The impact of the supply chain structure on bullwhip effect

Dominguez R., Cannella S., Framinan J.M. 2015. The impact of the supply chain structure on bullwhip effect. Applied Mathematical Modelling, 39 (23-24), 7309-7325. DOI: https://doi.org/10.1016/j.apm.2015.03.012 1 The impact of the supply chain structure on bullwhip effect Roberto Dominguez1*, Salvatore Cannella1,2, Jose M. Framinan1 1 Industrial Management, School of Engineering, University of Seville, Ave. Descubrimientos s/n, Seville, E41092, Spain; tel: (+34) 954488137; e-mail: [email protected] (*corresponding author). 2 School of Industrial Engineering, Pontificia Universidad Católica de Valparaíso, Ave. Brasil, 2362807, Chile. Dominguez R., Cannella S., Framinan J.M. 2015. The impact of the supply chain structure on bullwhip effect. Applied Mathematical Modelling, 39 (23-24), 7309-7325. DOI: https://doi.org/10.1016/j.apm.2015.03.012 2 Abstract: The aim of this paper is to study how the structural factors of supply chain networks, (i.e. the number of echelons, the number of nodes and the distribution of links) impact on its dynamics performance (i.e. bullwhip effect).To do so, we systematically model multiple structures according to a robust design of experiments and simulate such structures under two different market demand scenarios. The former emulates a stationary condition of the market, while the latter reproduce the extreme volatility and impetuous alteration of the market produced by the current economic recession. Results contribute to the scientific debate on supply chain dynamics by showing how the advocated number of echelons is not the only structural factor that exacerbates the bullwhip effect. In particular, under a sudden shock in market demand, the number of nodes and the divergence of the supply chain network affect the supply chain performance. Keywords: supply chain management; multi-agent systems; simulation; demand amplification; complex supply chain; ANOVA. 1 Background and Motivation Bullwhip Effect (BWE) refers to a progressive increase in order (demand) variance as order information passes upstream in a Supply Chain (SC) (Chatfield and Pritchard, 2013). BWE is the responsible of inefficiencies in terms of total costs increase, profitability deterioration, increased inventory holding costs, and higher cost of capital (Hassanzadeh et al., 2014; Li and Liu, 2013; Turrisi et al., 2013; Li, 2013). Nowadays, about two-thirds of firms are affected by the BWE (see e.g. Bray and Mendelson, 2012; Shan et al., 2013). Thus, BWE continues to be one of the most widely investigated phenomena in modern-day SC management research (Nepal et al., 2012; Zotteri, 2013). Among the streams of research dealing with BWE, an important one has focused on showing its existence and on identifying its possible causes (Sucky, 2009). Among the different root causes that have been identified (please see Section n. 2), the ‘number of echelons’ or ‘number of channel intermediaries’ (Disney and Lambrecht, 2008) is considered a root cause that explicitly depends on the structure of the SC. In fact, there is a common agreement on the existence of a positive correlation between the reduction of the intermediate stages in the SC and the reduction of the BWE (Disney et al., 2004; Paik and Bagchi, 2007; Disney and Lambrecht, 2008; Bottani and Montanari, 2010; Yang et al., 2011; Sodhi and Tang, 2011). For this reason, the reduction of channel intermediaries and the adoption of reduced SCs (such as the direct channel, or “the Dell model”) (Disney and Lambrecht, 2008) have been promoted as effective strategies to mitigate BWE. Dominguez R., Cannella S., Framinan J.M. 2015. The impact of the supply chain structure on bullwhip effect. Applied Mathematical Modelling, 39 (23-24), 7309-7325. DOI: https://doi.org/10.1016/j.apm.2015.03.012 3 However, SCs are usually networks or global networks (Corominas, 2013). Hence, the number of echelons only represents an indicator of the structure of the Supply Chain Network (SCN). The structure of the SCN, defined as the arrangement of the various SCN nodes (Giard and Sali, 2013) is a critical decision for managers that is becoming increasingly complex (Von Massow and Canbolat, 2014). In general, three main factors determine the structure of the SCN and consequently also the material flow from the raw materials stage to the final customer stage (Suchy, 2009): (1) the number of echelons, (2) the number of facilities at each echelon, and (3) the number of links between the locations. These elements may have a dramatic effect in terms of cost, customer satisfaction, ability to respond to market changes, and ability to innovate and bring new products to the market (Von Massow and Canbolat, 2014). Among these three elements, published works have only explicitly investigated the impact of one of them, i.e. the number of echelons in the BWE. Probably the reason is that most scientific works dealing with the BWE are confined to the classical single-echelon, dyadic, or serially-linked configurations (Sucky, 2009; Bhattacharya and Bandyopadhyay, 2011; Giard and Sali, 2013). In these configurations it is not possible to assess the impact of the aforementioned structural factors on the BWE, with the mere exception of the serially-linked configuration, where it is possible to quantify only the effect of the number of echelons. However, recent studies (see e.g. Dominguez et al., 2014 and Dominguez et al., 2015) show how different