sterman et. al. 1989), is unstable in many situations. To ensure stability of the echelon-stock policy subject to delays, we apply the pole-location approach as described by Nise (2008) which is also frequently applied to supply chain problems in the context of linear control theory (see Hoberg et al. 2007a,b). Here, the stability of a system can be determined by locating the poles of the transfer function, that is, the zeros of the denominator of the transfer function . A system is stable if all poles zi are within the unit circle of the complex plane (|zi | < 1), marginally stable if all poles zi are on or within the unit circle (|zi | ≤ 1) with at least one pole on the unit circle (|zi | = 1), and unstable if at least one pole zi is outside the unit circle (|zi | > 1). For the order transfer function GIS O2 (z), the zeros of the denominator are determined by solving the characteristic equation (z − 1 + α1 )(z − 1 + α2 )z τ +2 = 0. Accordingly, the poles are z1 = 1 − α1 , z2 = 1 − α2 and z3 = z4 = ...zτ +2 = 0. z3 ...zτ +2 are clearly inside the unit circle. For z1 (and accordingly for z2 ), we receive |z1 | < 1, if |1 − α1 | < 1.That means 1 − α1 < 1 and α1 − 1 < 1 or α1 > 0 and .α1 < 2. In a situation with α1 = 0 or α1 = 2, then |z1 | = 1, which means that the order policy is marginally stable. In a situation with α1 < 0 or α1 > 2, then |z1 | > 1, which would mean that order policy is unstable. The same holds for α2 with z2 . Accordingly we receive that the order transfer function is stable for exponential smoothing factors 0 < α1,2 < 2 and marginally stable for order transfer function for exponential smoothing factors 0 ≤ α1,2 ≤ 2. However, For the inventory transfer function, GIS I2 (z), we find the poles at z1 = 1 − α1 , z2 = 1 − α2 , z3 = 1 and z4 = z5 = ...zτ +2+L2 = 0. Considering that, the exponential smoothing factors are selected 0 < α1,2 < 2, we have a remaining pole at z3 = 1 . Accordingly, the inventory transfer function is marginally stable. In addiotion, we found that the pole z3 cancels out in the course of the calculation for all considered parameter sets and we find the inventory transfer function to be even stable for 0 < α1,2 < 2.

14

4

Analyzing the Effect of Information Delays

Various performance measures can be used for analyzing the supply chain performance. As Lee et al. (1997a,1997b) have shown, order variability is an important driver of supply-chain costs: High order variability causes companies to hold excess capacity, to apply overtime production, or to use premium shipping. Therefore, we consider performance measures that relate to the increase in order variability along the supply chain. We focus on the performance measures at echelon 2, since the behavior of echelon 1 is not affected by the information delay. While these performance measures can be mathematically handled, performance measures for the inventory at echelon 2 are difficult to acquire. However, our numerical studies indicate that high inventory variability accompanies high order variability such that we can forgo the performance measures for the inventory. We use three popular approaches from linear control theory to quantify order variability in the case of stationary demand: the frequency response plot, the H∞ -norm, and the H2 -norm (as applied by Dejonckheere et al. 2003, 2004) . The frequency response plot fully describes the policies’ order behavior in response to any stationary customer demand. The H∞ -norm corresponds to worst-case amplification, which measures the maximum amplification of any demand. The H2 -norm corresponds to white-noise amplification, which measures how a supply chain responds to normally distributed demand.

4.1

Frequency-response Plot

The frequency-response plot (see Franklin 1997) shows the response of a system to sinusoidal inputs with any frequency. While customer demand in real life will rarely be composed of a single sinusoidal input, any demand stream can be composed by adding up multiple sinusoids with different frequencies and amplitudes. Let us consider the sinusoidal input u(T ) = sin(ωT + φ0 ). Accordingly, a sinusoidal input with frequency ω = 0 corresponds to constant demand and a sinusoidal input with frequency ω = π corresponds to a demand that is alternating between each two periods. The steady-state response to a sinusoidal input signal will have the same frequency ω as the input signal but in general, its amplitude A and phase φ differ from those of the input. Thus, the output signal will be of the following form: y(T ) = A sin(ωT + φ0 + φ), which we refer to as the sinusoidal steady-state response. Here the amplitude response for the order transfer function at echelon 2 yields the amplification of customer variability at that echelon. Unfortunately, the form

15

of the order transfer function complicates the analysis: The information delay triggers several terms with z τ in the transfer function; thus, the degree of the numerators and denominators polynomials is driven by the information delay. However, most analytical approaches are only valid for a given degree. Nevertheless, interesting results can be received for given information delays. The amplitude response A(ω) of a linear, time-invariant system with transfer function G(z) is defined as the amplitude (magnitude or gain) of the frequency response with |G(ω)|

= =

|G(z)|ejωT  Re2 (G(ejwT ) + Im2 (G(ejwT )).

For a given supply chain, we can derive the frequency-response plot. Proposition 1 The order amplification of the echelon-stock policy with smoothing factors α1 = α2 = 0.2 and an information delay τ = 2 in response to any demand in a supply chain with lead times L1 = L2 = 4 and inventory-cover times C1 = C2 = 1 is:  880 cos(ω)3 − 904 cos(ω)2 − 1360 cos(w) + 1385 |GO2 (ω)| = 41 − 40 cos(ω)

(24)

which is defined between ω = 0 and ω = π. See Appendix for all proofs. Figure 4 shows the frequency-response plot for the orders at echelon 2 for this setting. The height of the curve indicates the degree to which a demand signal at a particular frequency would be amplified. As a comparison the order amplification of the installation-stock (IS) policy and the echelon-stock policy without information delay (τ = 0) is given. As already known, the amplification for the echelon-stock policy without information delay is in general significantly lower than for the installation-stock policy. However, the frequency response plot shows that for the echelon-stock policy with an information delay of τ = 2 periods this situation changes. For all frequencies the order amplification is higher in the presence of a delay. In addition, there is a significant peak in the frequency response plot for medium frequencies. Here, the performance of the echelon-stock policy deteriorates so significantly, that even the installation-stock policy outperforms the echelon-stock policy.

