Capacity restrictions and supply chain performance: Modelling and analysing load-dependent lead times

“Capacity restrictions and supply chain performance: Modelling and analyzing load-dependent lead times”, by S. Cannella, R. Domínguez, B. Ponte, & J. M. Framinan. Article accepted by the International Journal of Production Economics. DOI: 10.1016/j.ijpe.2018.08.008 1 Capacity restrictions and supply chain performance: Modelling and analysing load-dependent lead times Abstract Several studies have reported that capacitated supply chains may benefit from an improved dynamic performance as compared to unconstrained ones. This occurs as a consequence of the capacity limit acting as a production smoothing filter. In this research, the relationship between capacity restrictions and the operational performance of supply chains is investigated from a novel perspective, i.e. we assume that the influence of capacity constraints on lead times depends on the supply chain’s responsiveness. Under these circumstances, the experiments show that there is a negative effect of capacity constraints on supply chain performance, both in terms of process efficiency (internal) and customer satisfaction (external). Nonetheless, the magnitude of this impact greatly depends on the responsiveness of the firm, market conditions and adopted policies for inventory management. More specifically, the combination of tight capacity restrictions and low responsiveness significantly contributes to decrease supply chain performance, which may be very damaging for the dynamic behavior of the system. In this sense, efficient capacity planning proves to be essential to prevent the supply chain from entering into a vicious circle. Keywords Bullwhip Effect; capacity constraints; CT-TP curve; lead time; responsiveness; supply chain dynamics. “Capacity restrictions and supply chain performance: Modelling and analyzing load-dependent lead times”, by S. Cannella, R. Domínguez, B. Ponte, & J. M. Framinan. Article accepted by the International Journal of Production Economics. DOI: 10.1016/j.ijpe.2018.08.008 2 1. Introduction. 1.1. Context. Supply chains are severely impacted by the dynamic phenomenon known as the Bullwhip Effect (Lee et al., 1997), which amplifies the variability of orders as they pass through the various echelons of the system. The first records of this phenomenon by Procter & Gamble date back to the 1910s (Schisgall, 1981). Some decades later, Forrester (1958) initiated its theoretical analysis via system dynamics simulation in the MIT. However, it was not until the 1990s when the Bullwhip Effect became a major stream of research within the management literature, which is possibly a consequence of the new business environment drawn by globalisation. In this increasingly competitive scene, supply chain management has become a key success factor for firms (Buchmeister et al., 2014), while the Bullwhip Effect, which creates a climate of instability in production and distribution systems, tends to decrease the firms’ operational and financial performance (Disney and Lambrecht, 2008). A direct consequence of the Bullwhip Effect is the generation of highly variable production schedules (Disney and Towill, 2003). Consequently, companies need to invest in high capacity to meet peaks in demand, while this capacity will be later underutilised when demand decreases. In this sense, the Bullwhip Effect closely relates to a traditional problem in the operations management field, namely, the capacity choice (Cachon and Lariviere, 1999). Under these circumstances, the relationship between capacity constraints and the dynamic performance of supply chains has been explored by several authors. Interestingly, some works have observed a positive impact of capacity restrictions on supply chain performance that emerges from these acting as a production smoothing mechanism (Evans and Naim, 1994; Cannella et al., 2008; Chen and Lee, 2012; Ponte et al., 2017). Nonetheless, this operational improvement generally occurs at the expense of decreasing the inventory performance of the system, which in turn results in a decreased customer service level (Evans and Naim, 1994; Nepal et al., 2012; Hussain et al., 2016; Ponte et al., 2017). 1.2. Problem statement. Despite the evident interest on capacitated supply chains, none of these prior studies have introduced into the analysis a common cause-effect relationship in real-world “Capacity restrictions and supply chain performance: Modelling and analyzing load-dependent lead times”, by S. Cannella, R. Domínguez, B. Ponte, & J. M. Framinan. Article accepted by the International Journal of Production Economics. DOI: 10.1016/j.ijpe.2018.08.008 3 supply chains: the impact of capacity constraints on the lead time (Upasani and Uzsoy, 2008; Orcun et al. 2009, Fransoo and Lee, 2013; Kacar et al., 2016), as high orders usually increase the time that manufacturers, with constrained capacities, need to replenish these orders (Sterman, 2006; Boute et al., 2007). In other words, decreasing capacity tends to increase lead times throughout the supply chain. Note that this situation may lead to a ‘vicious circle’: The Bullwhip Effect increases the lead times in the supply chain, which in turn causes higher Bullwhip Effect due to the need of protecting against demand uncertainty for longer periods of time. This circle has been previously identified (Disney and Lambrecht, 2008), but has been barely explored in the literature (Childerhouse et al., 2008). To the best of the author’s knowledge, only three studies explore supply chain performance explicitly assuming load-dependent lead times, namely, Helo et al. (2000), Boute et al. (2009) and Framinan (2017). Helo et al. (2000) employ a lead time factor defined as the ratio of the backlog to the capacity. Boute et al. (2009) use a time queueing model to estimate the lead time distribution. Framinan (2017) models a case in which capacity is directly linked