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Comparing validity of risk measures on newsvendor models in open innovation perspective
 Sungyong Choi^{1},
 KyungBae Park^{2}Email authorView ORCID ID profile and
 SangOh Shim^{3}
https://doi.org/10.1186/s4085201700788
© The Author(s). 2017
 Received: 30 September 2017
 Accepted: 30 November 2017
 Published: 3 January 2018
Abstract
In the era of Industry 4.0, firms are facing with greater uncertainty. Accordingly, it is important to select quality risk measures to analyze newsvendor problems under risk. Then, open innovation can be a good remedial option for such riskaverse newsvendors because open innovation can offset the profit losses from risk aversion by sharing revenues in supply chains. To find such risk measures in newsvendor problems, we review various risk measures of riskaverse inventory models and existing articles in inventory management literature. Then we provide a logical reasoning and axiomatic framework to evaluate validity of each risk measure in newsvendor problems  consistency to the four axioms in coherent risk measures. In this framework, the underlying assumptions and managerial insights to the newsvendor problems are examined for each risk measure. Consequently, exponential utility function and coherent measures of risk are selected as two plausible risk measures to analyze multiproduct riskaverse newsvendor models.
Introduction
In the era of Industry 4.0, firms are facing with greater uncertainty. Accordingly, we cannot always expect that similar outcomes may be repeated in random situations. The first few outcomes may turn out to be very bad such that they might be unacceptable losses. Then, open innovation can be a good remedial option for such riskaverse newsvendors because open innovation can offset the profit losses from risk aversion by sharing revenues in supply chains (refer to Yoon and Jeong (2017)).
In the literature of inventory management, the (single or multiproduct) newsvendor model, initiated by Arrow et al. (1951), is a wellknown classical stochastic inventory replenishment problem in supply chain management literature. In this model, there may exist perishable products with random demand in a singleselling season. Then a newsvendor should decide his optimal ordering quantity for each product in this singleperiod model before demand realization. Because the product demand is only given as a probability distribution, the objective function is represented as a random outcome. If the newsvendor orders too much for any product, all the leftover items are sold at a discounted price; if the newsvendor orders too little, it will lose sales opportunity.
The original model by Arrow et al. (1951) maximizes the expected value of profits without resource constraints and demand substitution. Then the multiproduct model is decomposable into multiple singleproduct models in each product and has a simple analytical closedform optimal solution for each product. This solution is known as a fractile, described with overage and underage profits, of the arbitrary (cumulative) demand distribution function. Thus, it can characterize the optimal solution effectively with underage and overage profits as well as its solvability as a closedform solution. Owing to its simple solution with tradeoff analysis between underage and overage profits, it has many applications in industries such as overbooking problems or facility capacity problems.
Since Arrow et al. (1951), many variations of multiproduct newsvendor models have been studied in literature. Hadley and Whitin (1963) add a resource constraint and suggest solution methods using Lagrangian multipliers. Van Ryzin and Mahajan (1999) study a multiproduct newsvendor with demand substitution. In both of Hadley and Whitin (1963) and van Ryzin and Mahajan (1999), multiproduct models is not decomposable, so we need to consider all the products simultaneously. In that sense, a multiproduct newsvendor model considers heterogeneous expectations in each product at a time and such setting has been found quite common in literature (refer to Lee and Lee (2015)).
Again Arrow et al. (1951) and its variations focus on maximizing the expected (random) profit. That is, the newsvendor selects his optimal solution based on the expected value of the random outcome. Thus, the original model and its variations can be said to be expectedvalue optimization models and also equivalently riskneutral models under uncertainty. However, risk neutrality guarantees the best decision only on average. It may be justified by the Law of Large Numbers. However, we cannot expect that the actual single realization is sufficiently close to its expected value. In fact, when the single realization is very much deviated from its expected value, riskneutral models will lose their validity. Then riskaverse decision making can be a good alternative, instead of riskneutral decision making.