4.2

Worst-case Order Amplification

For a given supply chain, we can calculate the worst-case order amplification. The worst-case order amplification is achieved at a certain frequency ω of the end-customer demand. Formally, we apply the H∞ -norm

16

Figure 4: Frequency response plot for echelon-stock policy with information delays (L1 = L2 = 4, C1 = C2 = 1).

17

Figure 5: H∞ -Order amplification for various information delays at echelon 2 (L1 = L2 = 4, C1 = C2 = 1, and α1 = 0.20). concept from hardy space. Here, the H∞ -norm of a system with the transfer function G(z) is defined as   G(z = eiw ) = sup G(eiw ) . ∞ w

For the given supply chain with lead times L1 = L2 = 4, cover times C1 = C2 = 1 and an information delay τ = 2, the worst-case order amplification can be calculated. To simplify calculations, we use |GO2 (ω)|

2

rather than |GO2 (ω)| and receive the first derivative as d |GO2 (ω)| dω

2

= =

d 880 cos(ω)3 − 904 cos(ω)2 − 1360 cos(w) + 1385 dω 41 − 40 cos(ω) 8800 cos(ω)3 − 18050 cos(ω)2 + 9266 cos(w) + 45 . 1600 cos(ω)2 − 3280 cos(ω) + 1681

Setting the first derivative to zero and solving the equations, we identify five possible locations of a maximum that relate to the worst-case order amplification at echelon 2: ⎧ 0 ⎪ ⎪ ⎪ ⎨ 1.5756 −1.5756 . ω0 = ⎪ ⎪ −0.2320 − 0.3339i ⎪ ⎩ −0.2320 + 0.3339i Since complex or negative numbers are not valid, it is sufficient to check the second-order condition for the first two zeros. We receive: 18

704000 cos(ω)5 − 2165200 cos(ω)4 + 1868150 cos(ω)3 +320788 cos(ω)2 − 1111305 cos(ω) + 383506 d |GO2 (ω)| . = 8 dω 2 64000 cos(ω)3 − 196800 cos(ω)2 + 201720 cos(ω) − 68291 2

2

Accordingly, we find a local minimum at ω 0 = 0 and a maximum at ω 0 = 1.5756 which is in line with Figure 4. Moreover, we can calculate the worst-case order amplification as |GO2 (ω)|∞ = 5.812. Similarly, we can identify the worst-case order oscillation for other situations. Figure 5 shows the worst-case order amplification for various information delays in a supply chain with lead times L1 = L2 = 4, cover times C1 = C2 = 1 and a smoothing factor at echelon 1 of α1 = 0.20, while the smoothing factor at echelon 2 is varied. We find the worst-case order amplification to be higher for situations with information delays rather than for situations without information delay. In addition, we find the worst-case order amplification to be increasing in the exponential smoothing factor as it could be expected. In contrast to intuition, the length of the information delay does not drive the magnitude of the worst-case order amplification, i.e., the worst-case order amplification with an information delay τ = 1 is higher than the worst-case order amplification with an information delay τ = 2. Here, the length of the delay drives the inventory system into a resonance situation. Next, we review a third performance measure which relates to the more common amplification of normally-distributed demand.

4.3

White-Noise Order Amplification

We use the subscript 2 of the norm because the H2 -norm concept from the Hardy space is used to compute the white-noise amplification (Doyle et al. 1992 and De Bruyne et al. 1995). The white-noise order amplification is the ratio of the standard deviation of the orders to the standard deviation of the demand in response to normally distributed demand. Likewise, the white-noise inventory amplification is the ratio of the standard deviation of the inventory level to the standard deviation of demand. The H2 -norm of a stable system with transfer function G(z) is   G(z = eiw ) = 1 2 2π



2π 0

G(eiw ) 2 dw.

(25)

Due to the complex form of the transfer function GES O2 , we omit analytical results at this stage and focus on numerical insights. Figure 6 shows the white-noise order amplification O2 for a supply chain with with the parameters lead time L1 = L2 = 4, cover time C1 = C2 = 1, information delay τ = 2, smoothing factor α1 = 0.2, and varying smoothing factors α2 for three inventory policies. We realize that the order amplification of the echelon-stock policy with delays in response to normally distributed demand is higher than for the echelon-stock policy without delays (i.e. lower performance) but typically lower compared to

19

Figure 6: H2 -Order amplification for different inventory policies.

20

L1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4

L2 2 2 2 2 2 4 4 4 4 4 2 2 2 2 2 4 4 4 4 4

α1 0.2 0.3 0.4 0.3 0.3 0.2 0.3 0.4 0.3 0.3 0.2 0.3 0.4 0.3 0.3 0.2 0.3 0.4 0.3 0.3

α2 0.3 0.3 0.3 0.2 0.4 0.3 0.3 0.3 0.2 0.4 0.3 0.3 0.3 0.2 0.4 0.3 0.3 0.3 0.2 0.4

O(ESτ )2 τ =1 τ =2 τ =4 5.57 4.32 4.34 6.20 4.69 4.73 6.90 5.09 5.19 5.41 4.06 4.09 7.07 5.46 5.50 6.20 4.87 4.89 6.83 5.19 5.24 7.51 5.53 5.65 5.80 4.33 4.37 7.94 6.22 6.27 6.77 5.24 5.26 7.73 5.82 5.88 8.79 6.48 6.61 6.72 5.05 5.09 8.85 6.79 6.85 7.40 5.78 5.80 8.35 6.30 6.37 9.39 6.89 7.04 7.10 5.30 5.35 9.72 7.53 7.60