to current/past orders and/or demand (among others). All three studies suggest that capacity imitations may increase operational costs, including those related to the Bullwhip Effect, in countertendency with other studies --such as the aforementioned Evans and Naim, (1994); Cannella et al., (2008); Chen and Lee, (2012); Ponte et al., (2017) -- assuming the rejection of orders in excess of a capacity threshold. These works have provided significant insights regarding modelling and analysis of capacitated supply chains with load-dependent lead time. However, all three studies have been developed under a number of rather restrictive assumptions and thus, the impact of capacity has been studied only for a limited number of market and decision-making scenarios. As a summary, we can conclude that the impact of load-dependent lead times and capacity constrained supply chains has been understudied in the literature of supply chain dynamics (see Section 2). 1.3. Objective. Motivated by the above-mentioned considerations, we argue that a possible avenue to improve the understanding of the effect of capacity constraints in supply chain performance can be given by the two following objectives: (1) To explicitly model and analyze the effect of capacity and load-dependent lead time on the basis of empirically-driven assumptions/observations “Capacity restrictions and supply chain performance: Modelling and analyzing load-dependent lead times”, by S. Cannella, R. Domínguez, B. Ponte, & J. M. Framinan. Article accepted by the International Journal of Production Economics. DOI: 10.1016/j.ijpe.2018.08.008 4 (2) To explore the impact of capacity restrictions under a variety of scenarios, including supply chain responsiveness, variability of market demand, and replenishment decisions. To fulfill objective (1) we model the capacity using a cycle time-throughput (CT-TP) curve, a load-dependent lead time model commonly adopted in industry for the estimation of cycle times. More specifically, the CT-TP curve empirically quantifies the relationship between the average cycle time and the throughput rate (Ankenman et al., 2011). By doing so, we provide novel insights regarding the relation between capacity constraints and the operational performance of supply chains by investigating how load-dependent lead times influence the behavior of capacitated supply chains. To fulfill objective (2) we explore the capacitated supply chain for different levels of supply chain responsiveness, variability of market demand and replenishment decisions. More specifically, supply chain responsiveness is considered as the ability of the system in delivering the same product within a shorter lead time. The influence of market demand and replenishment decisions are modelled by considering different levels of turbulence in a customer demand (i.e., coefficient of variation), order policies (i.e., classical order-up-to (OUT) and smoothing OUT) and customer service level (i.e., safety stock factors). We adopt a performance measurement system aimed at capturing both operational costs (i.e., demand amplification and inventory instability) and customer satisfaction (i.e., percentage of delivering products to customer) Our methodological approach is based on modelling and simulation techniques and supported by inferential statistics (Kleijnen et al. 2008, Evers and Wang 2012.). The capacitated supply chain is modelled and implemented using difference equation (Riddals et al., 2000). The experiments have been carried out according to a full factorial design and their results have been examined by analyzing the main effect of the capacity and its interaction with the other four analyzed factors (i.e., responsiveness, coefficient of variation of customer demand, proportional controller of the OUT and safety stock factor). The results suggest that, in a capacitated supply chain with load-dependent lead time, capacity constraints significantly impact supply chain performance, and that this impact depends on the responsiveness of the supply chain (lead time increase), market conditions (demand variability), and on the replenishment decisions (safety stock factor and proportional controller). In view of the results obtained, we derive three main implications for researchers and “Capacity restrictions and supply chain performance: Modelling and analyzing load-dependent lead times”, by S. Cannella, R. Domínguez, B. Ponte, & J. M. Framinan. Article accepted by the International Journal of Production Economics. DOI: 10.1016/j.ijpe.2018.08.008 5 practitioners regarding (1) the assumption on capacity in supply chains, (2) the investments on capacity and higher responsiveness of production-distribution system, and (3) the use of some inventory management decisions for limiting the potential negative effect of capacity restrictions. The rest of this article is structured as follows. Section 2 reviews the most relevant literature on the subject. In Section 3 we detail the capacitated supply chain model, with an especial emphasis on the lead time function, and define the key performance metrics. Section 4 describes the experiments and presents the main results and findings. In Section 5 we discuss the managerial implications reflected from our results. Finally, Section 6 concludes and paves avenues for future research. 