Robust Optimization Approaches in Inventory Management: Part B - The Comparative Study

Abstract

This comparative study, constituting Part B of our extensive investigation into robust optimization (RO) in inventory management, builds on the foundational insights from Part A’s survey. It conducts a thorough analysis of various RO formulations and algorithms, emphasizing their practical application, efficacy, and computational considerations in inventory management contexts. This study is meticulously structured to address the pivotal inquiries identified in Part A, encompassing the effective representation of uncertainty, the selection of optimal decision criteria, the influence of decision rules on inventory management performance, computational challenges, and the adaptability of these methods to evolving technological and market conditions. This research juxtaposes theoretical findings with empirical data, offering a comprehensive evaluation of the strengths, limitations, and practical implications of each robust inventory model discussed in the literature. This investigation not only complements the survey in Part A but also serves as a standalone, in-depth contribution to robust inventory management.

Keywords:

• Inventory management
• Robust optimization
• Decision rules
• Model uncertainty
• Comparative study

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1 Introduction

Inventory management is a critical component of supply chain operations, essential for balancing inventory holding costs with the risks of stockouts. In today’s increasingly volatile market environment, traditional inventory management models, like dynamic programming (DP) and stochastic programming (SP), are often found lacking due to their reliance on deterministic or precisely known random parameters. This gap has highlighted the growing relevance of robust optimization (RO) in the field. RO adeptly tackles uncertainties in parameters, delivering feasible and robust solutions amidst unpredictable market conditions. The core strength of RO lies in its provision of strategies that are not only optimal under specific assumptions but also maintain resilience when these assumptions are challenged (Ben-Tal et al., 2009).

The extensive survey presented in Part A of this study established a solid foundation for comprehending the varied applications and impacts of RO in the realm of inventory management. This exploration encompassed an array of RO and Distributionally Robust Optimization (DRO) methodologies, assessing their efficacy across different contexts (representation of uncertainty, decision criteria, decision rules, etc.) and the essential trade-offs inherent in their utilization. Crucial insights were gleaned about the representation of uncertainty, the selection of decision criteria, and the adaptability of decision rules within multi-stage decision frameworks. These insights play a pivotal role in guiding the focus of this comparative study, offering a detailed perspective on the current state and evolution of RO strategies in inventory management.

Building on the insights from Part A, this comparative study aims to delve deeper into the practical applications and implications of different RO approaches in inventory management. Our primary objectives are to:

1. Evaluate and Compare Different RO Approaches: We aim to critically analyze and compare various RO techniques, mainly focusing on their efficacy, computational tractability, and adaptability to different inventory management scenarios.

2. Decision Criteria, Rules, and Uncertainty Representation: A key focus will be on how different decision criteria, rules, and methods of representing uncertainty influence the effectiveness and applicability of the RO approach.

3. Practical Implications and Future Directions: We seek to provide concrete recommendations for practitioners in inventory management, highlighting which RO approaches are best suited for specific operational contexts. This includes, but is not limited to 1) customizing RO models to address specific industry challenges, such as managing asymmetric or heavy-tailed demand distributions and multimodal demand scenarios, 2) striking an optimal balance between robustness and cost-efficiency by setting appropriate uncertainty levels or employing target-attainment criteria, and 3) incorporating machine learning and artificial intelligence to enhance model accuracy and efficiency. Furthermore, we aim to identify areas where future research could be most beneficial, while considering the evolving nature of supply chain challenges.

This comparative study is structured to systematically address these objectives, offering a detailed analysis that complements and extends the foundational knowledge established in Part A. Through this approach, we aim to provide a comprehensive and practical guide for both researchers and practitioners in the field of inventory management.

The remainder of this paper is organized as follows: Section 2 delineates the criteria and metrics employed for comparison. Section 3 offers an in-depth comparative analysis of various robust inventory models, focusing on their analytical results and theoretical contributions. Subsequently, Section 4 expands this analysis to encompass a comprehensive evaluation of the numerical outcomes obtained from these models. Section 5 highlights potential avenues for future research and their associated challenges. Finally, Section 6 concludes the paper, synthesizing the principal findings and insights derived from the study.

2 Criteria for comparison

In evaluating the diverse RO approaches, we establish criteria grounded in modeling framework and practical applicability. These criteria include adaptability to varying degrees of uncertainty, computational efficiency, scalability to complex inventory scenarios, and balancing optimality and robustness. This multi-dimensional framework allows for a holistic assessment of each RO approach.

The selection of specific RO approaches for in-depth analysis in this study is heavily influenced by the insights derived from the survey conducted in Part A. Our focus is on approaches that have demonstrated a significant influence on inventory management practices, especially those that offer novel solutions to prevalent challenges in the field. This encompasses approaches that exhibit promising computational efficiency in managing high-dimensional uncertainty, as evidenced in works like Bertsimas and Thiele (2006) and Mamani et al. (2017). We also consider approaches that incorporate advanced algorithms or sophisticated modeling techniques, such as those proposed by Bienstock and Özbay (2008) and Zhang et al. (2024). Additionally, models with innovative formulations, like those by Solyalı et al. (2016), Xin and Goldberg (2021), and Wagner (2018), are included in our analysis. In total, 28 representative references have been selected for examination, comprising 14 focused on the robust newsvendor problem and another 14 addressing multi-period inventory management challenges.

The comparative analysis is divided into two primary segments: analytical and numerical comparisons.

1. Analytical Comparisons: Here, we juxtapose the modeling frameworks of selected RO approaches, examining their theoretical underpinnings and the extent to which they address the nuances of inventory management. This includes an evaluation of how different models represent uncertainty, their decision criteria, and the implications of these choices on the overall performance of the inventory system.

2. Numerical Comparisons: This segment delves into the solution methodologies employed by the RO approaches and their numerical efficacies. We assess the computational demands of each approach and their performance in real-world or simulated scenarios. This analysis is crucial in understanding the practical viability of these approaches in dynamic inventory management environments.

Through this structured methodology, the study aims to provide a comprehensive and nuanced understanding of the strengths and limitations of various RO approaches in inventory management, guiding practitioners and researchers in their application and further development.

Notations: In accordance with the abbreviations and symbols used in Part A, we adhere to the same nomenclature for consistency. The symbols ${\Omega }_{\text{BT}}$ and ${\Omega }_{\text{CLT}}$ represent the budget uncertainty set and the central limit theorem (CLT) uncertainty set, respectively. Similarly, ${\mathcal{A}}_{\text{MB}}$ and ${\mathcal{A}}_{\mathrm{W}}$ denote the moment-based and Wasserstein ambiguity sets. The variables ${\tilde{d}}_t$ and $d_t$ correspond to uncertain demand and its realization in period t, with $\tilde{d}$ representing random demand in the newsvendor problem. Additionally, ${\overline{d}}_t$ and ${\widehat{d}}_t$ indicate the nominal value of ${\tilde{d}}_t$ and its deviation, respectively. The vector ${\mathbf{d}}_t$ signifies cumulative demand up to period t, while ${\mu }_t$ and ${\delta }_t$ represent the mean and standard deviation of ${\tilde{d}}_t$. The acronyms LDP and PLDR refer to linear and piece-wise linear decision rules, respectively.

3 Modeling Frameworks and Analytical Contributions

This section comprehensively assesses prominent robust inventory formulations, emphasizing their modeling frameworks and analytical outcomes. We categorize the literature into two primary groups: single-period newsvendor and multi-period inventory management models.

Numerous robust modeling approaches in inventory management lead to computationally tractable problems. Some of these frameworks present analytical solutions for optimal decision-making or delineate the structure of the optimal replenishment policy. In contrast, others introduce tractable reformulations or approximations, converting the robust counterpart into a deterministic problem, eliminating uncertainty through a worst-case analysis.

3.1 Robust Newsvendor Formulations

In Table 1, we offer a systematic comparison of diverse robust newsvendor formulations, with a focus on their modeling frameworks and analytical outcomes. A comprehensive comparative analysis is presented as follows:

$•$ Single-Product Formulations The robust classical single-product newsvendor model can typically be formulated in two versions, depending on the primary objective: either maximizing profit or minimizing cost. The formulations for these two objectives are developed as follows: $\mathrm{P}:={\max }_q{\min }_{\mathcal{F}\in \mathcal{A}}\mathbb{E}[p\min (\tilde{d},q)-cq]$, or $\mathrm{C}:={\min }_q{\max }_{\mathcal{F}\in \mathcal{A}}\mathbb{E}\mathrm{t}[h\max \{q-\tilde{d},0\}+l\max \{\tilde{d}-q,0\}]$, where p, c, h, and l represent the unit profit, ordering cost, overage cost, and shortage cost, respectively. q and $\tilde{d}$ denote the ordering quantity and random demand, respectively. $\mathcal{F}$ denotes the underlying probability distribution of $\tilde{d}$, and $\mathcal{A}$ is the ambiguity set that encompasses all potential probability distributions of $\tilde{d}$. Note that these two variations of the newsvendor model are equivalent after some parameter transformation (Yue et al., 2006).

In the investigation of the single-product robust newsvendor problem, pivotal analytical results include closed-form solutions for the optimal ordering quantity $q^{*}$ and upper and lower bounds for the total cost C (or profit). Our survey reveals that many researchers use Scarf’s rule (Scarf, 1958) and the critical fractile as benchmarks, seeking to refine or adapt them based on their specific decision-making contexts.

In a classical newsvendor model with a known demand distribution $\mathrm{F}$, the optimal ordering quantity is commonly referred to as the critical fractile. Specifically, it is either the $\frac{l}{l+h}$-quantile or the $1-\frac{c}{p}$-quantile of the demand distribution, depending on model variations. However, complete knowledge of the probability distribution is rarely available in practice. To address this issue, distribution-free newsvendor models have been investigated since the seminal work by Scarf (1958).

Scarf (1958) assumed that only the mean $\mu$ and standard deviation $\delta$ of the demand $\tilde{d}$ are specified within the ambiguity set $\mathcal{F}$. Consequently, the optimal ordering quantity is determined as follows:(1) $$q^{*}=\mu +\frac{\delta }{2}\frac{1-2a}{{[a(1-a)]}^2},$$(1)

where $a=c/p$. Equation (1) is often known as Scarf’s rule. Meanwhile, the upper bound for C is given by:(2) $$\overline{\mathrm{C}}=\frac{(h-l)}{2}(q-\mu )+\frac{(l+h)}{2}{[{\delta }^2+{(q-\mu )}^2]}^{1/2}$$(2)

The key step in deriving Equation (2) involves deducing the upper bound (Gallego and Moon, 1993) based on the Cauchy-Schwarz inequality, as follows:(3) $$\mathbb{E}|\tilde{d}-q|\le {\left[\mathbb{E}{[\tilde{d}-q]}^2\right]}^{1/2}={[{\delta }^2+{(q-\mu )}^2]}^{1/2}$$(3)

Min-Max Regret Model: Scarf’s rule and related methods offer robust, but often overly conservative, solutions for distribution-free problems (Gallego and Moon, 1993; Gallego et al., 2001). To mitigate this, Yue et al. (2006) incorporated min-max regret criteria into the robust newsvendor model, following Scarf’s early work Scarf (1958). They identified that the worst-case demand distribution, ${\mathcal{F}}^{*}$, varies with the order quantity q and can be represented by a two-point distribution. Yue et al. (2006) then devised a set of two-point probability density functions to represent all distributions in the ambiguity set $\mathcal{A}$. They showed that the cost function C is piecewise-linear in q, allowing for closed-form maximum regret calculations for each q. A convex optimization model was also developed to determine the optimal $q^{*}$ that minimizes maximum regret.

The min-max robust newsvendor formulation has been further extended by Perakis and Roels (2008). Building on the work of Scarf (1958); Gallego and Moon (1993); Yue et al. (2006), Perakis and Roels (2008) takes into account not just the moments (mean, variance) of the demand distribution $\mathcal{F}$, but also its shape characteristics (range, symmetry, unimodality). The distributionally robust min-max regret model in this work can be simplified as: $\mathrm{R}:={\min }_q{\max }_x\{{\max }_{\mathcal{F}\in \mathcal{A}}{{\rm Y}}_{\mathcal{F}}(x,\tilde{d})-{{\rm Y}}_{\mathcal{F}}(q,\tilde{d})\}$, where ${{\rm Y}}_{\mathcal{F}}(q,\tilde{d})={\mathbb{E}}_{\mathcal{F}}[p\min (q,\tilde{d})-cq]$. x represents the ordering quantity determined by the decision-maker. The objective is to minimize the maximum regret, expressed as ${\max }_{\mathcal{F}\in \mathcal{A}}{\max }_x\{{{\rm Y}}_{\mathcal{F}}(x,\tilde{d})-{{\rm Y}}_{\mathcal{F}}(q,\tilde{d})\}$, under a given q and a probability distribution $\mathcal{F}$.