To overcome drawbacks of riskneutral models, various risk preferences have been studied in literature. Lee et al. (2016) argues that degree of ambiguity may affect decision makers’ risk preferences. More specifically, consumers tend to be more riskaverse with more ambiguous situations and vice versa. In riskaverse models, inventory managers consider the variability of the outcome as well as its expected value. That is, under risk aversion, a riskaverse inventory manager may prefer more stable outcome even if the outcome is worse on average. Schweitzer and Cachon (2000) conducted two empirical experiments to show risk preferences of inventory managers. By the experiments, they showed that inventory managers may be riskaverse for short lifecycle or highvalue products. Therefore, risk aversion can capture the decision making of inventory managers at a different angle from risk neutrality and both of them are consistent with rational decision makers. Because risk aversion significantly affects the optimal choices of inventory managers, it is a very interesting and important factor to analyze the optimal choices of inventory managers. In particular, risk aversion has a very good fit to conservative decision makers. Some good industrial examples are energy, environment and sustainability where risk measurement is very important.
This paper aims to extend the series of previous works, Choi and Ruszczyński (2008), Choi et al., 2011and Choi and Ruszczyński (2011). In those three papers, they conducted the extensive literature review for various risk measures used in the inventory management literature and categorized the risk measures into four typical approaches. Then they selected coherent measures of risk as quality risk measures in Choi and Ruszczyński (2008) and Choi et al. (2011) and an exponential utility function in Choi and Ruszczyński (2011), respectively. In each paper, a logical justification was given for using such specific risk measure selected. Then the optimal policy of the newsvendor models was studied by providing several analytical propositions and numerical insights. On the other hand, we examine such logical justifications in those papers more comprehensively and deeply. As a result, we provide a logical reasoning and then axiomatic framework to compare the validity of such risk measures in multiproduct newsvendor models by analyzing the underlying assumptions and managerial insights.
In order to find plausible risk models in newsvendor problems, we focus on the measures based on risk aversion. Then we consider risk neutrality for a reference purpose only. For this purpose, the wellknown Prospect Theory and loss aversion, initiated by Kahneman and Tverski (1979), are not considered in this paper. The Prospect Theory assumes that people are riskaverse for their gains, but riskseeking for their losses. It can explain why sometimes people may buy lottery and insurance together, which was not explained by expected utility theory. This situation may be consistent to individual decision makers, but not inventory managers in a company because inventory managers do not have to carry the products incurring losses. Loss aversion is a concept introduced first by Kahneman and Tverski (1979). It refers to the tendency of an individual decision maker who prefers avoiding losses to obtaining gains. However, in a successive work in the Prospect Theory, Tverski and Kahneman, 1992revealed that loss aversion does not occur in routine transactions (refer to Novemsky and Kahneman (2005)), which describe typical inventory decisionmaking situations.
The remainder of this paper is organized as follows: First, we briefly review the four typical approaches in §1. Second, we conduct a literature review in riskaverse inventory models in § 2. Third, we discuss the validity of risk measures for newsvendor problems in §3. Forth, we show newsvendor problem formulations in §4. Lastly, we conclude this paper by summarizing the main results and suggesting some extensions of the paper in §5.
Risk measures
Due to the aforementioned reasons in §1, riskaverse newsvendor models have been recently studied very actively with various risk measures in inventory management literature. Choi et al. (2011) had an attempt to categorize the risk measures of riskaverse inventory models in inventory Management literature. Then the authors summarize the typical approaches of risk measures into four groups. They are expected utility theory, stochastic dominance, chance constraints and meanrisk analysis. Although these four categories of risk measures are different from each other, they are closely related and consistent to some extent. In this paper, we continue to use this fourgroup classification in Choi et al. (2011).
Expected utility theory
In the utility function approach, inventory managers optimize the expected value of their utility function, instead of the expected outcomes. Then the optimization model of utility function approach can be represented as follows:
Consider an optimization model where the decision vector x affects a random performance measure, ϕ_{ x }. Here, for all x ∈ ℵ with ℵ being a vector space, ϕ_{ x } : Ω → ℝ is a measurable function on a probability space \( \left(\varOmega, \mathcal{F},P\right) \) where Ω is the sample space, \( \mathcal{F} \) is a σ–algebra on Ω and P is a probability measure on Ω. Then, the modern theory of the expected utility by von.
where ϕ_{ x } is an (measurable) outcome function. From now on, ϕ_{ x } denotes a profit function in this paper. When the performance measure is defined as a profit function, a riskaverse decision maker is consistent to the secondorder stochastic dominance and he has a concave and nondecreasing utility function. Since Eeckhoudt et al. (1995), an approach of utility functions has been popular in riskaverse newsvendor models. In Eeckhoudt et al. (1995), nondecreasing and concave utility function are used to analyze riskaverse newsvendor models.