2. Literature review: the capacity-constrained supply chain. Many studies in the supply chain field assume unconstrained production, distribution, and storage capacities, which can be interpreted more as a necessity than as an attempt to model real-world systems (Shukla and Naim, 2017). Although “one of the relevant features of the global enterprise business network is the constrained capacity of production plants and distribution centres” (Ciancimino and Cannella, 2009), some common techniques in this area —such as control engineering or stochastic analysis— present serious difficulties when dealing with nonlinearities (Grubbström and Wang, 2000; Riddalls and Bennett, 2002). For this reason, the works investigating the implications of capacity limits on the supply chain response are relatively scarce and most of them have been carried out using simulation techniques, as highlighted by Cannella et al. (2008) and Ponte et al. (2017). This section is devoted to review these studies, whose the most relevant information is included in Table 1. The first column shows the reference to the article. Then, it follows the methodological approach, the variable over which the constraint was placed, and the assumptions on lead time modelling. Finally, we summarize their main conclusions regarding the impact of capacity restrictions on supply chain performance. Table 1 shows that the dynamic analysis of the capacity-constrained supply chain was initiated by Tang and Naim (1994). They compared eight different three-echelon supply chains that only differ in the capacity assumptions, and discovered that the unconstrained system did not produce the best response. The operational “Capacity restrictions and supply chain performance: Modelling and analyzing load-dependent lead times”, by S. Cannella, R. Domínguez, B. Ponte, & J. M. Framinan. Article accepted by the International Journal of Production Economics. DOI: 10.1016/j.ijpe.2018.08.008 6 Table 1. Relevant research on the operational impact of capacity constraints on supply chains. [Ref] Authors (Year) Methodological approach Capacity limit Lead time assumption Key findings [i] Evans and Naim (1994) Simulation (differential equations modelling) Order rate Independent • Capacity limits tend to decrease inventory service levels, but they generally lead to an improved dynamic performance. • Overall, the unconstrained system does not always produce the best response. [ii] de Souza et al. (2000) Simulation (system dynamics) Production Independent • Capacity constraints have a major impact on supply chain dynamics and costs. • The supply chain response is seriously damaged by capacity shortages; hence capacity planning becomes essential. [iii] Helo (2000) Simulation (system dynamics) Production Load-dependent • Reduced capacity damages the agility of the supply chain. • Capacity utilization can be used as a substitute for inventory. [iv] Vlachos and Tagaras (2001) Mathematical analysis & simulation (differential equations modelling) Order rate Independent • Capacity limits have a negative impact on system performance especially when lead times are long. [v] Wilson (2007) Simulation (system dynamics) Transportation (short-term) Independent • Short-term capacity loss due to transportation disruption results in a reduced fill rate, but it may generate a dynamic improvement in the system. “Capacity restrictions and supply chain performance: Modelling and analyzing load-dependent lead times”, by S. Cannella, R. Domínguez, B. Ponte, & J. M. Framinan. Article accepted by the International Journal of Production Economics. DOI: 10.1016/j.ijpe.2018.08.008 7 [vi] Cannella et al. (2008) Simulation (differential equations modelling) Order rate Independent • The Bullwhip Effect is mitigated when capacity is considered in the supply chain. • An increment of production capacity does not necessarily imply an improvement in customer service. [vii] Boute et al. (2009) Mathematical analysis Production Load-dependent • Inflexible capacity results in stochastic lead times, thereby increasing the inventory requirements and supply chain costs. [viii] Ciancimino and Cannella (2009) Simulation (differential equations modelling) Production Independent • The “rogue” dampening of the Bullwhip Effect provoked by capacity constraints increases supply chain risk, as it may lead to satisfy at a higher cost a “false” demand. [ix] Juntunen and Juga (2009) Simulation (discrete event) Transportation Independent • An increase in the transportation capacity does not necessarily translate into an improved customer service. [x] Hamdouch (2011) Network equilibrium Production and shortage Independent • Capacity restrictions do not only affect supply chain dynamics but also market response. By impaction on the impact on the price of the product, demand forecasting becomes more complex, which will add to the generation of the Bullwhip Effect. [xi] Nepal et al. (2012) Simulation (differential equations modelling) Production Independent • While capacity limits do not significantly impact order variability, they increase inventory variability. [xii] Chen and Lee (2012) Mathematical analysis Order rate Independent • Imposing a finite capacity to supply chains smooths the order variability. [xiii] Spiegler and Naim (2014) Simulation (system dynamics) Transportation Independent • Capacity limitations negatively impact both inventory and backlog costs, although there is a positive impact on the “Capacity restrictions and supply chain performance: Modelling and analyzing load-dependent lead times”, by S. Cannella, R. Domínguez, B. Ponte, & J. M. Framinan. Article accepted by the International Journal of Production Economics. DOI: 10.1016/j.ijpe.2018.08.008 8 ‘backlash’ effect (the reflection of the Bullwhip Effect on transportation). [xiv] Hussain et al. (2016) Simulation (system dynamics) Order rate Independent • Tight capacity constraints (in relation to the mean demand) result in high inventory shortfalls. • Increasing capacity is a necessary solution for agile manufacturing. [xv] Ponte et al. (2017) Simulation (differential equations modelling) Order rate Independent • When capacity reduces, order variability decreases at the expense of an increase in inventory variability —and hence a reduced fill rate. • Overall, capacity limitations stop unnecessarily large orders being issued and this has some economic value. [xvi] Shukla and Naim (2017) Simulation (system dynamics) Transportation and production Independent • Detecting capacity constraints is essential to improve the dynamic performance of supply