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• DOI: 10.1016/j.ejor.2010.11.024
• Corpus ID: 9186425

The newsvendor problem: Review and directions for future research

• Yan Qin , Ruoxuan Wang , +2 authors Michelle M. H. Şeref
• Published in European Journal of… 1 September 2011
• Business, Economics

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Please note you do not have access to teaching notes, behavioral perspective of newsvendor ordering decisions: review, analysis and insights.

Management Decision

ISSN : 0025-1747

Article publication date: 20 March 2020

Issue publication date: 22 January 2021

The traditional newsvendor model has focused on deriving the optimal order quantity that minimises the balance between stocking too much or too less number of products. However, the managers make inventory decisions based on intuitions and shortcuts, which may involve human errors and biases. The effect of cognitive biases and heuristics influencing the inventory ordering decisions in newsvendor settings is highlighted. The advancement of research associated to the newsvendor biases is reviewed to appreciate the behavioral aspects of the minds underlying this process.

Design/methodology/approach

The use of experimental and non-experimental methods to investigate the ordering behaviour of newsvendors is described and we present a framework of the existing literature and highlight the research gaps to point to future research possibilities and priorities.

The proposed framework gives a systematic approach to confirm the existence of a substantial scope of research opportunities and points to specific areas for further research. It synthesizes the existing results of behavioral newsvendor research and will act as a key reference paper. In addition, it will help the practitioners and software tool vendors to comprehend the behavioral perspective of newsvendor preferences and design strategies to mitigate this effect. The insights will be helpful for academicians, researchers and practitioners working in the areas of experimental economics, behavioral economics, behavioral operations, bounded rationality theory, newsvendor modelling and supply chain contracts.

Originality/value

A summary of literature in this evolving area of research is very scarce. Considering the impact of behavioral economics on managerial decisions in the contemporary world, it is highly important to have an educational summary which can act as a tool for the practitioners and researchers in the area of behavioral operations management.

• Behavioral economics
• Behavioral operations
• Experimental economics
• Newsvendor ordering
• Bounded rationality
• Cognitive biases and heuristics

Acknowledgements

The author thanks the reviewers and the editor for their valuable feedback and suggestions to improve the quality of the paper.

Yamini, S. (2021), "Behavioral perspective of newsvendor ordering decisions: review, analysis and insights", Management Decision , Vol. 59 No. 2, pp. 240-257. https://doi.org/10.1108/MD-07-2019-0975

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• Open access
• Published: 03 January 2018

Comparing validity of risk measures on newsvendor models in open innovation perspective

• Sungyong Choi 1 ,
• KyungBae Park   ORCID: orcid.org/0000-0003-0900-609X 2 &
• Sang-Oh Shim 3  

Journal of Open Innovation: Technology, Market, and Complexity volume  4 , Article number:  1 ( 2018 ) Cite this article

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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 risk-averse 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 risk-averse 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 multi-product risk-averse 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 risk-averse 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 multi-product) newsvendor model, initiated by Arrow et al. ( 1951 ), is a well-known classical stochastic inventory replenishment problem in supply chain management literature. In this model, there may exist perishable products with random demand in a single-selling season. Then a newsvendor should decide his optimal ordering quantity for each product in this single-period 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 multi-product model is decomposable into multiple single-product models in each product and has a simple analytical closed-form 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 closed-form solution. Owing to its simple solution with trade-off 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 multi-product 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 multi-product newsvendor with demand substitution. In both of Hadley and Whitin ( 1963 ) and van Ryzin and Mahajan ( 1999 ), multi-product models is not decomposable, so we need to consider all the products simultaneously. In that sense, a multi-product 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 expected-value optimization models and also equivalently risk-neutral 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, risk-neutral models will lose their validity. Then risk-averse decision making can be a good alternative, instead of risk-neutral decision making.

To overcome drawbacks of risk-neutral 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 risk-averse with more ambiguous situations and vice versa. In risk-averse models, inventory managers consider the variability of the outcome as well as its expected value. That is, under risk aversion, a risk-averse 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 risk-averse for short life-cycle or high-value 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., 2011 and 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 multi-product 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 well-known Prospect Theory and loss aversion, initiated by Kahneman and Tverski ( 1979 ), are not considered in this paper. The Prospect Theory assumes that people are risk-averse for their gains, but risk-seeking 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, 1992 revealed that loss aversion does not occur in routine transactions (refer to Novemsky and Kahneman ( 2005 )), which describe typical inventory decision-making 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 risk-averse 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, risk-averse 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 risk-averse 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 mean-risk 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 four-group 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.

von Neumann and Morgenstern ( 1944 ) derives, from simple axioms, the existence of a nondecreasing utility function, which transforms (in a nonlinear way) the observed outcomes. That is, each rational decision maker has a nondecreasing utility function u (∙) such that he prefers random outcome ϕ 1 over ϕ 2 if and only if [ u ( ϕ 1 )]> \( \mathbb{E}\left[u\left({\phi}_2\right)\right] \) , and then he optimizes, instead of the expected outcome, the expected value of the utility function. Therefore, the decision maker solves the following optimization model.

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 risk-averse decision maker is consistent to the second-order 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 risk-averse newsvendor models. In Eeckhoudt et al. ( 1995 ), nondecreasing and concave utility function are used to analyze risk-averse 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 first-order, the second-order, the higher-orders 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 lower-orders are stronger relations in the sequence. In the sequence of the relations, the second-order 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 second-order 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 risk-averse inventory models.

Chance constraints

Chance constraints add some constraints on the probabilities that measure the risk such as:

where η is a fixed target value and α   ∈  (0, 1) is the maximum level of risk of violating the stochastic constraint, ϕ x  ≥  η . Then, we consider the following optimization model.

subject to P ( ϕ x  ≥  η ) ≥ 1 −  α

In finance, chance constraints are very popular as the name of VaR (Value-at-Risk). For consistency to stochastic dominance, VaR is a relaxed version of the first-order stochastic dominance, but might violate the second-order stochastic dominance.

Mean-risk analysis

Mean-risk analysis provides efficient solutions and quantifies the problem in a clear form of two criteria: the mean (the expected value of the outcome) and the risk (a scalar measured variability of the outcome). In mean-risk analysis, one uses a specified functional \( r:\aleph \to \mathbb{R} \) , where ℵ is a certain space of measurable functions on a probability space \( \left(\varOmega, \mathcal{F},P\right) \) to represent variability of the random outcomes, and then solves the problem:

Here, λ is a nonnegative trade-off constant between the expected outcome and the scalar-measured value of the variability of the outcome. This allows a simple trade-off 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 trade-off analysis between mean and risk. However, for some popular dispersion statistics used as risk measures, the mean-risk 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 mean-variance (or mean-standard deviation) model and coherent risk measures.

Mean-variance model

Since the seminal work of Markowitz ( 1952 ), mean-variance model has been actively used in the literature and it used the variance of the return as the risk functional, i.e.

Since its introduction, many authors have pointed out that the mean-variance 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 mean-variance 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 mean-risk 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(α ϕ 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 ).

An optional axiom in coherent measures of risk is law-invariance. A coherent measure of risk ρ (∙) is called law-invariant , if the value of ρ ( ϕ 1 ) depends only on the distribution of ϕ 1 , that is ρ ( ϕ 1 )= ρ ( ϕ 2 ) if ϕ 1 and ϕ 2 have identical distributions. Acerbi ( 2004 ) summarizes the meaning of this property that a law-invariant coherent measure of risk gives the same risk for empirically exchangeable random outcomes . Law-invariance looks so obvious that it is no wonder even if most risk practitioners take it for granted. However, it also implies that, for a coherent measure of risk ρ which is not law-invariant, ρ ( ϕ 1 ) and ρ ( ϕ 2 ) may be different even if ϕ 1 and ϕ 2 have same probability distribution. This apparent paradox can be resolved by reminding the formal definition of random variables. Actually, one needs to determine simultaneously “probability law” and “field of events” endowed with a σ -algebra structure to define a random variable. Thus, the two random variables with same probability distributions can be different and may have different values of ρ . An example of the coherent measure of risk which is not law-invariant is the so-called worst conditional expectation WCE α defined in Artzner et al. ( 1999 ).

The infimum of conditional expectations \( \mathbb{E}\left[{\phi}_1|A\right] \) 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 law-invariant and coincides with a famous risk measure CVaR (Conditional Value-at-Risk). For more technical details, see Acerbi and Tasche ( 2002 ), Delbaen ( 2002 ) and Kusuoka ( 2003 ).

Because coherent measures of risk are any functionals to satisfy the four axioms above, their functional forms are not determined uniquely. The two popular examples are obtained to put deviation-from-quantile, r β [∙] with λ  ∈  [0, 1], or semideviation of order k  ≥ 1, σ k [∙] with λ   ∈  [0, 1/ β ], into r [∙], variability of the outcome:

The optimal η in the eq. (5) is the β -quantile of ϕ 1 . Then CVaR is a special case of mean-deviation-from-quantile when λ  = 1/ β . All these results can be found at Ruszczyński and Shapiro ( 2006a ) and Choi ( 2009 ) with a sign adjustment.

Literature review

Choi et al. ( 2011 ) also provided a comprehensive literature review in risk-averse inventory models since the seminal works of Lau ( 1980 ) and Eeckhoudt et al. ( 1995 ). In this paper, we provide a summary of literature of the risk-averse newsvendor models studied after Choi et al. ( 2011 ) at Table  1 where we classify and tabulate the literature by model types (as columns) and risk measures used (as rows). The key research question from the literature is the impact of degree of risk aversion to the optimal ordering quantity with parametric and comparative static analysis. A common finding from literature is that higher degree of risk aversion results in fewer optimal ordering quantities because higher ordering quantity implies higher variability of the profits. Then, risk-averse newsvendor tends to decrease ordering quantity to avoid higher risk.