In this paper, we select an exponential utility function among various nondecreasing and concave utility functions. Choi and Ruszczyński (2011) point out that.
Exponential utility function is a particular form of a nondecreasing and concave utility function. It is also the unique function to satisfy constant absolute risk aversion (CARA) property. For those reasons, exponential utility function has been used frequently in finance and also in the supply chain management literature such as Bouakiz and Sobel (1992) and Chen et al. (2007).
Stochastic dominance
Stochastic dominance is the sequence of the partial orders defined on the space of random variables in a nested way such as the firstorder, the secondorder, the higherorders than the second and so on. This sequence of relations allow pairwise comparison of different random variables (see Lehmann (1955) and Hadar and Russell (1969)) and lowerorders are stronger relations in the sequence. In the sequence of the relations, the secondorder stochastic dominance is consistent to risk aversion.
Then an important property of stochastic dominance relations is its consistency to utility functions. That is, a random variable ϕ_{1} dominates ϕ_{2} by a stochastic dominance relation is equivalent that the expected utility of ϕ_{1} is better than that of ϕ_{2} for all utility functions in a certain family of utility functions. For the first and secondorder stochastic dominance relations, this property is represented as follows:
\( {\phi}_1{\succcurlyeq}_{(1)}{\phi}_2\iff \mathbb{E}\left[u\left({\phi}_1\right)\right]\ge \) \( \mathbb{E}\left[u\left({\phi}_2\right)\right] \), for every nondecreasing U[∙].
\( {\phi}_1{\succcurlyeq}_{(2)}{\phi}_2\iff \mathbb{E}\left[u\left({\phi}_1\right)\right]\ge \) \( \mathbb{E}\left[u\left({\phi}_2\right)\right] \), for every nondecreasing and concave U[∙]
In spite of such nice properties, stochastic dominance does not have a simple computational method unfortunately for its implementation by itself. Thus, it has been mainly used as a reference criterion to evaluate the legitimacy of riskaverse inventory models.
Chance constraints
In finance, chance constraints are very popular as the name of VaR (ValueatRisk). For consistency to stochastic dominance, VaR is a relaxed version of the firstorder stochastic dominance, but might violate the secondorder stochastic dominance.
Meanrisk analysis
Here, λ is a nonnegative tradeoff constant between the expected outcome and the scalarmeasured value of the variability of the outcome. This allows a simple tradeoff analysis analytically and geometrically.
In the minimization context, one selects from the universe of all possible solutions those that are efficient: for a given value of the mean he minimizes the risk, or equivalently, for a given value of risk he maximizes the mean. Such an approach has many advantages: it allows one to formulate the problem as a parametric optimization problem, and it facilitates the tradeoff analysis between mean and risk. However, for some popular dispersion statistics used as risk measures, the meanrisk analysis may lead to inferior conclusion. Thus, it is of primary importance to decide a good risk measure for each type of the decision models to be considered. The two important examples are meanvariance (or meanstandard deviation) model and coherent risk measures.
Meanvariance model
Since its introduction, many authors have pointed out that the meanvariance model is, in general, not consistent with stochastic dominance rules. It may lead to an optimal solution which is stochastically dominated by another solution. Thus, to overcome drawbacks of meanvariance model, the general theory of coherent measures of risk was initiated by Artzner et al. (1999) and extended to general probability spaces by Delbaen (2002).
Coherent measures of risk
Coherent measures of risk are extensions of meanrisk model to put different variability measures r[∙] (e.g. deviation from quantile or semideviation) instead of variance. A formal definition of the coherent measures of risk is presented by following the abstract approach of Ruszczyński and Shapiro (2005 and 2006a).
Let \( \left(\varOmega, \mathcal{F}\right) \) be a certain measurable space. A uncertain outcome is represented by a measurable function ϕ_{ x } : Ω → ℝ. We specify the vector space \( \mathcal{Z} \) of the possible functions of ϕ_{ x }; in this case it is sufficient to consider \( \mathcal{Z}={\mathcal{L}}_{\infty}\left(\varOmega, \mathcal{F},P\right) \).