SCN configurations with the same number of echelons may have different SCN performances. Thus, there is a need to assess the impact of all SCNs structural factors on performance. To the best of our knowledge, the potential relation between key structural factors and the BWE is almost unknown, with the exception a few anecdotic evidences, which, however, do not provide information on the impact of the different factors in the BWE (see e.g. Sodhi and Tang, 2011). Motivated by these considerations, the aim of this paper is to quantify the impact of the SCN structure (i.e. the number of stages, the number of facilities at each stage, and the number of links between the locations) on the BWE. To do so, we simulate the dynamic response of several configurations in one of the most widely used SCN typologies in real businesses: the divergent SCN (Beamon and Chen, 2001). This configuration is characterized by a tree-like structure, where every stock point in the system receives supply from exactly one higher level stock point, but can supply to one or more lower level stock points (Hwarng et al., 2005). Consumer-oriented industries, such as cell phone manufacturers, appliances, Dominguez R., Cannella S., Framinan J.M. 2015. The impact of the supply chain structure on bullwhip effect. Applied Mathematical Modelling, 39 (23-24), 7309-7325. DOI: https://doi.org/10.1016/j.apm.2015.03.012 4 electronics, and computer industries often adopt this typology of SCN (Hung, 2011). To identify the structural factors having a statistically significant impact on the BWE, we perform a full factorial set of experiments by varying these factors under identical SCN operational parameters (e.g. lead times, safety stock factors, demand forecast factors, etc.). Furthermore, in order to increase the robustness of the analysis, we adopt the framework for studying the BWE proposed by Towill et al. (2007). More specifically, we adopt two input demand patterns, i.e. the variance lens and the shock lens. The former aims at inferring on the performance of SCNs for a stationary input demand. The latter aims at inferring on the performance of SCNs for an unexpected and intense change in the end customer demand. This type of demand has been adopted in theoretical BWE studies in order to model the extreme volatility and impetuous alteration of the market produced by the current economic recession (Cannella et al., 2014a). The simulation platform used in our work is SCOPE (Domínguez and Framinan, 2013), a multi-agent system (MAS) based software platform for the simulation of complex SCNs. Results for the variance lens show that the factor ‘number of echelons’ has a high impact on the BWE while the number of nodes and the divergence of the SCN have a low impact, which is in line with the results found by other authors. However, for the shock lens, in addition to the number of echelons, the number of nodes and the divergence of the SCN also have a significant impact on the BWE. More specifically, as the levels of the structural factors increase, the BWE increases with different trends. In fact, BWE quickly (exponentially) increases as the SCN shifts from a low number of echelons to a high number of echelons, but the increase is smoother with the number of facilities in each echelon and with the divergence of the SCN. Also, there is an important interaction between the number of echelons and the divergence of the SCN in this scenario. Finally, we prove how BWE is very sensitive to the structure of the SCN under a sudden shock in customer demand. The rest of the paper is organized as follows: Section 2 presents a literature review. Section 3 describes the structural elements of SCNs and the inherent structural characteristics of divergent SCNs. In Section 4 the SCN model is presented. Section 5 briefly describes the software platform used for computer simulation. Section 6 includes the design of experiments and Section 7 shows the results numerical analysis. Finally, Section 8 presents the implications of the research and Section 9 contains the conclusions and future research lines. Dominguez R., Cannella S., Framinan J.M. 2015. The impact of the supply chain structure on bullwhip effect. Applied Mathematical Modelling, 39 (23-24), 7309-7325. DOI: https://doi.org/10.1016/j.apm.2015.03.012 5 2 Literature Review The identification of the root causes of the BWE is an important stream in SCN literature and has long been of interest for industrial practitioners and academics (Lin et al., 2014b). In this context it is possible to distinguish two schools of thought, i.e.: the System Thinking school, and the Operations Managers’ school (Miragliotta, 2006). The former, focused on the behavioral causes, is mainly interested in the ‘‘systemic’’ nature of the SCN, reflecting a holistic perception of the causes of the BWE. The Operations Managers’ school focuses on the operational causes. Thus, it concentrates on single elements rather than on the whole system. Both schools have largely contributed in suitably defining causes and remedies for the BWE. Thanks to these efforts, during the last decades several classification frameworks have been proposed. Undoubtedly Lee et al. (1997) provided the seminal work that defined the BWE and identified the well-known five causes (Disney and Lambrecht, 2008; Zotteri, 2012). A further relevant framework was proposed by Geary et al. (2006). The authors identified 10 published causes of BWE, based on the works by Mitchell (1924), Wikner et al. (1992), and Lee et al. (1997). Bhattacharya and Bandyopadhyay (2011) identify 19 causes, 16 of them operational and 3 behavioral. Operational causes include demand forecasting (Syntetos et al., 2009; Trapero et al., 2012), order batching (Potter and Disney 2006), price fluctuation (Ma et al., 2013; Lu et al., 2012), rationing and shortage gaming, lead time, inventory policy, replenishment policy, improper control system (Disney and Towill 2003; Syntetos et al., 2011), lack of transparency (Cannella et al., 2014b; Hussain et al., 2012), number of echelons (Disney et al., 2004; Paik and Bagchi, 2007), multiplier effect, lack of synchronization (Ciancimino et al., 2012), misperception of feedback (Gonçalvez et al., 2005), local optimization without global vision (Disney and Lambrecht, 2008), company processes (Holweg et al, 2005, Cannella et al. 2014c) and capacity limits (Crespo-Marquez, 2010). The behavioral causes cover neglecting time delays in making ordering decisions (Wu and Katok, 2006), lack of learning and/or training (Akkerman and Voss, 2013, Bruccoleri et al., 2014), and fear of empty stock/customers’ baulking behavior (Croson and Donohue, 2006; Lin et al., 2014a). A recent classification of the BWE causes is provided by Lin et al. (2014b). However, all previous works have not considered the different factors of SCN structure as potential drivers of the BWE, with the mere exception of one factor: the number of echelons. To the best of the authors’ knowledge, the first framework that explicitly considers the SCN Dominguez R., Cannella S., Framinan J.M. 2015. The impact of the supply chain structure on bullwhip effect. Applied Mathematical Modelling, 39 (23-24), 7309-7325. DOI: https://doi.org/10.1016/j.apm.2015.03.012 6 structure as a root cause of the BWE is Giard and Sali (2013). The authors perform an extended literature review, classifying approximately 50 articles published in major journals. In their work, authors identify 7 root causes, being the “SCN structure” one among them. Furthermore, they classify each paper according to the adopted SCN configuration. More specifically, they identify the following 5 configuration:  Dyadic: single customer with a single supplier.  Serial: a succession of nodes in which each node has at most one predecessor and one successor.  Convergent (assembly): are assembly-type configurations in which each node in the SCN has at most one successor, but may have any number of predecessors.  Divergent or arborescent (distribution): if each node has at most one predecessor, but any number of successors.  General: is a general configuration that does not fall into any of the preceding configurations. Table 1 reports an adapted version of literature effort provided by Giard and Sali (2013). By analyzing this table, it can be noted that most studies have exclusively adopted the classical serially-linked SCN (Bhattacharya and Bandyopadhyay, 2011). Other studies have adopted the convergent configuration. However, none of these works explores the SCN structure as a potential driver for the BWE. Analogously, the few studies adopting the divergent configuration do not focus on the relation of between the structural factors and the BWE. According to Giard and Sali (2013), the only two works considering the SCN structure as a potential driver of the BWE are the framework of Geary et al. (2006) and the simulation study of Wangphanich et al. (2010). The framework of the former authors merely identifies the well-known "number of echelon" as a root cause of the BWE. Analogously, the latter authors, in their analysis of a multi-product SCN do not report any insight on how these structural factors influence the performance of the SCN. In fact, they focus on the dynamic response of a fixed SCN structure: a 3-echelon divergent SCN under different order policies and information sharing strategies. This finding stimulates the need of further structured studies on the relation between the structural factors of the SCN and BWE. Dominguez R., Cannella S., Framinan J.M. 2015. The impact of the supply chain structure on bullwhip effect. Applied Mathematical Modelling, 39 (23-24), 7309-7325. DOI: https://doi.org/10.1016/j.apm.2015.03.012 7 Table 1. SCN configuration and root causes (adapted from Giard and Sali, 2013). Kelepouris et al. (2007) Jung et al. (2007) Huang et al. (2005) Geary et al. (2006) Ganesh et al. (2008) Özelkan and Çakanyildirim (2009) Disney and Towill (2003) Dejonckheere et al. (2004) Dejonckheere et al. (2003) Dejonckheere et al. (2002) Croson and Donohue (2005) Childerhouse et al. (2008) Cheng (2009) Chen and Lee (2009) Chen et al. (2006) Chatfield et al. (2004) Chan et al. (2004) Cachon et al. (2007) Cachon and Lariviere (2001) Cachon and Fisher (2000) Campuzano et al(2009) Ben-Tal et al. (2009) Bayraktar et al. (2008) Balan et al. (2009) Bailey and Francis (2008) Agrawal et al. (2009) Articles X X X X X X X X X X X Dyadic SCN Configuration X X X X X X X X X X X Serial X X Divergent Convergent X X Network X X X X X X X X X X Reliability of Forecasts Causes of the BWE X SC Structure X X X X X Demand Variability X X X Pricing Policies X X Shortage Risk X X Lot-sizing Policy X X X X X X Lead Time Variability X X X X X X X X X X X Control Model X X X X X X X X X Shared Information X X Human Factor Zhao and Xie (2002) Zhang (2004) Yee (2005) Wu and Cheng (2008) Wright and Yuan (2008) Wong et al. (2009) Wangphanich et al. (2010) Wang et