Analysing bullwhip effect in supply networks under information sharing and exogenous uncertainty

: This paper analyses the bullwhip effect in single product supply network topologies, operated with linear and time-invariant inventory management policies and shared supply network information, considering exogenous uncertainty. Information sharing is determined as the degree of coordination across the supply network. Exogenous uncertainty (e.g., transportation delay) cannot be governed by any supply network members. We characterise the stream of orders placed at any stage of the network assuming the customer demand is ergodic. In fact, this paper gives exact formulae to predict the magnitude of bullwhip effect in any shared supply network information topologies. The mentioned formulae is explored by means of mathematical method called frequency domain analysis (FDA) and the relevant analyses are progressed by Fourier transform method. The main contribution of the present work is defined as considering information sharing and exogenous uncertainty simultaneously in supply networks and using Fourier transforms.


Introduction
Supply chain includes a set of organisations which cooperate together to provide final products and services for end customers and creating value in the chain (Kouvelis et al., 2006).
The concept of bullwhip effect (BWE) first is presented by Forrester (1961). Burbidge (1991) discussed problems with causes in detail and introduced an inventory control model regarding demand amplification. BWE is considered as a phenomenon in supply chains where the amplification in the order sequences is usually greater than that of the downstream of a chain (Lee et al., 1997). BWE was presented by macroeconomic data (Blinder, 1986;Blanchard, 1983;Kahn, 1987). They showed effect increases the cost of operating the supply chain network. Regarding BWE literature review and observations on huge extra operational costs records for suppliers, analysing BWE and trying to reduce it, is so important (Dejonckheere et al., 2003). Refer to BWE literature, the research areas are classified in seven branches, detecting the relevant causes and recommending some solutions (Dejonckheere et al., 2003(Dejonckheere et al., , 2004Geary et al., 2004;Kim, 2008;Su and Wong, 2008;Zarandi et al., 2008), drawing demand models (Bayraktar et al., 2008;Hien Duc et al., 2008;Zarandi et al., 2008) , detecting the effects of demand amplifications (Haughton, 2009), analysing the BWE (Kouvelis et al., 2006;Ouyang and Li, 2010;Sucky, 2008), information sharing (IS) (Hsieh et al., 2007;Jaksic and Rousjan, 2008), channel alignment (Zhang, 2004) and operational efficiency (Bayraktar et al., 2008;Hsieh et al., 2007;Jaksic and Rousjan., 2008;Miragliotta, 2006;Ozelkan and Cakanyildirim, 2009). Dejonckheere et al. (2004) studied the BWE avoiding solutions and characterised a model for measuring it. A supply chain includes multiple customers and multiple markets, so serial supply chains cannot accommodate these issues and as a result network structure is necessary for interactions potential competitions or collaborations among suppliers and customers. In network topology of a supply chain each supplier's ordering decisions may be influenced directly by orders from multiple 'neighbours', or indirectly via network wide IS. It is considered that any member in supply network may undertake the role of being the customer of upstream members or the supplier of downstream members or the both (Ouyang and Li, 2010).
There is a large number of research related to the BWE in serial supply chain network (Table 1). This table includes techniques for reducing BWE, inventory management policies, BWE measurement and type of analysis BWE. IS is the practice of making strategic and operation information available for other partners of the network and as some degree of coordination across the supply network, and it creates visibility along the network and helps suppliers to plan their policies (Prajogo and Olhager, 2012). So, one of the BWE causes can be realised as decentralised supply network where information is not shared among suppliers. Miragliotta (2006) showed IS can decrease BWE from 0% to 35% dependent on the supply network stages interest and customer demand scenario. One of the particular IS strategies in supply networks can be pointed as sharing customer demand information [e.g., point of sales (POS) data] across the network to increase supply network visibility. One of the other BWE causes which makes the propagation of order amplification strengthen is known as uncertainty. The exogenous uncertainty instances can be transportation delays, disasters, and natural accidents whose occurrence is not under control of the supply network members. This paper focuses on exogenous uncertainties (e.g., transportation delay) which cannot be controlled by any network members and are independent of the suppliers' inventory management policy. Darvish et al. (2014) explained a model for analysing bullwhip effect for a single-product supply network topology considering exogenous uncertainty and time-invariant inventory. Miragliotta (2006) applied variance ratio greater than 1 to detect BWE which is defined as the ratio between the supply variance at the upstream and the demand variance at the downstream. Ouyang and Daganzo (2006) showed how to reduce BWE by introducing advance demand information (ADI) into the ordering schemes of supply chains. Ouyang and Daganzo (2008) analysed the BWE in single-echelon supply chain driven by arbitrary customer demands, considering exogenous uncertainty introducing the concept of uncertainty as exogenous to the suppliers. Ouyang (2007) analysed the BWE in multi-stage supply chains operated with linear and time-invariant inventory management policies and shared supply chain information. Such information includes past order sequences and inventory records at all supply chain stages. Ouyang and Li (2010) analysed BWE in supply chain networks operated with linear and time-invariant inventory policies management which can be applied for any arbitrary supply network and stationary customer demand process.
The few studies based on the non-serial SCN or network supply chain modelling assumption investigating the dynamics of IS and demand amplification phenomenon. The best of our knowledge, there is a lack of consistent studies for evaluating the BWE with the IS and exogenous uncertainty, in general, in supply network. The aim of this paper is analysing the impact of bullwhip reduction strategies on a supply network with IS and exogenous uncertainty and comparing this impact with the effect of these techniques with conditions when there is no IS. In fact, the paper proposes a model for supply network as a Markovian chain system under stochastic dynamic parameters and exogenous uncertainty to diagnose BWE and presents network members complex reactions. The presented model provides the basis for predicting the presence of BWE and bound its magnitude in any network topologies operated with linear and time-invariant inventory management policies.
IS is supposed as centralised supply network definition where information is shared among network members. Exogenous uncertainty is defined as transportation delay which any network member cannot control it. Supply network is allowed to include any member which can play the role of being the customer of up-streams or the role of being the supplier of down-streams or the both. The customer demand process is known and ergodic. The mentioned formula is come by the means of frequency domain analysis (FDA) method (Ouyang and Daganzo, 2008).
FDA has been used to study properties of time series for a long time (Brockwell and Davis, 1998).A system control framework was recently introduced to study the BWE in the frequency domain (Ouyang and Li, 2010). As a result, this paper aims at studying the supply network system in equilibrium state and then analysing different equations (as state deviation) to propose a robust field for measuring the BWE by the means of FDA and significantly, by Fourier Transform method, under IS. The rest of the paper is organised as follows: Section 2 presents a review of supply network concept and system dynamic parameters; Section 3 describes the steady state in supply network with IS and exogenous uncertainty; Section 4 presents the model of supply chain network with IS and exogenous uncertainty; Section 5 presents the BWE and the FDA for the BWE metric; Section 6 demonstrates numerical examples and in Section 7 the conclusions and future research lines are pointed out.