Yang et al. ( 2008 ) consider a single-product risk-averse newsvendor with a capacity constraint for ordering quantity. They select two risk measures, CVaR (Conditional Value-at-Risk) and VaR (Value-at-Risk), for their models. As a result, they provide closed-form optimal solution with both risk measures and confirm their results with numerical examples. Chen et al. ( 2009 ) study a single-product newsvendor of stochastic price-dependent 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 risk-neutral models of stochastic price-dependent demand. Özler et al. ( 2009 ) consider a multi-product newsvendor with a Value-at-Risk constraint. They also consider a single-product newsvendor as a special case. For a single-product system, they obtain the closed-form optimal ordering quantity which is the same result of Gan et al. ( 2004 ). Their biggest contribution to the literature is that for a two-product 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 multi-variate exponential demands.

Discussion of validity of risk measures in newsvendor models

Four axioms in coherent measures of risk.

The four axioms (Convexity, Monotonicity, Translation Equivariance and Positive Homogeneity) in coherent measures of risk have attractive features and implications to analyze newsvendor problems and thus the axioms make coherent measures of risk worth considering. Although these four axioms are briefly discussed at the previous studies in literature, none of them had an attempt to consider all of the four axioms in a comparative sense. More specifically, based on the four axioms, we develop them as an axiomatic framework to analyze the validity of newsvendor problems. The optional axiom, Law-Invariance, does not have a practical meaning, so we do not consider it in this section.

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 second-order 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 multi-product 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

In this subsection, we compare the validity of various risk measures in newsvendor problems by our axiomatic framework based on the four axioms of coherent risk measures. The axiomatic approach provides a clear standard to evaluate risk measures in risk-averse newsvendor models. (Table  2 ).

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 )).

Mean-variance and mean-standard deviation model have been very well-known since the seminal work of Markowitz ( 1952 ). The mean-variance model satisfies the Translation Equivariance axiom only. Mean-standard deviation model satisfies additionally Positive Homogeneity as well as Translation Equivariance, but not Convexity and Monotonicity.

Since its introduction, many authors have pointed out that the mean-variance and mean-standard deviation models are, in general, not consistent with stochastic dominance rules, nor the Monotonicity axiom. Because both models consider over-performance and under-performance equally, they are not so-called downside risk measures and may lead to an optimal solution which is stochastically dominated by another solution. Thus, to overcome drawbacks of mean-variance model, the general theory of coherent measures of risk was initiated by Artzner et al. ( 1999 ) and extended. We provide a specific and simple counterexample that a mean-variance model violate the monotonicity axiom in Table  3 .

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 mean-risk 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 mean-variance 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 second-order 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 risk-averse decision making. Choi et al. ( 2011 ) provide a numerical example where they compare solutions of a single-product newsvendor model with coherent measures of risk, exponential utility function and mean-variance. 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.

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 mean-risk 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 risk-averse 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 risk-averse 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 multi-period 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 multi-period case.

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Choi, S., Park, K. & Shim, SO. Comparing validity of risk measures on newsvendor models in open innovation perspective. J. open innov. 4 , 1 (2018). https://doi.org/10.1186/s40852-017-0078-8

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The newsvendor model revisited: the impacts of high unit holding costs on the accuracy of the classic model

• Shaolong Tang 1 ,
• Stella Cho 1 ,
• Jacqueline Wenjie Wang 2 &
• Hong Yan 3  

Frontiers of Business Research in China volume  12 , Article number:  12 ( 2018 ) Cite this article

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The newsvendor problem has been applied in various business settings. It is often assumed that the decision variable, i.e., order-up-to level, has no impacts on the holding costs for average inventory cycled in a given period, which is the difference between beginning and ending inventory levels on hand in that period. The average holding cost for this portion of inventory is conveniently and approximately calculated as half the product of the unit holding cost and the expectation of the demand in one period if it is assumed that the inventory is approximately evenly consumed. It is a good approximation when the unit holding cost is significantly lower than the unit backorder cost as this optimal solution to inventory level is able to guarantee a low probability of understocking. However, if this condition does not hold, the approximation may deviate from the actual cost and cannot measure the expected holding cost for this portion of inventory. This paper examines the impact of the cycle stock holding cost on the newsvendor model and the conditions under which this portion of cost is not negligible.

Introduction

The newsvendor model is fundamental for stochastic inventory management theories, which has been studied and applied in various business settings (e.g., Erlebacher, 2000 ; Mieghem, 2007 ; Olivares et al., 2008 ; Petruzzi et al., 2009 ; Krishnan et al., 2010 ). Consider a typical single period newsvendor model. An optimal inventory level is determined to minimize the expected cost, usually including the ordering cost, and the expected overstocking and shortage costs. Ordering too many items can incur overstocking costs, while ordering too few can cause shortage costs. In this typical model, it is often assumed that the decision variable, i.e., order-up-to level, has no impacts on the holding cost for the average inventory consumed in a specific period, which is the difference between beginning and ending inventory levels at hand in that period. Consumed stocks are items that are sold or used by the holder in a particular period. In the classic model, the average holding cost for this portion of inventory is conveniently and approximately calculated as half the product of the unit holding cost and the expectation of the demand in that period, if it is assumed that the inventory is approximately evenly consumed. It is a good approximation when the unit holding cost is much lower than the unit shortage cost because the optimal inventory level of the classic newsvendor model can lessen the possibility of understocking. However, if this condition does not hold, the approximation may deviate from the actual cost and cannot measure the expected holding cost for this portion of inventory accurately. In fact, the unit holding cost for certain products may experience a quick increase. Factors that determine the unit inventory holding cost usually include storage space costs, handling and service costs, risk costs and capital costs. One or more of the above factors may lead to a high unit holding cost. For example, luxury products usually have a high capital cost due to their high ordering costs, and a high risk cost for insurance. Goods that require specific storage conditions, e.g., isoperibol, refrigeration, temporary control or air conditioning may also incur a high unit holding cost. For instance, in Qingdao, a city in North China which is well-known as a seafood distribution center, the rent for cold warehouses for seafood has continuously increased for the past two years. This is mainly because cold storage units are being charged heavily for pollution violations by the local Environmental Protection Bureau. With the ratification of the Paris Climate Agreement in 2015, many countries have committed to cutting greenhouse gas emissions significantly by 2030. Carbon charging has been implemented in several pilot markets in China including Qingdao and will be implemented nation-wide in the near future. Prices for CO 2 emitted by energy consumption for storage are likely to continue its increase as a way to address global warming. Indeed, such pricing has already been in effect in a number of industries in the European Union for many years. These kinds of policies push up the unit holding cost for storage, which in turn affects the accuracy of the traditional newsvendor model. Other factors can also drive up holding costs. In modern warehouses armed with advanced technologies, to smooth the process, many goods are labeled with RFID tags. As is well-known, RFID can greatly enhance the efficiency and effectiveness of warehouse management, although it incurs much higher costs compared to traditional bar coding systems. Overall, holding costs have been experiencing a sharp increase recently in some specific industries.

To examine the inaccuracy of the classic newsvendor model when the unit holding cost is high, we analyze two scenarios with different inventory levels at the end of one period in Figs.  1 and 2 . Figure  1 shows the scenario where overstocking occurs at the end of the period, and the consumed inventory used to meet customers’ demand is the difference between the beginning and ending inventory levels. In contrast, Fig.  2 shows the scenario where understocking takes place at the end of the period, the consumed inventory equals the beginning inventory level and there is unsatisfied demand at the end of the period. The unit holding cost is defined as the cost for one item held in the warehouse for the whole period. In the first scenario (Fig.  1 ), all demand in the period is satisfied, and the average consumed inventory approximately equals half the demand if it is assumed that the inventory is approximately evenly consumed. The approximation works well herein. In contrast, in the second scenario (Fig. 2 ) with unsatisfied demand, inventory has been used up before the end of the period. Then it is not appropriate to take half the demand as the approximation of the average consumed inventory. In this scenario the approximation is not very accurate and the degree of deviation greatly depends on the critical ratio of the newsvendor model. We can also observe that the decision variable, i.e., order-up-to level, does influence the holding cost for consumed inventory.

The situation where overstock occurs at the end of one period

The situation where out-of-stock occurs at the end of one period (with notation)

In the literature, most prior research has not considered the holding cost for consumed inventory in the cost (objective) function and has employed the approximation. There are two possible reasons. First, if the unit holding cost is much lower than the unit shortage cost, the optimal inventory level that is derived from the cost function considering ordering, overstocking and shortage costs can lessen the possibility of understocking, and the approximation is an accurate one to measure the holding cost for consumed goods. In this situation, whether this portion of the holding cost is added to the cost function does not affect the optimal inventory level much. A detailed explanation is also shown in the numerical experiment of this paper. Second, the newsvendor model, without considering the holding cost for consumed goods, can provide a compactly analytic solution, which is convenient for further analysis such as multi-echelon problems and multi-tier supply chain problems. For many cases where the ratio of unit holding cost to unit shortage cost is low, the approximation is accurate and skipping the holding cost for consumed inventory in the cost function is acceptable. However, when the unit holding cost is not much lower or even higher than the unit shortage cost, it may cause a non-ignorable deviation from the actual optimal inventory level that minimizes the sum of the expected ordering cost, shortage cost and holding cost for consumed and overstocked items. In such cases, we cannot ignore the consumed inventory holding cost in the cost function for simplification and should seriously consider it when we determine the optimal inventory level.

