Abstract
We investigate two main sources of information inaccuracies (i.e., errors and delays) in demand information sharing along the supply chain (SC). Firstly, we perform a systematic literature review on inaccuracy in demand information sharing and its impact on supply chain dynamics. Secondly, we model several SC settings using system dynamics and assess the impact of such information inaccuracies on SC performance. More specifically, we study the impact of four factors (i.e., demand error, demand delay, demand variability, and average lead times) using three SC dynamic performance indicators (i.e., bullwhip effect, inventory variability, and average inventory). The results suggest that demand error has a negative impact on SC performance, which is exacerbated by the magnitude of the error and by low demand variability scenarios. In contrast, demand delay produces a nonlinear behavior in the supply chain response (i.e., a short delay may have a negative impact and a long delay may have a positive impact), being influenced by the supply chain configuration.
Introduction
Information sharing is an effective collaboration strategy widely analyzed in the supply chain (SC) managementrelated literature, as it improves SC performance and reduces the noxious bullwhip effect [1] [2] [3] [4] [5] [6] . This phenomenon can be defined as the amplification of order variability upstream in the SC, and it is known to cause excessive inventory, grossly inaccurate demand forecasts, low capacity utilization, and poor customer service level [3, 4] . Since the lack of market visibility has been traditionally identified as one of the main causes of the bullwhip effect, sharing market demand information to the upstream members of the SC becomes an effective strategy for bullwhip effect reduction [7] .
Introduction
Nevertheless, implementing information sharing (IS) in SCs presents some serious obstacles due to numerous barriers [8] . Perhaps one of the most severe barriers is the quality of the shared information, which can be defined as the degree to which the information exchanged between organizations meets the needs of the organizations [9] . Without reliability or validity, information has no value for the receiving partner [10, 11] , since such information might be erroneous. While most of the studies about IS in SCs assume that the information shared between members is fully accurate, in practice, information is frequently distorted, either willfully or unintentionally, as it travels through the SC [12] . Therefore, unless each party can verify the authenticity of the other parties' information, SC members may spread false information for their own benefit [13] .
Introduction
Errors may be present in the transmission of market demand information and inventory levels, both of them having negative consequences on SC performance. Demand errors occur when the real market demand of a product differs from the level of sales recorded in the information system. As reported by Kwak and Gavirneni [14] , the detrimental impact of errors in demand information outweighs the beneficial impact of IS when the variance of information errors exceeds the variance of end-customer demands. Inventory errors occur when the level of actual inventory of a product differs from that recorded in the information system. The value of transmitting inventory information to other SC partners may be undermined by these errors, which are often present in SCs [15] . In turn, the presence of inventory errors at downstream echelons causes demand information distortion and may undermine the expected benefit of IS and the effort in information technology investment [16] .
Introduction
In addition to the quality of the information, another important obstacle to IS implementation is the synchronization of the information transmission process. A bad synchronization leads to delayed information regarding market sales and/or inventory levels (i.e., the information transmitted upstream is received one or more periods later). The impact of such information delays on SC performance remains unclear in the literature; for example, Hoberg and Thonemann [17] state that the presence of unsynchronized processes or inadequate communication structures hinders the widespread availability of real-time information, thus harming the overall SC performance; Hosoda and Disney [18, 19] show that while the first level of a SC can benefit from shorter time delays, the benefit perceived by the second level of the SC is minor or even detrimental.
Introduction
After a thorough review of the related literature, we notice that (1) the impact of transmitting erroneous demand information has not been addressed from a SC dynamic perspective, (2) the impact of demand information delay on SC performance presents contrasting results, and (3) most of the related literature focuses on two-echelon SCs. Motivated by these considerations, in this work, we aim to fill these gaps by analyzing the dynamics of a collaborative four-echelon SC where demand information is transmitted with errors and/or delayed. The SC is modelled using a system dynamics approach ( [20] , Syntetos 2011, [21, 22] ). The results show that both the bullwhip effect and inventory variability are exacerbated in the presence of errors in market demand information, particularly for low demand variability scenarios. The impact of information delays depends on the magnitude of the delay, so the performance of the SC deteriorates for short delays and improves for long delays. Finally, we found that the impact on the average SC inventory of erroneous/delayed market demand IS is lower than on the bullwhip effect and inventory variability.
Introduction
The remainder of the paper is structured as follows: in Section 2, we perform a systematic literature review. Then, we present the SC model in Section 3. In Section 4, we describe the design of experiments and show the results obtained. Section 5 summarizes findings and managerial implications, while Section 6 includes the conclusions of the research.