O(ES)2 3.22 3.22 3.22 2.44 4.05 3.87 3.87 3.87 2.86 4.94 3.87 3.87 3.87 2.86 4.94 4.52 4.52 4.52 3.28 5.83

O(IS)2 4.18 5.24 6.36 4.18 6.36 5.39 6.75 8.19 5.16 8.44 5.16 6.75 8.44 5.39 8.19 6.64 8.70 10.89 6.64 10.89

Table 1: Numerical results for H2 -order amplification of different supply chains. the installation-stock policy (i.e. better performance). In addition, the variability increases with α2 as to be expected. To broaden the scope, we present numerical results for different supply chain settings with alternative values of L1 ,L2 ,α1 and α2 (keeping C1 = C2 = 1) as shown in Table 1. Results show the white noise order amplification O2 for the echelon-stock policy with different delays, the echelon-stock policy without delays and the installation-stock policy. For further insights, we define the relative performance of the white noise order amplification of the echelon-stock policy with delays compared to the installation-stock policy as ΔIS with ΔIS = (O(ESτ )2 − O(IS)2 )/ O(IS)2 . Likewise, we define ΔES which refers to the white-noise order amplification of the echelon-stock policy with delays compared to the echelon-stock policy without delays ΔES = (O(ESτ )2 −O(ES)2 )/ O(ES)2 . The summarized results are shown in Table 2. Negative percentages indicate a deterioration of the echelon-stock policy with delays below the benchmark policy. Unlike to the initial example in Figure 6, we find that there are situations, where the echelon-stock policy with delays is outperformed by the simple installation-stock policy. Accordingly, there is a strong need to align the policy to cope with information delays.

5

Countering the Effect of Information Delays

In the previous section, we have shown that the performance of the echelon-stock policy can drop significantly in the presence of information delays. However, once the length of the information delay is disclosed to the supply chain members they can take measures to incorporate this delay into the inventory policy (see also 21

L1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4

L2 2 2 2 2 2 4 4 4 4 4 2 2 2 2 2 4 4 4 4 4

α1 0.2 0.3 0.4 0.3 0.3 0.2 0.3 0.4 0.3 0.3 0.2 0.3 0.4 0.3 0.3 0.2 0.3 0.4 0.3 0.3

α2 0.3 0.3 0.3 0.2 0.4 0.3 0.3 0.3 0.2 0.4 0.3 0.3 0.3 0.2 0.4 0.3 0.3 0.3 0.2 0.4

τ =1 -73% -92% -114% -121% -75% -60% -76% -94% -103% -61% -75% -100% -127% -135% -79% -64% -85% -108% -116% -67%

ΔES τ =2 -34% -45% -58% -66% -35% -26% -34% -43% -51% -26% -35% -50% -67% -76% -38% -28% -39% -52% -61% -29%

τ =4 -35% -47% -61% -67% -36% -26% -35% -46% -53% -27% -36% -52% -71% -78% -39% -28% -41% -56% -63% -30%

τ =1 -33% -18% -8% -29% -11% -15% -1% 8% -12% 6% -31% -15% -4% -25% -8% -12% 4% 14% -7% 11%

ΔIS τ =2 -3% 10% 20% 3% 14% 10% 23% 32% 16% 26% -2% 14% 23% 6% 17% 13% 28% 37% 20% 31%

τ =4 -4% 10% 18% 2% 13% 9% 22% 31% 15% 26% -2% 13% 22% 5% 16% 13% 27% 35% 19% 30%

Table 2: Relative H2 -order amplification performance of different supply chains. Hosoda and Disney 2012b). One effect of the information delay is the wrong perception of the echelon stock: Once information on the inventory at the downstream partner arrives at the upstream echelon, new customer demand has already been observed and additional shipments have been shipped to the downstream echelon. Thus, the actual echelon-stock is not considered correctly and orders are placed based on outdated inventory information. In this section, we address this issue and present an approach to estimate the actual echelon stock. This calculation of the reference echelon stock is incorporated into the echelon-stock policy, i.e., to receive the compensated echelon-stock policy (CES). Note, that the compensation is a heuristic approach and we do not believe that it is optimal in any sense (see also Section 6).

5.1

Compensating for Information Delays

Due to the information delay, information about latest customer demand and inventory data at the downstream echelon do arrive tardy at the upstream echelon and the echelon stock at the upstream echelon is not calculated correctly. To compensate for the information delay, its effect can be encountered by estimating an adjusted echelon stock. Accordingly, a reference echelon-stock is now calculated based on the latest available information at the upstream echelon. The reference echelon stock considers two important types of information: customer demand at the downstream echelon and shipments to the downstream echelon. Therefore, we introduce a shipping adjustment SADJ and a demand adjustment DADJ. The shipping adjustment SADJ considers the orders that have been shipped from the upstream echelon to the downstream echelon. The upstream inventory position at echelon 2 is directly reduced when the goods