Closed-form solutions for Problem R have been established based on various known partial information about $\mathcal{F}$, such as range, mean, mode, and median. Additionally, the authors demonstrated that distributions maximizing entropy should be selected as the benchmark distribution for $\tilde{d}$ to optimize performance under the min-max regret criteria. This finding bridges the research gap left by Yue et al. (2006).

Remark 1 .

The min-max regret newsvendor approach treats uncertainty more as an opportunity than a risk, leading to less conservative solutions in robust newsvendor models using this criterion (e.g., Yue et al. (2006); Perakis and Roels (2008); Lin and Ng (2011)) compared to those based on Scarf (1958); Gallego and Moon (1993). However, it requires accurate estimation of regret for each decision, demanding precise knowledge of uncertain parameters’ probability distributions, which is often challenging to obtain in practice.

Conditional Value at Risk (CVaR) Measurement: The CVaR metric, capturing both the probability and magnitude of loss, is widely used in newsvendor problems. However, optimizing CVaR for continuous variables often requires complex calculations, such as $\mathbb{E}({(·)}^{+})$ (see Equations (19) and (20) in Part A). To simplify, Qiu et al. (2014) assumed discrete demand $\tilde{d}=\{d_1,d_2,\dots ,d_n\}$, with $\text{Pr}\{\tilde{d}=d_i\}={\omega }_i$, ${\sum }_{i=1}^n{\omega }_i=1$, and ${\omega }_i\ge 0$ for $i=1,2,\dots ,n$. The probability vector $\omega =\{{\omega }_1,{\omega }_2,\dots ,{\omega }_n\}$ is uncertain and lies within an uncertainty set $\Omega$, leading to a reformulated CVaR newsvendor model: ${\max }_{\gamma }{\min }_{\omega \in \Omega }\gamma -\frac{1}{\alpha }{\sum }_{i=1}^n{\omega }_i{[\gamma -p\min \{q,d_i\}+cq]}^{+}$, where $\alpha$ represents the confidence level of CVaR, and the goal is to find the optimal pair ( $q^{*}$, ${\alpha }^{*}$) to maximize CVaR ${}_{\alpha }$ (q)= ${\max }_{\gamma }\gamma -\frac{1}{\alpha }{\sum }_{i=1}^n{\omega }_i{[\gamma -p\min \{q,d_i\}+cq]}^{+}$.

Note that for a finite number of possible demand quantities n and a tractable uncertainty set $\Omega$, this CVaR newsvendor model can be reformulated as a convex optimization model. Specifically, when employing a box uncertainty set, it can be transformed into a general convex optimization problem through the duality of linear programming. When using an ellipsoidal uncertainty set, the robust counterpart can be converted into a second-order cone program via the Lagrange dual function. However, the assumption of a finite number of realizations for the random demand $\tilde{d}$ may not align well with practical scenarios, potentially limiting the applicability of this model.

Despite the increasing computational complexity, CVaR newsvendor models inspired by the DRO methodology employ ambiguity sets to represent uncertainty. This methodology is more in tune with real-world decision-making scenarios, as evidenced by Hanasusanto et al. (2015a); Rahimian et al. (2019); Lee et al. (2021).

Rahimian et al. (2019) employ a $\varphi$-divergence function, specifically the variation distance, to construct the ambiguity set. This set includes all potential probability distributions of random demand. Furthermore, they use the maximum allowable distance within the ambiguity set, represented by the value of the variation distance, as a metric to assess the level of robustness. The model is given by:(4) $$\underset{q}{\min }\left\{\underset{P\in {\mathcal{A}}_{\rho }}{\sup }{\mathbb{E}}_P\left[h\max \{q-\tilde{d},0\}+l\max \{\tilde{d}-q,0\}-p\tilde{d}\right]\right\},$$(4)

where ${\mathcal{A}}_{\rho }=\left\{P:\frac{1}{2}{\int }_{\Omega }\left|P(x)-Q(x)\right|dx\le \rho ,{\int }_{\Omega }P(x)dx=1,P\ge 0\right\}.$

The ambiguity set ${\mathcal{A}}_{\rho }$ comprises all distributions P that lie within a specified variation distance $\rho$ from the nominal distribution Q. The support set $\Omega$ is defined as $\Omega :=\{\tilde{d}�\underset{¯}{d}\le \tilde{d}\le \overline{d}\}$, which is an interval containing all possible realizations of $\tilde{d}$ when it follows the nominal distribution Q. Here, $\underset{¯}{d}$ and $\overline{d}$ are the lower and upper bounds of $\tilde{d}$, respectively.

Intriguingly, the inner supremum of Equation (4) can be equivalently reformulated as the following CVaR measure:(5) $$\begin{array}{c}\underset{P\in {\mathcal{A}}_{\rho }}{\sup }{\mathbb{E}}_P[C(q,\tilde{d})]\hfill \\ =\rho {\text{CVaR}}_1[C(q,\tilde{d})]+(1-\rho ){\text{CVaR}}_{\rho }[C(q,\tilde{d})],\hfill \end{array}$$(5)

where $C(q,\tilde{d}):=h\max \{q-\tilde{d},0\}+l\max \{\tilde{d}-q,0\}-p\tilde{d}$ and $\tilde{d}$ follows the nominal distribution Q with $\tilde{d}\in \Omega$.

Note that the variation distance $\rho$ relates the level of robustness (i.e., the size of the ambiguity set ${\mathcal{A}}_{\rho }$) to the level of risk-aversion, as demonstrated by the equivalence shown in Equation (5). Furthermore, Equation (5) effectively transforms the distributionally robust newsvendor model (4) into a classical robust model that employs CVaR measurements. When $\rho =0$, the DRO model (4) reduces to a risk-neutral stochastic optimization model under the nominal probability distribution Q. Conversely, when $\rho =1$, model (4) becomes a classical RO problem related to the CVaR formulation proposed in Qiu et al. (2014).

The model proposed by Rahimian et al. (2019) offers several advantages over the one by Qiu et al. (2014). While both works propose a linear combination similar to Equation (5) to bridge the risk-averse CVaR measurement with the risk-neutral expected objective, the weighted parameter $\rho$ serves different roles in each model. In Qiu et al. (2014), $\rho$ is a user-defined parameter that can range between zero and one. In contrast, $\rho$ in Equation (5) quantifies the size of the ambiguity set, thereby linking the level of robustness to risk aversion. Furthermore, model (4) does not require the assumption of a discrete distribution for random demand, thereby significantly expanding its applicability.

The $\varphi$-divergence-based ambiguity set utilized in Rahimian et al. (2019) has been critiqued in a closely related work by Lee et al. (2021). This critique specifically questions the comprehensiveness of $\varphi$-divergence-based ambiguity sets, arguing that they may inadvertently exclude certain distributions that the decision-maker intends to consider. For example, an ambiguity set constructed using $\varphi$-divergence and based on a discrete empirical distribution will only include discrete distributions sharing the same support as the empirical distribution. This becomes problematic when the historical data is generated from a continuous distribution (e.g., normal, uniform, lognormal), as the data-generating distribution would be excluded from the $\varphi$-divergence-based ambiguity set.

In response to the limitations associated with $\varphi$-divergence-based ambiguity sets, Lee et al. (2021) proposed a distributionally robust newsvendor model that employs Wasserstein’s ambiguity set to overcome these drawbacks. The authors successfully derived closed-form solutions for both risk-neutral and CVaR objectives within the framework of their DRO model. However, one notable gap in their work is the lack of exploration into the relationship between these two types of models, a topic that has been addressed in previous studies such as Qiu et al. (2014) and Rahimian et al. (2019).

Remark 2 .

The CVaR-based robust newsvendor model excels in tail risk management, aligning with organizational risk strategies (Qiu et al., 2014; Rahimian et al., 2019; Lee et al., 2021). However, it brings complexities in computation, higher data needs, and possibly increased holding costs. While effective for risk minimization in extreme scenarios, it may not support goals like profit maximization or customer satisfaction (Wang et al., 2022; Ng et al., 2012; van der Laan et al., 2022). On the other hand, the classic maximum-profit model prioritizes profitability but can neglect key risks, potentially undermining long-term stability and performance.

Heavy-tailed Distributions: Heavy-tailed distributions are increasingly gaining attention in the field of inventory management, especially in the context of robust newsvendor models. These distributions are characterized by tails that are not exponentially bounded, which leads to a higher likelihood of extreme values occurring.

Distributionally robust newsvendor models often employ moment-based ambiguity sets, focusing specifically on known mean and standard deviation or covariance of the demand distributions. This approach is well-documented in seminal works such as Gallego and Moon (1993), Yue et al. (2006), and Perakis and Roels (2008). However, a significant limitation of this approach is that covariance metrics are unable to capture the asymmetry of some demand distributions. As a result, models that do not consider the skewness or heavy-tailed nature of a demand distribution may perform suboptimally in real-world scenarios where such characteristics are prevalent.

To address this limitation, Natarajan et al. (2018) introduced a class of second-order partitioned statistics to model the asymmetry in demand distributions. Specifically, in the case of a single-product inventory system, one of these specialized statistics is semivariance, which is a well-known risk measure. With the known mean $\mu$ and standard deviation $\delta$, the second-order partitioned statistics, normalized semivariance, s, is defined as:(6) $$s=\frac{{\mathbb{E}}_{\mathcal{F}}[\max {\{\tilde{d}-\mu ,0\}}^2]-{\mathbb{E}}_{\mathcal{F}}[\max {\{\mu -\tilde{d},0\}}^2]}{{\delta }^2}.$$(6)

Note that $s\in [-1,1]$ as ${\delta }^2={\mathbb{E}}_{\mathcal{F}}[\max {\{\tilde{d}-\mu ,0\}}^2]+{\mathbb{E}}_{\mathcal{F}}[\max {\{\mu -\tilde{d},0\}}^2]$ . The value of the normalized semivariance s serves as an indicator of the skewness of the random demand $\tilde{d}$ relative to its mean $\mu$. Specifically, a positive value of $s>0$ suggests that the deviations from the mean are concentrated above the mean, indicating a positively skewed distribution. Conversely, a negative value $s<0$ implies that the deviations are concentrated below the mean, indicating a negatively skewed distribution. It is worth noting that for all symmetric distributions (e.g., normal, uniform), $s=0$.

The distributionally robust newsvendor problem with known $\mu$, $\delta$, and s is given by:(7) $$\begin{array}{cc}{\mathrm{P}}^{\text{MVS}}:\hfill & =\underset{q}{\max }\underset{\mathcal{F}\in \mathcal{A}}{\inf }p{\mathbb{E}}_{\mathcal{F}}[\min (\tilde{d},q)]-cq\hfill \end{array}$$(7) $$\begin{array}{c}s.t.{\mathbb{E}}_{\mathcal{F}}\left[\tilde{d}\right]=\mu ,\hspace{1em}{\mathbb{E}}_{\mathcal{F}}\left[{(\tilde{d}-\mu )}^2\right]={\delta }^2,\hfill \\ {\mathbb{E}}_{\mathcal{F}}[\max {\{\tilde{d}-\mu ,0\}}^2]-{\mathbb{E}}_{\mathcal{F}}[\max {\{\mu -\tilde{d},0\}}^2]=s{\delta }^2,\hfill \\ {\mathbb{E}}_{\mathcal{F}}\left[1\right]=1,\hspace{1em}\mathcal{F}(\tilde{d})>0,\hspace{1em}\forall \tilde{d}\ge 0\hfill \end{array}$$

Formulation (7) is a semi-infinite linear optimization problem, similar to the one presented by Perakis and Roels (2008). However, problem (7) incorporates an additional second-order moment s that captures the asymmetry of the demand distribution $\mathcal{F}$. Leveraging the strong duality theorem, which ensures that the primal and dual moment problems achieve the same optimal value, Natarajan et al. (2018) provided closed-form solutions for (7).

It is noteworthy that the optimal solution to Problem (7), denoted as $q^{*}$, is an increasing function of the critical fractile $(1-\frac{c}{p})$. This is in line with real-world practice, where a higher service level necessitates an increased order quantity q to mitigate the risk of stockouts.