Inaccuracy in Demand Information Sharing: Systematic Literature Review
In order to depict the state of the art on demand information sharing with errors and/or delays, we perform a systematic review of the literature, focusing specifically on studies that analyze the dynamic behavior of the SC. This systematic review is conducted according to the framework provided by Denyer and Tranfield [23] , that is, (1) question formulation, (2) locating studies, (3) study selection and evaluation, (4) analysis and synthesis, and (5) result reporting. Therefore, in the first step, we define the following research question for the literature review: how does erroneous/delayed demand information sharing impact on the dynamic of a multi-echelon SC?
Inaccuracy in Demand Information Sharing: Systematic Literature Review
In the second step, in order to locate the relevant studies, we use three well-established scientific databases, that is, Scopus, ABI/Inform Global, and Web of Science. In each database, we perform an identical search with the following string: "supply chain * " AND ("demand error * " OR "information error * " OR "false demand" OR "false information" OR "demand delay * " OR "information delay * ").
Inaccuracy in Demand Information Sharing: Systematic Literature Review
The resulting queries provided a total of 34 different papers in the English language. Table 1 summarizes these papers, which are classified by scientific databases.
Inaccuracy in Demand Information Sharing: Systematic Literature Review
This third step concerns the explicit elucidation about the selection/exclusion criteria of the review. In the following, we report the exclusion criteria used in this review and the excluded articles:
Inaccuracy in Demand Information Sharing: Systematic Literature Review
(i) Lack of specific insights: articles [2] , [3] , [8] , [9] , [12] , [20] , [23] , [26] , [28] , [29] , [31] , [32] , and [33] are excluded from further analysis due to lack of specific insights on the potential impact of the demand error and the demand delay on the dynamics of SC.
Inaccuracy in Demand Information Sharing: Systematic Literature Review
(ii) Nonrelated: articles [13] , [14] , [15] , [29] , and [31] merely contain the search string's terms in the keywords, title, or abstract. However, they are not related to the research question in Step 1.
Inaccuracy in Demand Information Sharing: Systematic Literature Review
(iii) Similar research results: article [6] is excluded as it reports similar research results as in [5] . Thus, it does not contain additional contributions.
Inaccuracy in Demand Information Sharing: Systematic Literature Review
In Step 4, after the application of the exclusion criteria, the remaining 15 papers are reviewed in detail and crosstabulated in order to identify their key features ( Table 2) . To this end, articles are categorized according to the following criteria: type of shared information, type of information distortion (i.e., demand error and/or demand delay), adopted methodology, number of echelons in the SC, customer demand patterns, replenishment policy, forecasting model, and indicator for SC dynamic performance (i.e., bullwhip effect, average inventory, and inventory variability).
Inaccuracy in Demand Information Sharing: Systematic Literature Review
By analyzing Table 2 , in the following, we summarize the assumptions and observations arising from the literature gaps:
Inaccuracy in Demand Information Sharing: Systematic Literature Review
(i) The majority of the studies focuses on a two-echelon SC structure. Moreover, the three studies adopting a four-echelon structure (see, e.g., [37, 40, 45, 53] ) do not provide specific insights on the dynamics of SC under demand information sharing with delays/ errors (in the following, demand delays/errors).
Complexity
(ii) The market demand structure adopted in studies on demand error is the normal distribution (iid), while the autoregressive model is the preferred demand pattern in works on demand delay.
Complexity
(iii) The most used replenishment policy is the classical order-up-to, and the most used forecast policy is the exponential smoothing.
Complexity
(iv) The erroneous demand is usually modeled as a percentage of the customer demand. Articles dealing with demand delays assume a constant and deterministic time delay.
Complexity
Analogously, in the following, we provide a summary of the main findings related to the impact of demand and errors and delays on SC dynamics:
Complexity
(1) Impact on the bullwhip effect: no work deals with the impact of the demand error on the bullwhip effect. In contrast, four studies focus on the demand delay and the bullwhip effect (see [17, 19, 40, 43] ). However, the works report conflicting views on how demand delay affects SC performance.
Complexity
(2) Impact on average inventory: four studies, using different methodological approaches and modelling assumptions, prove that demand errors increase the average inventory level for all SC members (see [14, 36, 38, 45] ). According to these studies, demand errors increase forecast errors, which leads to the need of holding higher safety stocks in order to achieve the target service level. Analogously, two studies agree on the fact that demand delays increase (3) Impact on inventory variability: the work of Lu et al. [52] provides some insights on the impact of demand error on inventory variability, which may increase or decrease depending on the demand shock, on information reliability and if errors occur during or before the replenishment. Moreover, four studies [18, 19, 35, 42] show that demand delays increase inventory variability.