System dynamics of supply network
A general supply network is illustrated in Figure 1. Network members consist of three sets of various members, including a set of primitive suppliers (A 1 ), a set of intermediate suppliers (A 2 ), and a set of final customers (A 3 ). The set of intermediate suppliers serves the set of final customers with determined product and services, and the set of primitive suppliers serves the set of intermediate suppliers with products and services. Inventory stream starts from the set of primitive suppliers to the set of intermediate suppliers and then the set of final customers, in supply network. It is clear that the order stream is reverse in supply network. The concept of primitive suppliers is pointed not to have anymore up-streams in supply network and the concept of final customers is applied not to have anymore down-streams. The set of intermediate suppliers is introduced as system main part in which any member of the set of intermediate suppliers plays the role of both order placer (customer of up-streams) and order receiver (supplier of down-streams). While primitive suppliers just play the role of order receivers (as system output) and final customers just play the role of order placers (as system input). A directed arc (i, k) ∈ N is applied to show that partner i ∈ A 2 ∪ A 3 orders from k ∈ A 1 ∪ A 2 . This paper assumes that A 1 , A 2 and A 3 are disjoint and the entire network includes four disjoint subsets of arcs (Ouyang and Li, 2010): Refer to have part A 2 as the system main part, this paper try to identify system dynamics on part A 2 basis. So, for a generic supplier i ∈ A 2 , its inventory position x i (t) (including in-transit inventory) and in-hand inventory y i (t) satisfy the orders placed and received. Linear and time-invariant policies definition comes as follow: if (i, k) ∈ N, supplier i orders u ik (t) items from k at discrete times t = …, -2, -1, 0, 1, 2, …, and receives the items after a constant lead time, l ik = 0, 1, 2, … Some researches assuming the up-streams always have in-stock items (Lee et al., 2000;Chen et al., 2012;Ouyang andDaganzo, 2006, 2008;Ouyang, 2007;Ouyang and Li, 2010).
Equations (5) to (6) define the system dynamics for the supply network. The order quantity by each member is related on member's order policy. Consider situations that complete information are shared between members. At time t, the complete information set for the entire network includes the inventory records x i , y i ∀i ∈ A 2 up to period t, and orders u rs ∀(r, s) ∈ N up to period t -1 (Ouyang and Li, 2010): If information is shared across the network, every supplier may determine its order quantities based on any subset of β i (t). So, the impressive role of order policies in supply network order quantities introduces them as system dynamic. As in Ouyang andDaganzo (2006, 2008), Ouyang (2007) and Ouyang and Li (2010), this paper focuses on linear and time-invariant policies (Wu and Katok, 2006;Zarandi et al., 2008;Zhang, 2005Zhang, , 2004. So, the most general linear and time-invariant expression of policy for u ik (t), C P respectively indicate how u ik (t) is determined based on supplier r's inventory history, x r , y r , and its past orders u rs (Zhang, 2004).
A general definition of ( ), C P shows that these equations may improve any shared (or local) information so can represent any possible LTI ordering polices and these equations are denoted by: In IS as a remedy to reduce the BWE has been mentioned by several authors notably (Lee et al., 1997). We consider situation that information are shared between suppliers; therefore supplier may instead use the moving average of customer demand to forecast future demand. Therefore the policy for calculation of ( ) B p and is same as before, but for calculation of ( ) rs ik C p we have: ( 1 1 ) This part shows system dynamics (5) to (8) as a basis for model introduced in next part.