22

are shipped. However, due to the information delay on the inventory position of the downstream echelon, the inventory position at the downstream echelon is not immediately increased accordingly. By incorporating information on the inventory shipped from echelon 2, the reference echelon stock can synchronized accordingly. The adjustment is required for the duration of the information delay. The demand adjustment DADJ considers the customer demand that has potentially been observed at the downstream echelon 1 and has affected the inventory at this echelon. In the case of an echelon stock policy with uncompensated information delays this demand is neither immediately reported nor is the demand effect on the inventory at echelon 1 reported immediately to the upstream echelon 2. The information is only made available after the information delay. In the case of an echelon stock policy with uncompensated information delays there is no adjustment the estimate the currrent downstream inventory. However, in the case with information delays the latest demand forecast can be used to reduce the inventory at echelon 1 in the absence of additional information. Therefore, the reference echelon stock is adjusted accordingly for the periods within the information delay.. Figure 7 shows the corresponding block diagram with the adjusted signal flow at echelon 2. The set of basic equations in Equations (2) to (16) remains largely untouched. The estimate of the echelon stock 2 at the upstream echelon is now calculated using four variables. First, the current installation stock at echelon 2. Second, the delayed information about the installation stock at echelon 1. Third, the shipping adjustment that considers the goods that have been shipped to echelon 1 but are not reported yet in the downstream installation stock. Finally, the demand adjustment that is uses as the latest demand forecast to estimate the decrease in the downstream installation stock compared to the reported information. Equation (12) is altered accordingly to incorporate both adjustments and to calculate the reference echelon-stock as follows: ES2 (z) = IS2 (z) + IS1 (z) · z −τ + SADJ2 (z) − DADJ2 (z)

(12a)

The variable SADJ2 refers to the shipping adjustment at echelon 2. The reference echelon position is increased by orders that have recently been shipped from echelon 2 to echelon 1 and periods that are not affected by the information delay are taken out. To reflect the shipping adjustment, Equation (26) is added as follows: SADJ2 (z) =

z(1 − z −τ ) O2 (z) z−1

(26)

The variable DADJ2 refers to the demand adjustment at echelon 2, i.e. the latest demand forecast that is considered for the periods with information delay - earlier demand forecasts are disregarded. To

23

z-

D O1

2 z z-1+

O2 2

F2 L1+C1+L2+C2 TES 2 ES 2

+ ¯

ES 2

DADJ2

+ ¯

+ +

+¯

z-L2

z z-1

I2

+¯

+ ¯

SADJ2

z z-1

z-1

z z-1

z z-1

PI2 + + +

+ ¯ +¯

z-1

z-1

z-

IS 1 O1

Echelon 2

Figure 7: Block diagram of compensated echelon-stock policy at echelon 2. reflect the demand adjustment, Equation (27) is added as follows: DADJ2 (z) =

z(1 − z −τ ) F2 (z) z−1

(27)

While the transfer functions for echelon 1 remain untouched, the transfer function of the compensated echelon-stock policy CES at echelon 2 can be calculated as follows: GCES = O2

GCES I2

z − 1 − α2 z −τ +1 + (zα2 − α2 )(C1 + C2 + L1 + L2 ) + 2zα2 (z − 1 + α2 )z τ +1

(z − 1 + α1 )z 1−L2 −τ [z − 1 − α2 z −τ +1 + (α2 z − α2 )(C1 + C2 + L1 + L2 ) + 2zα2 ] −(z − 1 + α2 )z[(α1 z − α1 )(C1 + L1 ) + z(α1 + 1) − 1] = z 2 (z − 1 + α1 )(z − 1 + α2 )

(28)

(29)

Note again, that for no information delay with τ = 0, both transfer functions simplify to the transfer function of the standard echelon-stock policy. 24

Figure 8: Frequency response plot of compensated echelon stock policy with τ = 2

5.2

Stability of Compensated Policy

In line with section 3.4, we analyze the stability of the compensated echelon stock policy based on the pole location-approach. Proposition 2 summarizes the stability of the inventory system. Proposition 2 An inventory system subject to a delay τ that operates under an compensated echelon stock policy is stable for exponential smoothing factors 0 < α1 < 2 and 0 < α2 < 2. Proof. For the order transfer function GCES O2 (z), the zeros of the denominator are determined by solving the characteristic equation (z − 1 + α2 )z τ +1 = 0. Accordingly, the poles are z1 = 1 − α2 and z2 = z3 = ...zτ +2 = 0. z3 ...zτ +2 are clearly inside the unit circle. For z1 ,we receive |z1 | < 1, if |1 − α2 | < 1. We receive (z), the zeros of the denominator are stability, for 0 < α2 < 2. For the inventory transfer function GCES I2 determined by solving the characteristic equation z 2 (z − 1 + α1 )(z − 1 + α2 ) = 0. Accordingly, the poles are z1 = 1 − α1 , z2 = 1 − α2 and z3 = z4 = 0. Again, we re receive stability, for 0 < α1 < 2 and 0 < α2 < 2.

25

L1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4

L2 2 2 2 2 2 4 4 4 4 4 2 2 2 2 2 4 4 4 4 4

α1 0.2 0.3 0.4 0.3 0.3 0.2 0.3 0.4 0.3 0.3 0.2 0.3 0.4 0.3 0.3 0.2 0.3 0.4 0.3 0.3

α2 0.3 0.3 0.3 0.2 0.4 0.3 0.3 0.3 0.2 0.4 0.3 0.3 0.3 0.2 0.4 0.3 0.3 0.3 0.2 0.4

O(ES)∞ τ =1 τ =2 τ =4 7.25 5.21 6.75 8.29 5.88 7.70 9.47 6.62 8.76 7.38 5.30 6.86 9.32 6.74 8.67 7.95 5.74 7.41 9.00 6.33 8.35 10.18 7.01 9.39 7.82 5.52 7.25 10.32 7.50 9.61 8.84 6.35 8.23 10.41 7.39 9.66 12.18 8.57 11.27 9.24 6.69 8.59 11.74 8.47 10.91 9.55 6.87 8.89 11.12 7.82 10.31 12.88 8.91 11.89 9.68 6.89 8.98 12.74 9.19 11.84