al. (2005) Viswanathan et al. (2007) Thonemann (2002) Sucky (2009) Springer and Kim (2010) Ryu et al. (2009) Pereira et al. (2009) Ouyang and Li (2010) Ouyang (2007) O´Donell et al. (2009) Nienhaus et al. (2006) Moinzadeh (2002) Liu et al. (2009) Li and Zhang (2008) Li and Gao (2008) Lee et al. (2004b) Lee et al. (2004a) Lee et al. (1997) Lau et al. (2004) Lau et al. (2005) Articles X X X X Dyadic SCN Configuration X X X X X X X X X X X Serial X X X X X X X Divergent Convergent X X Network X X X X X X X Reliability of Forecasts Causes of the BWE X X SC Structure X X X X Demand Variability X X X X Pricing Policies X X X Shortage Risk X X X Lot-sizing Policy Lead Time Variability X X X Control Model X X X X Shared Information X Human Factor 3 The Divergent SCN configuration This work focuses on the analysis of divergent SCNs. In this section, the structural elements of a SCN are described, and then, the inherent characteristics/constraints of this specific configuration are formalized. The SCN structure arises from the connected facilities that work together in order to supply products or services. In a SCN, each link represents the flow of materials and information that makes possible the functions of procurement, processing (or Dominguez R., Cannella S., Framinan J.M. 2015. The impact of the supply chain structure on bullwhip effect. Applied Mathematical Modelling, 39 (23-24), 7309-7325. DOI: https://doi.org/10.1016/j.apm.2015.03.012 8 manufacturing), storage and distribution. For any given SCN, each functional level comprises an echelon, and there may be numerous facilities within each echelon (Beamon and Chen, 2001). This definition of the SCN structure is in accordance with the growing literature on complex networks, in which the SCN is modeled as a network by a set of “nodes” that represent autonomous business units (firms or facilities), and a set of “connections” (links) that link these firms together in demand-supply relationships for the purposes of creating products or services (Hearnshaw and Wilson, 2013; Gerschberger et al., 2012; Wen et al., 2012; Kim et al., 2011; Li et al., 2010a; Li et al., 2010b; Li et al., 2009; Choi et al., 2001). Hence, in line with the literature, we formalize the structure elements of a SCN as follows:  Echelons: the number of echelons is denoted by 𝑖∈(1,𝐸), with E the total number of echelons in the SCN. Echelons are numbered downstream starting from the suppliers, which are in echelon i = 1.  Nodes: a generic node j in echelon i is denoted by 𝑛𝑖𝑗. The number of nodes in a specific echelon i is 𝑁𝑖. The total number of nodes in the SCN is: ∑𝑁𝑖 𝐸𝑖=1 =𝑁.  Links: a link between nodes 𝑛𝑖𝑗 and 𝑛𝑖′𝑗′ is denoted by 𝑙(𝑛𝑖𝑗,𝑛𝑖′𝑗′) and the total number of links is L. There are two commonly used indicators to measure the degree of linkage in a SCN, namely the connection degree and the cluster coefficient (see e.g. Wen et al., 2012; Kim et al., 2011; Xuan et al., 2011; Li et al., 2010a; Barbási et al., 2002) . The connection degree 𝐷𝑖𝑗 is defined as the sum of a node’s links (Li et al., 2010a). The number of suppliers linked with node 𝑛𝑖𝑗 is the in-degree (𝑑𝑖𝑖𝑗), and the number of customers linked with a node 𝑛𝑖𝑗 is the out-degree (𝑑𝑜𝑖𝑗) (Kim et al., 2011; Xuan et al., 2011). The sum of the in-degree and the out-degree is the connection degree: 𝐷𝑖𝑗=𝑑𝑖𝑖𝑗+𝑑𝑜𝑖𝑗. The clustering coefficient C is the probability that two nearest neighbors of a node are also nearest neighbors of one another (Li et al., 2010a). Given node 𝑛𝑖𝑗 linked to 𝑘𝑖 other nodes in the system, if these 𝑘𝑖 nodes form a fully connected clique, there are 𝑘𝑖(𝑘𝑖−1)/2 links between them. Let us denote by 𝜆𝑖 the number of links that connect the selected 𝑘𝑖 nodes to each other. The clustering coefficient for node 𝑛𝑖𝑗 is then 2𝜆𝑖/𝑘𝑖(𝑘𝑖−1) (Barbási et al., 2002). The number of nodes, the number of echelons, and the structure of the material and information flows (links) has given rise to a structural classification scheme of SCNs based on the material relationship between nodes (Beamon and Chen, 2001). Up to now, most of the literature on the BWE topic has analyzed the classical serial SCN (see Sections n.1 and 2). In Dominguez R., Cannella S., Framinan J.M. 2015. The impact of the supply chain structure on bullwhip effect. Applied Mathematical Modelling, 39 (23-24), 7309-7325. DOI: https://doi.org/10.1016/j.apm.2015.03.012 9 this SCN, the number of nodes in each echelon is limited to one (𝑁𝑖=1), and hence, the number of nodes and echelons in the SCN is the same (𝑁=𝐸). The connection degree is also limited: each node supplies to one node in the successor echelon (𝑑𝑜𝑖𝑗=1) and it is supplied by one node in the predecessor echelon (𝑑𝑖𝑖𝑗=1), thus limiting the total number of links to 𝐿=𝑁−1. Summing up, the structure of the serial SCN configuration is very restrictive: by selecting the quantity of one of the structural elements above mentioned (echelons, nodes or links), the SCN structure is defined, thus limiting the analysis of the influence of the SCN structure on the BWE to the number of echelons. In this paper, we focus on the divergent SCN configuration, which is less restrictive than the serial configuration. The inherent structural restrictions of divergent SCNs are described and formalized next: 1. The number of nodes in each echelon is equal or greater than the number of nodes in its predecessor, i.e.: 𝑁𝑖≥𝑁𝑖−1. Furthermore, in order to exclude the serial SCN, the total number of nodes is constrained to 𝑁≥𝐸+1. 