Compared to the distributionally robust newsvendor model that only considers known Mean-Variance (MV) statistics (Scarf, 1958; Gallego and Moon, 1993), formulation (7), that incorporates known Mean-Variance-Semivariance (MVS) statistics, yields a significantly narrower range of expected profit across various values of s. Specifically, the gap between the worst-case and best-case profits in the MVS model is substantially smaller than that in the MV model. This implies that the MVS model offers more robust performance in terms of profit variability.

Furthermore, under heavy-tailed demand distributions such as exponential or lognormal, the regret associated with the MVS model is remarkably lower than that of the MV model. Here, regret refers to the loss in profit incurred when ordering is based on the distributionally robust solution instead of the optimal solution for the true underlying distribution. This suggests that the MVS model is particularly effective in scenarios where the demand distribution exhibits heavy tails.

The heavy-tail distribution behaviors of distributionally robust newsvendor models are further explored in the work by Das et al. (2021). The authors extended the model by incorporating higher-order moments, beyond the second moment, to address the conservatism inherent in MV) models. The moment-based ambiguity set defined in this work is given by:(8) $${\mathcal{A}}_{M,k}=\{\mathcal{F}\in \mathbb{M}({ℝ}_{+}):{\int }_0^{\infty }d\mathcal{F}(x)=1,{\int }_0^{\infty }xd\mathcal{F}(x)=m_1,{\int }_0^{\infty }{x}^{k}d\mathcal{F}(x)=m_k\},$$(8)

where $m_1$, $m_k$ represent the first and $k^{th}$ moments of $\mathcal{F}$.

Under the ambiguity set (8), the distributionally robust newsvendor model is defined as follows:(9) $${\mathrm{P}}^{\text{MK}}:=\underset{q}{\min }(\frac{c}{p}q+\underset{\mathcal{F}\in {\mathcal{A}}_{M,k}}{\sup }{\mathbb{E}}_{\mathcal{F}}{[\tilde{d}-q]}^{+})$$(9) $$\begin{array}{c}s.t.{\int }_0^{\infty }d\mathcal{F}(x)=1,{\int }_0^{\infty }xd\mathcal{F}(x)=m_1,{\int }_0^{\infty }{x}^{k}d\mathcal{F}(x)=m_k,\mathcal{F}\in \mathbb{M}({ℝ}_{+}),\hfill \end{array}$$

Note that formulation (9) aims to maximize the worst-case expected profit. With simple mathematical transformations, this formulation can be converted into a form similar to that of (8). Furthermore, the moment problem in (9) is more general than those in Perakis and Roels (2008); Natarajan et al. (2018). Specifically, k in this formulation can take any real value greater than one, not necessarily restricted to integers.

Similarly, formulation (9) can be resolved based on the strong duality theorem. The dual formulation of the inner supremum $\underset{\mathcal{F}\in {\mathcal{A}}_{M,k}}{\sup }{\mathbb{E}}_{\mathcal{F}}{[\tilde{d}-q]}^{+}$ is given as:(10) $$\begin{array}{c}\mathit{inf}{y}_0+y_1{m}_1+y_k{m}_k\hfill \end{array}$$(10) $$\begin{array}{cc}s.t.y_0+y_1\tilde{d}+y_k{\tilde{d}}^k\ge 0,\hfill & \forall \hspace{1em}\tilde{d}\ge 0\hfill \\ y_0+y_1\tilde{d}+y_k{\tilde{d}}^k\ge \tilde{d}-q,\hfill & \forall \hspace{1em}\tilde{d}\ge 0\hfill \end{array}$$

where $y_0$, $y_1$, and $y_k$ correspond to the first three constraints of the model (9), respectively.

Das et al. (2021) indicates that the computational tractability of problem (9) depends on the value of the ordering quantity q. Specifically, for $q\in \left[0,\frac{k-1}{k}{\left(\frac{m_k}{m_1}\right)}^{\frac{1}{k-1}}\right]$, a closed-form solution can be obtained. Additionally, for any finite k, there exists a certain range of small critical fractiles $\left(1-\frac{c}{p}\right)$ where $q=0$ is the optimal solution for the moment-based ambiguity set. For large values of q, the authors provided both lower and upper bounds for the worst-case expected profit.

Das et al. (2021) provided theoretical proof that the optimal solution of formulation (9) is also optimal for its stochastic optimization counterpart under a known heavy-tailed distribution with tail parameter k. This links the solution of the distributionally robust newsvendor model, addressing worst-case behaviors, to heavy-tailed distributions modeling extreme events. In contrast, Natarajan et al. (2018) showed through simulations that their robust solution performs well under heavy-tailed distributions. However, due to increased model complexity with larger k, Das et al. (2021) only offered closed-form solutions for cases with small q. This underscores a trade-off in DRO where better solutions from additional distribution information often result in higher model complexity.

Remark 3 .

The distributionally robust newsvendor model that considers heavy-tailed distributions performs well in high-service-level cases, characterized by a high critical fractile $1-\frac{c}{p}$ (Natarajan et al., 2018; Das et al., 2021) . While additional distribution information improves the quality of the solution, it simultaneously results in increasing computational challenges.

$•$ Multiple-Product Formulations The multi-item robust newsvendor model can be approached by solving a series of its single-item counterparts, without taking into account the correlation among demands (Gallego and Moon, 1993). However, given information on correlations of demands, the multi-item model turns out to be much harder to solve than the single-item inventory problems.

Hanasusanto et al. (2015a) investigated a distributionally robust newsvendor model that accounts for multimodal demand ambiguity. Their study employs a linear combination of CVaR and expected cost objectives, akin to the approaches in Qiu et al. (2014) and Rahimian et al. (2019). The model that accommodates a worst-case mean-risk objective is defined as follows:(11) $$\underset{\mathbf{q}}{\min }\hspace{1em}\rho \underset{\mathbb{F}\in \mathcal{A}}{\sup }{\text{CVaR}}_{\alpha }(C(\mathbf{q},\tilde{\mathbf{d}}))+(1-\rho )\underset{\mathbb{F}\in \mathcal{A}}{\sup }{\mathbb{E}}_{\mathbb{F}}(C(\mathbf{q},\tilde{\mathbf{d}})),$$(11)

where $C(\mathbf{q},\tilde{\mathbf{d}})$ represents the total cost incurred by the newsvendor for selling n products. The ordering quantity vector is denoted by $\mathbf{q}=(q_1,\dots ,q_n)$, and the demand vector is $\tilde{\mathbf{d}}=({\tilde{d}}_1,\dots ,{\tilde{d}}_n)$. The parameter $\rho \in [0,1]$ is selected by the decision-maker to reflect their level of risk aversion. It is worth noting that, unlike the previously mentioned $\mathcal{F}$, which denotes the probability distribution of a single random variable $\tilde{d}$, $\mathbb{F}$ here represents the distribution of the random demand vector $\tilde{\mathbf{d}}$.

The probability distribution $\mathbb{F}={\sum }_{i=1}^m{\omega }_i{\mathbb{F}}_i$ is assumed to be a mixture of m distinct distributions ${\mathbb{F}}_1,{\mathbb{F}}_2,\dots ,{\mathbb{F}}_m$ with known mixture probabilities ${\omega }_1,{\omega }_2,\dots ,{\omega }_m$, subject to ${\sum }_{i=1}^m{\omega }_i=1$. Moreover, each ${\mathbb{F}}_i$ is only known to be supported on a non-degenerate ellipsoid ${\Xi }_i$ with a known mean ${\mu }_i$ and a known covariance matrix ${\Sigma }_i>0,$ $i=1,\dots ,m$. Each ellipsoidal support set ${\Xi }_i$ is represented as(12) $${\Xi }_i=\left\{\tilde{\mathbf{d}}\in {ℝ}^n:(\tilde{\mathbf{d}}-{\mathbf{v}}_i){\mathbf{\Lambda }}_i^{-1}(\tilde{\mathbf{d}}-{\mathbf{v}}_i)\le {\theta }_i^2\right\},$$(12)

where ${\mathbf{v}}_i\in {ℝ}^n$, n-dimension symmetric definite matrice ${\mathbf{\Lambda }}_i$ and ${\theta }_i\in {ℝ}_{+}$ determine the center, shape and size of ${\Xi }_i$, respectively.

The ambiguity set $\mathcal{A}$ that emcompasses all possible distribution $\mathbb{F}$ is given by:(13) $$\mathcal{A}=\sum _{i=1}^m{\omega }_i\mathcal{A}({\Xi }_i,{\mu }_i,{\mathbf{\Sigma }}_i),$$(13)

where $\mathcal{A}({\Xi }_i,{\mu }_i,{\mathbf{\Sigma }}_i)=\left\{\mu \in {\mathcal{M}}_{+}:{\int }_{\Xi }\mu =1,{\int }_{\Xi }\tilde{\mathbf{d}}\mu =\mu ,{\int }_{\Xi }\tilde{\mathbf{d}}{\tilde{\mathbf{d}}}^{\mathrm{T}}\mu =\mathbf{\Sigma }+\mu {\mu }^{\mathrm{T}}\right\}$ denotes the set of all distributions supported on $\Xi$ that share the same mean $\mu$ and covariance matrix $\mathbf{\Sigma }$.

Note that the ambiguity set defined in Equation (13) is specifically tailored to represent the ambiguous demand for a class of products, denoted by $i=1,\dots ,n$. These products have strongly correlated demands, as indicated by ${\int }_{\Xi }\tilde{\mathbf{d}}{\tilde{\mathbf{d}}}^{\mathrm{T}}\mu =\mathbf{\Sigma }+\mu {\mu }^{\mathrm{T}}$. Additionally, they are subject to fashion trends, represented by $\mathbb{F}={\sum }_{i=1}^m{\omega }_i{\mathbb{F}}_i$. For example, if a particular product has a high component in the mean vector ${\mu }_i$, it can be considered popular in the corresponding state (e.g., season, trend) i.

Despite the aforementioned merits, such as capturing demand correlation and fashion trends, Hanasusanto et al. (2015a) demonstrated that problem (11) with the ambiguity set defined in Equation (13) is generally NP-hard. Specifically, the problem can be reformulated as a semi-definite programming problem, albeit with matrix inequalities of exponential size. Nevertheless, by employing a quadratic decision rule, which can be interpreted as applying Scarf’s bound in Equation (2) for each item i, $i=1,\dots ,n$, problem (11) can be conservatively approximated as a semi-definite program with polynomial-sized constraints that can be efficiently solved.

In the work by Natarajan et al. (2018), the multi-product newsvendor problem is also addressed. To tackle the complexities arising from correlated demands, the authors extend the concept of semivariance, as defined in Equation (6), by introducing second-order statistics across multiple demand partitions. Specifically, they assume that the probability density function $\mathcal{F}$ of $\tilde{d}$ has a support partitioned into $m+1$ intervals, denoted as $D_0,\dots ,D_t$. Each interval $D_i$ is defined as $[d_i,d_{i+1}]$ for $i=0,\dots ,t-1$, and $D_t=[d_t,\infty )$. Under this setting, the formulation (7) can be redefined as follows:(14) $$\begin{array}{cc}{\mathrm{P}}^{t-\text{PART}}:\hfill & =\underset{q}{\max }\underset{\mathcal{F}\in \mathcal{A}}{\inf }p{\mathbb{E}}_{\mathcal{F}}\left[p\min (\tilde{d},q)\right]-cq\hfill \end{array}$$(14) $$\begin{array}{c}s.t.{\mathbb{E}}_{\mathcal{F}}\left[\tilde{d}\right]=\mu ,\hspace{1em}{\mathbb{E}}_{\mathcal{F}}\left[{\tilde{d}}^2·1_{\{\tilde{d}\in D_i\}}\right]={\delta }_i^2,\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}\forall i=0,\dots ,t,\hfill \\ {\mathbb{E}}_{\mathcal{F}}\left[1\right]=1,\hspace{1em}\mathcal{F}(\tilde{d})>0,\hspace{1em}\forall \tilde{d}\ge 0,\hfill \end{array}$$

where ${\delta }_i$ represents the second moment of the partitioned demand $\tilde{d}\in D_i$.

With $t=0$, Equation (14) reduces to the Mean-Variance (MV) model studied by Scarf (1958) and Gallego and Moon (1993), which only assumes information about the mean and standard deviation. The Mean-Variance-Semivariance (MVS) model, represented by Equation (7), is a special case of Equation (14) when $t=1$.