Steady state with IS and exogenous uncertainty
This paper assumes that linear and time-invariant policies are stable. All suppliers of a supply network use proper policies where the place orders of suppliers have the same sizes over time, so the supplier inventories tend to equal equilibrium values that are independent of the initial conditions. For this condition we have: 1 place orders: x r x 3 in-stock inventory: .
x r y In the steady state the whole orders received by supplier i is equal with the whole orders: Refer to above sentences, system dynamics with complete information with relationship steady-state inventory positions and the equilibrium order quantity can be written as: The target of this model is formulated according to studied exogenous uncertainty which can be transportation delays, disasters, and natural accidents whose occurrence is not under the control of the supply network members. We define supply chain uncertainty as the standard deviation of the difference between the actual and expected amount. Therefore, when exogenous uncertainty is occurred, there is a deviation from steady state conditions. The major parameters can be expressed in terms of deviations from their value at the equilibrium condition that which are on-hand inventory, Inventory position and order policy. In order to show order sequences, the inventories and orders equations can be expressed in terms of deviations from their value at equilibrium: ( 1 6 ) The system dynamics with complete information and exogenous uncertainty by subtracting equations (5) to (8) and (14) to (16) can be represented by the following equations: As a result by considering this assumption, for downstream demand and upstream order sequence we have:  ,