2. A node 𝑛𝑖𝑗 can supply to any number of nodes in the successor echelon (𝑑𝑜𝑖𝑗≥1), but can be supplied only by one node from the predecessor echelon (𝑑𝑖𝑖𝑗=1) (Beamon and Chen, 2001). 3. Nodes in the same echelon are not linked. Hence, the network clustering coefficient C is zero. This is consistent with most cases in real-world SCNs (e.g. divergent SCN), that is, entities in the same echelons normally have no demand-supplier relations (Li et al., 2010a). This constraint, together with the previous restriction, limits the total number of links to the total number of nodes minus one: 𝐿=𝑁−1. By observing the above constraints, it can be noted that N is greater than E in divergent SCNs and thus, echelons are allowed to contain more than one node. Furthermore, for a given E, there is no upper bound for N. Thereby, any distribution of nodes across the SCN satisfying restriction (1) is allowed. In addition, nodes can supply to any number of nodes downstream, as indicated by restriction (2). Hence, there might be nodes with a high connection degree while others with low connection degree, resulting in SCNs with different degree distributions. In a first attempt to measure the impact of the SCN structure on the BWE, in this paper we do not consider the connection degree as a factor, and for this reason, the divergent SCNs under analysis have homogeneous degree distributions: all nodes in the same Dominguez R., Cannella S., Framinan J.M. 2015. The impact of the supply chain structure on bullwhip effect. Applied Mathematical Modelling, 39 (23-24), 7309-7325. DOI: https://doi.org/10.1016/j.apm.2015.03.012 16 𝑁(50,202) distribution. In the shock lens scenario, a 𝑁(50,202) distribution suffer an average increment of 100% in a certain time period (see Table 3), turning into a 𝑁(100,202). These demand patterns are applied to every customer in the SCN. A set of the above mentioned 2,700 experiments have been run using the variance lens and another identical set have been run suing the shock lens, making a total of 5,400 experiments. To isolate the effects of the structural factors on the BWE, other characteristics which are known to be BWE initiators, with the exception of the stochastic demand and its forecast, are not included in the SCN model. The selection of the parameter’s values of the SCNs has been done according to Chatfield et al. (2004) (see Table 3). The simulation horizon is set to 900, with the first 400 periods used as a warm-up used to set up the system. Table 3. Model’s parameters. P Periods of forecasting 15 Z Safety factor 2 (service level of 97.72%) R Review interval 1 L Lead time 4 simTime Simulation time 900 wUP Warm-up 400 vL Variance Lens 𝑁(50,202) ∀ 𝑡 sL Shock Lens 𝑁(50,202) 𝑡∈ [0-549] 𝑁(100,202) 𝑡∈ [550-900] In order to measure the BWE, we found several key performance indicators in the literature. The order rate variance ratio, first proposed by Chen et al. (2000) and often computed as the ratio of the order variance in a generic node and the order variance of the customer, is by far the most widely used indicator to detect the BWE (Cannella, 2014; Cannella et al., 2013). This metric is appropriate in situations where customer demand is stochastic, following a probability distribution, e.g. the variance lens scenario (Towill et al., 2007). However, as we also consider the shock lens scenario, a peak of orders metric has been chosen to measure the extreme swings in order patterns (Towill et al., 2007). Since the dynamics of the order pattern at the first echelon presents the “worst-case” scenario, the BWE registered at this echelon is analyzed (Hussain et al. 2012). Hence, we measure the BWE as the maximum change in orders placed by nodes in the first echelon. In order to obtain this measure, we have to note that, in a divergent SCN, echelons are allowed to contain more than one node, and thereby it is necessary to find an aggregate measure. Therefore, the sum of orders of every node j in the Dominguez R., Cannella S., Framinan J.M. 2015. The impact of the supply chain structure on bullwhip effect. Applied Mathematical Modelling, 39 (23-24), 7309-7325. DOI: https://doi.org/10.1016/j.apm.2015.03.012 17 echelon i (𝑂𝑖𝑗 𝑡) are considered, resulting in an aggregate order pattern for the echelon i: 𝐴𝑂𝑖𝑡= ∑𝑂𝑖𝑗 𝑡 𝑛𝑖 𝑗=1 . Thus, we formalize the peak of orders in echelon one as follows: 𝑃𝑒𝑎𝑘𝑂1=max(𝐴𝑂1𝑡)−min(𝐴𝑂1𝑡)∀𝑡∈[𝑤𝑈𝑃,𝑠𝑖𝑚𝑇𝑖𝑚𝑒] (6) 7 Results and numerical analysis In order to identify the statistically significant factors, two ANOVAs are performed separately for the variance lens and the shock lens, and both scenarios are analyzed. The independent variables are factors E, N, and DivF, while the dependent variable is the level of order instability at the first echelon (𝑃𝑒𝑎𝑘𝑂1) in the SCN. Systems are often driven primarily by some of the main effects and low-order interactions, say, two-factor interactions, while higher order interactions are negligible for all practical purposes (Hinkelmann and Kempthorne, 1994). ‘Main effect’ refers to the effect of a structural factor on the BWE when the factor’s value changes from one level to another. Interaction refers to the effect of changes in a particular structural factor value as the values of another factor change. Since high-order interactions are often minimal, only information on the main effects and low-order interactions is analyzed for each scenario. After analyzing the variance and the shock lens scenarios, a comparison between both of them is performed. Variance Lens ANOVA results are presented in Table 4, where the degree of freedom (DOF) of each factor, F-ratios, p-values, and partial 𝑅2 are shown. When all factors are considered together, the model is statistically significant with a 95% confidence level. The value of 𝑅2 is 0.891, indicating that 89.1% of the variation in 𝑃𝑒𝑎𝑘𝑂1 can be explained by the structural factors. Furthermore, it can be seen that all structural factors are statistically significant, as well as the interaction between echelons and nodes. Figure 2 shows the main effects of structural factors (E, N, DivF) by plotting the mean 𝑃𝑒𝑎𝑘𝑂1 values for each level of the factor (Low, Medium, High). These values are calculated for a given structural factor by averaging the results obtained for all levels of the other structural factors (i.e. 