Recall that given the mean $\mu$ and covariance matrix $\mathbf{\Sigma }$, Hanasusanto et al. (2015a) show that the multi-item distributionally robust newsvendor model is NP-hard. To tackle this complexity, Natarajan et al. (2018) reformulates the multi-item version of Equation (14) as a semi-definite program with constraints of exponential size, aligning with the approach used in Hanasusanto et al. (2015a). Unlike the application of Scarf’s bound to individual items, the authors offer a computationally tractable lower bound for the model, leveraging the approximation technique developed in Natarajan and Teo (2017).

In contrast to the approaches taken by Hanasusanto et al. (2015a) and Natarajan et al. (2018), Zhang et al. (2021) employ classical RO methods to address the multi-item newsvendor model, while accounting for demand correlations. In this work, demand correlations are modeled as substitutions among different items. Both the demand and substitution rates are assumed to be uncertain and are constrained by a budget uncertainty set introduced in Section 2 of Part A.

The substitution rate ${\mathrm{\varrho }}_{i,j}$ represents the proportion of unmet demand for product j that is substituted by product i. The substitution rate ${\mathrm{\varrho }}_{i,j}$ is constrained to lie in the interval [0, 1], with ${\mathrm{\varrho }}_{i,i}=0$. Then the effective demand of product i, $i=1,2,\dots ,n$ is given by:(15) $${\tilde{d}}_i^{{}^{\prime }}(\mathbf{q})={\tilde{d}}_i+\sum _{j=1}^n{\mathrm{\varrho }}_{i,j}\max \{{\tilde{d}}_j-q_j,0\}.$$(15)

The robust multi-item newsvendor model with substitution is defined as follows:(16) $$\underset{\mathbf{q}}{\max }\underset{\tilde{\mathbf{d}}\in \Omega ,\mathrm{\varrho }\in {\Omega }^{{}^{\prime }}}{\min }\left\{\sum _{i=1}^n{p}_i\min \left\{q_i,{\tilde{d}}_i^{{}^{\prime }}\right\}-c_i{q}_i\right\}$$(16)

where $\Omega$ and ${\Omega }^{{}^{\prime }}$ here, are budget uncertainty sets.

Zhang et al. (2021) demonstrates that the optimization problem described in Equation (16) is both nonconvex and nonsmooth, making it intractable through conventional methods. To address this, they reformulate the problem as a mixed-integer linear program (MILP) with an exponential number of constraints, which can be solved using a branch-and-cut algorithm for intermediate-sized instances. Additionally, they provide a computationally tractable conservative approximation of the model.

Remark 4 .

Robust and distributionally robust multi-item newsvendor models that consider demand correlations are generally NP-hard, as shown in the literature (Hanasusanto et al., 2015a; Natarajan et al., 2018; Zhang et al., 2021). The common approach to tackle this complexity is deriving conservative approximations, providing effective bounds on worst-case performance. Incorporating correlation information notably reduces the over-conservatism of robust solutions, contrasting with earlier works like Gallego and Moon (1993), which focused on marginal distributions and effectively solved separate Scarf (1958) problems.

3.2 Multi-period Robust Inventory Formulation

The multi-period robust inventory management formulation builds upon the single-period newsvendor model, incorporating temporal dynamics to capture the complexities of multi-period inventory scenarios. Table 2 presents a comprehensive comparative analysis that emphasizes both the modeling frameworks and the analytical results.

In Table 2, we carry out a systematic comparison of diverse robust multi-period inventory formulations, focusing on their modeling frameworks and analytical outcomes. In the realm of multi-period robust inventory problems, key analytical contributions encompass closed-form solutions for the optimal ordering quantity ${\mathbf{x}}_t^{*}$ and the optimal robust version of the base-stock policy $(s,S)$, as well as computationally tractable exact reformulations or conservative approximations of these robust models. The detailed comparative analysis is presented as follows:

$•$ Robust Version of Optimal $(s,S)$ Policies Our review indicates that many researchers employ the $(s,S)$ policy (Clark and Scarf, 1960) and its various extensions (Zipkin, 2000) as benchmarks, aiming to modify or adapt them to suit robust decision-making contexts.

The $(s,S)$ policy, commonly known as the base-stock policy, sets a lower stock threshold s and an upper stock limit S. When the inventory level drops below s in any given period, an order is placed to replenish the stock up to S. Scarf et al. (1960) established that base-stock policies are optimal for single-installation stochastic inventory models when the probability distribution of random parameters is known. Clark and Scarf (1960) extended this to serial supply chains without capacity constraints, demonstrating that the optimal ordering policy for multi-echelon systems can be decomposed based on echelon stock levels. Federgruen and Zipkin (1986) expanded the analysis to single-stage capacitated systems, and Rosling (1989) further extended it to assembly systems. However, all these works assume full knowledge of the underlying probability distributions, which is barely available in practice. Another significant limitation is that these studies rely on dynamic programming, which is subject to the well-known “curse of dimensionality”.

To mitigate these limitations, Bertsimas and Thiele (2006) employed a RO approach to address multi-period inventory issues in both single-installation and supply chain contexts. Leveraging the budget uncertainty set introduced in Bertsimas and Sim (2004) to model demand uncertainty, the proposed RO approach does not assume any specific distribution and remains computationally tractable as the problem dimension increases.

The foundational model presented in Bertsimas and Thiele (2006) aligns closely with the formulation style of model (29) of Part A, incorporating the budget uncertainty set specified in equation (3) of Part A. Data uncertainty predominantly impacts the constraints related to holding and backlogging costs, as elaborated below,(17) $$z_t\ge h_t(I_1+\sum _{i=1}^t(x_i-\widetilde{d_i})),z_t\ge -l_t(I_1+\sum _{i=1}^t(x_i-\widetilde{d_i})),{\tilde{d}}_t\in {\Omega }_{\text{BT}},\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}t\in \mathcal{T}$$(17)

where $I_t$: initial stock at period t, $x_t$: quantity ordered at the start of period t, $\tilde{d}$: demand during period t, $c_t$: unit ordering cost in period t, $h_t$: unit holding cost in period t, $l_t$: unit backlogging cost in period t.

Revisiting the budget uncertainty set outlined in equation (3) of Part A, the demand ${\tilde{d}}_t$ for $t\in \mathcal{T}$ can be formulated as ${\tilde{d}}_t={\overline{d}}_t+{\widehat{d}}_t{\eta }_t$, where $-1\le {\eta }_t\le 1$ and ${\sum }_{i=1}^t{\eta }_i\le {\Gamma }_t$ for $t\in \mathcal{T}$. The core concept behind reformulating the robust model into its deterministic equivalent involves maximizing the right-hand side of the constraints presented in equation (17) over the set of permissible scaled deviations. For the t-th pair of holding/shortage constraints, this amounts to solving the auxiliary linear programming problem:(18) $$\underset{{\eta }_i}{\max }\left\{\sum _{i=1}^t{\widehat{d}}_i{\eta }_i:\sum _{i=1}^t{\eta }_i\le {\Gamma }_t,0\le {\eta }_i\le 1,\forall i\right\}$$(18)

Given that the linear programming problem in equation (18) is both feasible and bounded, strong duality ensures that its optimal cost is equal to the optimal cost of its dual problem. By substituting the dual of the auxiliary problem from equation (18) into the constraints specified in equation (17), the deterministic model is as follows:(19) $$\begin{array}{cc}\min \hfill & \sum _{t=1}^T(f_t{y}_t+c_t{x}_t+z_t)\hfill \end{array}$$(19) $$\begin{array}{cc}s.t.z_t\ge h_t(I_1+\sum _{i=1}^t(x_i-\overline{d_i})+u_t{\Gamma }_t+\sum _{i=1}^t{r}_{i,t}),\hfill & t\in \mathcal{T}\hfill \\ z_t\ge -l_t(I_1+\sum _{i=1}^t(x_i-\overline{d_i})-u_t{\Gamma }_t-\sum _{i=1}^t{r}_{i,t}),\hfill & t\in \mathcal{T}\hfill \\ u_t+r_{i,t}\ge {\widehat{d}}_t,\hspace{1em}{u}_t\ge 0,\hspace{1em}{r}_{i,t}\ge 0\hfill & t\in \mathcal{T},\forall i\le t\hfill \\ 0\le x_t\le M{y}_t,\hspace{1em}{y}_t\in \{0,1\},\hfill & t\in \mathcal{T}\hfill \end{array}$$

where M: is a large positive number, $f_t$: fixed ordering cost, ${\overline{d}}_t$ and ${\widehat{d}}_t$: nominal and maximum permissible deviations of uncertain demand ${\tilde{d}}_t$ in period t, respectively. $u_t,r_{i,t}$ are dual variables of problem (18).

A notable strength of model (19) is that its computational complexity is on par with its nominal version (i.e., the model is devoid of uncertainty ${\tilde{d}}_t={\overline{d}}_t$ $t\in \mathcal{T}$). Furthermore, this foundational model can be easily extended to accommodate both capacitated systems and supply chain contexts. Bertsimas and Thiele (2006) established that the optimal $(s,S)$ policy in the robust inventory formulation (19) aligns with the optimal $(s,S)$ policy for its corresponding nominal problem, but with a modified demand given by: $d_t^{\prime }={\overline{d}}_t+\frac{l-h}{l+h}(D_t-D_{t-1}),$. Here, $D_t=u_t^{*}{\Gamma }_t+{\sum }_{i=1}^t{r}_{i,t}^{*}$ represents the deviation of the cumulative demand from its nominal value $\overline{d}$ at time t, with $u^{*}$ and $r^{*}$ being the optimal u and r variables in equation (19).

The robust base-stock policy received further scrutiny in the work of Bienstock and Özbay (2008), where a decomposition approach was introduced to solve the true min-max problem inherent in the robust inventory formulation to optimality. In this study, a time-independent base-stock level is considered in the robust inventory model, which can be described as: $R^{*}={\min }_{S\ge 0}R(S)$ with(20) $$R(S)=\underset{\mathbf{x}}{\max }\sum _{t=1}^T(c_t{x}_t+\max \{h_t{I}_{t+1},-l_t{I}_{t+1}\})$$(20) $$\begin{array}{c}s.t.x_t=\max \{S-I_t,0\},\hspace{1em}{I}_{t+1}=I_t+x_t-{\tilde{d}}_t,\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}{\tilde{d}}_t\in \Omega ,\hspace{1em}t\in \mathcal{T},\hfill \end{array}$$

where S represents the optimal “order-up-to” level that the inventory manager must determine before demand is realized.

The problem $R(S)$ is termed the adversarial problem (Bienstock and Özbay, 2008), and was designed to compute the true worst-case cost. In contrast to the work by Bertsimas and Thiele (2006), which utilizes an epigraph reformulation for the holding/shortage cost term $\max \{h_t{I}_{t+1},-l_t{I}_{t+1}\}$ (refer to Equation (17)), this model directly tackles the term in its unaltered form. It is noteworthy that while epigraph reformulation offers computational tractability, it may yield more conservative solutions by relocating sum terms from the objective function to the constraints. Specifically, the worst-case values of $\max \{h_t{I}_{t+1},-l_t{I}_{t+1}\}$ for each period t can occur under different realizations of ${\tilde{d}}_t$ for $t\in \mathcal{T}$ (Bienstock and Özbay, 2008; Gorissen and Den Hertog, 2013).

Solving the adversarial problem (20), which includes non-convex terms such as $\max \{h_t{I}_{t+1},\backslash \backslash -l_t{I}_{t+1}\}$, presents significant computational challenges. To alleviate this, problem (20) can be initially reformulated as either a dynamic programming or a mixed-integer programming model, and subsequently solved using algorithms based on Benders’ decomposition. Moreover, the authors theoretically demonstrated that adversarial problem (20) can be solved within a finite number of iterations, contingent on the planning horizon T and the uncertainty budget, particularly when the uncertain demands ${\tilde{\mathbf{d}}}_t$ reside in ${\Omega }_{\text{BT}}$. Alternatively, near-optimal solutions can be obtained when ${\tilde{\mathbf{d}}}_t$ falls within a particular box uncertainty set that is characterized by sporadic demand peaks (e.g., periods where demand reaches a predetermined high value).

Diverging from the approaches of Bertsimas and Thiele (2006) and Bienstock and Özbay (2008), See and Sim (2010) examined the robust base-stock policy within a dynamic framework. The authors utilized historical demand data to refine the parameters of replenishment policies more precisely. Furthermore, their robust inventory model accommodates correlated demands, thereby relaxing the assumption of independently distributed demand prevalent in the works of Bertsimas and Thiele (2006) and Bienstock and Özbay (2008).