The rest of this paper is organized as follows. Section “ Literature Review ” reviews the related literature on newsvendor models. In Section “ Newsvendor Models with Consumed Inventory Holding Cost ”, the model that includes the consumed inventory holding cost is constructed and analyzed. In Section “ Numerical Experiments ”, numerical experiments are conducted to analyze the cases where the holding cost for consumed inventory becomes non-ignorable. Section “ Conclusion ” concludes the whole paper and extends the discussion.

Literature review

The newsvendor model is one of the fundamental models for Operations Research and Management Science. Porteus ( 1990 ) summarizes the typical newsvendor model for one-period and multi-period cases. The ordering cost, and overstocking and understocking costs are considered in the cost function and the unsatisfied demand at the end of one period is either lost or backlogged. The critical ratio that is the optimal probability of not stocking out is the ratio of the unit underage cost to the sum of the unit underage and overage costs. Arrow et al. ( 1951 ) give the first derivation of the optimal inventory level and reorder point as a function of the demand distribution, the cost of making an order, and the overstocking and understocking costs. Arrow and Karlin ( 1958a , 1958b ) analyze the optimality of the base stock policy for one stage inventory models with uncertain demand. The optimal inventory level can be determined by examining the derivative of the objective function. Concerning the batch ordering problem of the newsvendor model, Veinott ( 1965a ) shows that a base stock policy, with which the stock is replenished to a certain level if possible, is still optimal. For a single period model with convex objective function and setup cost for ordering, an ( s , S ) policy is optimal. Karlin (1958a) and Porteus ( 1971 , 1972 ) examine the conditions under which a generalized ( s, S ) policy is optimal. Porteus ( 1971 , 1972 ) and Heyman and Sobel ( 1984 ) further show that an ( s, S ) policy will still be optimal if the objective function is a quasi- K -convex for some K that is less than the fixed ordering cost. Hadley and Whitin ( 1963 ) consider the newsvendor model with a single linear constraint on the initial inventory levels for multiple products. Evans ( 1967 ) analyzes a newsvendor model with a linear constraint on the total order and shows that the optimal policy is a base stock policy. Veinott ( 1965b ) examines the optimal policy for the multi-period problem and shows that a myopic base stock policy is optimal under a set of conditions and can be found as a solution to the single period newsvendor model with modified unit overstocking and understocking costs. Li and Lin ( 2006 ) examine the affecting factors for information sharing and information quality in supply chain management. Ferguson et al. ( 2007 ) analyze an extension of the economic order quantity model where the cumulative holding cost is a nonlinear function of time. Berling ( 2008 ) examines the problem with a stochastic holding cost. He considers a single-item inventory model with a fixed set-up cost and stochastic purchase price, and assumes the holding cost is the product of interest rates and purchase price. Shi and Yan ( 2017 ) analyze features of the heterogeneity of consumer preferences and address the multiple equilibrium of retail formats. In sum, prior research has analyzed the optimal solutions of the basic newsvendor model and has provided extensions that consider the setup cost, batch ordering, linear constraints on the total order and initial inventory levels for multi-product cases, etc. The ordering cost, overstocking and understocking costs are considered in the cost function. To our knowledge, no research addresses the consumed inventory holding cost in the objective function and its impact on the optimal inventory level, while this portion of holding costs does occur. This research aims to narrow this gap. Correia et al. ( 2013 ) develop performance measures for a multi-period, two-echelon supply chain network.

Newsvendor models with consumed inventory holding cost

The newsvendor model with instantaneous receipt.

In this section, we first formulate the newsvendor model with instantaneous receipt by considering inventory holding costs for consumed inventory in the cost function. At the beginning of the period, an order-up-to inventory level is determined to minimize the sum of the expected cost. At the end of the period, there are two possible scenarios: overstocking or understocking. Figures 1 and 2 show the changes in inventory level when overstocking or understocking occurs respectively. The unsatisfied demand is backlogged if there is understocking. The stocking out cost is the backordering cost. The inventory level determined at the beginning of the period affects the overstocking and understocking costs, as well as the holding cost for consumed inventory during the period. The model is described by the following parameters and variables:

h the unit inventory holding cost;

π the unit backordering cost if shortage occurs;

x the random demand in the period, which follows a normal distribution;

μ the mean of the demand distribution in the period;

σ the standard deviation of the demand distribution in the period;

I the order-up-to inventory level at the beginning of the period;

C ( I ) the inventory cost when the order-up-to level is I;

φ (•) the probability density function of the demand in the period;

Φ(•) the standard normal distribution function;

E (•) the expected value of a random variable.

Then the objective function of the newsvendor model with instantaneous receipt is

The first and second terms in the objective function represent the expected overstocking and understocking costs. The third term is the expected inventory holding cost for consumed inventory if there are items left at the end of the period. This situation is also shown in Fig. 1. The fourth term is the expected consumed inventory holding cost if out-of-stock occurs, which can be derived from the plane-geometry as follows. In Fig. 2, it can be shown that the holding cost for consumed stock is

We can show that \( \frac{t}{T}=\frac{I}{x} \) Then we have

By rearranging Program P1, we get

The sum of the first three terms in Program P2 is the total inventory cost if we use half the product of the unit holding cost and the expectation of the demand in the period to approximate the consumed inventory holding cost, which is usually used in classic newsvendor models. With this approximation, the decision variable I affects the first two terms (i.e., overstocking and understocking costs), but has no impact on the third term, and the optimal inventory level is determined by minimizing the sum of the first two terms, and can be expressed as.

However, when we consider the consumed inventory cost in a more accurate way, there is another term \( h{\int}_I^{+\infty}\left({I}^2/x-x\ \right)\ \varphi (x)\ dx/2 \) in the cost function, which is also affected by the decision variable I . Then the optimal inventory level for Program P2 is different with the one given in (3.1). We take the first-order and second-order derivatives of the objective function with respect to I in Program P2, and we have.

(3.3) shows that the second-order derivative is greater than 0, and therefore the objective function is convex. It should be noted that, due to the second and third terms of the first-order derivative in (3.2), a closed form solution cannot be derived. Denote the solution of C ′ ( I ) = 0 by I ∗ , which is the optimal inventory level of Program P2. From (3.2) and (3.3), we can obtain the following theorems.

Theorem 1 The optimal solution to P2, i.e., I ∗ , is less than I ∗ ′ in (3.1).

Proof: We need to prove I ∗ ′  >  I ∗ . Substituting I ∗ ′ into (3.2), we have.

\( {C}^{\prime}\left({I}^{\ast \prime}\right)={hI}^{\ast \prime }{\int}_{I^{\ast \prime}}^{+\infty}\frac{1}{x}\varphi (x)\ dx \) as \( -\pi +\left(h+\pi \right){\int}_0^{I^{\ast \prime }}\varphi (x)\ dx=0 \) .

Noting \( {C}^{\prime}\left({I}^{\ast \prime}\right)={hI}^{\ast \prime }{\int}_{I^{\ast \prime}}^{+\infty}\frac{1}{x}\varphi (x)\ dx>0 \) , C ′ ( I ∗ ) = 0 and C (•) is convex, we can obtain.

I ∗ ′  >  I ∗ .

In words, for a single-period newsvendor model, when the expected overstocking and shortage costs at the end of the period and the consumed inventory cost are considered in the cost function, the optimal inventory level for this cost function decided at the beginning of the period is lower than the optimal solution that only minimizes the sum of the expected overstocking and shortage costs at the end of the period.

Theorem 2 The optimal solution to P2, i.e., I ∗ , is a decreasing function of the unit holding cost h .

Proof: From (3.2), we obtain

From this implicit function, we can show

Then the optimal inventory level I ∗ for this cost function decreases with the unit holding cost.

Theorem 3 The optimal solution to P2, i.e., I ∗ , is an increasing function of the unit backordering cost π .

Then the optimal inventory level I ∗ for this cost function increases as the unit backordering cost.

In section “ Numerical Experiments ”, we conduct the numerical experiments to get I ∗ , and the expected average inventory cost per period C ( I ∗ ), then compare it with C ( I ∗ ′ ).

The newsvendor model without instantaneous receipt

In this section, we further consider the newsvendor model without instantaneous receipt for an infinite planning horizon. The demand in each period follows a normal distribution with mean μ and variance σ 2 . Suppose there is a delay of L periods between ordering goods and receiving them. At the beginning of each period, goods ordered L periods ago arrive. During the whole period, the on-hand inventory is consumed approximately evenly to meet customers’ demands, and the unsatisfied demand is backlogged if stocking out occurs. Demands are independent over time. The order-up-to level I m is determined at the end of the period. We still let h and π denote the unit holding and shortage cost per period, and use the following parameters to formulate the model:

D 1 the demand in one period;

D L the sum of demand in L periods;

D L  + 1 the sum of demand in L  + 1 periods;

φ 1 (•) the probability density function of the demand in one period;

φ L (•) the probability density function of the demand in L periods;

φ L  + 1 (•) the probability density function of the demand in L  + 1 periods;

I m the order-up-to level in one period;

C m ( I m ) the expected average inventory cost when the order-up-to level is I m .

Then the newsvendor problem without instantaneous receipt can be modeled as

In Program P3, the objective function is the expected average inventory cost including overstocking costs, shortage costs and consumed inventory holding costs. Similar to Program P1, the first two terms are the expected overstocking and shortage costs. The third term is the expected inventory holding cost for consumed inventory if there are items left at the end of the period. The last term is a little complicated, which represents the expected holding cost for consumed inventory if there is unsatisfied and backlogged demand in the period. The value of this term is determined by the on-hand inventory level at the beginning of the period that is the difference between the order-up-to level decided L periods ago and the demand satisfied during the preceding L periods, and the demands that will be experienced in the coming period. Then there are two sources of uncertainty in the last term and a double integral is needed to quantify them.