chains. [xvii] Framinan (2017) Mathematical analysis Order rate Lead time Independent Load-dependent • If capacity refers to the rejection of orders in excess of a threshold, the effect of capacity in the Bullwhip Effect is to dampen it. • If capacity constraints induce some variability in the lead times, the effect in the Bullwhip Effect is linked to the way the lead times and the demand are forecast “Capacity restrictions and supply chain performance: Modelling and analyzing load-dependent lead times”, by S. Cannella, R. Domínguez, B. Ponte, & J. M. Framinan. Article accepted by the International Journal of Production Economics. DOI: 10.1016/j.ijpe.2018.08.008 9 improvement was explained in subsequent works as a reduction in the variability of the orders issued throughout the supply chain (Wilson, 2007; Cannella et al., 2008; Ciancimino et al., 2009; Chen and Lee, 2012; Spiegler and Naim, 2014; Ponte et al., 2017), which clearly has an economic value for the members of this system. Interestingly, Hamdouch (2011) underlined that capacity limits also impact on market response through the selling price of products, which decreases the accuracy of demand forecasting —a second route through which capacity constraints impact on the generation of the Bullwhip Effect. However, such decrease in order variability was found to be achieved at the expense of an increase in inventory variability (Vlachos and Tagaras, 2001; Ciancimino and Cannella, 2009; Nepal et al., 2012; Ponte et al., 2017). A direct consequence is that customer satisfaction tends to decrease as capacity tightens (de Souza et al., 2000; Wilson, 2007; Boute et al., 2009; Spiegler and Naim, 2014; Hussain et al., 2016; Ponte et al., 2017). In addition, a negative impact of capacity constraints on the agility of the supply chain was observed by Helo (2000) and Hussain et al. (2016). Although the previous insights represent the main stream of research, some works reach different conclusions. For example, both Cannella et al. (2008) and Juntunen and Juga (2009) show that an increased capacity does not always result in an improved customer service level, while Nepal et al. (2002) do not observe a significant impact of capacity constraints on order variability. These contributions illustrate that the impact of capacity restrictions on supply chain performance heavily depends on the assumptions made. All in all, the aforementioned works highlight the significant impact of capacity constraints on the performance of supply chains, which cannot be ignored. In this sense, the overall system must be analyzed and these constraints must be detected (Shukla and Naim, 2017). Table 1 also underscores the fact that the majority of these studies (14 out of 16) consider the lead time to be an independent parameter. This simplified approach to model the lead time is by far the most common assumption in the literature on supply chain dynamics (see e.g. Lee et al. 1997, Chen et al. 2000, and Dejonckheere et al. 2003). Indeed, as mentioned in Section 1, we have found only three studies dealing capacity constraint supply chain and load-dependent lead time, i.e., Helo et al. (2000), Boute et al. (2009) and Framinan (2017). Helo (2000) concludes that idle capacity is not always non-productive: In this sense, he highlights that “cost efficiency and fast delivery are trade-off performances which cannot be maximized “Capacity restrictions and supply chain performance: Modelling and analyzing load-dependent lead times”, by S. Cannella, R. Domínguez, B. Ponte, & J. M. Framinan. Article accepted by the International Journal of Production Economics. DOI: 10.1016/j.ijpe.2018.08.008 16 decision parameters known as proportional controller β. If β=1, the policy is equivalent to the classical OUT, whereas if β < 1, we are emulating a smoothing replenishment rule. As β decreases, so the amount of the gaps between target and currents levels of inventory and work in progress do. The rationale for the smoothing order policy is to limit the tiers’ over-reaction/under-reaction for changes in demand as orders are essentially “smoothed”. Thus, by varying β we can explore if shifting from a classical OUT to a smoothing replenishment rule may impact the performance of the analyzed capacity constraints supply chain. In each time period, the producer performs the following sequence of actions: i. Receive ordered/processed products Tht (Eq. (2)). These receptions correspond to orders placed several periods ago whose lead time depend on past work in process according to Eq. (3). 11 / t t t Th W L −− = (2) 11t t t t W W O Th −− = + − (3) The throughput/receipt process is modelled according to Little’s law (1968) (Eq. (2)) according to other relevant studies dealing with the dynamics of supply chains (see e.g., Sterman, 2000; Wikner, 2003; Deif and ElMaraghy, 2007; Chaudhari, 2011, among others). Equation (3) states that the work in progress Wt at time t is increased by Ot-1 the order placed by the producer at time t-1 and is decreased by Tht the throughput received at time t. ii. Receive and satisfy the customer demand dt. The fulfilled (or satisfied) demand d*t (Eq. (4)) can be obtained as the minimum value between the inventory available (i.e. the sum of the inventory in the previous period and the receipts) and the demand, as long as this is greater than zero. ( )   1 max min 0*,, t t t t d I Th d − =+ (4) iii. Update the inventory of final products It (Eq. (5)). Note that the inventory position at time t is increased by the throughput at time t and decreased by the quantity dt sent to the final customer at time t. Thus, It can be positive, “Capacity restrictions and supply chain performance: Modelling and analyzing load-dependent lead times”, by S. Cannella, R. Domínguez, B. Ponte, & J. M. Framinan. Article accepted by the International Journal of Production Economics. DOI: 10.1016/j.ijpe.2018.08.008 17 representing storage or products at the end of the period, or negative, representing stock-outs. 1t t t t I I Th d − = + − (5) iv. Compute forecast ˆt d of the future demand (Eq. (5)) and a lead time forecast ˆt L (Eq. (6)). 