A coherent measure of risk is a functional \( \rho :\mathcal{Z}\to \mathbb{R} \) satisfying the following axioms:
Convexity: ρ(αϕ_{1} + (1 − α)ϕ_{2}) ≤ αρ(ϕ_{1}) + (1 − α)ρ(ϕ_{2}), for all \( {\phi}_1,{\phi}_2\in \mathcal{Z} \) and all α ∈ [0, 1];
Monotonicity: If \( {\phi}_1,{\phi}_2\in \mathcal{Z} \) and ϕ_{1} ≽ ϕ_{2}, then ρ(ϕ_{1}) ≤ ρ(ϕ_{2});
Translation Equivariance: If a ∈ ℝ and \( {\phi}_1\in \mathcal{Z} \), then ρ(ϕ_{1} + a) = ρ(ϕ_{1}) − a;
Positive Homogeneity: If t ≥ 0 and \( {\phi}_1\in \mathcal{Z} \), then ρ(tϕ_{1}) = tρ(ϕ_{1}).
The infimum of conditional expectations \( \mathbb{E}\left[{\phi}_1A\right] \) is taken on all the events A with probability larger than α in the σ–algebra \( \mathcal{A} \). However, under certain conditions on nonatomic probability space, this risk measure becomes lawinvariant and coincides with a famous risk measure CVaR (Conditional ValueatRisk). For more technical details, see Acerbi and Tasche (2002), Delbaen (2002) and Kusuoka (2003).
The optimal η in the eq. (5) is the βquantile of ϕ_{1}. Then CVaR is a special case of meandeviationfromquantile when λ = 1/β. All these results can be found at Ruszczyński and Shapiro (2006a) and Choi (2009) with a sign adjustment.
Literature review
Summary of the literature on riskaverse newsvendor models
Risk Measures \ Model Types  Singleproduct Newsvendor  Multiproduct Newsvendor  

Utility Function  Choi and Ruszczyński (2011)  Choi and Ruszczyński (2011)  
Stochastic Dominance  It is a reference criterion. Thus, it is not directly applicable for implementation.  
Chance Constraints (or Valueatrisk)  Özler et al. (2009)  Özler et al. (2009)  
Meanrisk Analysis  Coherent  Meandeviation from quantile or meansemideviation  Chen et al. (2009) Yang et al. (2008)  N/A 
General Coherent Risk Measures  Choi and Ruszczyński (2008)  Choi and Ruszczyński (2011)  
NonCoherent  Meanvariance  N/A  N/A  
Meanstandard deviation  N/A  N/A 
Yang et al. (2008) consider a singleproduct riskaverse newsvendor with a capacity constraint for ordering quantity. They select two risk measures, CVaR (Conditional ValueatRisk) and VaR (ValueatRisk), for their models. As a result, they provide closedform optimal solution with both risk measures and confirm their results with numerical examples. Chen et al. (2009) study a singleproduct newsvendor of stochastic pricedependent demand with CVaR. That is, their models are joint models of ordering quantity and price. The key research questions are to characterize the optimal order quantity and prices and to conduct comparative statics analysis with respect to model parameters for additive and multiplicative demand cases. In addition, they compare their results with those in the corresponding riskneutral models of stochastic pricedependent demand. Özler et al. (2009) consider a multiproduct newsvendor with a ValueatRisk constraint. They also consider a singleproduct newsvendor as a special case. For a singleproduct system, they obtain the closedform optimal ordering quantity which is the same result of Gan et al. (2004). Their biggest contribution to the literature is that for a twoproduct system, they obtain the mathematical formulation of mixed integer programming where the objective function is nonlinear and the constraints are mixed linear and nonlinear functions. Then, they conducted their numerical analysis to confirm their analytical results under multivariate exponential demands.
Discussion of validity of risk measures in newsvendor models
Four axioms in coherent measures of risk

Convexity axiom means that the global risk of a portfolio should be equal or less than the convex combination of its partial risks. Because lower measured risk is better in coherent measures of risk, this axiom is consistent with the diversification effects.