O(CES)∞ τ =1 τ =2 τ =4 3.76 3.77 3.76 3.82 3.87 3.83 3.88 3.98 4.00 2.83 2.98 3.19 4.94 4.81 4.92 4.47 4.42 4.47 4.53 4.55 4.53 4.59 4.65 4.59 3.28 3.41 3.58 5.94 5.78 5.91 4.47 4.42 4.47 4.53 4.55 4.53 4.59 4.65 4.59 3.28 3.41 3.58 5.94 5.78 5.91 5.18 5.11 5.17 5.24 5.22 5.23 5.29 5.33 5.30 3.72 3.84 3.97 6.94 6.75 6.91

τ =1 -93% -117% -144% -160% -89% -78% -99% -122% -139% -74% -98% -130% -165% -182% -98% -84% -112% -143% -160% -84%

ΔCES τ =2 -38% -52% -66% -78% -40% -30% -39% -51% -62% -30% -44% -62% -84% -96% -47% -34% -50% -67% -79% -36%

τ =4 -79% -101% -119% -115% -76% -66% -84% -104% -102% -63% -84% -113% -145% -140% -84% -72% -97% -125% -126% -71%

Table 3: Numerical results for H∞ -order amplification of different supply chains.

5.3

Quantifying the Benefit of Compensating for Information Delays

Similar to the uncompensated echelon-stock policy we can now analyze the performance of the compensated policy with regard to the frequency response plot, the worst-case order amplification and the white-noise order amplification.

5.3.1

Frequency-Response Plot

As in Section 4.1 we can calculate the frequency response plot for the compensated echelon-stock policy for any demand input in a given supply chain. Proposition 3 In a supply chain with lead times L1 = L2 = 4 and inventory-cover times C1 = C2 = 1 for an echelon-stock policy with smoothing factors α1 = α2 = 0.2 and an information delay τ = 2, we receive a demand response of the compensated echelon-stock policy as  680 cos(3ω) + 8206 cos(2ω) − 39600 cos(w) + 30715 . GO2CES (ω) = 800 cos(2ω) − 3280 cos(ω) + 2481

(30)

Figure 8 shows the frequency response plot. We find the frequency response of the compensated policy to be much closer to the frequency response of the echelon-stock policy without delays compared to the non-compensated echelon-stock policy in Figure 4.

26

L1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4

L2 2 2 2 2 2 4 4 4 4 4 2 2 2 2 2 4 4 4 4 4

α1 0.2 0.3 0.4 0.3 0.3 0.2 0.3 0.4 0.3 0.3 0.2 0.3 0.4 0.3 0.3 0.2 0.3 0.4 0.3 0.3

α2 0.3 0.3 0.3 0.2 0.4 0.3 0.3 0.3 0.2 0.4 0.3 0.3 0.3 0.2 0.4 0.3 0.3 0.3 0.2 0.4

τ =1 -9% -10% -11% -10% -10% -7% -8% -10% -9% -8% -7% -8% -10% -9% -8% -6% -7% -8% -8% -7%

ΔES τ =2 -8% -9% -10% -10% -9% -7% -8% -9% -8% -7% -7% -8% -9% -8% -7% -6% -7% -7% -7% -6%

τ =4 -8% -9% -9% -9% -8% -6% -7% -8% -8% -6% -6% -7% -8% -8% -6% -5% -6% -6% -7% -5%

τ =1 16% 32% 44% 36% 30% 23% 38% 48% 40% 37% 19% 38% 50% 42% 35% 28% 44% 55% 47% 43%

ΔIS τ =2 16% 33% 44% 36% 31% 23% 38% 49% 40% 37% 20% 38% 50% 42% 35% 28% 45% 55% 47% 43%

τ =4 17% 33% 45% 36% 31% 24% 39% 49% 40% 38% 20% 39% 51% 43% 36% 28% 45% 56% 47% 44%

Table 4: Relative H2 -order amplification performance of different supply chains. 5.3.2

Worst-Case Order Amplification

We can calculate the worst-case order amplification to any demand frequency. For a supply chain with our   standard parameters, we find a worst-case amplification O2CES ∞ = 3.731 at ω = 0.971. Comparing the performance measures for the compensated and uncompensated policies we realize that the worst-case order amplification can be significantly improved by using the new concept. Table 3 highlights the results for a set of supply chain situations. ΔCES refers to relative performance of the compensated policy compared to non-compensated policy. We find a reduction in the worst-case order amplification in all settings. On average, the improvement is 90% compared to the non-compensated policy.

5.3.3

White-Noise Order Amplification

Due to the form of the order transfer function of the CES policy, we can calculate a closed-form equation for the white-noise order amplification in a situation with a given information delay. Proposition 4 For any supply chain with τ = 2, we receive the white-noise order amplification (i.e. normally distributed demand) as  2α22 (T1 + T2 )2 + (6α2 − 2α22 + 4)α2 (T1 + T2 ) + (6α2 − 4α22 + 7)α2 + 2 . O2 (CES)2 = 2 − α2

(31)

We can identify several familiar properties based on Proposition 3, e.g. the white-noise order amplification is increasing with the smoothing factor as well as lead times and cover times. We can compare the white-noise order amplification to the white-noise order amplification for the installation-stock policy as 27

presented by Hoberg et al. (2007a):  O2 (IS)2 =

2 + 3α1 + 4α1 T1 + 2α21 (T12 + T1 ) 2 + 3α2 + 4α2 T2 + 2α22 (T22 + T2 ) . 2 − α1 2 − α2

(32)

For simplicity, we set α1 = α2 = α and show the superiority of the compensated echelon-stock policy compared to the installation-stock policy. Proposition 5 For any supply chain with an information delay τ = 2 and arbitrary values for L1 , L2 , C1 , C2 and α, we receive that the white-noise order amplification (i.e. in response to normally distributed demand) of the compensated echelon policy outperforms the installation-stock policy. We can show similar properties for other delays τ . Our numerical results in Table 4 indicate that the relative improvement ΔIS over the installation-stock policy typically ranges between 15% and 50% while the relative detoriation compared to the echelon-stock policy without delay ΔES only ranges between 5% and 10%. This poses a significant improvement in the commonly analyzed setup with normally distributed demand.