Complexity
As a summary, the impact of demand errors on SC dynamic performance is almost unknown. On the other hand, the effect of demand delays has not received particular attention in terms of the inventory variance ratio. Furthermore, these few related studies present conflicting results.
Supply Chain Model
In this section, we present the main assumptions and the mathematical formalization of the SC model. To perform a structured analysis, we adopt the information sharing SC model presented by Cannella [21] via system dynamics simulation (Syntetos et al. 2001) and extend it to the case of delayed or erroneous customer demand information. More specifically, in contrast to the classical SC, in which the information flow consists of the mere upstream transmission of members' orders, in the information exchange SC, the information flow consists of the upstream transmission of members' orders and on sharing the market demand information. Consequently, a member receives data regarding the replenishment from the downstream adjacent member and regarding the forecast on the market demand [21] . The following assumptions are adopted in the SC model:
Supply Chain Model
(i) In a single-product, K-stage production-distribution serial system, each echelon in the system has a single successor and a single predecessor. The generic echelon's position is represented by index i. Echelon i = 1 denotes the manufacturer and i = K + 1 the final customer.
Supply Chain Model
(ii) In the nonnegative condition of order quantity, products delivered cannot be returned to the supplier.
Supply Chain Model
(iii) In each echelon, the backlog will be fulfilled as soon as on-hand inventory becomes available. Thus, orders not fulfilled in time are backlogged so that inventory remains a positive or null value.
Supply Chain Model
(iv) Production-distribution capacity is unconstrained. No quantity limitations in production, buffering, and transport are considered.
Supply Chain Model
(v) Customer demand is assumed to be independent and identically distributed (i.i.d).
Supply Chain Model
(vi) Each member generates the order quantity on the basis of the available data about incoming orders and received information on market demand and according to its own ordering parameters (i.e., inventory level and in-transit orders).
Supply Chain Model
(vii) In line with related studies (see, e.g., [14] ), demand error is modelled by a multiplicative error structure, since the magnitude of the error may depend on the quantity requested.
Supply Chain Model
(viii) Analogously, according to the related literature, demand delay is modeled as a shift in the time dimension of the information about market demand.
Supply Chain Model
(ix) The replenishment rule employed is a periodicreview order-up-to (R, S) [54] . More specifically, each member adopts a specific typology of orderup-to named Automatic Pipeline, Variable Inventory, and Order-Based Production Control System (APVIOBPCS) [2] .
Supply Chain Model
(x) Products delivered cannot be returned to the supplier. By adopting this assumption, we implement a more reliable modelling assumption for the bullwhip effect analysis [1] .
Supply Chain Model
The mathematical nomenclature is reported in Table 3 . The full model is reported in Table 4 .
Experiment and Simulation Output
In this section, we discuss the design of experiments, the experimental factors (independent variables), and the outcome of the experiments (dependent variables), as well as the parameters of the model and simulation conditions. Then, we present a statistical analysis of the output data using ANOVA.
Design of Experiments.
To infer on the dynamics of a multiechelon SC, we consider a serial system composed of four echelons (K = 4), that is, retailer (i = 4), wholesaler (i = 3), distributor (i = 2), and manufacturer (i = 1). The experimental factors of our experiment are (1) the level of demand error (measured as a percentage of the market demand), (2) the demand information delay (measured in time periods), (3) the average lead time, and (4) the standard deviation of customer demand. We test the statistical significance of the impact of these experimental factors by performing a full factorial set of experiments.
Design of Experiments.
The demand error factor is represented by three levels (0%, ±10%, and ±20%). The first level, that is, 0%, emulates the situation where the members receive in each period nonerroneous data on the observed customer demand. The other two levels emulate erroneous demand information, where the percentage of error changes every time period according to a uniform distribution. Analogously, demand delay is set to three levels and varies from 0 periods (it is equivalent to a scenario where customer demand information is timely shared among partners) to 5 Complexity demand is characterized by two levels (i.e., 10 and 30 units of product), as well as the average lead time (i.e., 2 periods and 5 periods). Thus, we analyze a total of 36 (3 × 3 × 2 × 2) scenarios [55] .
Design of Experiments.