In the Monotonicity axiom, ϕ_{1} ≽ ϕ_{2} means that ϕ_{1} is always preferred to ϕ_{2} for all possible scenarios. Thus, this axiom means that if portfolio 1 always has better values than portfolio 2 under all possible scenarios, then the measured risk of the portfolio 1 should be less than the measured risk of portfolio 2. By satisfying this axiom, coherent measures of risk are consistent with the secondorder stochastic dominance.

Translation Equivariance axiom means that the existence of a constant cost (or gain) is equivalent to equally decreasing (or increasing) the vendor’s performance measure.
Thus, fixed parts can be separated equivalently from the vendor’s random performance measure at every possible state of nature. Thus, this axiom allows one to draw a comparison between the only random parts of different random performance measures and thus rank risk properly (see Artzner et al. (1999)). However, this axiom is contradictory to initial endowment effects (refer to Choi et al. (2011)).

Positive Homogeneity axiom guarantees that the optimal solution is invariant to rescaling of units such as currency (e.g., from dollars to pounds) or considering the total profit or the average profit per product. In addition, this axiom guarantees no diversification effects in a limiting case when the multivariate demand has a perfect positive correlation (see Choi et al. (2011)).
These features are derived regardless of any specific problem formulations in multiproduct newsvendor problems. That is, these features and implications can be directly applied in any type newsvendor problems with different formulations to evaluate the validity of risk measures.
The axiomatic framework
Comparison between risk measures in newsvendor problems
Risk measures \ Model types  Consistency to the four axioms for coherent risk measures  

Convexity  Monotonicity  Translation Equivariance  Positive Homogeneity  
Utility Function  Yes  Yes  No  No  
Stochastic Dominance  It is a reference criterion. Thus, it is not directly applicable for implementation.  
Chance Constraints (or Valueatrisk)  No  Yes  Yes  Yes  
Meanrisk Analysis  Coherent  Meandeviation from quantile or meansemideviation  Yes for all of the four axioms  
General Coherent Risk Measures  Yes for all of the four axioms  
NonCoherent  Meanvariance  No  No  Yes  No  
Meanstandard deviation  No  No  Yes  Yes 
Stochastic dominance is a reference criterion to give pairwise comparison between different random outcome. Thus, it is not directly implemented for its application.
Chance constraints have been actively used in finance historically. In financial terms, they are intuitive and easy to understand. However, they generally violate Convexity, which implies that chance constraints may penalize diversification instead of encouraging it. Historically, the Convexity has been a controversial axiom in finance literature due to the popularity of VaR in financial markets. However, such situations may be justified in finance literature such as insurance industry, but very different from that in newsvendor problems. In fact, the Convexity axiom is especially valid in newsvendor models. Each product is very likely to have some nonzero value in newsvendor models because very small amounts will be sold almost always for each product (refer to Choi et al. (2011) and Choi and Ruszczyński (2011)).
Meanvariance and meanstandard deviation model have been very wellknown since the seminal work of Markowitz (1952). The meanvariance model satisfies the Translation Equivariance axiom only. Meanstandard deviation model satisfies additionally Positive Homogeneity as well as Translation Equivariance, but not Convexity and Monotonicity.
A counterexample to show imperfection of meanvariance model
ω _{1}  ω _{2}  

ϕ _{1}  −1  3 
ϕ _{2}  −1  −1 
In Table 3, we set up Ω = {ω_{1}, ω_{2}} and P(ω_{1}) = P(ω_{2}) = 0.5. Then larger value is always preferred to smaller value in this table. Each random variable ϕ_{1} and ϕ_{2} has a value for any possible states of nature, ω_{1} and ω_{2}, and ϕ_{1}(ω) is always better than ϕ_{2}(ω) for all ω ∈ Ω. Thus, ϕ_{1} dominates ϕ_{2} by the rule of statewise dominance and this table is a good example where an efficient solution (in the sense from meanrisk analysis) is dominated by another solution. Clearly, ϕ_{1} may be preferred to ϕ_{2}. However, \( \mathbb{E}\left({\phi}_2\right)1\bullet \mathbb{V}\mathrm{ar}\left({\phi}_2\right)=1>3=\mathbb{E}\left({\phi}_1\right)1\bullet \mathbb{V}\mathrm{ar}\left({\phi}_1\right). \) This implies that ϕ_{2} is more preferable to ϕ_{1} under meanvariance criterion, which is inconsistent with the Monotonicity axiom.