𝑃𝑒𝑎𝑘𝑂1 for Low DivF is calculated by averaging the results obtained for Low, Medium and High levels of E and N when DivF was Low). In the subsequent analysis, all 𝑃𝑒𝑎𝑘𝑂1 values are divided by 104 (1E4). Dominguez R., Cannella S., Framinan J.M. 2015. The impact of the supply chain structure on bullwhip effect. Applied Mathematical Modelling, 39 (23-24), 7309-7325. DOI: https://doi.org/10.1016/j.apm.2015.03.012 18 Table 4. ANOVA results in Variance Lens scenario. Factors DOF F-ratio p-value 𝑹𝟐 (percent) Model 17 1290.495 <0.001 89.1 Echelons 2 10776.161 <0.001 88.9 Nodes 2 111.484 <0.001 7.7 Divergence 1 141.558 <0.001 5.0 Echelons * Nodes 4 3.196 0.013 0.5 Echelons * Divergence 2 1.498 0.224 0.1 Nodes * Divergence 2 2.130 0.119 0.2 Looking at the main effects in Figure 2 and in Table 4, it can be noted that the most significant factor is the number of echelons: SCNs with higher number of echelons show higher BWE, following an exponential trend. This result is in line with numerous works that already have identified the number of echelons as one of the most influential in contributing to the BWE (Bottani and Montanari, 2010; Paik and Bagchi, 2007; Chatfield et al., 2004; Disney et al., 2004, among others). In fact, by adding echelons to a SCN, the number of decision points increase, contributing to a higher distortion of the demand. Thus, each SCN member faces a more fluctuating order pattern (Paik and Bagchi, 2007). This behavior can be observed in Figure 2: SCNs with a low number of echelons show low values of 𝑃𝑒𝑎𝑘𝑂1, but this indicator abruptly increases when moving to SCNs with medium and high number of echelons. Figure 2. Main effects in Variance Lens scenario. The number of nodes and the divergence of the SCN are both significant, but with a lower impact on the BWE as compared to the number of echelons. Considering that each node 0 10 20 30 40 50 60 EL EM EH NL NM NH DivFL DivFH PeakO1/1E4 Dominguez R., Cannella S., Framinan J.M. 2015. The impact of the supply chain structure on bullwhip effect. Applied Mathematical Modelling, 39 (23-24), 7309-7325. DOI: https://doi.org/10.1016/j.apm.2015.03.012 19 distorts the demand signal due to inventory policies, forecast rules and lack of coordination, demand distortion is higher when increasing the number of nodes in the SCN and hence, BWE increases. More specifically, by increasing the total number of nodes in a given SCN, we are in fact increasing the number of nodes per echelon (see Figure 3a). In this situation, nodes may have to fill the demand of a higher number of nodes and hence, they have to face a higher variability of orders and, consequently, BWE increases. However, under a stationary market demand nodes are able to make proper forecasts and fill the incoming orders with a high customer service, showing the SCN a stable behavior. Thus, an increase in the number of nodes per echelon has a low impact on the BWE. The impact of the divergence of the SCN on the BWE is of similar magnitude as the impact of the number nodes. In a SCN with low divergence (see e.g. Figure 3b), nodes are uniformly distributed along the echelons (the number of nodes per echelon (∑𝑛𝑖𝑗𝑗) is close to the average (N/E)). In this situation, demand is also uniformly distributed among the different nodes, thus limiting the amplification effect. However, when the divergence of the SCN increases (∑𝑛𝑖𝑗𝑗 is far from N/E), there are one or more critical echelons in which the number of nodes abruptly increases and therefore, there are few nodes supplying a high number of nodes downstream in these echelons, as it can be seen in Figure 3b. This situation increases the variability of orders received by these nodes and hence, increases the BWE. But as for the number of nodes, the SCN shows a stable behavior due to a stationary market demand and thus, changing the divergence of the SCN has a low impact on the BWE. (a) (b) Figure 3. Increasing N (a) and divF (b) in a divergent SCN. Dominguez R., Cannella S., Framinan J.M. 2015. The impact of the supply chain structure on bullwhip effect. Applied Mathematical Modelling, 39 (23-24), 7309-7325. DOI: https://doi.org/10.1016/j.apm.2015.03.012 20 Finally, there is one significant interaction between the number of echelons and the number of nodes, although it has a low impact on the overall BWE and so it is not described. Shock Lens ANOVA results are summarized in Table 5. When all factors are considered together, the model is statistically significant at a 95% confidence level with an overall 𝑅2 of 0.892, indicating that 89.2% of the variation in 𝑃𝑒𝑎𝑘𝑂1 can be explained by the structural factors considered. Furthermore, all factors