Model representation
In previous section system dynamic for a network supply with IS and all order sequences in the network is presented. Regarding to equations (17) to (21), the system is represented by following equations: Then using equations (18) and (19)  ( 2 3 ) where Ψ(P) and Φ(P) are polynomials respectively referred to follow equations: Equation (25) shows the effect of deviation of order placed by supplier i from (25) is just represented at state t.
To identify system dynamics, the paper satisfies relevant equation in terms of multiple states, i.e., {t, t -1, t -2, …}. At first, defines (K + 1) × 1 column vector for two unique members {i, k} in supply network. These vectors are calculated as: The system dynamics of a supply network with IS and exogenous uncertainty with matrix presentation can now be written in terms of multiple states: where R IK and S RS are matrix including the effect of multiple states introduced in (25). To augment the state, the paper represents system dynamics (25) for entire supply network not just for two unique members of supply network, it obtains: where R IK and S RS are square matrices {(K + 1) × (K + 1)} including some sub-matrices, i.e., {R IK , S RS } as their elements. The paper assumes now that the state space of Markovian chain has multiple dimensions that capture the stochastic status at all supplier stages, represented in matrix pairs: It should be mentioned that the number of state space members is equivalent to the number of matrix pairs in supply network which affect on order relationships. Then system dynamics (32), with exogenous uncertainty is as follow.
Regarded introduced order state, R IK(t) and S RS(t) including the matrices of R IK and S RS at multiple stages of state space. In other hands, the transaction probability matrix of supply network is defined as following: ; Pr | , , ( 3 8 ) The stochastic order relationship equation (34) is useful to model any supply network topologies under exogenous uncertainty. Figure 3 shows matrix presentation of system dynamic a supply network with exogenous uncertainty. In previous section, the inventories and orders equations can be expressed in terms of deviations from their value at equilibrium. A general definition of ( ), C p shows that these equations may improve any shared (or local) information.
In this section it is defined that R IK(t) and S RS(t) are square matrix which including the effect of multiple states for these deviations from their value at equilibrium. Therefore and S RS (t) come as following: a With IS and exogenous uncertainty: b Without IS and with considering exogenous uncertainty: , 1 , where the relations of any columns and rows are shown by out-space symbols of each matrices. The element in which across first row and second column (S R,S-1 ), shows the effect amount of deviation of orders placed by supplier R from S -1 on the mentioned supplier (i.e., supplier i) order deviation at relevant state. In this paper, we consider situations with IS.
Respectively, based on previous points, any elements of matrices (39) to (42) can be expressed as: The matrices (43) and (44) can play role as any elements of matrices (39) to (42) at multiple defined states (focus on one supplier to analyse the BWE magnitude in that stage). Figure 4 illustrates stochastic order relationship equation for supply network with Exogenous uncertainty and IS.

BWE measurement
In this section a metric is defined for measuring BWE. Lee et al. (1997) presented an amplification effect as the phenomenon where orders to the suppliers tend to have larger variance than the sales to the buyers. Authors proposed a quantitative measurement for BWE as the ratio of variance of order quantity at the echelon under consideration to the variance of demand of the end customer. Five roots are defined for these measurement tools for amplification effect in supply chains and BWE that they are demand forecast updating, lead time, batch ordering, supply shortages and price variations. This paper, for robust analysis, adopts the worst case expected root mean square errors (RMSE) amplification factor (W) across all possible customer demand sequences (Ouyang andDaganzo, 2006, 2008;Ouyang, 2007;Ouyang and Li, 2010). This RMSE is calculated by: To have order deviation of down streams (46) and up streams (47): The BWE magnitude can be represented by Dejonckheere et al. (2003), Gaalman and Disney (2006), Geary et al. (2004) and Kouvelis et al. (2006): By this mean, the condition W ≤ 1 guarantees that the RMSE is not amplified (i.e., BWE dose not arise) under any customer ordering scenario (Ogata, 1987;Ouyang et al., 2010).

FDA for BWE metric
FDA has been used to study properties of time series for a long time (Brockwell and Davis, 1998). There are some techniques for FDA which expressively which they are orthogonal decomposition, transformation and superposition (Ouyang and Daganzo, 2009). This paper presents relevant transformation techniques beyond the exogenous uncertainty and also considering IS. The supply network can be regarded as multiple-input such as final customer demands and multiple-output system, such as supplier orders. The BWE magnitude (W) can be expressed equivalently in the frequency domain. Any realisation of customer demands

iwt rs
A w e w π π i ∈ − = − With regarding to Fourier transform definition and also by considering that system dynamics are linear and time-invariant, it is known that for any harmonic component of any customer demand (as system input), [A rs (w) iwt ], the resulting orders placed by each supplier of the linear and time-invariant network (as system output) [ ( ) ] rs iwt ik A w e are also harmonic. Fourier transform presentation of all received orders by A 1 can be written as follows.
where first set shows received orders by A 1 from A 2 which they have received from A 3 in one back-ward step and second set shows received orders by A 1 from A 3 directly. So, it is clear that the union of two mentioned sets represents all received orders by A 1 . Also, Fourier transform representation of all placed orders by A 3 is introduced by: iwt rs w e A rs N N w π π ∀ ∈ ∈ − ∪ ( 5 0 ) b Then, regarded the definition of matrices (26) to (28), the equation (29) is expressed by z-transform as in (62): Transfer function matrix for system dynamics of supply network

