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.

$$ \max \mathbb{E}\left[u\left({\phi}_x\right)\right],\kern0.5em \mathrm{for}\kern0.5em x\in X $$

(1)

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:

$$ P\left({\phi}_x\ge \eta \right)\ge 1-\alpha $$

(2)

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.

$$ \max \mathbb{E}\left[{\phi}_x\right] $$

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:

$$ \min \left\{-\mathbb{E}\left[{\phi}_x\right]+\lambda r\left[{\phi}_x\right]\right\},\kern0.5em \mathrm{for}\kern0.5em x\in X\kern0.5em \mathrm{where}\kern0.5em \uplambda \in {\mathrm{\mathbb{R}}}^{+}\cup \left\{0\right\} $$

(3)

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.

$$ r\left[{\phi}_x\right]=\mathbb{V}\mathrm{ar}\left[{\phi}_x\right]=\mathbb{E}\left[{\left({\phi}_x-\mathbb{E}\left[{\phi}_x\right]\right)}^2\right] $$

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).

$$ {WCE}_{\alpha }=-\operatorname{inf}\left\{\mathbb{E}\left[{\phi}_1|A\right]:A\in \mathcal{A},P\left[A\right]>\alpha \right\} $$

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:

$$ {\sigma}_k\left[{\phi}_1\right]=\mathbb{E}{\left[{\left\{{\left(\mathbb{E}\left[{\phi}_1\right]-{\phi}_1\right)}^{+}\right\}}^k\right]}^{\frac{1}{k}} $$

(4)

$$ {r}_{\beta}\left[{\phi}_1\right]=\min \mathbb{E}\left[\max \left(\left(1-\beta \right)\left(\eta -{\phi}_1\right),\beta \left({\phi}_1-\eta \right)\right)\right],\kern0.5em \mathrm{for}\ \eta \in \mathrm{\mathbb{R}}\kern0.5em \mathrm{with}\kern0.5em \beta \in \left(0,1\right) $$

(5)

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.