By rearranging Program P3, we can get

For the third and fourth terms in Program P4, we cannot differentiate the objective function with respect to I m and get the first-order and second-order derivatives analytically. Numerical experiments are conducted in next section to get the optimal solution for Program P4.

In contrast, if we use the approximation only, i.e., half the product of the unit holding cost and the expectation of the demand in the period, to represent the holding cost for consumed inventory, the order-up-to level I m only affects the expected average overstocking and shortage costs (the first two terms in Program P4). In such a scenario, the optimal order-up-to level is.

By substituting (3.4) into the objective function of Program P4 for I m , we can get the inventory cost when the order-up-to level is \( {I_m^{\ast}}^{\prime } \) . Denote the optimal solution of I m in Program P4 by \( {I}_m^{\ast } \) . In the next section, we first conduct numerical experiments to get \( {I}_m^{\ast } \) , then compare the difference between \( {C}_m\left({I}_m^{\ast}\right) \) and \( {C}_m\left({I_m^{\ast}}^{\prime}\right) \) in different situations.

Numerical experiments

In section “ Newsvendor Models with Consumed Inventory Holding Cost ”, we formulate models considering the consumed inventory holding cost in the cost function with and without instantaneous receipt. Now numerical experiments are designed to get their optimal solutions that cannot be derived analytically, and examine the difference of the average inventory costs for the cases with and without the consumed inventory cost considered in the objective function. In experiments, we set the unit holding cost h  = 2.5, and unit shortage cost π  = 8. We set the expected value and the standard deviation of the demand in one period μ  = 100 and σ  = 20, respectively. For the model without instantaneous receipt, we set the leadtime L  = 4.

We first obtain the optimal solutions I ∗ and \( {I}_m^{\ast } \) for Programs P2 and P4 with numerical experiments. By changing one parameter and holding others constant, we examine the impact of the factor on the differences of I ∗ and I ∗ ′ , \( {I}_m^{\ast } \) and \( {I_m^{\ast}}^{\prime } \) , C ( I ∗ ) and C ( I ∗ ′ ), \( {C}_m\left({I}_m^{\ast}\right) \) and \( {C}_m\left({I_m^{\ast}}^{\prime}\right) \) respectively. MATLAB 7.0.1 is employed to calculate the integrals in the cost functions.

The single-period model is studied first. Note that in (3.3) the second-order derivative of the objective function in Program P2 is greater than 0, and thus the objective function is convex. To get the optimal solution of Program P2 ( I ∗ ), we just find the value of I that make C ′ ( I ) in (3.2) equal 0 with numerical experiments. Figures 3 and 4 examine the differences of I ∗ and I ∗ ′ as well as C ( I ∗ ) and C ( I ∗ ′ ) respectively when we change the unit holding cost h from 1 to 9 and keep all the other parameters constant. In Fig. 3 , I ∗ is always less than I ∗ ′ , which is consistent with Theorem 1. It also shows that the difference between I ∗ ′ and I ∗ increases as the unit holding cost increases. It can be explained as follows. As the unit holding cost increases, both I ∗ ′ and I ∗ decrease (Theorem 2), and the scenario in Fig. 2 will have a greater chance to occur. Then the inventory cost for consumed products weighs more in the cost function. Since the deviation between I ∗ ′ and I ∗ is mainly caused by this term, the difference tends to increase with the unit holding cost. Figure 4 shows that the percentage difference of C ( I ∗ ′ ) and C ( I ∗ ), i.e.,[ C ( I ∗ ′ ) −  C ( I ∗ )]/ C ( I ∗ ), increases as the unit holding cost increases. Figure 4 is consistent with Fig. 3 , which shows that as the unit holding cost increases, the scenario in Fig. 2 will have more of a chance to occur and the inventory cost for consumed products in the cost function plays a more important role. In Figs.  3 and 4 , with a low unit holding cost, the difference of the cases with and without the consumed inventory cost in the objective function, in terms of the inventory level and expected average inventory cost, is not significant. In such a scenario, I ∗ ′ does not deviate from I ∗ much and is a good approximation of I ∗ for it has the analytical form expressed in (3.1). It can greatly facilitate the analysis of more complicated situations. However, we can also observe in Figs. 3 and 4 that when the unit holding cost is relatively high, the deviation of I ∗ ′ from I ∗ becomes non-ignorable. In this situation, I ∗ ′ is not suitable as the approximation of I ∗ otherwise it can cause the expected average cost C ( I ∗ ′ ) to deflect from the actual minimum expected inventory cost C ( I ∗ ) significantly. Numerical experiments may help in such a situation to derive the optimal inventory level as well as the minimum inventory cost. In Fig.  5 , let the standard deviation of the demand distribution vary under different unit holding costs, and fix the values of all the other parameters. It shows that with a higher unit holding cost, the volatility of the demand has an upward bending relation with the percentage difference of C ( I ∗ ′ ) and C ( I ∗ ). In contrast, with a lower unit holding cost, the impact of the demand volatility on this percentage is much lower or even ignorable. The insensitivity of the percentage difference to the demand variation for a low unit holding cost also demonstrates that I ∗ ′ is a suitable approximation of I ∗ in such situations. The inverse U-shape of the curves in Fig. 5 can be explained as follows. Note that the difference between I ∗ and I ∗ ′ is mainly determined by the fourth term in Program P2. When the degree of demand uncertainty is low, the first, second and fourth terms in P2are relatively small, compared to the third term. In this situation, the small weight of the fourth term in the total cost function only leads to a small difference between I ∗ and I ∗ ′ . Consequently, the percentage cost difference is also small. Then the percentage cost difference initially increases as the degree of demand uncertainty increases from a very low level. When the degree of demand uncertainty goes beyond a certain high level, the weight of overstocking and understocking costs, i.e., the first two terms in P2, tend to be larger. Then the fourth term in P2 in this situation tends to be less important in the cost function. It leads to smaller differences between I ∗ and I ∗ ′ and lower percentage cost differences.

The inventory level vs. the unit holding cost for the model with instantaneous receipt

The percentage difference of C ( I ∗ ′ ) and C ( I ∗ ) vs. the unit holding cost for the model with instantaneous receipt

The percentage difference of C ( I ∗ ′ ) and C ( I ∗ ) vs. the standard deviation of the demand distribution under different unit holding costs for the model with instantaneous receipt

Next we examine the model without instantaneous receipt. Due to the form of the objective function in Program P4, we cannot derive the closed-form optimal solution. Thus we first conduct numerical experiments to get the optimal solution of Program P4, and examine the impacts of factors on the differences of \( {I}_m^{\ast } \) and \( {I_m^{\ast}}^{\prime } \) as well as \( {C}_m\left({I}_m^{\ast}\right) \) and \( {C}_m\left({I_m^{\ast}}^{\prime}\right) \) respectively. In general, numerical results for the model without instantaneous receipt are similar to those with instantaneous receipt. More specifically, Figs.  6 and 7 reveal similar patterns to Figs. 3 and 4 , respectively. In Figs. 6 and 7 , with a low unit inventory holding cost, the deviations of the cases with and without the consumed inventory cost considered in the objective function, in terms of the inventory level and expected average inventory cost, are trivial, while they are noteworthy with a high unit holding cost. In Fig.  8 , let the standard deviation of the demand distribution vary under different unit holding costs, with other things constant. It shows some interesting patterns that cannot be revealed by examining the objective function of Program P4. A group of wave curves under different unit holding costs are graphed in this figure, and the amplitude of fluctuation increases as the unit holding cost increases. It indicates that the demand uncertainty has alternate positive and negative impacts on the difference of \( {C}_m\left({I}_m^{\ast}\right) \) and \( {C}_m\left({I_m^{\ast}}^{\prime}\right) \) in the model without instantaneous receipt.

The inventory level vs. the unit holding cost for the model without instantaneous receipt

The percentage difference of C ( I ∗ ′ m ) and C ( I ∗ m ) vs. the unit holding cost for the model without instantaneous receipt

The percentage difference of C ( I ∗ ′ m ) and C ( I ∗ m ) vs. the standard deviation of the demand distribution under different unit holding costs for the model without instantaneous receipt

Moreover, to exam whether above results are robust, we have further conducted.

In numerical experiments for different values of \( \frac{\pi }{h+\pi } \) , the shapes of curves are very close to the above results.

Our numerical experiments show that there exist differences in optimal inventory levels with and without the holding cost for consumed stock in the total cost function. A suboptimal solution is obtained if the cost for consumed stock is excluded in the cost function when making decisions. The numerical experiments indicate that this inventory level difference and associated total cost difference increase as the unit holding cost increases and can be affected by the degree of demand uncertainty. These findings provide some useful insights for practitioners. When the unit holding cost is relatively high compared to the unit backordering cost, there is more chance to have understock at the end of the period. That is, the scenario in Fig. 2 will have a greater chance to happen. In this situation, the decision on inventory level will remarkably affect the holding cost for consumed items. Our analysis shows that the inventory level considering the holding cost for consumed stock will be lower than the one determined by the model further taking into account such holding costs. In other words, the former one is inferior to the latter one when the total inventory cost function includes understocking costs, overstocking costs and holding costs for consumed stock. The difference in the inventory levels will lead to associated cost differences. Such cost difference also increases with the unit holding cost, and can be affected by the degree of demand uncertainty. Since the unit inventory holding cost usually covers storage space costs, handling and service costs, risk costs and capital costs, one or more such factors may cause a high unit holding cost. For example, luxury products or products requiring specific storage conditions may have a high unit holding cost. When inventory decisions are made for these products, the holding cost for consumed items should be included in the total cost function to obtain the optimal inventory level. Otherwise, a suboptimal solution will be obtained.