1 ˆ (1 ) ˆt t t ddd  − = + − (6) 1 ˆ (1 ) ˆt t t LLL  − = + − (7) Equations (6) models the exponential smoothing demand forecasting rule adopted for computing the Order Ot and the target inventory TIt, and Eq. (7) models the exponential smoothing lead time forecast rule adopted for computing the the target work in progress TWt. The value of α and θ reflects the weight given to the most recent observations. v. Update the target inventory TIt (Eq. (8)) and the target work in progress TWt (Eq. (9)), respectively. ˆ tt TI d  = (8) ˆ ˆ t t t TW L d= (9) In Equation (7) TIt the target inventory at time t is expressed as the product of the forecast demand of final customers ˆt d at time t times the safety stock factor ε, also known in literature as time to cover inventory (Sterman, 2000). In Equation (8), the target work in progress TWt at time t is computed as the product of the demand forecast ˆt d and the lead time forecast ˆt L at time t. vi. Place a replenishment/production order Ot (Eq. (10)). ( )   ˆ max ;0 t t t t t t O d TW W TI I  = + − + − (10) Equation (10) models both classical OUT (if β=1) and the smoothing replenishment rule (if 0<β<1) (Disney and Lambrecht 2008). The order quantity is the sum of three components: (I) the forecast on the order from customer ˆt d , (II) a fraction of the work in progress gap β(TWt – Wt), and (III) “Capacity restrictions and supply chain performance: Modelling and analyzing load-dependent lead times”, by S. Cannella, R. Domínguez, B. Ponte, & J. M. Framinan. Article accepted by the International Journal of Production Economics. DOI: 10.1016/j.ijpe.2018.08.008 18 the inventory gap β(TIt – It). Finally, the logical operator “max” models the non-negative condition of order quantity. 4. Experiments, numerical results and analysis. In this section, we first describe the experimental design by defining the factors to be analysed, the experimental outcomes (dependent variables), and the parameters of the model and simulation conditions. Then, we present a statistical analysis of the output data using ANOVA in order to assess the significance of the impact of the factors on the dependent variables. 4.1. Experimental design. The main factor considered in the analysis is the capacity saturation factor (φ), although we also include five additional factors to study the impact of the capacity saturation in different scenarios: one related to the responsiveness of the manufacturing system (i.e., the responsiveness factor η), two related to the order policy setting (i.e., the safety stock factor, ε, and the proportional controller, β), one related to the lead time forecasting method (i.e., the lead time forecast smoothing factor, θ), and a exogenous factor related to customer demand (i.e., the coefficient of variation of customer demand, 𝜎𝑑/𝜇𝑑). Since the main factor under analysis is φ, we select six levels for this factor and three levels for the other factors, as shown in Table 3. We adopt a full factorial design, so in total we perform 1458 different experiments. To set up the customer demand we assume that the quantities ordered represent a standard product with repetitive demand and thus, we use a normal distribution (see e.g., Chatfield 2013). To generate a more realistic representation of the production-distribution system, we assume that, when capacity saturation has been largely exceeded (i.e, work in progress is very high), lead-time cannot exceed by four times the stationary lead time value (δ) (see Table 3). The dependent variables are three common metrics assessing the operational performance and the customer service level. More specifically, the operational performance is measured using the Order Rate Variance Ratio (ORVrR) and the Inventory Variance Ratio (IVrR), while the customer service level is measured via the Fill Rate (FR). The reduction of ORVrR and IVrR reflects improved cost “Capacity restrictions and supply chain performance: Modelling and analyzing load-dependent lead times”, by S. Cannella, R. Domínguez, B. Ponte, & J. M. Framinan. Article accepted by the International Journal of Production Economics. DOI: 10.1016/j.ijpe.2018.08.008 19 effectiveness of supply chain members’ operations, while the increase of FR indicates lower stock-out costs. Table 3. Experimental design. Model parameters Value Customer demand (d) 𝑁(𝜇𝑑=100, 𝜎2) units per period Stationary lead time (δ) 2 periods Demand smoothing forecasting factor (𝛼) 0.2 Independent variables Values Capacity saturation factor (φ) 1.25 1.50 1.75 2.00 2.25 2.50 Responsiveness factor (η) 1.0 2.0 4.0 Customer demand variability (𝜎𝑑/𝜇𝑑) 0.15 0.30 0.45 Lead time smoothing forecast factor (θ) 0.1 0.5 1.0 Safety stock factor (ε) 0.5 1.0 3.0 Proportional controller (β) 0.3 0.6 1 Dependent variables Order Rate Variance Ratio (ORVrR) Inventory Variance Ratio (IVrR) Fill Rate (FR) ORVrR was proposed by Chen et al. (2000) and is the most common demand amplification measure in the literature (Disney and Lambrecht, 2008). It is defined as the ratio between the demand variance at the downstream and at the upstream stages (Miragliotta, 2006) (Eq. 13). 2 2 O d ORVrR   = (11) IVrR was proposed by Disney and Towill (2003) to measure net stock instability, as it quantifies the fluctuations in actual inventory against the fluctuation in demand, see (Eq. 14). An increased inventory variability results in higher holding and backlog costs, and increasing average inventory costs per period (Disney and Lambrecht, 2008). 