are found to be statistically significant, as well as two of the interactions. Similarly to the variance lens analysis, Figure 4 shows the main effects of the structural factors by plotting the average 𝑃𝑒𝑎𝑘𝑂1 for each level of the factors (Low, Medium, High). Note that all values appear divided by 104 (1E4). Table 5. ANOVA results in Shock Lens scenario. Factors DOF F-ratio p-value 𝑹𝟐 (percent) Model 17 1305.427 <0.001 89.2 Echelons 2 10231.880 <0.001 88.4 Nodes 2 439.303 <0.001 24.7 Divergence 1 693.143 <0.001 20.5 Echelons * Nodes 4 1.877 0.112 0.3 Echelons * Divergence 2 65.821 <0.001 4.7 Nodes * Divergence 2 7.909 <0.001 0.6 In view of the main effects in Figure 4 and the data from Table 5, it is noticeable that the most significant factor on the BWE is the number of echelons. In addition, the number of nodes and the divergence of the SCN have also a significant impact on the BWE. SCNs with higher number of echelons show higher BWE, following an exponential trend. The shock in the average demand causes an unexpected multi stock-out at the retailer level. Nodes at this level react by placing orders larger than usual to the upstream nodes, which fall in a stock-out situation too. This effect is amplified from one echelon to another, increasing the fluctuation of orders through the SCN and causing the high 𝑃𝑒𝑎𝑘𝑂1 values observed in Figure 4. Dominguez R., Cannella S., Framinan J.M. 2015. The impact of the supply chain structure on bullwhip effect. Applied Mathematical Modelling, 39 (23-24), 7309-7325. DOI: https://doi.org/10.1016/j.apm.2015.03.012 21 Figure 4. Main effects in Shock Lens scenario. The number of nodes and the divergence of the SCN show now a significant and relative high impact on the BWE, as it can be deducted from partial 𝑅2 in Table 5. In this case, the shock in the market demand makes a SCN with higher number of nodes to be more vulnerable to the BWE: the distortion of the demand signal caused by each node due to inventory policies, forecast rules and lack of coordination, becomes more important under a shock in demand than under a stationary demand. Furthermore, nodes filling demand from a higher number of nodes downstream are now affected by the unexpected shock wave transmitted upstream by the shock in demand and thus have to face a higher variability, consequently increasing the BWE. In case of the divergence of SCNs in the shock lens scenario, the presence of critical echelons due to a high divergence leads to a situation in which there are few nodes supplying a high number of nodes downstream in these echelons and thus, they are very sensitive to the shock wave transmitted upstream by the shock in demand, showing a higher BWE. There are two significant interactions in this scenario. The most important one takes place between the number of echelons and the divergence of the SCN. An “interaction plot” is used to determine the severity of this interaction. Due to the exponential nature of the obtained interaction curves, we have used logarithms to transform them into linear curves in order to make their interpretation clearer. In Figure 5a it can be seen that the linearized interaction curves are not parallel, which means that an interaction occurs between both factors. The DivFH curve shows a higher slope than the DivFL curve. Therefore BWE is more sensitive to the number of echelons in SCNs with high divergence than in SCNs with low divergence. In addition, BWE is more sensitive to the divergence in SCNs with high number of echelons than in SCNs with low number of echelons 0 100 200 300 400 500 600 700 800 EL EM EH NL NM NH DivFL DivFH PeakO1/1E4 Dominguez R., Cannella S., Framinan J.M. 2015. The impact of the supply chain structure on bullwhip effect. Applied Mathematical Modelling, 39 (23-24), 7309-7325. DOI: https://doi.org/10.1016/j.apm.2015.03.012 22 It is important to notice that although Figure 5a may indicate that there is not a strong interaction between the number of echelons and the divergence of the SCN, it is necessary to consider that the interaction has been plotted over Ln(𝑃𝑒𝑎𝑘𝑂1). In order to clarify how important this interaction is, the original 𝑃𝑒𝑎𝑘𝑂1values are shown in Figure 5b, where it can be appreciated that both curves are different. Figure 5c plots the percentage increase of 𝑃𝑒𝑎𝑘𝑂1between the scenarios DivFl and DivFH. In this figure it can be seen a clear increasing trend as the number of echelons increase, thus confirming a strong interaction between these two structural factors. In addition, using a single variable test we have verified the significance of the interaction for each level of the structural factors: the impact of the different levels of the divergence of the SCN on the BWE have been tested for each level of the number of echelons and vice-versa, obtaining that all contrasts are statistically significant (p<0.001). (a) (b) (c) Figure 5. Interaction between E and the DivF in Shock Lens scenario. Another significant interaction occurs between the number of nodes and the divergence of the SCN, but it has a very low impact on the overall BWE and so it is not described. 