The random demand considered in See and Sim (2010) was developed around a factor-based model where the demand in each period is an affine function of zero mean random factors $\tilde{\zeta }\in {ℝ}^N$ as: ${\tilde{d}}_t(\tilde{\zeta })=d_{t,0}+{\sum }_{i=1}^N{d}_{t,i}{\tilde{\zeta }}_i,t=1,2,\dots ,T,$ where $d_{t,i}=0$ $\forall i\ge N_t+1$, and $1\le N_1\le N_2\le \dots \le N_T=N$.

In a factor-based demand model, random factors ${\tilde{\zeta }}_i$, $i=1,2,\dots ,N$, are realized sequentially over time. By time period t, the factors ${\tilde{\zeta }}_i$ for $i=1,2,\dots ,N_t$ have already been revealed. As the model progresses to period $t+1$, additional factors ${\tilde{\zeta }}_i$ for $i=N_t,N_t+1,\dots ,N_{t+1}$ become available. It is important to note that $N_t$ represents the cumulative number of factors affecting the random demand ${\tilde{d}}_t$, and this number is non-decreasing. This implies that ${\tilde{d}}_t$ is influenced by all previously realized random factors, capturing the demands correlations over time.

The factor-based demand model offers a versatile framework that can seamlessly integrate business factors, along with time-series forecasting elements, such as trend, seasonality, and cyclic variations. Notably, widely-used demand processes like Autoregressive Moving Average (ARMA) can be reformulated within the context of a factor-based demand model. Furthermore, the assumption of independently distributed demand, as seen in the works of Bertsimas and Thiele (2006) and Bienstock and Özbay (2008), can be considered a special case of the factor-based demand model, described as: ${\tilde{d}}_t(\tilde{\zeta })=d_{t,0}+{\tilde{\zeta }}_t,t=1,2,\dots ,T$, where ${\tilde{\zeta }}_t$ are independently distributed.

In an effort to bridge the gap between RO and Stochastic Optimization, the robust formulation presented in See and Sim (2010) aims to approximate the expected values in the objective function of the stochastic problem using effective upper bounds. The stochastic optimization model is abbreviated as follows:(21) $$\min \sum _{t=1}^T\left(\mathbb{E}\left[c_t{x}_t({\tilde{\mathbf{d}}}_{t-1})\right]+\mathbb{E}\left[h_t\max \left\{I_{t+1}({\tilde{\mathbf{d}}}_t),0\right\}\right]+\mathbb{E}\left[l_t\max \left\{-I_{t+1}({\tilde{\mathbf{d}}}_t),0\right\}\right]\right)$$(21) $$\begin{array}{c}s.t.I_{t+1}({\tilde{\mathbf{d}}}_t)=I_t({\tilde{\mathbf{d}}}_t)+x_t({\tilde{\mathbf{d}}}_{t-1})-{\tilde{d}}_t,\hspace{1em}0\le x_t({\tilde{\mathbf{d}}}_{t-1})\le V_t,\hspace{1em}t\in \mathcal{T},\hfill \end{array}$$

where $V_t$ denotes the ordering capacity, and $x_t({\tilde{\mathbf{d}}}_{t-1})$ is the ordering quantity, which is a function of the previous demand ${\tilde{\mathbf{d}}}_{t-1}$.

Due to the functional nature of the decision variables, $x_t({\tilde{\mathbf{d}}}_{t-1})$ and $I_t({\tilde{\mathbf{d}}}_t)$, problem (21) becomes an optimization problem with an infinite number of variables and constraints, rendering it generally intractable. To approximate the optimality of problem (21), See and Sim (2010) introduced a novel replenishment policy, called the truncated linear replenishment policy, which is inspired by the base-stock policy. It takes the form: $x_t^{\text{TLRP}}({\tilde{\mathbf{d}}}_{t-1})=\min \{\max \{q_{t,0}+{\mathbf{q}}_t^{\prime }\tilde{\zeta },0\},V_t\}$, where $q_{t,0}$ and ${\mathbf{q}}_t^{\prime }$ serve as parameters of the function mapping random factors ${\tilde{\zeta }}_t$ to the ordering quantity $x_t^{\text{TLRP}}({\tilde{\mathbf{d}}}_{t-1})$.

Note that $x_t^{\text{TLRP}}({\tilde{\mathbf{d}}}_{t-1})$ follows the same structure as the base-stock policy, as described as: $x_t({\tilde{\mathbf{d}}}_{t-1})=\min \{\max \{S_t-I_t({\tilde{\mathbf{d}}}_{t-1}),0\},V_t\}$ where $S_t$ are the base-stock levels.

Building on the upper bounds for $\mathbb{E}[{(·)}^{+}]$ established in earlier work Chen and Sim (2009), problem (21) can be approximated through a robust formulation incorporating the truncated linear replenishment policy. This approximation is formulated as a second-order cone programming problem, enabling efficient solutions. Furthermore, the authors demonstrated that this innovative policy surpasses linear and static decision rules, thereby significantly enhancing the approximation quality of the stochastic optimization problem (21).

In recent years, DP has been seamlessly integrated into robust optimization frameworks, offering valuable insights. Shapiro (2011) clearly outlines the natural formulation of DP equations for adjustable multi-stage robust optimization. In the realm of robust inventory management, Qiu et al. (2017) has formulated dynamic robust counterpart models for single-product, multi-period inventory management challenges, accommodating both box and ellipsoidal demand uncertainty. Furthermore, the corresponding robust $(s_t,S_t)$ policies have been derived.

Contrary to the aforementioned literature that treats demands as continuous random variables, Qiu et al. (2017) assume that the stochastic demand ${\tilde{d}}_t$ is discrete and belongs to a countable set of non-negative numbers, denoted as ${\tilde{d}}_t\in \{d_t^1,d_t^2,\dots ,d_t^{K_t}\}$. Here, $K_t$ is a positive integer, and $d_t^k$ for $k=1,2,\dots ,K_t$ represents a possible value of ${\tilde{d}}_t$, termed as a demand scenario. Moreover, the probability of each demand scenario $d_t^k$ is denoted by $P_t^k=\text{Pr}\{{\tilde{d}}_t=d_t^k\}$, and the probability vector is represented as ${\mathbf{P}}_t={(p_t^1,p_t^2,\dots ,p_t^{K_t})}^{\prime }$. It is important to note that ${\mathbf{P}}_t$ is not assumed to be explicitly specified; rather, it is assumed to belong to a predefined uncertainty set.

Subsequently, the inventory problem is formulated as a robust DP model, abbreviated as follows:(22) $$J_t(I_t)=\underset{x_t\ge 0}{\min }\left\{f_t{y}_t+c_t{x}_t+G_t{(I_{t+1})}^{\prime }{\mathbf{P}}_t+{\tilde{J}}_{t+1}{(I_{t+1})}^{\prime }{\mathbf{P}}_t\right\},\hspace{1em}t=1,2,\dots ,T,$$(22)

where $G_t(I_{t+1})$= $(G_t(I_{t+1},d_t^1),G_t(I_{t+1},d_t^2),\dots ,G_t(I_{t+1},d_t^{K_t}))$ and $G_t(I_{t+1},d_t^k)$ $k=1,2,\dots ,K_t$ is given by: $G_t(I_{t+1},d_t^k)=-r_t\min \{I_t+x_t,d_t^k\}+h_t\max \{x_t+I_t-d_t^k,0\}+l_t{h}_t\max \{d_t^k-x_t-I_t,0\},$ in which $r_t$ represents the unit selling price.

Similarly, the vector ${\tilde{J}}_{t+1}(I_{t+1})$ — defined as $(J_{t+1}(I_{t+1},d_t^1),J_{t+1}(I_{t+1},d_t^2),\dots ,J_{t+1}(I_{t+1},\mathrm{ }{d}_t^{K_t}))$ — represents the optimal expected costs spanning periods $t+1$ through T. Assuming that ${\mathbf{P}}_t$ falls within either a box or an ellipsoidal uncertainty set, the dynamic robust counterpart of problem (22) can be transformed into linear and second-order cone programs, respectively. As a result, robust $(s_t,S_t)$ policies can be efficiently obtained by solving these tractable models. Also, the optimality of the $(s_t,S_t)$ policy under demand distribution uncertainty has been confirmed by Qiu et al. (2017).

The necessity for complete knowledge of the underlying probability distribution, combined with the curse of dimensionality, significantly restricts the practical application of classical DP (Clark and Scarf, 1960; Federgruen and Zipkin, 1986; Rosling, 1989). Nevertheless, when augmented with the RO technique, the robust DP approach effectively addresses these limitations, enabling efficient multi-period decision-making. This enhancement broadens the utility of DP in tackling multi-period inventory issues under data uncertainty.

From our survey it is concluded that the dominant approach in the literature is to formulate the multi-period robust inventory model using a worst-case objective decision criterion. Yet, for decision-makers with specific requirements, other criteria might be more appropriate. For example, Lim and Wang (2017) proposed a TRO framework aimed at aiding inventory managers in maximizing the probability of meeting a set performance target. Additionally, a target-oriented base-stock policy was established in their study.

The probability of achieving a target is represented by the feasibility of the solutions, specifically through the adjustable level of demand variability. To accommodate this, an adjustable uncertainty set is defined as: ${\Omega }_t(\Gamma ):=\left\{{\tilde{d}}_t\right|{\overline{d}}_t-\Gamma {\widehat{d}}_t\le {\tilde{d}}_t\le {\overline{d}}_t+\Gamma {\widehat{d}}_t\}\hspace{1em}t\in \mathcal{T},$ where $\Gamma \in [0,1]$ is the uncertainty set parameter, dictating the permissible deviation of ${\tilde{d}}_t$ from its nominal value ${\overline{d}}_t$.

The TRO approach is designed to maximize the sizes of all adjustable uncertainty sets, ensuring that any demand realization within these sets will not exceed a pre-specified cost target, denoted as $\gamma$. Specifically, the TRO model can be succinctly represented as follows:(23) $$\begin{array}{cc}\mathcal{U}:=\hfill & \max \hspace{1em}\Gamma \hfill \end{array}$$(23) $$\begin{array}{cc}s.t.\sum _{t=1}^T(f_t{y}_t+c_t{x}_t({\mathbf{d}}_{t-1})+z_t)\le \gamma \hfill & {\tilde{d}}_t\in {\Omega }_t(\Gamma ),t\in \mathcal{T}\hfill \\ z_t\ge h_t(I_1+\sum _{i=1}^t{x}_i({\mathbf{d}}_{i-1})-\sum _{i=1}^t{\tilde{d}}_i)\hfill & {\tilde{d}}_t\in {\Omega }_t(\Gamma ),t\in \mathcal{T}\hfill \\ z_t\ge -l_t(I_1+\sum _{i=1}^t{x}_i({\mathbf{d}}_{i-1})-\sum _{i=1}^t{\tilde{d}}_i)\hfill & {\tilde{d}}_t\in {\Omega }_t(\Gamma ),t\in \mathcal{T}\hfill \\ 0\le x_t({\mathbf{d}}_{t-1})\le M{y}_t,\mathrm{ }{y}_t\in \{0,1\},\mathrm{ }0\le \Gamma \le 1\hfill & {\tilde{d}}_t\in {\Omega }_t(\Gamma ),t\in \mathcal{T}\hfill \end{array}$$

It is noteworthy that the authors have demonstrated that the static rule (introduced in Part A) $x_t({\mathbf{d}}_{t-1})=x_t$ is optimal for problem (23). Consequently, there is no necessity to consider more intricate decision rules, which significantly reduces the computational complexity. Furthermore, for the single product and single period case where $\tilde{d}\in [\overline{d}-\underset{¯}{d},\overline{d}+\overline{d}]$, a target-oriented base-stock policy was developed, represented as follows:(24) $$x^{*}=\max \left\{\overline{d}+{\Gamma }^{*}\frac{(l\overline{d}-h\underset{¯}{d})}{(l+b)}-\overline{d}+\frac{\gamma }{c},0\right\},$$(24)

where ${\Gamma }^{*}$ is the optimal solution of problem (23) when $T=1$.

Note that from equation (24), it is evident that the ordering quantity is directly proportional to the prespecified cost target $\gamma$. This observation is in alignment with the findings of Chen et al. (2015), which investigated the impact of a predefined performance target (i.e., cost) on newsvendor decisions.

Unlike the robust base-stock policies derived from models focusing on optimizing the worst-case objective, such as those presented in (Bertsimas and Thiele, 2006; Bienstock and Özbay, 2008; Qiu et al., 2017), the target-oriented base-stock policy (24) correlates the order-up-to level with the cost budget. This policy is particularly suitable for inventory managers subject to stringent budget constraints.

Remark 5 .