As both demand error and delay may produce a polyhedral impact on the behavior of the whole supply chain, we adopt three complementary performance metrics for evaluating three different dimensions of the internal process efficiency of the SC at systemic level. By using the bullwhip slope (BwSl) ( * ), we may assess the dynamics of the members' orders. Using the inventory variance slope (InvSl) ( * * ), we may assess the dynamics of inventories, and finally, using the systemic average inventory (SAI) ( * * * ), we can assess the impact on inventory requirements. A reduction in all three metrics reflects improved cost effectiveness of members' operations. Table 5 provides more details on the performance metrics adopted [56] .
Design of Experiments.
The values adopted for the parameters of the model are common values used in SC dynamics literature (see, e.g., [20, 57] ). Table 6 shows the summary of the experimental design.
Design of Experiments.
We determine the simulation parameters (i.e., run length, number of replications, and warm-up period) by analyzing the results of a pilot test. As a result, we use a simulation time of 1000 periods with a warm-up of 200 periods. In order to obtain the consistency of the estimations in terms of the width of confidence intervals, we perform 30 replications of each experiment. The SC is modelled and simulated using Vensim PLE [58] , a widely used system dynamics software tool, running on a PC with Intel Core i7 CPU @ 2.00 GHz and 8 GB memory. The simulation output is obtained by adopting the Euler-Cauchy method with the order of accuracy Δt = 0 25.
Statistical Analysis (ANOVA)
. We present here the results of the experiments by performing an ANOVA on the simulation output using the statistical software Minitab. Only information regarding the main effects and first-order interactions is presented. Tables 6-8 show the results of the ANOVA for each one of the three performance measures. Table 8 reports the percentual variations in the performance indicators for demand error and demand delay when shifting from the condition of perfect and timely shared customer demand to the different levels of erroneous/delayed demand. Additionally, percentual variations due to the interactions between demand delay and demand error with demand variability and lead time are also reported in Table 9 . Finally, Figure 1 shows the main effects of demand error and demand delay on the performance measures.
Statistical Analysis (ANOVA)
Results show that all the main factors are statistically significant at the 95% confidence level (p < 0 05). Regarding the first order interactions, they are all significant except the interaction between demand error and demand delay for the inventory metrics (p = 0 076 for the average inventory level and p = 0 484 for the inventory variability).
4.3.
Interpretation of the Results. We start by analyzing the impact of the demand error on SC performance. In Figure 1 , it can be seen that each performance metric analyzed shows a growing trend as the demand error increases. More specifically, BwSl and InvSl show a quasilinear trend with the level of demand error. Instead, the trend of SAI is different and does not appear to be linear. Such trend is characterized by an almost horizontal stretch (for the error in the range from 0% to 10%) and another stretch with a higher slope (from 10% to 20%). However, as we can see in Table 8 , the overall impact of demand error on SAI is almost insignificant (the percentage variation of SAI is very low with demand error, not exceeding 0.4% on average). This result is confirmed by the low F value obtained from the ANOVA ( Table 7) . Regarding the interaction of demand error with demand variability, we can notice in Table 8 that the impact of the error on BwSl and InvSl is much stronger when demand variability is lower. In this case, the percentage variations of BwSl and of InvSl are +79.4% and +68.4%, respectively. Instead, when demand variability is high, these are only +6.6% and +5.6%. We interpret these results as the demand variability masks the impact of demand error on both performance metrics. In fact, when demand variability is low, both BwSl and InvSl remain low as well. As such, in this scenario, the introduction of an additional source of uncertainty (such as the demand error) is more noticeable, having a significant impact on both indicators. On the other hand, when the demand variability is high, the BwSl and InvSl are also high. In this scenario, the negative consequences of introducing the additional source of uncertainty are less noticeable than in the previous scenario. Therefore, the impact of demand error on BwSl and InvSl is still present in the scenario with high demand variability, but its relative impact is lower. Regarding the interaction of demand error with lead time, we observe a weak interaction, since the impact of the demand error on BwSl and InvSl is very similar for the different levels of lead time. Now, we continue the analysis by focusing on the impact of demand delay on SC performance. In Figure 1 , we can see a particular trend of all the performance metrics with the increase of the demand delay. Such trend is characterized 
4.3.
Order quantity placed by upstream echelon (i = 1,
4.3.
Order quantity placed by upstream echelon (i = 1, … , K − 1) under erroneous demand
4.3.
Inventory
4.3.
Nonnegativity condition of order quantity
4.3.