In this work, we revisit the newsvendor model by considering the consumed stock holding cost in the cost function. In the classic model, we assume the inventory level has no impact on the holding cost for consumed stock during one period, where the average holding cost for this portion of inventory is usually approximated as half the product of the unit holding cost and the expectation of the demand in that period. It works when the ratio of unit holding cost to the unit shortage cost is low. However, without this condition, the approximation can deviate significantly from the actual expected cost. The higher unit holding cost can be attributed to the high capital cost, insurance cost, holding cost for specific storage conditions, and, more prominently, coming carbon charges. With today’s trends of price-based policies for reduction of greenhouse gas emissions, as well as the advance of technologies in warehouse management, holding costs have been rising significantly in some industries. To mitigate the potential shortcomings of the classic model, we add the holding cost for consumed inventory into the cost function. The models with and without instantaneous receipt are formulated and analyzed.

For the model with instantaneous receipt, analysis shows that the optimal inventory level is lower if the consumed inventory cost is considered in the cost function, other things being constant. The relationships between the optimal inventory level and the unit holding cost as well as the unit backordering cost are shown. Numerical experiments are conducted to examine the factors that affect this deviation. Other things being constant, the deviation increases as the unit inventory holding cost increases, for both models. The uncertainty of demand, which is measured by its standard deviation, has either positive or negative impacts on the deviation depending on the specific circumstance. Both analyses and numerical experiments indicate the deviation caused by ignoring the holding cost for consumed inventory is unacceptable in some circumstances. The approximation, i.e., half the product of the unit holding cost and the expectation of the demand in the period, can lead to the closed-form expression of optimal inventory level (or position) and facilitate analyses of more complicated problems. In such situations, well-designed numerical experiments and simulations should be employed to conduct further research.

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The research is partially supported by Natural Science Foundation of Guangdong (2015A030313823), Zhuhai Outstanding Discipline Project-Accounting and UIC college research grant.

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ST carried out models setup and analysis; SC helped draft the manuscript and made the figures in the draft; JWW helped conduct the literature review and drafted the manuscript; HY helped construct models and conducted numerical experiments. All four authors read and approved the final manuscript.

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Tang, S., Cho, S., Wang, J.W. et al. The newsvendor model revisited: the impacts of high unit holding costs on the accuracy of the classic model. Front. Bus. Res. China 12 , 12 (2018). https://doi.org/10.1186/s11782-018-0034-x

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The production decision of a large commodity or equipment manufacturing enterprise can be modeled as a newsvendor problem. Managers must determine the optimal production volume in advance to minimize the underage cost and the overage cost. However, the traditional newsvendor problem assumes the known demand distribution, which is not the case in practice. Data-driven approaches have become the hot research topic and opened up new avenues for such issues. Recent studies have considered demand-related features but have failed to address how to optimize production and inventory using informative textual reviews, not just numerical feature data. To address this issue, we propose a data-driven newsvendor model that leverages sentiment analysis on textual reviews using a deep learning model to solve the data-driven newsvendor problem by integrating estimation and optimization. Experiments on real data show that our proposed method reduces the average cost by approximately 14.18% compared to the most advanced deep neural network method, making it the best-performing method. Furthermore, our method is more suitable for situations where unit shortage costs are greater than unit overage costs. Finally, our method is robust in terms of sample size and can still obtain good results even with insufficient historical data.

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This work was supported by the National Social Science Fund of China [Grant Number, 19BGL229].

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Mapping the main research themes in digital human resources

• Laura García-Fernández 1 ,
• Marta Ortiz-de-Urbina-Criado   ORCID: orcid.org/0000-0001-7527-6798 1 &
• María-José García-López 1  

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The COVID-19 pandemic sped up the digitalization process and revolutionized the world of the digital employee. And today, advances in artificial intelligence are having a major impact on the field of Digital HR. In that context, further literature review work is needed on the term Digital HR to complement previous studies and lay the foundation for more pioneering literature on this topic. Then, the aim of this paper is to provide a framework for organizing the main themes discussed in the pioneering literature on digital HR by answering the following research question: What is the knowledge structure of the research in the field of digital human resources? An adaptation of the PRISMA model is used to structure the research design. Applying a mixed methodology, this paper uses a bibliometric technique to identify the main topics studied in Digital HR. Subsequently, in-depth analysis and logical reasoning are applied and a model is proposed based on four questions (how, what, where, who) in order to understand and develop research on digital HR. The RQ4 Digital-HR model constitutes a useful tool in academic, practical, professional, and social contexts. It is worth highlighting the importance of the inclusion of artificial intelligence in the daily processes of a company, and therefore in the progress of the proposed research topic.

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Introduction.

The world is witnessing constant change due to digitalization and its effects on companies and their staff. The pandemic accelerated the adoption of digital technologies, which had an immense impact on all business sectors and brought about permanent changes in the workplace (Gkinko and Elbanna, 2023 ). Many organizations started working in hybrid mode, combining digital ways of working with the traditional ways of working prior to the pandemic. Moreover, the use of digital technologies and the acceptance of more agile and flexible procedures and rules have changed the way in which work is being done (Mićić and Mastilo, 2022 ).

In general, our daily lives have been altered by technological advances, one of the most innovative being the advances in artificial intelligence, which is transforming the way people carry out their daily activities of work, communication, and decision making (Duke, 2022 ). The concept of artificial intelligence seems to be relatively recent, and in many cases, its true meaning or significance is unclear. However, it was not until the 2010s that the AI paradigm was reconfigured to be based on the classification and storage of massive data (Cetindamar et al. 2024 ).

Human resource management has evolved over time. Twenty-five years ago, its main focus was on implementing practices that promoted the development of organizations. However, the need for organizations to adapt to more competitive environments has forced businesses to adjust the traditional business management model, moving from strategic management to a more sustainable management approach (Villajos et al. 2019 ). One of the main factors influencing the adaptation of human resources management to the new sustainable management model has been digitalization (Le et al. 2024 ). Through digitalization, all employees, professionals, managers and business leaders, who are key to making the necessary changes to increase workplace productivity, can see their tasks facilitated through this phenomenon. In this context, human resources management develops practices that promote the welfare of the employee and the company (Le et al. 2024 ). Thus, in recent years, and especially during the COVID-19 pandemic, Digital Human Resources (Digital HR) has received a great deal of attention, particularly regarding Digital Employees and Digital Leaders. Advances in artificial intelligence are having a major impact on the field of Digital HR (Gkinko and Elbanna, 2023 ). Although there are still few publications on the acceptance of the effects of AI on workers and how the current increase in the use of digital technologies affects the skills and expectations of the digital workforce (Alan, 2023 ; Cetindamar et al. 2024 ).

In the academic context, there has been a surge in the literature on Digital HR. However, literature review studies on this topic are lacking, with most of them focusing on analyzing the digital workplace phenomenon (De Moraes et al. 2024 ; Marsh et al. 2022 ; Mićić and Mastilo, 2022 ), digital employee experience (Moganadas and Goh, 2022 ), and workforce training in digital workplaces (Patino and Naffi, 2023 ). Only two reviews address the issue of Digital HR more generally. Theres and Strohmeier ( 2023 ) conducted a meta-analysis to analyze theories applied in research on Digital HRM adoption and proposed a unified theory. Alan ( 2023 ) performed a co-word analysis, considering Electronic Human Resources Management (e-HRM) as the main term, and analyzed previous literature found in the Web of Science (WoS) for the period of 2012–2022.

Thus, further literature review work is needed on the term Digital HR that analyzes the literature published before and during the pandemic to complement previous studies and lay the foundation for more pioneering literature on this topic. The interest in analyzing changes during the pandemic is motivated by the fact that adapting to the new context requires new human resources actions that are closely related to the phenomenon of digitalization. Digital HR is a constantly evolving topic, and pioneering studies are fundamental to understanding this new phenomenon. Today, there is also the challenge of appropriately and ethically adopting artificial intelligence in the context of human resources (Cetindamar et al. 2024 ; Gkinko and Elbanna, 2023 ). In the face of a novel topic, it is important to gain an overview of the aspects studied, to understand the changes that have occurred around the pandemic, and to provide a logical framework of analysis by which to explore such phenomena.

This paper aims to provide an overview of the pioneering research landscape in the field of digital HR, filling in some of the existing research gaps. As a complement to Alan’s ( 2023 ) work, this research will focus on the topic of ‘digital HR’ and conduct a co-word analysis to identify the main themes studied. Moreover, a second step, which is not usually included in previous literature reviews on this topic, will be carried out to detect the applications of digital human resources. To this end, a model based on questions (how, what, where, who) is proposed to facilitate the understanding and development of digital HR research.

Thus, the aim of this study is to provide a framework for the organization of the main themes that are discussed in the pioneering literature on Digital HR. The research question addressed is What is the knowledge structure of the research in the field of digital human resources? To answer this question, the Background section is developed and a mixed methodology is applied, adapting the PRISMA process. A bibliometric technique is used to identify the main topics studied in Digital HR. Subsequently, in-depth analysis and logical reasoning are applied to propose a model and some lines of future research. Finally, the Conclusion section contains theoretical and practical implications, the study limitations, and future lines of research.

This paper is an original contribution. Literature reviews, and more on rapidly developing novel topics, play an important role in advancing research as they help to synthesize and organize existing knowledge and identify areas or topics for future research. This article proposes an integrative review (Patriotta, 2020 ) that offers another voice to guide and write new articles on digital human resources. Authors such as Post et al. ( 2020 ) have also highlighted the importance of literature reviews as they can serve several purposes such as helping researchers understand the research topic, discerning important and under-examined areas and connecting research findings from disparate sources to create new perspectives and phenomena. Moreover, the topic “Digital HR” calls for looking for models that help connect academic research with the business world. As Markman ( 2022 ) proclaims, academia is challenged to develop research that addresses current problems affecting people, business and society to make the world a better place. In that line, the RQ4 Digital-HR model constitutes a useful tool for academic, practical, professional, and social contexts.