The robust $(s,S)$ policy significantly addresses the shortcomings of its counterpart derived from the classical DP approach (Clark and Scarf, 1960; Zipkin, 2000). Unlike the classical approach, it does not require exhaustive knowledge of the underlying distribution and often leads to more tractable models (Bertsimas and Thiele, 2006; Bienstock and Özbay, 2008). Furthermore, the integration of dynamic decision rules (See and Sim, 2010), the combination of DP with RO techniques Qiu et al. (2017), and the adoption of target-attainment decision criteria Lim and Wang (2017), have enriched the research on robust base-stock policies, enhancing both their applicability and performance.

$•$ Closed-Form Solutions Owing to the computational tractability of RO formulations, analytical solutions for the optimal outcomes of several fundamental multi-period inventory models, have been successfully derived.

Mamani et al. (2017) investigated the identical inventory model to Bertsimas and Thiele (2006), with the alteration that the demand is presumed to belong to a CLT uncertainty set and is correlated and non-identically distributed. Furthermore, the study explored both symmetric and asymmetric uncertainty set cases, revealing distinct optimal ordering strategies for each.

A class of partial-sum uncertainty sets was introduced in Mamani et al. (2017), defined as:(25) $${\Omega }^{\text{PS}}=\left\{({\tilde{d}}_1,{\tilde{d}}_2,\dots ,{\tilde{d}}_T):L_t\le \sum _{j=1}^t{\tilde{d}}_j\le U_t,{\underset{¯}{d}}_t\le {\tilde{d}}_t\le {\overline{d}}_t,t=1,2,\dots ,T\right\},$$(25)

where the parameters $(L_t,U_t,{\underset{¯}{d}}_t,{\overline{d}}_t)$ are exogenous, for all t, $L_t-U_{t-1}\le {\tilde{d}}_t={\sum }_{j=1}^t{\tilde{d}}_j-{\sum }_{j=1}^{t-1}{\tilde{d}}_j\le U_t-L_{t-1}.$ When ${\underset{¯}{d}}_t={\overline{d}}_t+{\widehat{d}}_t$, ${\overline{d}}_t={\overline{d}}_t+{\widehat{d}}_t$, $L_t=-{\widehat{d}}_t{\Gamma }_t+{\sum }_{j=1}^t{\overline{d}}_j$, and $U_t={\widehat{d}}_t{\Gamma }_t+{\sum }_{j=1}^t{\overline{d}}_j$, partial-sum uncertainty set (25) can be written as:(26) $${\Omega }^{{}^{\prime }}=\left\{({\tilde{d}}_1,{\tilde{d}}_2,\dots ,{\tilde{d}}_T):\left|\sum _{j=1}^t\frac{{\tilde{d}}_j-{\overline{d}}_j}{{\widehat{d}}_j}\right|\le {\Gamma }_t,{\overline{d}}_t-{\widehat{d}}_t\le {\tilde{d}}_t\le {\overline{d}}_t+{\widehat{d}}_t,t=1,2,\dots ,T\right\},$$(26)

Notably, when compared with the budget uncertainty set, ${\Omega }_{\text{BT}}$, considered in Bertsimas and Thiele (2006); Bienstock and Özbay (2008), equation (26) provides robust solutions with a higher level of protection against demand uncertainty. Clearly, given ${\overline{d}}_t$, ${\widehat{d}}_t$, and ${\Gamma }_t$ for $t=1,2,\dots ,T$, it holds that ${\Omega }_{\text{BT}}\subset {\Omega }^{{}^{\prime }}$ due to the following absolute value inequality: $\left|{\sum }_{j=1}^t\frac{{\tilde{d}}_j-{\overline{d}}_j}{{\widehat{d}}_j}\right|\le {\sum }_{j=1}^t\left|\frac{{\tilde{d}}_j-{\overline{d}}_j}{{\widehat{d}}_j}\right|.$

Furthermore, the $(L_t,U_t,{\underset{¯}{d}}_t,{\overline{d}}_t)$ parameterization offers added flexibility to cover asymmetric uncertainty sets. For instance, by setting ${\underset{¯}{d}}_t=0$ for all t, the uncertainty set ${\Omega }^{{}^{\prime }}$ becomes an asymmetric set in terms of the nominal demand ${\overline{d}}_t$.

Leveraging the structure of the uncertainty set ${\Omega }^{\text{PS}}$, one can derive the closed-form expressions for the minimum and maximum cumulative demands, ${\underset{¯}{D}}_t$ and ${\overline{D}}_t$, for the initial t periods where $t=1,2,\dots ,T$. Significantly, Mamani et al. (2017) demonstrated that, given any parameterizations $(L_t,U_t,{\underset{¯}{d}}_t,{\overline{d}}_t)$ of ${\Omega }^{\text{PS}}$, the cumulative orders ${\sum }_{j=1}^t{x}_j^{*}$ of the robust optimal ordering quantity $(x_1^{*},x_2^{*},\dots ,x_T^{*})$ adhere to a newsvendor-type optimality condition. This scenario is comparable to instances where the cumulative demand up to period t is uniformly distributed within the interval $[{\underset{¯}{D}}_t,{\overline{D}}_t]$. As a result, a recursion for computing the robust ordering quantities is provided as follows:(27) $$\begin{array}{c}{x}_t^{*}:=\{\begin{array}{cccc}\hfill & \frac{l{\overline{D}}_t+h{\underset{¯}{D}}_t}{l+h}-\sum _{j=1}^{t-1}{x}_j^{*}\hfill & \hfill & \mathrm{ }\text{for}\mathrm{ }t=1,\dots ,T-k,\hfill \\ \hfill & 0\hfill & \hfill & \mathrm{ }\text{for}\mathrm{ }t>T-k,\hfill \end{array}\hfill \end{array}$$(27)

where k is a integer that satisfies $kl<c\le (k+1)l$.

To derive a non-recursive, closed-form expression for the robust ordering quantity, the authors introduced a specific parameterization of $(L_t,U_t,{\underset{¯}{d}}_t,{\overline{d}}_t)$ inspired by the Central Limit Theorem. Assuming the known mean ${\mu }_t$, standard deviation ${\delta }_t$, and the covariance matrix $\mathbf{\Sigma }$ of the uncertain demands, the parameters are defined as: $L_t={\sum }_{j=1}^t{\mu }_j-{\Gamma }_t\sqrt{{\mathbf{e}}_t^{\prime }{\mathbf{\Sigma }}_t{\mathbf{e}}_t}$, $U_t={\sum }_{j=1}^t{\mu }_j+{\Gamma }_t\sqrt{{\mathbf{e}}_t^{\prime }{\mathbf{\Sigma }}_t{\mathbf{e}}_t}$ ${\underset{¯}{d}}_t=\max \{{\mu }_t-{\widehat{\Gamma }}_t{\delta }_t,0\}$, and ${\overline{d}}_t={\mu }_t+{\widehat{\Gamma }}_t{\delta }_t$. ${\mathbf{\Sigma }}_t$ is the submatrix composed of the first t rows and columns of $\mathbf{\Sigma }$ and ${\mathbf{e}}_t$ is a t-dimensional vector of all ones. Subsequently, the CLT uncertainty is defined as follows:$${\Omega }_{\text{CLT}}=\left\{({\tilde{d}}_1,{\tilde{d}}_2,\dots ,{\tilde{d}}_T):-{\Gamma }_t\le \left|\sum _{j=1}^t\frac{{\tilde{d}}_j-{\mu }_j}{\sqrt{{\mathbf{e}}_t^{\prime }{\mathbf{\Sigma }}_t}{\mathbf{e}}_t}\right|\le {\Gamma }_t,\max \{{\mu }_t-{\widehat{\Gamma }}_t{\delta }_t,0\}\le {\tilde{d}}_t\le {\mu }_t+{\widehat{\Gamma }}_t{\delta }_t,t=1,2,\dots ,T\right\},$$

where ${\Gamma }_t$ ${\widehat{\Gamma }}_t$ are tunable parameters that determine the demand variability.

Building upon the CLT uncertainty set, non-recursive closed-form ordering quantities are derived for both symmetric and asymmetric uncertainty sets, considering capacitated inventory constraints, in both static and dynamic settings.

Remark 6 .

Closed-form solutions enable decision-makers to comprehend the inventory management strategy more effortlessly and implement it more readily, eliminating the need for optimization software, in contrast to computational solutions, as exemplified by Ben-Tal et al. (2005); Bienstock and Özbay (2008); Qiu et al. (2022), among others.

$•$ Tractable Approximations and Reformulations Owing to potential stock surpluses or shortages from preceding periods and the challenges posed by the adversarial problem (Bienstock and Özbay, 2008), multi-period robust inventory management problems manifest greater complexity than their single-period newsvendor counterparts. The addition of setup/fixed costs, along with the associated binary variables determining order placements in specific periods, further complicates the resolution of these multi-period models. To address these intricacies, researchers have proposed tractable approximations and reformulations (Ardestani-Jaafari and Delage, 2016; Rodrigues et al., 2021).

Ardestani-Jaafari and Delage (2016) delved into the RO problems characterized by objective functions that consist of sums of piecewise linear functions over a polyhedral uncertainty set. Such problems are renowned for their intractability. The authors introduced innovative conservative approximations based on the relaxation of an embedded MILP. These were subsequently compared with ARO methods that utilize linear decision rules.

The robust problem examined in this study shares the same structure as the one investigated by Bienstock and Özbay (2008). The foundational formulation is presented as follows:(28) $$\underset{\mathbf{x}\in \mathbf{X}}{\min }\underset{\mathbf{d}\in \Omega }{\max }\sum _{i=1}^N{H}_i(\mathbf{x},\mathbf{d}),$$(28)

where $\mathbf{X}\in {ℝ}^n$ represents a bounded polyhedral set of feasible solutions for the decision $\mathbf{x}$, $\Omega \in {ℝ}^m$ denotes the set containing possible perturbations $\mathbf{d}$, and for each i, the cost function $H_i(\mathbf{x},\mathbf{d})$ is piecewise linear and convex with respect to both $\mathbf{x}$ and $\mathbf{d}$. This is analogous to the cost function of the aforementioned adversarial problem (20) .

Problem (28) is typically deemed intractable due to the computational challenges associated with solving the inner maximization problem (often referred to as the adversarial problem). While there are specific cases where an efficient solution method can be identified (Bienstock and Özbay, 2008), Gorissen and Den Hertog (2013) posits that, in general, this problem can be addressed by solving a MILP problem as follows, which is known to be NP-hard.(29) $$\begin{array}{cc}\underset{\mathbf{d}\in \Omega }{\max }\hfill & \mathcal{L}(\mathbf{d},\mathbf{x})+\sum _{i=1}^N{y}_i\hfill \end{array}$$(29) $$\begin{array}{cc}s.t.y_i\le {\mathcal{L}}_{i,k}(\mathbf{d},\mathbf{x})+M(1-s_{i,k}),\hfill & \forall i,\forall k,\hfill \\ \sum _{k=1}^K{s}_{i,k}=1,\hspace{1em}{s}_{i,k}\in \{0,1\},\hfill & \forall i,\forall k,\hfill \end{array}$$

where $\mathcal{L}$ and ${\mathcal{L}}_{i,k}$ are bilinear functions with respect to $\mathbf{d}$ and $\mathbf{x}$. Auxiliary variables $y_i$ are introduced to linearize the piecewise linear terms ${\max }_k\{{\mathcal{L}}_{i,k}(\mathbf{d},\mathbf{x})\}$ present in the objective function $H_i(\mathbf{x},\mathbf{d})$.