6

Conclusion

In this paper we have investigated how information delays affect the well-known echelon-stock policy. An echelon-stock policy leverages two types of information, i.e., point-of-sale information and down-stream inventory information, to gain visibility on downstream stocks and to optimize stock levels and replenishment orders. All data is subject to possible delays, e.g. due to information processing or unsynchronized information handling. To study the impact of these delays, we used different performance measures: frequency response plot, worst-case order amplification, and white-noise order amplification. We found a significant deterioration in the performance of the echelon-stock policy subject to information delays: the introduction of any delay will decrease the performance of the echelon-stock policy. In certain situations, the echelon-stock policy performs even worse than the simple installation-stock policy that does not require extensive systemwide information. However, we find a mixed picture regarding the impact of the length of the information delay. Longer information delays do not automatically further deteriorate the performance compared to shorter delays, but their impact depend on the setup of the supply chain. We further proposed an adjusted echelon-stock policy to compensate for possible information delays. If the supply chain partners are aware of the delay they can incorporate this knowledge to reduce their impact. We found that the proposed compensation does enable the echelon-stock policy to significantly improve and to outperform the installation-stock policy. 28

Our insights have relevant managerial implications for practitioners. Even with today’s advanced IT systems, real time information is not always available and difficult to set up. Frequently, managers also have to consider the cost of system-wide real-time synchronization of inventory and POS data as costs can be substantial; both for setting up real-time data gathering as well for sharing the data across the different stages of the supply chain. This is particularly relevant, if different legal entities or separate companies are involved, since the legal, technical or process-related challenges can be significant and costly. Our analysis provides insights into the benefits of sharing sales and inventory data in a supply chain if the data is not fully up-to-date. Our findings highlight that compensating for known information delays is possible and that the performance achieved with a compensated echelon stock policy is superior to that of the standard installation stock policy. Accordingly, managers should be inclined to use the advanced policy. We have identified three practical cases where information are delayed: (i) the service parts supply chain of an equipment manufacturer, (ii) the end-to-end hardwood supply chain, and (iii) a fast moving consumer goods (FMCG) company that receives point-of-sale data from retailers. These cases could be used for putting the theoretical insights into practice and for validating the potential benefits of the compensated echelon stock policy. A first case where our approaches could be used is the service parts supply chain of an equipment manufacturer that we have worked with. The main production site of the manufacturer is in Germany and equipment is installed and serviced worldwide. Parts are shipped from the German central warehouse to regional warehouses, from where service technicians retrieve the parts for their trunk stock. For service, they retrieve parts from the trunk stock, but do not record the retrieval electronically. Information about the part usage is transmitted when the technician returns to the regional warehouse to replenish the truck stock. From the regional warehouse, the information is transmitted to central planning one day after it is entered. The average total information delay is three days and our research indicates that considering the delay in the inventory model would improve performance considerably. Since information delays are rather static this company could directly apply the compensated echelon stock policy. However, certain parameters must be adopted since the company applies different forecasting tools. By analyzing the different demand patterns it can also be understood how order amplification can be avoided. Delays such as the one experienced by the equipment manufacturer are not uncommon and can be substantial, in particular, as multiple companies are involved in a supply chain. Stiess (2010), for instance, considers the hardwood supply chain and analyses information delays from retailer to raw material supplier. He finds delays between a few minutes, for instance, in order processing, up to several days, for instance, as inventory information are only shared in regular time intervals. The total information delay from the retailer

29

to the raw material supplier adds up over one hundred days. Our research indicates that considering these delays in the inventory model would improve performance considerably. In order to test the benefits of the compensated echelon stock policy all information delays have to be carefully identified and mapped similarly to Stiess (2010). Next, the methodology needs to reconsider the forecasting approaches that are applied in practice and needs to align them with the proposed model. Finally, the analysis needs to identify the most relevant performance metric that are applicable and performance must be optimized for the specific case (i.e. H2 -amplification, H∞ -amplifcation). The third example relates to a FMCG company that receives point-of-sale and inventory information from several retailers with a large number of stores. Due to different IT systems that are synchronized in different intervals (e.g. overnight, weekly) there are frequently significant time lags in transmitting data to the FMCG company. The lengths of these delays are well known and could be compensated using our proposed approach. In addition to known delays, there are also random information delays that are not known in advance. Some of these random delays originate in the store, e.g., local internet connection errors or hardware failures at the store that prevent the timely transmission of sales and inventory data. Other delays are caused by inaccurate booking of arriving goods into the local inventory, e.g., if the logistics service provider does not correctly scan the packages upon delivery and packages need to be scanned later. An interesting research project would be to use advanced analytics methodologies to estimate the true demands and inventories if delays are not known in advance. However, this would require an empirical study applying a different methodology compared to the one used in this paper. While we found that there are certain limitations in linear control theory analytics due to the information delay function and its impact on the structure of transfer functions we could obtain several sound properties. Accordingly, future research can address various additional topics. First, there might be different delays in the system, e.g. POS data might see another delay compared to inventory data. Further research should evaluate which type of information delay has the strongest impact on the performance of the inventory policy. Second, there are different alternative approaches to compensate for information delays. We have presented an approach to forecast the downstream inventory position based on shipping information and to incorporate latest available demand forecasts. However, there should be different approaches to incorporate information into the inventory policy, in particular if there are different information delays with different impacts. This applies in particular if the length of the information delay is not known or varying. Since the proposed CES policy is a heuristic and we do not claim that it is optimal in any sense. There could be many alternative approaches available that could be investigated. Third, additional performance measures should be addressed in future research. So far, we have only focused on the impact of delays on the stationary per-

30

formance, i.e. the increase of order variability along the supply chain. However, additional research should further be reviewing the impact of the inventory policy on cost and on non-stationary performance. Here, the impact of information delays on performance measures like the reaction time, settling time or the ITAE would be interesting to evaluate, i.e. the responsiveness of the inventory policy to changes in the demand pattern.