The global pandemic has accelerated digital transformation in every sense, and the rise of digital technology in the workplace is unstoppable (Kalischko and Riedl, 2021 ). Technology plays a vital role in our day-to-day lives. Digitization has arrived, yet what that means or entails at a work and/personal level remains unclear. According to Kraus et al. ( 2022 ), it is necessary to have a fundamental understanding of literature reviews as independent studies. Therefore, the key texts must be identified that lay the foundations of Digital Human Resources Management (Digital HRM) before undertaking a bibliometric study.

Main concepts

Few papers over the last decades have provided a clear, agreed-upon definition of the term “Digital Human Resources Management” that is shared by the scientific community. Most papers have only superficially addressed the whole social and economic context that affects the new confection of digital employee models. Moreover, papers have tended to narrow their focus to a specific aspect of human resources management (Alan, 2023 ; Costa et al. 2022 ), digital employee experience (Moganadas and Goh, 2022 ), and job performance (Kalischko and Riedl, 2021 ; Marsh et al. 2022 ), analyzing the situation individually and rather than as a whole. Therefore, the starting point for this study is to introduce some of the terms or concepts commonly used in previous literature on digital HR. Two widely used terms are “digital worker” and “digital employee”. A key resource in any company is the employee, the one who can contribute to superior and solid performance over time (Moganadas and Goh, 2022 ). For example, Fuchs ( 2014 ) defines digital employees as the workforce required for the existence, use, and application of digital media. Other studies define digital employees as those employees whose work is performed primarily using digital resources (Nelson, 2018 ). IBM ( 2024 ) states that “in the past, the term ‘digital worker’ described a human employee with digital skills, but more recently, the market has defined it as a category of software robots, which are trained to perform specific tasks or processes in partnership with their human colleagues.”

Another concept used is “digital workplace.” As management has adapted to new technologies, the workplace has also had to adapt. This new leadership style brings with it concepts such as flexibility, which in this context refers to the non-limitation of the workspace to a specific physical location. This new digital workplace refers to the set of technologies that employees use to perform their functions (Marsh, 2018 ) and includes, among others, the intranet, communication tools, e-mail, CRM, etc. It also refers to a set of procedures and rules that maximize productivity and improve collaboration, communication, and knowledge management (Mićić et al. 2022 ). Some researchers use the term “digital labor”, which initially referred to the unpaid work performed by consumers online during leisure time. However, this term is now used to describe all work in which digital technology plays a role (Jarrett, 2022 ). The term has also been used to describe employees who work independently, receiving low wages and no social security, in business models supported by digital platforms, such as Uber (Fumagalli et al. 2018 ), or to describe the workforce that uses other business models that are also based on digital platforms, such as Facebook or Google, and that capture information to transform it into big data (Fuchs and Sevignani, 2013 ).

In that context, another important concept is digital platform. Digital platforms are transforming almost every industry today (Reuver et al. 2018 ). They are continuously evolving and becoming increasingly complex. These digital platforms are the ones that facilitate online communities of consumers (Reuver et al. 2018 ). While there are several definitions of digital platforms that refer to the codes, software, and hardware of which they are composed, for this study the most suitable definition of digital platform would be the environment in which companies combine all the information available from their stakeholders to generate or co-create value (Karhu et al. 2018 ). According to Murati ( 2021 ), a digital platform is an open infrastructure that exercises a facilitator role or a high level of control and influence over providers and users.

The meeting point of each one of these concepts is the term Human Resource Management, which is understood as the processes that involve activities from recruitment to salary management and that are carried out simultaneously (Alan, 2023 ). All of these processes have been equipped with more technology and innovative methods over time. Thus, the concept has evolved to Digital Human Resource Management (HRM), understood as the set of software, hardware, and digital resources designed to automate the HR function (Jani et al. 2021 ; Marler and Parry, 2016 ), or in other words, to develop consistent, efficient and high-quality HR practices through the use of digital transformation and new technologies (Bondarouk and Brewster, 2016 ).

Previous literature reviews

Previous literature reviews established a set of definitions that, despite using common concepts, have left nuances that have yet to be fully addressed in subsequent works. Most of the work that reviews previous literature has focused on studying digital workplaces. Mićić and Mastilo ( 2022 ) conducted a systematic literature review on the digital transformation of the workplace and employees’ workplace preferences. The search terms used were “digital workplace”, “COVID-19”, and “innovation”, and the search was limited to English language papers published after 2010. The benefits of digital workplace transformation are analyzed and the critical success factors and significant challenges are identified.

Marsh et al. ( 2022 ) studied the application of digital technologies in the workplace with a particular focus on their dark side. They conducted an integrative literature review and limited the search to papers published between January 2007 and June 2020 that were written in English and carried out in Western countries only (in the United States, Europe, Canada, Australia, Latin America, and New Zealand). De Moraes et al. ( 2024 ) conducted a systematic review of the literature on the design of digital workplaces. Their main results include a definition of digital workplace and a four-phase model with guidelines for designing digital workplaces. Patino and Naffi ( 2023 ) conducted a systematic review of training approaches and resources for workforce development in digital workplaces. Using the PRISMA model, they analyzed articles published between 2020 and 2022. Their paper offers research-based perspectives and recommendations for employee training in highly digitalized workplaces.

Another aspect that has been studied is the experience of the digital worker. Moganadas and Goh ( 2022 ) discuss the concept of digital employee experience (DEX). They conducted a comprehensive literature review on DEX by analyzing the content of academic publications and professional reports. They used the Scopus and Google Scholar databases to identify “DEX” or “digital employee experience” in their title, abstract, and keywords and found 17 articles between 2016 and 2022. To complement these papers, they included grey literature to identify studies that addressed a similar topic, such as digital transformation, digital workplace, and employee experience.

Finally, a few papers have reviewed the literature on human resource management in a digital environment. Theres and Strohmeier ( 2023 ) analyzed the phenomenon of digital HRM. In their paper, they present an overview of the theories applied in digital HRM adoption research and propose a unified theory. To test their theory, they performed a combination of meta-analysis and structural equation modelling. Alan ( 2023 ) presented a systematic bibliometric analysis of electronic human resource management (e-HRM) by conducting a literature search in the Web of Science (WoS) for the period of 2012–2022.

Methodology

Figure 1 presents the methodological process used in this study. The methodological design used includes two parts. In the first, a multi-step process has been followed to perform the bibliometric analysis: sample selection, filtering of documents and keywords, and co-word analysis. In the second, a reflexive analysis was carried out. To facilitate the understanding of the process followed, the PRISMA 2020 statement has been adapted, which has been designed primarily for systematic reviews of studies (Moher et al. 2010 ; Page et al. 2021 ). The adaptation of the PRISMA process provides a more transparent view of the methodology used and the analyses carried out.

Own elaboration based on the PRISMA model.

The Scopus database was used. There is an open debate regarding whether Scopus or WoS is superior. Both have advantages and disadvantages (Stahlschmidt and Stephen, 2020 ). The Scopus database was chosen for this paper because it offered a larger sample of documents than did WoS. Although the research on Digital HR began over 35 years ago, most of the articles have been published in the last three years, demonstrating the impact of the COVID-19 pandemic on this topic. Until the year 2016, contributions were sporadic, and it is not until a year later, in 2017, that the research begins to approach 25 articles per year. Of the total of articles (347), 56% (196) were published between 2020 and 2022, with 2021 being the most important year, when a total of 82 articles (25%) were published.

A co-word analysis in conjunction with the SciMat program was used to identify the various themes covered in the literature on Digital HR (Cobo et al. 2012 ). Of the many tools that enable co-word analysis, SciMat was chosen for its ability to carry out the analysis with simplicity and rigor. Moral-Muñoz et al. ( 2019 ; 2020 ) describe the various tools that are available for bibliometric analysis and comment on SciMat as being a valid tool for co-word analysis. SciMat was suitable for achieving the objective of this paper because it analyzes the keywords of selected articles and calculates the strategic diagrams and networks for each thematic group. Moreover, SciMat incorporates all the necessary elements (methods, algorithms, and measurements) for performing a co-word analysis and obtaining its visualizations (Cobo et al. 2012 ).

Regarding the strategic diagram, centrality and density are calculated for each thematic group (Cobo et al. 2018 ). Centrality is a measure of the importance of a theme in the development of a field of knowledge. Density reflects the strength of a network’s internal relationships, thus identifying the level of development of that theme. The strategic diagram classifies the themes into four groups (Cobo et al. 2018 ). In the upper-right quadrant are the motor themes, which comprise themes that have strong centrality and high density. In the upper-left quadrant are the well-developed and/or isolated themes. The themes in the lower-left quadrant are presented as emerging or disappearing themes, while in the lower-right quadrant are themes that are considered basic and transversal themes.

This section presents the results of the co-word analysis. The bibliometric technique is suitable for identifying the knowledge structure of a research topic. Given the volume of articles published between 2020 and 2022, two periods of analysis were carried out to compare the networks that emerged prior to and after the COVID-19 pandemic. Figure 2 shows the evolution of all the topics mentioned, their typology, and how, depending on the period, they transform into a new topic.

Results from SciMat, diagram composed of themes by number of documents for all the periods.

Table 1 presents the evolution that Digital HR research has experienced during these years.