Fourier transform
Extracting transfer function matrix for system dynamics of supply network with information sharing and with exogenous uncertainty

Figure 6
Assumed network topology The model framework and analysis results presented can be applied to supply networks with any general topologies.
In this example A 1 , A 2 and A 3 are disjoint and the entire network includes four disjoint subsets of arcs: In order to analyse BWE and for comparison effectiveness of model the paper introduces the assumed topologies in two main following categories: 1 with IS and exogenous uncertainty 2 without IS while considering exogenous uncertainty

With IS and exogenous uncertainty
In this section, we examine how the sharing of information reduces the variance of the order quantity at the upstream level or reduce BWE. It is supposed that electronic data interchange (EDI) is elected as IS strategy in entire network. So, the relevant information can be shared from POS to all over the supply network. Regarding to paper assumption, the lead-time is considered as constant amount (m). It is assumed that all network members use (S, s) ordering policy with IS and the demand prediction methodology is determined by moving-average of orders received in the two most recent periods. where they are mathematically obtained from following system dynamic metrics: where the symbol (#) is used to control the amount of matrices rows and columns not to front any problems in calculations. Regarding literature review and equations (78) to (80), following matrices are mathematically written: The frequency domain representation of entire network orders can be stacked into the following matrices: It should be considered that z is equivalent to e iw and in order to maintain the system integrity, it is supposed as the average of e tw (z = -0.001). Thus, focusing on intermediate supplier 3, the BWE magnitude is calculated as: where the amount of W, 2 determines not to face with BWE in supply network (W get lesser than 1) and there is BWE.

Without IS while considering exogenous uncertainty
Regarding model assumptions and the previous section, the lead-time is either considered as constant amount (m) in this mode. As before, it is assumed that all network members use (S, s) ordering policy, but with no IS and the demand prediction methodology is determined by moving-average of orders received in the two most recent periods.  9 2 ) where,it is simply obtained from following system dynamic metrics: ( 1) Here, z is equivalent to e iw and in order to maintain the system integrity, it is supposed as the average of e iw (z = -0.001). The magnitude of the BWE is calculated from formula (…) and estimated from simulations for a range of parameters. Thus, focusing on intermediate supplier 3, the BWE magnitude is calculated as: where the amount of W, 3 determines the presence of BWE in supply network (W get greater than 1) and its magnitude is equivalent to 1.03. This result implies that IS tend to reduce the BWE. The formula of this research turns out using a mathematical method called FDA for a supply network with exogenous uncertainty. The major target of this paper is analysing BWE considering exogenous uncertainty in supply networks with using Fourier transform in order to simplify the relevant calculations and applying IS and compare results with conditions that information are not shared. Formula in reference (Ouyang and Li, 2010) for calculating BWE has a complex structure so a simplification equation is used for calculating the BWE. According the results shows that IS reduces the BWE and output of the methodology confirms the efficacy of the IS rule in terms of bullwhip reduction in the supply chain network. This methodology gives exact formula for calculating the BWE. Therefore, this comparison shows that the approach which can derive exact solution with Fourier transform for calculation the BWE is superior to Ouyang and Li (2010) approach.

Conclusions and further research
This paper has presented a system control framework for analysing order sequences deviation and BWE in supply networks under IS and exogenous uncertainty. By presenting supply network concept and formulating it as Markovian chain, the paper derived robust analytical conditions that diagnose the system under Markovian uncertainties in which are assumed to be as exogenous to the suppliers and are independent of the suppliers' state while the information is shared among entire network.
The presented model provides the basis for developing exact formula for analysing the propagation of order amplification in any single-product supply network topologies operated with linear and time-invariant inventory management policies and shared supply network information considering exogenous uncertainty. The mentioned formula is explored by the means of FDA. The numerical examples which were divided into two main categories have shown how the presented framework enables supply managers (or any practitioners) to study the effect of various factors (e.g., network structure) under robust conditions (exogenous uncertainty) and IS, on the BWE in supply network. Also, refer to numerical examples analysis, it can be noted that IS reduces BWE even while considering exogenous uncertainty model in entire supply network. The analysis framework presented in this paper can be further extended by focusing on endogenous uncertainty (controllable conditions by network suppliers, e.g., ordering policy) as first direction. Second one is to consider variant lead-times in supply network and the third is to present nonlinear ordering policies for determining system control framework.