Unlimited raw material supply The BwSl is a concise measure of the bullwhip propagation in a given multiechelon structure. Essentially, the BwSl is the slope of the linear interpolation of the set of bullwhip values for a given SC. It provides information on potential unnecessary costs for suppliers, such as lost capacity or opportunity costs and overtime working and subcontracting costs of the whole SC.
4.3.
Similar to the BwSl, the InvSl is the slope of the linear interpolation of the set of the inventory variance ratio values for a given SC. It measures the net stock instability, as it quantifies the fluctuations in inventory. An increased inventory variance results in higher holding and backlog costs, inflating the average inventory cost per period.
4.3.
The SAI is the sum of the average inventory values of all nodes, that is, the mean of tier's inventory values over the interval T. This metric provides concise information on inventory investment. State variables at t = 0 W i 0 , I i 0 , B i 0 = λd 0 , εd 0 , 0 ∀i Safety stock factor ε 3
4.3.
Demand smoothing forecasting factor α 0.33 7 Complexity by an increasing stretch (from a demand delay of 0 periods to a demand delay of 2 periods) and a decreasing stretch (from a demand delay of 2 periods to a demand delay of 5 periods). Moreover, a decrease of each metric is experienced when moving from a demand delay of 0 periods to a demand delay of 5 periods. As we can see in Table 8 , SAI does not change much (in percentage) with the increase in demand error. Nevertheless, the percentage variations of BwSl and InvSl are much larger. Regarding the interaction of demand delay with demand variability, we notice that the impact of the delay on BwSl and InvSl is stronger when demand variability is lower. Instead, in the case of high demand variability, the impact of demand delay on such performance indicators is lower. Following a line of reasoning similar to that for the demand error, it seems that demand variability masks the effect of delayed information on both performance metrics. When the demand variability is low, both BwSl and InvSl are also low. As such, in this scenario, the positive/negative impact of delayed demand information on both performance metrics (measured as the relative decrease/increase of the indicator) is more significant than in the scenario with high demand variability, where both performance metrics have high values regardless of the presence of delayed information. Regarding the interaction of demand delay with lead time, we notice that the impact of short delays on BwSl and InvSl is stronger when lead time is shorter; nonetheless, long delays have a stronger impact on these performance indicators when lead time is longer.
Discussion
5.1. Findings. From the analysis of the simulation outcome, a number of insights on the impact of demand error/delay on the dynamics of SCs have been identified. More specifically, in general, demand error causes an increase in the amplification of orders and inventory variability upstream in the SC, leading to a decrease of the SC performance. This is due to the fact that, in the presence of errors, the forecasts and the orders are based on market demand data characterized by a larger variability. Thus, the variability of the forecasts and the orders increases as well, amplifying the bullwhip effect and inventory variability. In practice, demand errors add another source of uncertainty to the system, increasing the Another important finding about the demand error is the fact that its impact on orders and inventory variability is much stronger when the demand variability is low. This is because, when demand variability is low, an important part of the total variability of orders/inventory is caused by demand errors. Instead, when demand variability is high, demand errors slightly impact on the total variability of orders/inventory.
Discussion
Regarding the demand delay, we found that a long delay decreases in the variability of the orders and the inventory, leading to a better performance of the SC. This observation is in line with the analytical results of Zhang [43] , who demonstrates that, in the presence of delays, forecasts are characterized by a smaller variability because they are less sensitive to changes in demand and so they tend to gravitate closer to the average market demand.
Discussion
Consequently, the orders are also characterized by a smaller variability and thus a decrease of bullwhip effect and inventory variability is obtained. However, this behavior is observed for a long demand delay. Instead, for a short delay, the results obtained are very different. In fact, in this case, the introduction of the delay causes increments in order variability and inventory variability, decreasing the SC performance.
Discussion
Interestingly and perhaps in contrast to intuition, demand delay may cause either an increase or a decrease of SC performance. Although we do not have a conclusive explanation, it might be due to the fact that, under the conditions of this study, the impact of the forecast variability reduction provided by the short delay (see [43] ) is not enough to compensate the impact of the forecast error. A possible further reason behind this nonlinear behavior may lie in the assumptions adopted and the design of experiments (e.g., different combinations of delay with lead time). Note that this nonlinear behavior caused by different demand delays has been previously noticed by Hoberg and Thonemann [17] . Using control theory, they study the dynamics of several configurations of a two-stage inventory system subjected to information delays and observe a mixed picture regarding the impact of the length of information delay. More specifically, in terms of order amplification, longer information delays do not seem to necessarily imply a performance deterioration as compared to shorter delays, but their impact depends on the setup of the SC. Interestingly, although our work greatly differs from the analysis of Hoberg and Thonemann [17] regarding the adopted methodological approach (i.e., control theory versus system dynamics simulation), modelling assumptions (e.g., two-echelon versus four-echelon), and metrics (e.g., order amplification at echelon level versus bullwhip slope), we have noted a similar counter-intuitive effect of demand delay on the SC dynamics. In line with Hoberg and Thonemann [17] , we may consider that this counter-intuitive effect depends on the SC setting. However, we acknowledge that further focused studies need to be developed for better understanding the nature of this interesting behavior, as it may have important implication in real-life SC.