Main themes studied in Digital HR

Regarding the total period (1984–2022), previous research focused on “digital workplace” and “digital platform” and “digital employee” as the motor themes. “Digital labor” appears as an emerging topic and as something remarkable. Despite not being connected to the human resources area, this entire digitization process is linked to the topic “enterprise bots”, a concept that had previously been highly developed in scientific fields. During the pre-COVID period (1984–2019), the motor theme was the “digital workplace”. During the COVID period (2020–2022), the motor themes were “digital employee” and “digital workplace”. Lastly, the emerging theme for all the periods is “digital labor”.

During the first period (1984–2019), only one theme, “digital workplace”, is positioned as a motor theme. It makes sense that after the 4 th industrial revolution, developed between 1950 and 1970, a study period would begin regarding how this digitization has affected the workspaces as well as how to continue innovating and improving them. Companies have needed workplaces to be transformed from a traditional perspective to a digital one (Colbert et al. 2016 ; Kaarst-Brown et al. 2018 ), since this change is key to organizational success (Colbert et al. 2016 , Köffer, 2015 ).

In the 2020–2022 period, two additional topics to those appearing in the previous period emerge. These are “digital employee” and “digital labor”, positioned as a motor and an emerging theme, respectively. These topics correlate with what occurred during the pandemic, which forced the digitalization of all types of situations. As a result, the research on this area has focused on the employee and, above all, the digitization of work that, as mentioned, appears as an emerging topic.

Based on these results and for the completion of the analyses, a manually and logical regrouping of themes was conducted in the SciMat program, and another strategic diagram was identified. Figure 3 , which presents the strategic diagram obtained from this new analysis, shows that the motor themes are “digital platform” and “workplace”. “Manufacturing”, “digital employee”, and “social media” are well-developed themes. “Digital”, “learning”, and “labor” are positioned as basic themes. Finally, the emerging theme is “artificial intelligence”. A logical knowledge structure of the study topics can be observed. On the one hand, what emerges are the themes related to the more digital aspect of work, namely “digital”, “digital platform”, and “digital employee”. On the other hand, there are items that refer to the more physical aspect, “labor” and “workplace”. Tangential to the most digital aspect linked to work is the communication channel or media used at work, “social media”. In turn, in the main sector where the research is applied or where the literature has further explored these issues, the theme of “manufacturing” is also well defined.

Results from SciMat, diagram composed of themes by a number of documents for all the periods.

Also evident in the diagram is the channel through which employees can progress in the work environment, through “learning”, a Human Resources practice that has been developed for some time but has become more crucial in recent years. Learning is the key to employees acquiring digital competencies and feeling comfortable in digital work environments. Finally, as an emerging topic, is everything related to “artificial intelligence”. This topic, of relatively recent creation, has all the works published in the year 2022 or later, and is creating a highly critical space in the business environment, and in this particular case, in everything related to Human Resources and how to implement it in departmental processes.

Thematic networks in Digital HR

It is interesting to know the thematic networks in which the most significant keyword (the one with the highest centrality) is placed at the center. The size of each node represents the number of documents containing that word, and the thickness of the line indicates the strength of the association between those topics.

Digital platform

Analysis of the “Digital Platform” subnetwork (Fig. 4 ) for the period 1984–2022, reveals four documents and a wide network of terms that correlate with each other. The most notable relationships are those related to health care. However, within our scope, there are several studies that focus on the use of digital platforms as a means of offering work in the “gig economy.” Taylor et al. ( 2017 ) define the concept of Gig Economy as the use of applications or platforms for work.

Results from SciMat cluster network for the digital platform.

The analysis also revealed the importance of collaborative work for the improvement of digital platforms, as shown through the connections between the terms “collaborative designs” and “co-creation”. The research also showed two important advances in what has been studied in recent years: the flexibility that this type of work facilitates (Soriano and Cabanes, 2020 ) and how these new jobs can change the lifestyle of digital employees (Graham et al. 2017 ).

Digital workplace

The “Digital workplace” network (Fig. 5 ) for the period 1984–2022 includes six documents and shows that the most important keywords in the cluster are “digital transformation” and “artificial intelligence”. Again, it is crucial for this network to talk about “collaboration”, as well as “cross-functional teams” and their “dynamic capabilities” that play a special role in developing the digital workplace. As Selimović et al. ( 2021 ) posit, the inclusion of the employee in the decisions on digital transformation is a key to its success. Moreover, the use of artificial intelligence, through the “chatbots” makes improvement in the workplace possible (Cetindamar et al. 2024 ). In both cases, the focus is placed on the inclusion of the employee as a key part of these processes. This demonstrates a strong relationship between digital and emotions in the cluster, since understanding how the use of technology affects employees’ emotions (Gkinko and Elbanna, 2022 ) is one of the most relevant topics in the current research.

Results from SciMat cluster network for the digital workplace.

The analysis of the “Digital workplace” network (Fig. 6 ) for the period 1984–2019, which contains 15 documents, reveals the strong presence of terms related to a collaborative work environment, such as “collaboration”, “cloud”, “cloud computing”; or even advancing further in the collaboration itself, it becomes necessary to talk about “digital platforms” or “crowdsourcing”, as means for it, being the key tools for developing the digital workplace (White, 2012 ; Attaran et al. 2019 ). Indeed, the most remarkable aspect of the network is the strong connection between “cloud computing” and “mobile working”. It must be considered that during these years, prior to the COVID-19 pandemic, the now-standardized option of mobile working was merely a practice applied by a few companies. Thus, it makes sense that during these years of strong digitization, research focuses on it. There is also one term, “artificial intelligence” (here also mentioned as “social software”), that researchers start to investigate during these years, since its use in the digital workplace is continuing to increase (Martensen et al. 2016 ) and, as could be seen throughout the paper, it will also become of vast importance for other networks.

For the last years (2020–2022), “Digital workplace” network (Fig. 6 ) contains the highest number of documents (16) and shows two remarkable themes “digital transformation” and “artificial intelligence”. In the figure can be seen a triangle formed by “emotions”, “emotions at work”, and “chatbot”, as employee users experience a connection emotion when using artificial intelligence (Gkinko and Elbanna, 2022 ), and there is an effort to understand how employees will accept these new systems in the enterprise context (Brachten et al. 2021 ). Moreover, in these recent years, the changes that companies must make to achieve the digitalization of the workplaces takes on a special relevance. Thus, it is not surprising that the investigation is linked to “organizational change” and “technology-adoption”. It should not be overlooked that none of these changes would be possible without including the “employees” in said decisions (Cetindamar et al. 2024 ).

There is a topic that remains throughout the analysis: “artificial intelligence” (Fig. 6 ). However, there is a positive evolution between the topics analyzed prior to the pandemic and those analyzed after it. In the first period (1984–2022), the topics were focused on how the workplace should be or what it should contain and how it should be digitized, as well as on platforms and software and everything related to the cloud. However, during the pandemic period, some changes were perceived, with the introduction of themes arising from having been forced to implement the digital work modality. These topics include technostress, emotions, and employees, as well as everything related to organizational change.

Digital employee

The “Digital employee” network (Fig. 7 ) for the period 1984–2022 contains 10 documents and shows, as previously mentioned, the relevance of the pandemic in this research. In this sense, it is crucial to understand how this situation affected the employee, the work itself, and the life experience of employees (Muszyński et al. 2021 ). Above all, it shows a strong connection between the concepts related to “Robotic Process Automation” (RPA), “digital automation process”, and “software robot”, as a means to increasing productivity in a company, leaving the routine tasks to RPA and assigning employees to perform more difficult tasks (Choi et al. 2021 ). Clearly, it is crucial to talk about the “digital competencies” that employees have or need to acquire to be included in the “new work” that globalization is forcing us to implement.

Results from SciMat cluster network for digital employees.

The “Digital employee” network, for the last years (2020–2022) (Fig. 7 ), with eight documents, shows that there have been no changes in recent years compared to what was already being studied. The only difference is that the research in these years does not focus on the new types of work, implemented post-pandemic, but studies how to improve the implementation of RPA (Costa et al. 2022 ) to achieve better economic results and an improved digital employee experience.

Digital labor

The “Digital labor” network (Fig. 8 ) for the years 1984–2022, contains nine documents and shows a star-shaped network characterized by the presence of keywords that only correlate with the cluster topic. The main theme of the cluster is the work itself with its main versions, with research on the best type of work being very common (Babapour Chafi et al. 2022 ): office work or digital work (commonly called digital nomadism). There is also an important connection with the information society.

Results from SciMat cluster network for digital labor.

“Digital Labor”, for the last years (2020–2022), provides six documents (Fig. 8 ). Studies related to the “gig economy” and the types of jobs related to digital platforms proliferate during this period. In addition, once the pandemic period was over, it was expected that employees would return to office work. Thus, there arises a need to understand which work model (remote, face-to-face, or hybrid) is more productive and which is more valued by the employee (Babapour Chafi et al. 2022 ).

A comparison of studies prior to the pandemic with those of recent years reveals that initially there were several issues related to digital work, whereas in recent years these have been reduced to two issues: office work or work through digital platforms.

Enterprise bots

The “Enterprise bots” network for the period 1984–2022 (Fig. 9 ) contains two documents that co-relate two concepts, virtual assistants and virtual agents, as being crucial to understanding the differences between them, and above all, to understanding the differences in use between the individual and the business context (Stieglitz et al. 2018 ). The focus was therefore on teaching an employer how to effectively introduce these systems in the company (Brachten et al. 2021 ).

Results from SciMat cluster network for enterprise bots.

The results of this study complement those of previous literature review studies. Alan ( 2023 ) focused on the term Electronic Human Resources Management (e-HRM) and conducted a review of the literature included in WoS from 2012 to 2022. Our study focused on the term “digital human resources” and used the Scopus database. Our study also included pioneering literature up to 2022 and an analysis of the differences between the pre-pandemic period (1984 to 2019) and the peak years of the pandemic (2020–2022).