Selection of risk measures in newsvendor problems
In summary, expected utility theory and coherent risk measures share the Convexity and Monotonicity axioms when a newsvendor has a nondecreasing and concave function. However, expected utility theory does not satisfy the Translation Equivariance and Positive Homogeneity. General coherent measures of risk are consistent to the first and secondorder stochastic dominance relations and satisfy all the four axioms. Thus, Translation Equivariance and Positive Homogeneity axioms are crucial to decide which one is better to use between utility function approach and coherent measures of risk.
The initial endowment effects, firstly theorized by Thaler (1980) in behavioral economics, mean that the initial states of the variables may affect the optimal decision. Sometimes the effects may have a significant role for inventory managers. Therefore, if a newsvendor takes initial endowment effects strongly, then coherent measures of risk may not be preferred by this newsvendor. Such effects can be captured by utility function approach, but not by coherent measures of risk. Thus, if newsvendors show initial endowment effects significantly, utility function approach is better to use to analyze the newsvendor problems. More specifically, exponential utility function is a particular form of a nondecreasing and concave utility function. It is also the unique function to satisfy constant absolute risk aversion property. For those reasons, exponential utility function has been used frequently in finance and also in supply chain management literature such as Bouakiz and Sobel (1992), Chen et al. (2007) and Choi and Ruszczyński (2011). However, the existence of initial endowment effects is still controversial (see Hanemann (1991) and Shogren et al. (1994)).
On the other hand, Positive Homogeneity implies invariance of the optimal solution from denomination of the currency to guarantee consistence to rational riskaverse decision making. Choi et al. (2011) provide a numerical example where they compare solutions of a singleproduct newsvendor model with coherent measures of risk, exponential utility function and meanvariance. They initially select parameters in each risk measure so that they have the same optimal solution when the unit of profit is measured as one dollar. Then they change the unit of profit continuously by denomination. Then the optimal solution with coherent measures of risk is unchanged, but the solutions significantly change with the other risk measures. Because utility functions are not compatible with Positive Homogeneity, they also have some drawbacks to analyze newsvendor problems.
In conclusion, considering relative advantages and disadvantages of using each type risk measure, exponential utility function approach and coherent measures of risk are two plausible risk measures to analyze newsvendor model by the consideration with the axiomatic approach.
Conclusion
In this paper, we have examined various risk measures in newsvendor problems. By focusing on the four axioms of coherent risk measures, we have compared the four typical approaches; expected utility theory, stochastic dominance, chance constraints and meanrisk analysis. As a result, an exponential utility function and coherent risk measures are selected as two quality risk measures for newsvendor problems.
It is natural that the newsvendors are riskaverse when they are in a preliminary transition stage to open innovation. Then, in order to handle risk aversion properly, it is necessary to find a quality risk measures for such cases. Due to this reason, we consider the validity of risk measures for the riskaverse newsvendor models when their levels of open innovation are relatively low. Finally, our contributions to literature are can be summarized as follows: First, we conduct an extensive and rigorous literature review in risk measures and newsvendor problems in a perspective of open innovation. Second, we discuss the relationship between two conflicting risk preferences, risk aversion and neutrality, with open innovation. Last, we provide an axiomatic framework to verify the validity of various risk measures used in real world as well as the literature of this research stream.
We believe that there is an important extension that can be addressed in this axiomatic framework. In this paper, we discuss meaning and implications of the four axioms in coherent risk measures in newsvendor models. For a multiperiod case, dynamic version of coherent risk measures were also analyzed in the literature (refer to Riedel (2004), Kusuoka and Morimoto (2004), Cheridito et al. (2006) and Ruszczyński and Shapiro (2006b)). Then, with appropriate adjustments, this axiomatic approach can be a good starting point of constructing another axiomatic framework to compare the validity of various risk measures for a multiperiod case.
Declarations
Acknowledgements
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Funding
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Availability of data and materials
The dataset(s) supporting the conclusions of this article is (are) included within the article.
Authors’ contributions
SC contributed to the overall idea, theoretical development, and writing of the manuscript; KBP contributed to discussions of risk measures on open innovation, as well as the preparation and publishing of the paper; SOS contributed to detailed writing and discussions of risk measures on sustainability. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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