5.2.
Insights for the Industry. Following the above considerations, we report next some brief recommendations for SC managers in the scenario of a collaborative multiechelon SC with market demand information sharing facing information errors and/or delays. First, as the demand error increases, the dynamic performance of the entire SC deteriorates. SC managers should work on eliminating or at least reducing these errors, mainly emphasizing the prevention of inaccuracies caused by human actions at the point-of-sale stage. Second, the negative impact of demand error on the dynamic performance is more noticeable when the demand variability is low. Therefore, error reduction can be more important in markets with relatively stable demand (as in the case of basic hardware, grocery, or pharmaceutical goods). However, the inventory levels may be more sensitive to demand error in turbulent markets and in SCs characterized by high lead times, making important to control these errors also in such 
5.2.
Demand error (λ L 1 )
5.2.
Demand delay
5.2.
Demand delay (λ L 1 )
5.2.
9 Complexity markets and SC settings, especially when the inventory holding costs are high.
5.2.
In contrast, the impact of demand delay on performance may depend on its length. While a short delay has a negative impact on performance, a long delay may have a positive impact on performance. Under both circumstances, managers should try to identify the length and causes of these inaccuracies and consider tuning the inventory control parameters for stabilizing the dynamics of the SC. Furthermore, the impact of demand delay may depend on the SC setting (e.g., lead time); thus, it may be relevant for SC managers to consider this potential error when the SC is characterized by high lead times.
Conclusions and Limitations
In this work, we have investigated the main aspects of inaccuracies (errors and delays) in demand information sharing along the SC. In particular, we analyze their causes and consequences, using previous studies and different research methodologies like simulation and statistical analysis. We have carried out a systematic literature review on inaccuracy in demand information sharing and its impact on SC dynamics. We have detected that the impact of demand errors and demand delay in the dynamics of SCs has been not specifically addressed and that more explorative studies were needed. Thus, we have used systems dynamics to develop a simulation model to assess the impact of such information transmission inaccuracies on SC performance. Through a factorial design of experiments, we have studied the impact of four factors (i.e., demand error, demand delay, demand variability, and average lead time) on three SC dynamic performance indicators (i.e., bullwhip effect, inventory variability, and average inventory). We have statistically analyzed the results obtained, from which we have extracted some findings and related managerial implications. More specifically, our simulation output suggests that demand error has a negative impact on SC performance and that its impact increases with its magnitude of the error; moreover, it has a higher impact when the market demand is stable. On the other hand, the impact of demand delay on SC performance depends on the length of the delay. In our study, a short delay 10 Complexity may have a negative impact while a long delay may have a positive impact. However, the final impact may also depend on the SC configuration. This work is limited by several assumptions, and therefore, it might be interesting for future works to extend it with different hypotheses or to investigate different aspects not covered in our paper. As previously stated, we believe that the impact of demand delay deserves more research to understand which factors-endogenous or exogenous to the SC-may produce the counter-intuitive effect observed in this work and, under different assumptions, also observed by Hoberg and Thonemann [17] . In this fashion, the adoption of analytical models, which may consider the uncertainty of the demand delay for different SC configurations and market demand, may shed light on this peculiar behavior. As the demand delay may produce better SC dynamics with respect to sharing timeless information on market demand, it can be particularly relevant to explore the causes of such behavior, as sharing demand delay can also be converted in an ad hoc bullwhip-dampening method. A starting point can be represented by structurally analyzing the interaction between the lead-time variability and demand delay.
Conclusions and Limitations
Further research should also focus on the impact of sharing timely erroneous data on customer demand. Our work, in line with other studies, reasserts that demand errors affect the internal process efficiency of the whole SC. However, our model does not take into account the intrinsic variability of several processes in SC, such as the delivery lead-time, neither other market scenarios nor SC configurations characterized by higher structural complexity (i.e., divergent and/or convergent SC).

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