Alan ( 2023 ) categorizes the research on this topic into three groups: the theoretical studies and theories used in the studies reviewed, empirical qualitative studies, and empirical quantitative studies. Alan ( 2023 ) presents summary tables for each category that include following information: related theoretical framework, related terms, studies, typology of study, aim of the study, and the main findings and propositions. In a complementary way, this current paper presents the studied themes and classifies them into four groups according to the strategic diagram and analyses the networks for each thematic group. Additionally, based on the results obtained in the previous section, a process of analysis and reflection was carried out to establish the roadmap of topics studied and to define the emerging and future topics. Four main research questions (how, what, where, who) are considered to propose a model (Fig. 10 ).

Own elaboration.

The RQ4 Digital-HR model presents four fundamental questions for understanding and developing research on digital HR -Research Questions for Digital HR.

The basis of the proposed model (Fig. 10 ) refers to “where” to apply it. The results of the analysis show that there is a sector where deep research on the subject has already been carried out, the manufacturing sector. However, the research should not stop there. Future research should take this model to other sectors of much greater complexity and scope, such as the service sector.

The pillars that support the model, “the what,” are on two levels: the advances in the digitalization of work on the one hand and, on the other hand, all the learning that a company can guarantee to its employees and that employees are able to assimilate. Clearly, the cross-cutting issue, “the who,” is the digital employee, the workforce member that drives the change, the one who is able to implement any Human Resources practice, and therefore the one who is able to assimilate business-driven change.

The roof of the model is “the how.” The implementation of all the changes we are forced to make is only possible through the implementation and improvement of three items in our daily processes: firstly, the digital platforms that we use every day at work, secondly, the management of information through the channels provided by the company, and thirdly—and most importantly—the inclusion of artificial intelligence in all our processes, as a means of improving productivity. This technology is transforming, and revolutionizing, the future of workspaces to make them more productive (Gkinko and Elbanna 2022 ), but again studies on the subject do not address how to accomplish this. Most of the texts reviewed on the subject focus on investigating some aspect of the implementation of artificial intelligence systems and their errors (Costa et al. 2022 ) or on employees’ acceptance of or trust in such technology (Gkinko and Elbanna, 2023 ). However, the work of Gkinko and Elbanna ( 2022 ) offers a starting point for how to incorporate this technology into a company, since the emotions of employees need to be taken into account when such tools are being created in order to facilitate their inclusion in day-to-day activities.

This makes it vital to focus the research on Digital HR, which requires researchers to collaborate to determine the what, where, how, and who. Based on the questions in the model, a further step has been taken to identify the aspects that would be interesting to analyze in future work to answer each of these fundamental questions.

WHAT: In reference to this, it is important to discover what digital processes should be introduced in our daily lives, what learning tools will help us to channel digitization, and what new labor trends can be found in the post-COVID stage.

WHERE: Future research should focus on analyzing what kind of workspaces exist in the labor market. This first line of research will undoubtedly lead towards the different sectors of activity, so it is also important to see what we know about the different sectors and their relationship with Digital HR, especially in the service sector, since globally it is the most present sector of activity.

HOW: Research should continue to determine how to do this and, as broken down in the table, it is important to address how to adopt digital platforms in the work environment, how to adapt existing social networks to the business context, and how to apply new artificial intelligence models.

WHO: As mentioned in this paper, digital HR is a transversal entity in all these lines of research. However, it is not left out of the future lines of research since it is essential to understand who this digital HR includes.

Finally, based on the results of this study (thematic groups) and the proposed model (questions), Table 2 presents a proposal for future research lines and questions.

The proposed model revolves around four fundamental research questions (Fig. 10 ). The importance of researchers developing the ability to formulate questions has an epistemological background expressed by Bachelard ( 1982 , p. 16) as ‘for a scientific spirit all knowledge is an answer to a question. If there was no question, there can be no scientific knowledge’. It should be noted that the quality of the questions asked is closely related to the prior knowledge they have about a given topic (Neber and Anton, 2008 ). Systematic questioning about different phenomena fosters meaningful learning by drawing on prior knowledge in a non-arbitrary and non-literal way (Moreira, 2000 ). Furthermore, knowing the background of a subject facilitates scientific modelling, an activity inherent to science, which can be understood as a process of constructing models for the purpose of apprehending reality (Giere, 1988 ) and providing answers to questions formulated about real facts or assumptions (Halloun, 1996 ). The model presented in Fig. 10 and developed in Table 2 , helps to logically order the themes studied in the previous literature and to propose emerging and current themes of great interest for the development of the literature on digital HR.

In addition, the model helps to sort out what company should focus on meeting the needs of its employees without leaving its own needs behind, first the basis, then the pillars and finally the roof. In this regard, it is important to start managing the workers’ workplaces to adapt them to the environment. Subsequently, the training needs of employees must be addressed, along with the necessary adaptations to enable them to function digitally. Thirdly, companies must develop internal and external communication systems that allow them to be in contact with all their stakeholders. Only having developed these points will be able to focus on meeting current social demands and introducing artificial intelligence in their daily work.

Therefore, if companies’ human resources departments understand this model and its order, will be able to act effectively and thus be more ethical and sustainable. On the other hand, acting in an inverted order will leave some of the pillars of the Triple Bottom Line uncovered, with the risks that this entails. The Triple Bottom Line (triple P´s) model is a model that calls for corporate commitment to measure its social (Person), environmental (Planet), and financial (Profit) impact. This is why it becomes necessary to have a human resources management model adapted to the current and changing context of the organization (Kramar, 2014 ).

In turn, for the employee, the implementation of a model will help them to prioritize their needs to be covered by the company so that, once managed, they can lead to a higher and better performance and thus achieve a high level of well-being at work. Ruiz-Palomino et al. ( 2019 ) explain that a good way to improve a company productivity is to promote corporate wellness and entrepreneurship.

Conclusions

This paper has answered the question What is the knowledge structure of pioneering research in the field of digital human resources? A mixed methodology was used to identify the main topics studied in Digital HR and to propose a model and some future research lines and questions.

This article presents an integrative review to generate ordered knowledge spaces, which as Patriotta ( 2020 ) explains serves to ‘put boundaries around an existing area of research in order to provide an organized sample of what is available and build a platform for future research’ (p. 1274).

Implications

Theoretical implications.

In terms of theoretical implications, this paper highlights the interest of extending current research to concretely define what the digitalization of work means as well as its implications and requirements. This will enable the discovery or even the proposal of new digital work models, incorporating those positions redefined as a result of the incorporation of artificial intelligence and thus make it possible to delimit digital HR. On the other hand, it is noted that the incorporation of new trends in the market must be reflected in the teaching/learning methods to achieve greater professionalization. In turn, a company will become the protagonist in designing these new training processes that are linked to its specific professional activity and the profile of its employees. On the other hand, emphasis can be placed on studying how to increase productivity through the application of artificial intelligence in routine tasks.

In addition, based on the proposed model, an expansion of research on Digital HR human resources is proposed, incorporating new lines and research questions, which will lead to a new categorization of workplaces according to their capacity to adopt digital models. This will lay the foundation for a new labor framework and the development of innovative capabilities in this regard. This broadening of research can be related to the development of thematic lines and professional sectors, especially in the service sector, thus accommodating the most important sector for the European economy. Future research can consider the development of new TAM (Technology Acceptance Model) models to measure the adoption of digital technologies in digital work environments. Social networks used in the work environment can also be investigated to detect those that best help to channel the processes of labor digitalization.

Practical implications

In terms of practical implications, the results obtained and the proposed model can be used to encourage the application of new technologies in the work environment; guarantee employees digital learning processes to increase productivity, facilitating this new learning process; and create policies and standards that include artificial intelligence and social networks in the business environment, thus standardizing their proper use to generate greater productivity and economic results. New and innovative workspaces can also be developed in order to integrate the improvements derived from digitalization and artificial intelligence. Information channels can also be developed to connect the new processes with stakeholders and adapt the work activity to the new demand of employees and the market, thus including, from its conception, digital natives in the entire process.

Social Implications

The results also have social implications. In this sense, the study of the digitization of human resources helps to adapt the usual performance of employees to the inclusion of new technologies in the business environment and to involve employees in training processes to promote professional and labor development. In turn, it can be used to involve employees in the development of new workspaces to maximize productivity, for the implementation of new work models designed by the company and the use of social networks as a means of labor communication. As a corollary, artificial intelligence can be considered as a tool to improve productivity, reducing the volume of monotonous or routine tasks and reinforcing those in which only the employee can provide real value.

Limitations and future research lines

The authors acknowledge the limitations of the methodology used in this study and call for further research to expand our understanding of the topic. Future studies could complement the co-word analysis with other bibliometric techniques such as co-citation analysis and develop theoretical and empirical models on the applications of digital human resources.

Data availability

Documents that support the findings of this study can be consulted in the Scopus database by following the search procedure indicated in the methodology section.

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This paper has been supported by Project PID2021-124641NB-I00 of the Ministry of Science and Innovation (Spain) and by research group in Open Innovation, Universidad Rey Juan Carlos (Spain). Open Access funding enabled and organized by Project V1313 “Sustainability Support”, signed under Article 60 of the LOSU between the Universidad Rey Juan Carlos (Spain) and the company Triple Sustainability SLU to carry out scientific-technical work and training activities.

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García-Fernández, L., Ortiz-de-Urbina-Criado, M. & García-López, MJ. Mapping the main research themes in digital human resources. Humanit Soc Sci Commun 11 , 1267 (2024). https://doi.org/10.1057/s41599-024-03795-8

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