Appendix Time-domain Equations

The set of basic equations is developed based on Figure 3 in the z-domain as

shown in section. Since the z-domain Equations (2)-(16) are frequently less intuitive to interprete, we also show the transformed equations in the time-domain for the uncompensated echelon stock policy at echelon 2, i.e., Equations (9)-(16). We assume that initial values are zero. Small letters indicate variables in the time-domain rather than the z-domain, e.g. ,inventory at echelon 2 is i2 (t) rather than I2 (z). Note, that the equations for echelon 1 are adjusted likewise. i2 (t)

=

t 

[o2 (k − L2 ) − o1 (k)]

(33)

k=1

pi2 (t)

t 

=

o2 (k)

(34)

k=t−L2 +1

is2 (t)

=

i2 (t) + pi2 (t)

(35)

es2 (t)

=

is2 (t) + is1 (t − τ )

(36)

f2 (t)

=

t 

α(1 − α)k · d(t − τ − k)

(37)

k=1

tes2 (t)

=

(L1 + L2 + C1 + C2 ) · f2 (t)

(38)

Δes2 (t)

=

tes2 (t) − es2 (t)

(39)

o2 (t)

=

Δes2 (t − 1) + f2 (t − 1).

(40)

For the compensated echelon stock policy, the equations for the echelon stock (12a), the shipping adjustment (28) and the forecast adjustment (29) in the time-domain are as follows: es2 (t) sadj2 (t)

= =

es2 (t) + is1 (t − τ ) + sadj2 (t) − dadj2 (t) t 

(41)

o2 (k)

(42)

f2 (k)

(43)

k=t−τ +1

dadj2 (t)

=

t  k=t−τ +1

Proof of Proposition 1 The order transfer functions of the echelon-stock policy with an information n4 z 4 + n 3 z 3 + n 2 z 2 + n1 z + n 0 delay τ = 2 has the form GES with n4 = α1 (T1 + 1) + 1, n3 = O2 (z) = d5 z 5 + d4 z 4 + d3 z 3 31

α1 α2 (T1 + 1) − α1 (2T1 + 1) + α2 (T1 + T2 + 2) − 1, n2 = α1 α2 (T2 + 1) − α2 (2T1 + 2T2 + 2) − 2, n1 = α1 2T1 + α2 (T1 + T2 − 1) − α1 α2 (2T1 + T2 + 1) + 3, d5 = 1, d4 = α1 + α2 − 2), d3 = α1 α2 − α1 − α2 + 1. Using Euler’s formula, we substitute z = cos(ω) + i sin(ω) with imaginary unit i and variable ω ∈ [0, π] and obtain n4 (cos(ω) + i sin(ω))4 + n3 (cos(ω) + i sin(ω))3 + n2 (cos(ω) + i sin(ω))2 + n1 (cos(ω) + i sin(ω)) + n0 GES (ω) = . O2 d5 (cos(ω) + i sin(ω))5 + d4 (cos(ω) + i sin(ω))4 + d3 (cos(ω) + i sin(ω))3 To separate the real and imaginary parts we expand the term by the complex conjugate of the denominator: [n4 (cos(ω) + i sin(ω))4 + n3 (cos(ω) + i sin(ω))3 + n2 (cos(ω) + i sin(ω))2 + n1 (cos(ω) + i sin(ω)) + n0 ]· [d5 (cos(ω) − i sin(ω))5 + d4 (cos(ω) − i sin(ω))4 + d3 (cos(ω) − i sin(ω))3 ] GES . O2 (ω) = 5 [d5 (cos(ω) + i sin(ω)) + d4 (cos(ω) + i sin(ω))4 + d3 (cos(ω) + i sin(ω))3 ]· [d5 (cos(ω) − i sin(ω))5 + d4 (cos(ω) − i sin(ω))4 + d3 (cos(ω) − i sin(ω))3 ] To simplify notation, we substitute the parameters L1 = L2 = 4, C1 = C2 = 1,and α1 = α2 = 0.2. After performing some trigonometric manipulations, we obtain 680i cos2 (ω) sin(ω) − 400i cos3 (ω) sin(ω) − 68i cos(ω) sin(ω) − 132 cos2 (ω)− 680 cos3 (ω) + 400 cos4 (ω) + 541 cos(ω) − 201i sin(ω) − 128 , (ω) = GES O2 −40 cos(ω) + 41 with real and imaginary parts Re(G(ω))

=

Im(G(ω))

=

−132 cos2 (ω) − 680 cos3 (ω) + 400 cos4 (ω) + 541 cos(ω) − 128 , −40 cos(ω) + 41 680 cos2 (ω) sin(ω) − 400 cos3 (ω) sin(ω) − 68 cos(ω) sin(ω) − 201 sin(ω) . −40 cos(ω) + 41

The absolute value of the order transfer function is: ES  GO (ω) = Re2 (G(ω)) + Im2 (G(ω)) 2

which yields Equation (24).

Proof of Proposition 2

Proof of Proposition 2 can be directly aligned to proof of Proposition 1 considering

the appropriate polynomial degree of the transfer function.