2 2 I d IVrR   = (12) “Capacity restrictions and supply chain performance: Modelling and analyzing load-dependent lead times”, by S. Cannella, R. Domínguez, B. Ponte, & J. M. Framinan. Article accepted by the International Journal of Production Economics. DOI: 10.1016/j.ijpe.2018.08.008 20 The Fill Rate (Eq. 17) is a key metric of customer satisfaction within supply chains, which measures the demand fulfilment experienced by the consumer. It can be obtained as the ratio of the mean fulfilled demand to the mean demand, see e.g. Ponte et al. (2017). *% t t d FR d = (13) The length of simulations runs is set to T=20,000 periods, ensuring reaching a steady state in the system, and data from the first 200 periods of each replication are removed as warm-up. In order to account for randomness and test the statistical significance of the experimental factors, we perform several replications of each experiment. According to Kelton et al. (2007), when the half-width of confidence interval is smaller than a user-specified value (e.g. within 10% of the mean, Yang et al. 2011), the number of replications is acceptable for statistical analysis. Due to the high length of the simulation runs, the obtained output randomness is very low, and we achieve this condition with a low number of replications (i.e. 5 replications). Finally, in order to be able to compare the different replications, we fix the random number generator, and thus the differences obtained are due to the parameter settings and not randomness. 4.2. Analysis of results. The statistical analysis of the simulation data is carried out via an Analysis of Variance (ANOVA), with a level of significance of p=0.05. Normality of data output was checked through Kolmogorov-Smirnov and Shapiro-Wilk tests. Table 4 shows the results of the ANOVA for the three performance metrics. Due to the numerous interactions among factors that result from the experimental design, and since the main factor in the scope of this analysis is the capacity factor φ, we show only the (first-order) interactions that involve this factor and do not focus on all other interactions from our analysis. Results show that all factors and interactions are statistically significant at 95% confidence level, so we reject the null hypothesis that there is no difference in means between the different groups. In the following we analyse the results using main effects and interaction plots together with the results obtained from ANOVA. We start our analysis by focusing on the main effects of φ on ORVrR, IVrR, and FR. Looking into Table 4 (F-value) it can be noticed that φ and η have a similar impact “Capacity restrictions and supply chain performance: Modelling and analyzing load-dependent lead times”, by S. Cannella, R. Domínguez, B. Ponte, & J. M. Framinan. Article accepted by the International Journal of Production Economics. DOI: 10.1016/j.ijpe.2018.08.008 21 on ORVrR and IVrR, being φ statistically more significant than η in FR. φ is also more significant than 𝜎𝑑/𝜇𝑑 for ORVrR and IVrR, while for FR the opposite happens. The impact of φ on all the three metrics is less significant when compared with ε and β (i.e., ε and β have a direct impact on how orders are placed and thus, they have a big impact on all the three metrics). Finally, θ has the lowest impact on all the three metrics. Table 4. ANOVA for ORVrR, IVrR and FR. ORVrR IVrR FR Source DF F-value p DF F-value p DF F-value p φ 5 1392,771 <0.001 5 1375,480 <0.001 5 511,604 <0.001 η 2 1511,993 <0.001 2 1456,671 <0.001 2 408,381 <0.001 𝜎𝑑/𝜇𝑑 2 1246,008 <0.001 2 742,976 <0.001 2 87046,411 <0.001 ε 2 21028,743 <0.001 2 23935,971 <0.001 2 90666,789 <0.001 β 2 133877,414 <0.001 2 3310,967 <0.001 2 1956,753 <0.001 θ 2 153,731 <0.001 2 53,489 <0.001 2 36,971 <0.001 φ * β 10 206,741 <0.001 10 5,025 <0.001 10 1,124 0.339 φ * ε 10 24,641 <0.001 10 3,345 <0.001 10 125,721 <0.001 φ * η 10 1140,242 <0.001 10 953,171 <0.001 10 292,303 <0.001 φ * θ 10 87,530 <0.001 10 70,620 <0.001 10 27,125 <0.001 φ * 𝜎𝑑/𝜇𝑑 10 143,932 <0.001 10 425,874 <0.001 10 242,198 <0.001 Adjusted R2 (%) = 98.1 Adjusted R2 (%) = 92.6 Adjusted R2 (%) = 98.6 Figure 4 shows the main effects of φ and confidence intervals at a 95% confidence level. It is observed that the performance of the system improves significantly if the capacity is high as compared to scenarios with tight capacity restrictions (i.e. increasing φ reduces ORVrR and IVrR and increases FR). This result implies that, if the capacity of the system is large, lead times tend to stabilize around the minimum stationary value, thus improving the performance of the supply chain. On the contrary, when there is a tight capacity limit, the system works under saturation and lead times tend to increase, which worsens the supply chain response. Interestingly, the supply chain performance stabilizes for a certain value of φ (i.e., additional increments in φ do not produce any performance improvement). This output suggests that, if the capacity becomes large enough, the capacity constrained system turns into an unconstrained supply chain. In the literature, such supply chains are known as the so-called exogenous supply chains (Hum and Parlar, 2014), in which lead times are independent from internal variables. Under the boundary “Capacity restrictions and supply chain performance: Modelling and analyzing load-dependent lead times”, by S. Cannella, R. Domínguez, B. Ponte, & J. M. Framinan. Article accepted by the International Journal of Production Economics. DOI: 10.1016/j.ijpe.2018.08.008 22 conditions of our simulation analysis, ORVrR, IVrR and FR stabilize, in average, for φ~1.75 (when the capacity is around 75% higher than the mean demand during lead times). ORVrR IVrR FR Capacity Figure 4. Main effects and 95% confidence intervals (φ). Figure 5 shows the main effects of the other experimental factors (i.e., η, 𝜎𝑑/𝜇𝑑, ε, β and θ) and confidence intervals at a 95% confidence level. Looking at the responsiveness factor η, a reduction of ORVrR and IVrR is obtained when η increases from η=1 to η=2, while no significant change can be observed when η increases from η=2 to η=4. In addition, an increase of η improves FR. Thus, by increasing this factor (i.e. increasing firm’s capability to respond to variations in the market), the