Expanding upon the findings of Gorissen and Den Hertog (2013), Ardestani-Jaafari and Delage (2016) proposed a novel linearization technique rooted in MILP to derive a conservative approximation of the adversarial problem: $\max {\sum }_{i=1}^N{H}_i(\mathbf{x},\mathbf{d}),\forall \mathbf{d}\in \Omega$. The uncertainty set $\Omega$ in this context is defined as a general polyhedral set, described as:$${\Omega }_{\text{Poly}}=\left\{\mathbf{d}\in {[-1,1]}^m|\mathbf{Ad}\le \mathbf{b},|\left|\mathbf{d}\right|{|}_1\le \mathbf{\Gamma }\right\}.$$

Note that the representation ${\Omega }_{\text{Poly}}$ simplifies to a budget uncertainty set, as utilized in Bertsimas and Thiele (2006); Bienstock and Özbay (2008), when $\mathbf{A}=\mathbf{0}$ and $\mathbf{b}=\mathbf{0}$. As a result, the adversarial problem considered in Ardestani-Jaafari and Delage (2016) is defined as:(30) $$\underset{\mathbf{d}\in {ℝ}^m}{\max }\sum _{i=1}^N\left\{\underset{k}{\max }{\mathbf{C}}_{i,k}^T\mathbf{d}+V_{i,k}\right\}$$(30) $$\begin{array}{c}s.t.\mathbf{Ad}\le \mathbf{b},\hspace{1em}\left|\right|\mathbf{d}|{|}_1\le \mathbf{\Gamma },\hspace{1em}|\left|\mathbf{d}\right|{|}_{\infty }\le 1,\hfill \end{array}$$

where ${\max }_k{\mathbf{C}}_{i,k}^T\mathbf{d}+V_{i,k}$ is a specific version of $H_i(\mathbf{x},\mathbf{d})$, and ${\max }_k{\mathbf{C}}_{i,k}^T\mathbf{d}$ shares a similar structure with ${\max }_k\{{\mathcal{L}}_{i,k}(\mathbf{d},\mathbf{x})\}.$ For clarity, equation (30) ignores the dependence of ${\mathbf{C}}_{i,k}$ and $V_{i,k}$ on $\mathbf{x}$.

In line with Gorissen and Den Hertog (2013), the authors devised an approximation scheme based on MILP to transform problem (30) into a solvable format. Their approach began with a conservative approximation, formulated as a linear program, by fractionally relaxing the integer variables inherent in the MILP representation of problem (30). Building on this, they further refined the approximation by establishing a mixed-integer semidefinite program (MISDP) that mirrors problem (30). This MISDP was subsequently converted into a semidefinite program via fractional relaxation.

The robust multi-period inventory management model explored by Ardestani-Jaafari and Delage (2016) can be viewed as a specific instance of problem (29), which is defined as follows:(31) $$\underset{\mathbf{x},\mathbf{y}}{\min }\underset{\mathbf{d}\in {\Omega }_{\text{Poly}}}{\max }\sum _{t=1}^T(f_t{y}_t+c_t{x}_t+\max \{h_t{I}_{t+1}(\mathbf{x},\mathbf{d}),-l_t{I}_{t+1}(\mathbf{x},\mathbf{d})\})$$(31) $$\begin{array}{cc}s.t.0\le x_t\le M{y}_t,\hspace{1em}{y}_t\in \{0,1\},\hfill & t\in \mathcal{T}\hfill \\ I_{t+1}(\mathbf{x},\mathbf{d})=I_1+\sum _{i=1}^t\left(x_i-{\tilde{d}}_i\right),\hfill & {\tilde{d}}_t\in {\Omega }_{\text{Poly}},\hspace{1em}t\in \mathcal{T}.\hfill \end{array}$$

Employing the MILP approximation method, problem (31) can be recast into its conservative approximation, either as a linear program or a semidefinite program, facilitating efficient solutions. Furthermore, when ${\Omega }_{\text{Poly}}$ is set to either $\Gamma =1$ or $\Gamma =T$, the approximation model corresponds precisely with problem (31). The authors also established that any robust multi-period inventory model utilizing LDR, as seen in Ben-Tal et al. (2005) and Qiu and Shang (2014), can be equivalently represented using this MILP approximation approach.

Recently, Rodrigues et al. (2021) provided a deeper insight into the adversarial problem and its implications for multi-period robust inventory management. They introduced a Lagrangian-Relaxation-Based method to build a novel conservative approximation of the adversarial problem. The inventory problem addressed in their work mirrors the form of problem (31), as discussed in Ardestani-Jaafari and Delage (2016). However, they assumed the uncertain demand falls into the budget uncertainty set, as described in Bertsimas and Thiele (2006); Bienstock and Özbay (2008). The adversarial problem in this study is given by:(32) $$\underset{\mathbf{I},\mathbf{u}}{\max }\sum _{t=1}^T(c_t{x}_t+\max \{h_t{I}_{t+1}(\mathbf{x},\mathbf{u}),-l_t{I}_{t+1}(\mathbf{x},\mathbf{u})\}\mathrm{)}$$(32) $$\begin{array}{cc}s.t.I_{t+1}(\mathbf{x},\mathbf{u})=I_1+\sum _{i=1}^t\left(x_i-{\overline{d}}_i-{\widehat{d}}_i\mathfrak{u}\right),\hfill & t\in \mathcal{T},\hfill \\ \sum _{i=1}^t\left|{\mathfrak{u}}_i\right|\le {\Gamma }_t,\hspace{1em}{\mathfrak{u}}_t\in [-1,1],\hfill & t\in \mathcal{T},\hfill \end{array}$$

where ${\mathfrak{u}}_t=\frac{{\tilde{d}}_t-{\overline{d}}_t}{{\widehat{d}}_t}$ represents the relative deviation of ${\tilde{d}}_t$ from its nominal value ${\overline{d}}_t$, with ${\tilde{d}}_t\in {\Omega }_{\text{BT}}$, $\mathbf{u}=({\mathfrak{u}}_1,{\mathfrak{u}}_2,\dots ,{\mathfrak{u}}_T)$.

The adversarial problem offers an advantage over the epigraph reformulation by avoiding scenarios where the worst-case values of $\max \{h_t{I}_{t+1},-l_t{I}_{t+1}\}$ occur under different realizations of ${\mathfrak{u}}_t$ for each period t. To address this, a Lagrangian relaxation approach is introduced, using multipliers to penalize varying realizations of ${\mathfrak{u}}_t$. This involves duplicating uncertainty variables and sets, making them independent per constraint, as explained in Section 4 of Part A (Ben-Tal et al., 2004; Marandi and Den Hertog, 2018). Uniformity constraints are then applied, and a Lagrangian dual model is derived, either as an MILP with setup costs or an LP without them. This approach aligns with the MILP approximation for robust inventory models using LDR, as demonstrated in Ardestani-Jaafari and Delage (2016).

The decomposition algorithms presented by Bienstock and Özbay (2008) excel in efficiently addressing the adversarial problem, albeit primarily in distinct scenarios. Conversely, the Lagrangian dual approach provides a versatile reformulation of the adversarial problem and demonstrates comparable efficacy to the decomposition algorithms. Notably, while the dualization method explored in Bertsimas and Thiele (2006) ensures tractability, it tends to yield more conservative outcomes than the Lagrangian dual model—a phenomenon theoretically validated by Rodrigues et al. (2021). Intriguingly, both the MILP relaxation approximation posited by Ardestani-Jaafari and Delage (2016), and the Lagrangian relaxation model, bear a connection to the LDR model. In essence, every robust multi-period inventory model employing LDR can be equivalently depicted using the MILP approximation. Furthermore, the Lagrangian relaxation approximation aligns with the LDR framework when all replicated uncertainty sets mirror the original set. Hence, under such circumstances, the Lagrangian relaxation model parallels the MILP model. Additionally, the efficacy of the MILP approximation can be enhanced by incorporating semidefinite constraints, leading to a more refined MISDP approximation. As a result, the MISDP model surpasses the Lagrangian relaxation model in performance under these conditions.

Remark 7 .

Solving the adversarial problem (20) optimally with decomposition algorithms (Bienstock and Özbay, 2008; Thorsen and Yao, 2017) can be impractical for many inventory challenges, and the dualization approach (Bertsimas and Thiele, 2006; Adida and Perakis, 2006) often results in overly conservative solutions. The MILP-based approximation (Ardestani-Jaafari and Delage, 2016) and the Lagrangian relaxation approximation (Rodrigues et al., 2021) provide viable alternatives, circumventing the adversarial problem’s complexities and the conservativeness of dualization. These methods balance solution quality with model tractability.

$•$ Time Consistency of Multi-period Models Recently, within the domain of multi-period distributionally robust inventory management, a subtle phenomenon, known as time inconsistency, has emerged. Notably, a replenishment policy considered optimal at the outset might lose its optimality when an inventory manager re-evaluates the policy at later stages. On the other hand, a policy that remains optimal across both formulations is termed as time consistent (Shapiro and Xin, 2020; Xin and Goldberg, 2021).

Xin and Goldberg (2021) explored a multi-period inventory control problem through a distributionally robust DP approach and provided formal definitions pertaining to time consistency. The distributionally DP equation is given by:(33) $$J_t(I_t)=\underset{x_t\ge I_t}{\inf }\left\{c_t(x_t-I_t)+\underset{{\mathcal{F}}_t\in {\mathcal{A}}_{\text{MB}}}{\sup }{\mathbb{E}}_{{\mathcal{F}}_t}\left[{\Psi }_t(x_t,{\tilde{d}}_t)+J_{t+1}(x_t-{\tilde{d}}_t)\right]\right\},\hspace{1em}t=1,2,\dots ,T,$$(33)

where ${\Psi }_t(x_t,{\tilde{d}}_t):=l_t\max \{d_t-x_t,0\}+h_t\max \{x_t-d_t,0\}$, $c_t$, $h_t$, and $l_t$ represents the unit ordering, holding, and backlogging costs for period t, respectively. The terms $I_t$ and $x_t$ denote the stock level at the onset of period t and the stock level post-ordering in period t, respectively, prior to the realization of demand ${\tilde{d}}_t$. The ordering quantity in period t is defined as $x_t-I_t$. Both $x_t$ and $I_t$ are recognized as the control variable and state variable, respectively, within the inventory control community.

The set ${\mathcal{A}}_{\text{MB}}$ corresponds to a moment-based ambiguity set, as detailed in Equation (5) of Part A. Given the boundary condition $J_{T+1}(·)=0$, the optimal value of DP formulation (34) is represented as $J_1(y_1)$. Additionally, dynamic equation (33) naturally defines a set of policies $x_t$ $t=1,2,\dots ,T$, being measurable selections $x_t\in {\mathfrak{V}}_t(I_t)$ from sets:(34) $${\mathfrak{V}}_t(I_t)=\arg \underset{x_t\ge I_t}{\min }\left\{c_t(x_t-I_t)+\underset{{\mathcal{F}}_t\in {\mathcal{A}}_{\text{MB}}}{\sup }{\mathbb{E}}_{{\mathcal{F}}_t}\left[{\Psi }_t(x_t,{\tilde{d}}_t)+J_{t+1}(x_t-{\tilde{d}}_t)\right]\right\},\hspace{1em}t=1,2,\dots ,T.$$(34)

Note that DP formulation (33) permits a decision-maker to re-evaluate the ordering policy at every stage t. Conversely, the multi-period inventory control problem, when subjected to demand ambiguity, can alternatively be framed using the classical DRO approach as follows:(35) $$\underset{\pi \in \Pi (I_1)}{\inf }\underset{\mathcal{F}\in {\mathcal{A}}_{\text{MB}}}{\sup }{\mathbb{E}}_{\mathcal{F}}\left[\sum _{t=1}^T\left(c_t\left(x_t({\mathbf{d}}_{t-1})-I_t({\mathbf{d}}_{t-1})\right)+{\Psi }_t\left(x_t({\mathbf{d}}_{t-1}),{\mathbf{d}}_t\right)\right)\right],$$(35)

where $\Pi (I_1)$ is the set of admissible policies.

In the implementation of model (34), the policy choice is not recalculated by the decision-maker. This characteristic differentiates it from model (33). As a result, model (35) is termed a ’multistage static formulation’, to distinguish it from the DP approach used in formulation (33). Moreover, OPT( $I_1$) denotes the set of optimal policies for problem (35).

Building upon the foundational formulations (33)-(35), the authors introduced the notion of time consistency as follows:

I. A policy $\pi \in \text{OPT}(I_1)$ is classified as type-I time consistent if it is robust with probability 1 (w.p.1) for equation (33). Specifically, for all $t\in [1,T]$, we have $x_t^{\pi }({\mathbf{d}}_{t-1})\in {\mathfrak{V}}_t(I_t({\mathbf{d}}_{t-1}))$. If at least one such policy $\pi$ exists, problem (33) is type-I weakly time consistent. If all policies $\pi \in \text{OPT}(I_1)$ meet this criterion, problem (33) is type-I strongly time consistent.

II. A policy $\pi \in \text{OPT}(I_1)$ is termed type-II time consistent if, for every $i\in [1,T-1]$ and $\mathcal{F}\in {\mathcal{A}}_{\text{MB}}$ w.p.1, there exists a policy ${\pi }^i\in {\text{OPT}}^i(I_{i+1}^{\pi }({\mathbf{d}}_i))$ such that $x_1^{{\pi }^i}=x_{i+1}^{\pi }({\mathbf{d}}_i)$. Here, ${\pi }^i$ and ${\text{OPT}}^i(I_{i+1}^{\pi }({\mathbf{d}}_i))$ represent the admissible policy and the set of optimal policies for a subproblem of (33) last spanning $(T-i)$ periods, respectively. The conditions for deeming problem (33) as weakly or strongly type-II time consistent are analogous to those in I.