Proof of Proposition 3

For an information delay τ = 2, the order transfer function of the compensated

echelon policy can be simplified to GCES O2 (z) =

(1 + 2α2 + α2 (T1 + T2 ))z 2 + (−1 − α2 T )z − α2 . z 5 + (α2 − 1)z 4

Jury (1982) shows how the H2 -norm of a transfer function k 

G(z) =

ni z i

i=0 k  i=0

32

di z i

can be computed as

 G(z)2 =

⎡

nk 0 0 .. . 0

nk−1 ak 0 .. . 0

nk−2 ak−1 ak .. . 0

... ... ... .. . 0

a0 a1 a2 .. . ak

det [Xk+1 + Yk+1 ]d , dk det [Xk+1 + Yk+1 ] ⎡

⎤

0

0 .. .

...

(44)

0

a0 .. .

⎤

⎥ ⎥ 0 ⎥ .. with Xk+1 ⎥. . 0 a0 ... ak−2 ⎥ ⎦ 0 a0 a1 ... ak−1 a0 a1 a2 ... ak det[Xk+1 + Yk+1 ] refers to the determinant of Xk+1 + Yk+1 and det[Xk+1 + Yk+1 ]d to the determinant of   1 k−1 k 2 di di+k−1 , ..., 2 di di+1 , 2 di . Here Xk+1 + Yk+1 with the last row replaced by 2dn d0 , 2 ⎢ ⎢ =⎢ ⎢ ⎣

⎢ ⎥ ⎢ ⎥ ⎥ and Yk+1 = ⎢ ⎢ ⎥ ⎢ ⎦ ⎣

i=0

the transfer function

GCES O2 (z)

i=0

i=0

has the form

G(z) =

n5 z 5 + n 4 z 4 + n 3 z 3 + n 2 z 2 + n 1 z + n 0 . d5 z 5 + d4 z 4 + d3 z 3 + d2 z 2 + d1 z + d 0

Thus, we can calculate ⎤ ⎡ 1 α −1 0 0 0 0 2 0 1 α − 1 0 0 0 2 ⎥ ⎢ ⎥ ⎢ 0 0 1 α2 − 1 0 0 ⎥, ⎢ Xk+1 = ⎢ ⎥ 0 0 1 α2 − 1 0 ⎦ ⎣ 0 0 0 0 0 1 α2 − 1 0 0 0 0 0 1 ⎤ ⎡ 0 0 0 0 0 0 0 0 ⎥ ⎢ 0 0 0 0 ⎥ ⎢ 0 0 0 0 0 0 ⎥, ⎢ Yk+1 = ⎢ ⎥ 0 0 0 0 0 0 ⎦ ⎣ 0 0 0 0 0 α2 − 1 0 0 0 0 α2 − 1 1 ⎤ ⎡ 1 α −1 0 0 0 0 2 1 α2 − 1 0 0 0 ⎥ ⎢ 0 ⎥ ⎢ 0 0 1 α2 − 1 0 0 ⎥ , and ⎢ Xk+1 + Yk+1 = ⎢ ⎥ 0 0 0 1 α − 1 0 2 ⎦ ⎣ 0 0 0 0 1 2α2 − 2 0 0 0 0 α2 − 1 2 ⎤ ⎡ 1 α −1 0 0 0 0 2 1 α2 − 1 0 0 0 ⎥ ⎢ 0 ⎥ ⎢ 0 0 1 α2 − 1 0 0 ⎥ ⎢ [Xk+1 + Yk+1 ]d = ⎢ ⎥ 0 0 0 1 α − 1 0 2 ⎦ ⎣ 0 0 0 0 1 2α2 − 2 0 0 0 c4 c5 c6 with c4 = −2α2 (1 + 2α2 + α2 T ), c5 = 2α2 (1 + α2 T ) − (2 + 2α2 T )(1 + 2α2 + α2 T ) and c6 = 2α22 + 2(−1 − α2 T )2 + 2(1 + 2α2 + α2 T ). Calculating the determinants, we receive   ES  |Xk+1 + Yk+1 |d  GO  = 2 2 d5 |Xk+1 + Yk+1 |  2α22 (T1 + T2 )2 + (6α2 − 2α22 + 4)α2 T + (6α2 − 4α22 + 7)α2 + 2  O2 (CES)2 = 2 − α2 Proof of proposition 4

Let α = α1 = α2 . Then, Equation (31) can be written as 

O2 (CES)2 =

2α2 (T1 + T2 )2 + (6α − 2α2 + 4)αT + (6α − 4α2 + 7)α + 2 2−α

33

(45)

and Equation (32) can be rewritten as    4T1 T2 (T1 + 1)(T2 + 1)α4 + (4T12 T2 + 4T1 T22 + 3T12 + 3T22 + 8T1 T2 + 3T1 + 3T2 )α3   +(9 + 4(T12 + T22 ) + 16(T1 + T2 + T1 T2 ))α2 + (8T1 + 8T2 + 12)α + 4 O2 (CES)2 = . (2 − α)2 The difference in the squared white-nose order amplifications of the inventory policies can be simplified to  IS  2  ES  2 c4 α 4 + c3 α 3 + c2 α 2  GO  −  GO 2  2 = , 2 2 (2 − α)2 with c4 = 4T12 T22 + 4T12 T2 + 4T1 T22 + 4T1 T2 − 2T1 − 2T2 − 4, c3 = 4T12 T2 + 4T1 T22 + 4T12 + 4T22 + 10T1 T2 + 8T1 + 8T2 + 7, c2 = 4T1 + 4T2 + 4T1 T2 + 2. c4 is positive since T1 , T2 ≥ 1 and c3 and c2 are non-negative.  ES       Accordingly the difference is non-negative, which implies that GIS O2 2 > GO2 2 .

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