system is able to mitigate the variance of the orders received, which translates into more stable production schedules and inventories and higher customer service levels. In other words, the dynamics of the system dramatically suffer from not being able to deal with high orders in a time-effective manner, since lead times are longer. However, the benefit of increasing the system’s responsiveness is limited, since once the system has reached enough responsiveness to handle demand variations, it is no longer benefited from higher responsiveness. In Figure 5, we also show the main effects of 𝜎𝑑/𝜇𝑑, ε, β and θ in order to provide a general picture of the simulation outputs. Note that impact of some of these factors has already been analysed in the literature (see e.g. Chatfield et al., 2004: Disney and Lambrecht, 2008; Ciancimino et al., 2012; Syntetos et al. 2011, among others). In this research, we are more interested in the interactions between these factors and the capacity saturation factor φ, which will be analysed later (see Table 4 and Figure 6). Nonetheless, we would like to remark that the results regarding the impact of these parameters on supply chain performance are in line with observations showed by previous related empirical and theoretical studies. This concordance supports the validation of our model. “Capacity restrictions and supply chain performance: Modelling and analyzing load-dependent lead times”, by S. Cannella, R. Domínguez, B. Ponte, & J. M. Framinan. Article accepted by the International Journal of Production Economics. DOI: 10.1016/j.ijpe.2018.08.008 23 ORVrR IVrR FR Responsiveness Demand variability Safety stock Proportional controller Lead time forecast Figure 5. Main effects and 95% confidence intervals (η, 𝜎𝑑/𝜇𝑑, ε and β). The ANOVA reports significant interactions between φ and the other experimental factors (η, 𝜎𝑑/𝜇𝑑, ε and β), as it can be seen in Table 4. Some of these interactions have an important impact on ORVrR, IVrR and FR (they show a high F-value) and, as such, the impact of φ on supply chain performance needs to be interpreted by considering the interactions with other factors (Figure 6). Overall, the interaction between φ and η is the most important one. When η is high (η=4), φ barely impacts on performance (only a small performance increase is observed by increasing φ from “Capacity restrictions and supply chain performance: Modelling and analyzing load-dependent lead times”, by S. Cannella, R. Domínguez, B. Ponte, & J. M. Framinan. Article accepted by the International Journal of Production Economics. DOI: 10.1016/j.ijpe.2018.08.008 24 φ=1.25 to φ=1.50). This happens because the system only achieves the capacity limit in a very low percentage of the time periods, which has a marginal impact on the overall results. Thus, a manufacturer with high responsiveness is able to avoid the amplification of the variability in the system, achieving the minimum ORVrR and IVrR or maximum FR with a low capacity (φ~1.50). On the contrary, as η decreases, the system becomes very sensitive to φ, and the capacity needed to achieve such minimum/maximum performance increases (for η=1 ORVrR and FR stabilizes at φ~1.75, and IVrR stabilizes at φ~2.00). It is also important to point out that a manufacturer operating with low capacity and low responsiveness benefits from a significant performance improvement by either increasing its capacity and/or its responsiveness. The next interaction in importance takes place between φ and 𝜎𝑑/𝜇𝑑, in particular for IVrR and FR. When 𝜎𝑑/𝜇𝑑 is low (𝜎𝑑/𝜇𝑑=0.15) φ barely impacts on IVrR and FR. However, as 𝜎𝑑/𝜇𝑑 increases, the supply chain becomes more sensitive to φ, and the capacity needed to achieve the minimum IVrR and maximum FR increases. More specifically, the capacity needed to achieve the minimum IVrR (maximum FR) is φ~1.75 (φ~1.50) when 𝜎𝑑/𝜇𝑑=0.30 and φ~2.00 (φ~1.75) when 𝜎𝑑/𝜇𝑑=0.45. Therefore, higher variability of demand requires higher capacity to avoid performance deterioration. Finally, ORVrR improves when φ increases from φ=1.25 to φ=1.50 for all the analysed values of 𝜎𝑑/𝜇𝑑. After that value, any increment of φ does not improve ORVrR, except if 𝜎𝑑/𝜇𝑑=0.45, where ORVrR can be reduced by increasing φ from φ=1.50 to φ=1.75. To sum up, a supply chain operating in a market characterised by high variability is more vulnerable to capacity limitations. The interaction between φ and ε is particularly important for the FR. If the safety stock factor is set to high value (ε=3.0) the system maintains a high stock level, which is enough to handle the variability of market demand and thus, a high FR can be obtained regardless the capacity of the manufacturer. As ε decreases, the system maintains lower stock levels and becomes more sensitive to the value of φ (i.e. the system needs a higher capacity to react to the incoming orders). Specifically, for low values of ε (ε=0.5, ε=1.0) the FR decreases when φ is below φ~1.75. In case of ORVrR and IVrR, performance deterioration occurs for any value of ε if φ is below a certain value (φ~1.50 and φ~1.75, respectively). However, this deterioration may be slightly different depending on the value of ε. To sum up, the customer service level of a “Capacity restrictions and supply chain performance: Modelling and analyzing load-dependent lead times”, by S. Cannella, R. Domínguez, B. Ponte, & J. M. Framinan. Article accepted by the International Journal of Production Economics. DOI: 10.1016/j.ijpe.2018.08.008 25 supply chain characterised by relevant capacity restriction strongly depend from the safety stock factor. ORVrR IVrR FR Figure 6. Interaction plots with capacity saturation factor (φ). Regarding the interaction between φ and β, the performance deterioration (ORVrR and IVrR) caused by a low capacity is higher for a classical OUT (β=1) than for a smoothing replenishment rule (β=0.67 and β=0.33). In fact, the smoothing “Capacity restrictions and supply chain performance: Modelling and analyzing load-dependent lead times”, by S. Cannella, R. Domínguez, B. Ponte, & J. M. Framinan. Article accepted by the International Journal of Production Economics. DOI: 10.1016/j.ijpe.2018.08.008 32 6. Conclusions and further directions In this paper we offer a novel perspective on capacitated supply chains by assuming load-dependent lead times. Based on empirical evidence, we model the manufacturing lead times as a nonlinear function depending on the current work in progress at the manufacturer and its capacity saturation limit and responsiveness. By doing so, we are able to move a step further in the understanding of the dynamic behaviour of capacitated supply chains by considering the response of the system after capacity saturation (i.e., the order size cannot be fulfilled in the average stationary lead time). In this situation, the lead time of the order to be fulfilled depends on the current work in progress and the responsiveness of the manufacturing system. In our analysis, we consider two factors to model the manufacturing lead time: the capacity saturation factor and the responsiveness factor. In order to determine how different manufacturing settings perform in different scenarios, we consider three additional factors, i.e. customer demand variability, safety stock factor, and proportional controller. Via discrete time difference equations approach, we perform a comprehensive simulation analysis to determine how these operational factors impact orders and inventory variability and customer service level. We show that, in contrast to other previous studies, increasing the capacity limit of the manufacturer has a positive effect on supply chain dynamics derived from maintaining a lower and constant lead time. On the contrary, reducing such capacity has a negative effect on performance. In the event of a manufacturer working under saturation (i.e., the current work in progress is higher than its capacity), the responsiveness of the manufacturing system plays an important role (e.g., a manufacturer with low responsiveness is not be able to efficiently manage the capacity saturation, resulting in long and variable lead times). We find a strong interaction between both factors, showing how the responsiveness of the manufacturer becomes more critical if she/he is working close to the capacity limit. Finally, we show that there are strong interactions among capacity, customer demand variability, safety stock factor, proportional controller and lead time forecasting factor. More specifically, we show that (1) a high demand variability, (2) a low safety stock factor, (3) a high proportional controller and (4) a highly reactive lead time forecast exacerbate the negative impact of a low capacity in terms of Bullwhip Effect. “Capacity restrictions and supply chain performance: Modelling and analyzing load-dependent lead times”, by S. Cannella, R. Domínguez, B. Ponte, & J. M. Framinan. Article accepted by the International Journal of Production Economics. DOI: 10.1016/j.ijpe.2018.08.008 33 The most important limitation of our work concerns the modelling assumption of load-dependent lead times. We assume that orders that do not exceed the capacity saturation limit are fulfilled in a constant lead time. However, modelling the capacity is still an issue, since all the complexities of a real manufacturing system cannot be captured by an implicit capacity model such as the CT-TP curve. Further studies should consider to explicitly modelling the capacity and/or to extend the current research by considering the stochasticity of lead-times and other methods for emulating the load-dependent lead time, e.g. clearing functions (Orcun, 2009; Mönch, 2012). Another limitation concerns the supply chain topology. We have studied a two-node supply chain. However, this assumption is a simplification of real supply chains, which often show more complex structures (Chatfield et al., 2013; Dominguez et al., 2015a). Thus, the capacity-constrained supply chain needs to be analyzed in more realistic structures, such as divergent, convergent, conjoined, or the closed loop supply chain. Also, upstream members may limit/amplify the effects of both capacity and responsiveness of downstream members on supply chain dynamics, e.g., how important is to increase the capacity/responsiveness of a downstream manufacturer if there is an upstream echelon with tight capacity/responsiveness? In addition, we note that we assumed specific boundary conditions and parameters for the experiments. Even if they have been selected and organized to provide general and reliable results, the impact of capacity in supply chain dynamics needs to be explored under other scenarios in order to contrast the obtained results (e.g. demand distributions, order policies, collaboration strategies, etc.). Finally, considering the impact of demand correlation on the link between capacity constraints and supply chain performance is also a line of research worth pursuing. Acknowledgements We are grateful to the anonymous referees and the editorial team as their suggestions allowed us to improve earlier versions of this manuscript. This research was supported by the Italian Ministry of Education, University and Research (Rita Levi Montalcini fellow), by the University of Seville (V PPIT-US), and by the Spanish Ministry of Science and Innovation, under the project PROMISE with reference DPI201680750P. “Capacity restrictions and supply chain performance: Modelling and analyzing load-dependent lead times”, by S. Cannella, R. Domínguez, B. Ponte, & J. M. Framinan. Article accepted by the International Journal of Production Economics. DOI: 10.1016/j.ijpe.2018.08.008 34 References Ankenman, B. E., Bekki, J. M., Fowler, J., Mackulak, G. T., Nelson, B. L., Yang, F. (2011). Simulation in production planning: an overview with emphasis on recent developments in cycle time estimation. 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