7 8 9 10 11 12 13 14 15 16 EL EM EH Ln(PeakO1) DivFL DivFH 0 200 400 600 800 1000 1200 1400 EL EM EH PeakO1/1E4 DivFL DivFH 0 50 100 150 200 250 300 350 400 EL EM EH (%) PeakO1 increase Dominguez R., Cannella S., Framinan J.M. 2015. The impact of the supply chain structure on bullwhip effect. Applied Mathematical Modelling, 39 (23-24), 7309-7325. DOI: https://doi.org/10.1016/j.apm.2015.03.012 23 A comparison between the variance lens scenario and the shock lens scenario In order to simplify the comparison between both scenarios, a summary of numeric results is provided in Table 6. This table shows the average of 𝑃𝑒𝑎𝑘𝑂1 (scaled by 104) and the 95% confidence intervals (CIs) given by their lower and upper bounds for all the 32 experimental points under analysis (18 for the variance lens and 18 for the shock lens). There are three important differences between the variance and the shock lens scenarios. First of all, the number of nodes and the divergence of the SCN have a higher impact on the BWE in the shock lens scenario than in the variance lens scenario (see 𝑅2 in Tables 4 and 5). This fact is confirmed by comparing the main effects of the number of nodes and the divergence of the SCN in both scenarios (see Figure 6): 𝑃𝑒𝑎𝑘𝑂1 curves show higher slopes in the shock lens scenario than in the variance lens scenario and hence, BWE is more sensitive to these factors in the former scenario than in the latter scenario. The higher number of nodes per echelon and/or the presence of critical echelons (SCNs with high divergence) make the SCN more vulnerable to an unexpected shock in demand and the consequent multi stock-out situation. Figure 6. A comparison of the main effects of E, N and DivF between Variance Lens and Shock Lens scenarios. 0 100 200 300 400 500 600 700 800 EL EM EH PeakO1/1E4 Shock Lens Variance Lens 0 50 100 150 200 250 300 350 400 450 NL NM NH DivFL DivFH PeakO1/1E4 Shock Lens Variance Lens Dominguez R., Cannella S., Framinan J.M. 2015. The impact of the supply chain structure on bullwhip effect. Applied Mathematical Modelling, 39 (23-24), 7309-7325. DOI: https://doi.org/10.1016/j.apm.2015.03.012 24 Table 6. Average 𝑷𝒆𝒂𝒌𝑶𝟏 and 95% confidence intervals from ANOVA. Lens Average 𝑷𝒆𝒂𝒌𝑶𝟏/𝟏𝑬𝟒 95% CI Lower Bound 95% CI Upper Bound Echelons Nodes Divergence Factor: Low Low Low Variance 0.11682 0.10521 0.12843 Shock 0.19005 0.16911 0.21099 Medium Variance 0.15991 0.14452 0.1753 Shock 0.32386 0.27305 0.37467 High Variance 0.22061 0.19993 0.24129 Shock 0.58536 0.52231 0.64841 Medium Low Variance 1.6834 1.50315 1.86365 Shock 3.8115 3.10436 4.51864 Medium Variance 2.09657 1.947 2.24614 Shock 8.14334 6.76431 9.52237 High Variance 2.51728 2.26373 2.77083 Shock 13.89944 11.5298 16.26908 High Low Variance 31.47346 28.24269 34.70423 Shock 116.54383 90.83327 142.25439 Medium Variance 39.51214 36.55075 42.47353 Shock 240.26322 194.30447 286.22197 High Variance 48.0766 43.90626 52.24694 Shock 431.59873 350.41371 512.78375 Echelons Nodes Divergence Factor: High Low Low Variance 0.14833 0.13362 0.16304 Shock 0.24597 0.21766 0.27428 Medium Variance 0.23406 0.20556 0.26256 Shock 0.65664 0.55743 0.75585 High Variance 0.30149 0.2749 0.32808 Shock 1.05754 0.89348 1.2216 Medium Low Variance 2.26011 1.97476 2.54546 Shock 10.84904 9.05746 12.64062 Medium Variance 3.45542 3.04776 3.86308 Shock 25.46908 21.4585 29.47966 High Variance 3.84292 3.40116 4.28468 Shock 48.33374 41.71319 54.95429 High Low Variance 47.77472 40.84654 54.7029 Shock 381.67474 320.13077 443.21871 Medium Variance 66.88626 55.88368 77.88884 Shock 1057.80539 885.78346 1229.82732 High Variance 83.14433 66.85235 99.43631 Shock 2042.64747 1655.25052 2430.04442 A second important difference between both scenarios is that the BWE is higher in the shock lens scenario in all cases at a 95% confidence level (see Table 6 and Figure 6). Furthermore, since the shock lens scenario presents higher values of 𝑃𝑒𝑎𝑘𝑂1 and higher slopes than the variance lens scenario, the discrepancies in terms of BWE between both scenarios increase as Dominguez R., Cannella S., Framinan J.M. 2015. The impact of the supply chain structure on bullwhip effect. Applied Mathematical Modelling, 39 (23-24), 7309-7325. DOI: https://doi.org/10.1016/j.apm.2015.03.012 25 the levels of the three structural factors increase. In order to quantify these discrepancies we employ a measure of the relative increase of the average BWE in the shock lens scenario over the average BWE in the variance lens scenario (see equation 7). ∆=(𝑃𝑒𝑎𝑘𝑂1𝑠ℎ𝑜𝑐𝑘𝐿𝑒𝑛𝑠−𝑃𝑒𝑎𝑘𝑂1𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒𝐿𝑒𝑛𝑠) 𝑃𝑒𝑎𝑘𝑂1𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒𝐿𝑒𝑛𝑠 ∗100 (7) By plotting ∆ for each level of the structural factors in Figure 7 we observe how the discrepancies between both scenarios show an increasing trend for each factor. This metric reveals an interesting behavior of the divergent SCN: as the structural complexity of the SCN increases (by means of the number of echelons, the number of nodes and/or its divergence) the differences between both scenarios increase and thus, the SCN becomes more vulnerable to unexpected shocks in market demand. Figure 7. BWE discrepancies between Variance Lens and Shock Lens scenarios. Finally, the third important difference between both scenarios refers to the interactions between the structural factors: in the variance lens scenario there is a significant interaction between the number of echelons and the number of nodes that however has a low impact on the overall BWE. In the shock lens scenario one of the two significant interactions found has an important impact on the overall BWE: the interaction between the divergence of the SCN and the number of echelons. 0 500 1000 1500 2000 Low Medium High Δ(%) E N DivF Dominguez R., Cannella S., Framinan J.M. 2015. 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