Furthermore, the authors delineated specific conditions under which problem (33) exhibits either weak or strong time consistency. The concept of time consistency is expanded upon in a subsequent study by Xin and Goldberg (2022). In this work, a distributionally robust multistage newsvendor model akin to that in Xin and Goldberg (2021) is examined. However, the demand is modeled as a martingale distribution with a predefined mean and support, allowing past demand realizations to be inherently integrated into the future uncertainty set structure. This modeling assumption diverges from Xin and Goldberg (2021), where demand is independent across periods.

Xin and Goldberg (2022) demonstrated that the distributionally robust multistage newsvendor problem, subject to martingale demand with only known mean and support, allows for a DP formulation, thereby establishing its type-I time consistency (Xin and Goldberg, 2021). Moreover, the authors derived a closed-form optimal robust policy and its corresponding worst-case distribution from this DP formulation. A comparison was made between their model and a similar setting where demand is independent across periods. The comparative insights revealed nuanced differences: for specific initial stock levels $I_1$ (i.e., $I_1=0$), the optimal value of the model with independent demand surpasses that of one with martingale demand. However, under other $I_1$ scenarios, the opposite holds true.

The integration of DP into a classical RO framework has further advanced the concept of time consistency. Revisiting Qiu et al. (2017), an optimal robust base-stock policy for a multi-period inventory management problem is derived using DP (Equation 23), characterized as type-I consistent following Xin and Goldberg (2021). Bandi et al. (2019) expands this research by examining a newsvendor network with uncertain, correlated demand. They introduce time-consistent replenishment policies using RO, ensuring optimality over subperiods and the entire planning horizon. A novel class of adaptive, periodic-affine policies is proposed, dividing the time horizon into subperiods linked by preceding surplus ( $I_t$). Order quantities are determined as a linear function of demands within each subperiod, contrasting with the LDR (Ben-Tal et al., 2004) and PLDR (Chen et al., 2008) methods that consider all past demands. This policy design reduces decision variables and computational time.

Remark 8 .

Time consistency is of paramount importance from an implementability standpoint. A replenishment policy that is deemed optimal at time 0 may fail to maintain its optimality upon re-evaluation at a later time, thus, it may not be adopted by decision-makers (Xin and Goldberg, 2021). Integrating the DP formulation into the (distributionally) RO framework amalgamates the strengths of both DP and RO, positioning it as a potent tool for addressing multi-stage optimization challenges with parameter uncertainty (Bandi et al., 2019; Xin and Goldberg, 2022). Although the folding-horizon approach (Solyalı et al., 2016) can yield time consistent policies, it imposes a significant computational load. This is because the model must be solved sequentially for each period, while only the decisions pertaining to the current period are executed.

4 Solution Approaches and Numerical Results

In the preceding section, we delineated a comparative analysis of diverse robust formulations, emphasizing their modeling intricacies and analytical contributions. In this section, we present a comprehensive examination of the solution methodologies and numerical efficacies of these formulations, serving as a complementary extension to Section 3. In this paper, we synthesize computational results from existing literature, rather than conducting new computational experiments. While this approach allows us to draw on a broad range of established studies and aggregate insights across various scenarios and methodologies, it also comes with certain limitations:

• Inconsistency in Methodologies: The computational results we synthesize may come from studies with different assumptions, models, and experimental setups. This variability can lead to inconsistencies in the results and may affect the comparability of findings.

• Publication Bias: Our synthesis relies on published studies, which may be subject to publication bias. Studies with positive or significant results are more likely to be published, potentially skewing our synthesis towards more favorable outcomes.

• Lack of Control Over Data Quality: Since we are not conducting the experiments ourselves, we have limited control over the quality and rigor of the computational experiments conducted in the original studies. This reliance on secondary data means that any errors or biases present in the original studies could influence our conclusions.

Despite these limitations, we believe that synthesizing computational results from the literature provides valuable insights into the state-of-the-art in robust optimization and inventory management. It allows us to highlight trends, identify gaps, and propose directions for future research based on a comprehensive view of existing knowledge.

While RO is known for its straightforward resolution, its application in inventory challenges, particularly multi-item robust newsvendor problems with demand correlation and multi-period robust inventory models addressing adversarial problems, is often NP-hard (Hanasusanto et al., 2015a; Natarajan et al., 2018; Ardestani-Jaafari and Delage, 2016; Rodrigues et al., 2021). To address these complexities, efficient algorithms like decomposition (Bienstock and Özbay, 2008) and cutting plane methods (Gorissen and Den Hertog, 2013) have been developed, showing promising results, as discussed in previous sections.

We classify the formulations with commonalities into four distinct categories: single-item and multi-item newsvendor problems and multi-period models, both with and without the inclusion of setup costs. For each category, we provide a detailed comparative analysis by synthesizing the numerical findings from existing literature. This will provide a comprehensive overview of the performance and applicability of each formulation in real-world scenarios.

4.1 Robust Single-Item Newsvendor Formulations

We provide a comprehensive summary of the practical guidelines, key findings, solution methodologies, data sets, run times, and experiments associated with various robust single-item newsvendor formulations in Table 3 . To maintain clarity and consistency, we have harmonized the problem parameters from different references to correspond with formulations (1), (2), and (3). This standardization ensures that our comparisons are coherent and that the conclusions drawn are grounded in a consistent framework.

Table 3 offers a comparison of various formulations. We have the following observations:

I. Computational Tractability: Single-item robust newsvendor models, including both classic and distributionally robust variants, are often reformulable into convex optimization problems, enabling efficient resolution through optimization solvers. Computational times are typically short, usually just seconds, even for larger instances, as shown by Zhang et al. (2024). In multi-market scenarios, effective heuristics can be derived due to the problem’s inherent simplicity (Lin and Ng, 2011), suggesting that computational complexity is not a major hurdle for single-item newsvendor models.

II. Performance Comparisons: Excluding Rahimian et al. (2019) and Qiu et al. (2014), the other studies contrast their methodologies with alternative approaches in the literature through numerical experiments. The primary insights from these comparisons can be summarized as follows: 1. The Wasserstein metric effectively addresses the distributionally robust newsvendor problem, especially in cost minimization and on-target service levels (Lee et al., 2021; van der Laan et al., 2022). It outperforms the moment-based model (Delage and Ye, 2010) and the $\varphi -$ divergence model (Ben-Tal et al., 2013), with notable cost advantages over KL-divergence and ${\chi }^2$-distance models, especially as shortage cost l increases (Lee et al., 2021).

2. For heavy-tailed demand distributions, higher-order moment information in the DRO newsvendor model reduces costs, achieving profits up to 63.63% higher than Scarf (1958) for service levels $\eta \ge 0.97$ (Das et al., 2021).

3. The shapley policy outperforms other feature-based newsvendor models, including kernel weights optimization (Ban and Rudin, 2019), random forests and K-Nearest Neighbors (Bertsimas and Kallus, 2020), and stochastic optimization forests (SOF) (Kallus and Mao, 2023), particularly for smaller sample sizes and lower $\frac{h}{l}$ ratios.

4. The Min-Max regret model significantly reduces opportunity costs, achieving maximum regret reductions of up to $28.59\%$, when compared to Scarf’s model (Yue et al., 2006). In multi-market scenarios, it reduces maximum regret by $111.62\%$ and $19.98\%$ compared to deterministic and classic robust models, respectively (Lin and Ng, 2011).

III. Methodology Innovations: The introduction of the “Shapley Policy” by Zhang et al. (2024), which operates independently of specific parametric classes and guarantees a zero suboptimality gap, represents a significant advancement in ARO research. This development suggests a promising direction for future studies in this field. Additionally, Rahimian et al. (2019) employs the variation distance to define the size of the ambiguity set in DRO models, positioning them between classic RO and SP models. This variation distance, which quantifies robustness, is directly related to the decision-maker’s risk aversion.

4.2 Robust Multi-Item Newsvendor Formulations

This subsection examines robust optimization methods for multi-item newsvendor models. In Table 4, we critically evaluate relevant studies, focusing on their solution methodologies and numerical substantiation. For direct comparisons, we standardize parameters akin to the single-item case, introducing additional parameters N and r, representing the number of items and salvage value, respectively. This analysis highlights the computational viability of these methods and their effectiveness in optimizing costs and mitigating risks. Key findings are summarized as follows:

I. Computational Tractability: Accounting for demand correlations significantly intensifies the complexity of the distributionally robust multi-item newsvendor problem, elevating it to NP-hard status (Hanasusanto et al., 2015a; Natarajan et al., 2018; Zhang et al., 2021). To navigate this complexity, approximation and heuristic methods are employed to derive near-optimal solutions. Classical robust multi-item newsvendor models with demand correlations can be solved efficiently, albeit at the cost of potentially overconservative outcomes (Ng et al., 2012). Without demand correlations, the problem reduces to an aggregation of single-item scenarios, amenable to established solution techniques (Gallego and Moon, 1993). Notably, Wang et al. (2022) demonstrates that even within a distributionally robust framework, multi-item newsvendor models without demand correlation can be efficiently addressed.

II. Performance Comparisons: Similar to the single-item scenario, empirical comparisons of multi-item newsvendor models focus on contrasting various (distributionally) robust variants and benchmarking these against other stochastic optimization methods like SP and SAA. Key insights include:

1. Natarajan et al. (2018) shows that adding asymmetry information to a moment-based DRO model improves cost minimization by up to 11%, as supported by real-world data.

2. In cases of demand correlations, especially for multimodal demands, DRO with moment information generally outperforms the SP method based on a known demand distribution (Hanasusanto et al., 2015a). The advantage of DRO over SP grows with increasing probability ambiguity, ranging from a 20% reduction to a 120% enhancement in performance. DRO consistently outperforms SP in multimodal demand scenarios.

3. In the co-production newsvendor domain, the RO model shows between 12% and 38% higher no-shortage probability than the SAA model, with similar runtime. However, the probability maximization approach, while offering service levels comparable to RO, requires about 30.86 times longer runtime, as demonstrated in Ng et al. (2012).

III. Methodology Innovations: As Natarajan et al. (2018) reports, adding asymmetry measures like semivariance to the distributionally robust newsvendor model significantly reduces profit ambiguity and regret for heavy-tailed distributions, outperforming the moment-based model of Scarf (1958). This approach avoids the need for higher-order moments required by Das et al. (2021), providing a practical alternative for both single and multi-item scenarios. The ambiguity set in Hanasusanto et al. (2015a) includes a mix of ambiguous distributions, effectively addressing multimodal demand scenarios common in real-world situations, such as new product launches, significant customer acquisitions, market entries, and fashion trends, as discussed in their work.

4.3 Formulations without Setup Cost, Adversarial Problems

In this subsection, we delve into an in-depth analysis of multi-period robust inventory models. The computational complexity of these models primarily stems from the inclusion of binary variables for setup costs and the approach to tackling the adversarial problem, either through direct resolution or epigraph reformulation. Accordingly, we categorize relevant literature into two groups based on these computational aspects.

Table 5 presents comparative results for models excluding setup costs and adversarial challenges. For consistency, we have standardized parameters from various studies to align with formulation (25). Key insights are summarized as follows:

I. Computational Tractability: When setup costs and adversarial challenges are excluded, multi-period robust inventory models become more computationally tractable. Analytical characterization of these models facilitates deriving closed-form solutions (Mamani et al., 2017; Wagner, 2018) and establishing structured ordering policies (Bertsimas and Thiele, 2006). Additionally, the application of tailored decision rules and corresponding algorithms allows for resolving complex multi-installation inventory problems within seconds, even for planning horizons of up to 20 periods (Bandi and Bertsimas, 2012).

II. Performance Comparisons: Extensive numerical analyses in Bertsimas and Thiele (2006) and Mamani et al. (2017) evaluate the robust optimization (RO) methodology against alternative approaches and compare different robust methods. Key findings include:

1. RO solutions effectively mitigate the impact of demand distribution misspecification. Bertsimas and Thiele (2006) demonstrates that RO solutions can yield costs up to 13% lower than DP methods when the actual demand distribution deviates from the assumed model.

2. The service-level ratio $\frac{l}{l+h}$ significantly influences the comparative efficacy of various RO models. Mamani et al. (2017) show that their robust ordering policy outperforms the one from Bertsimas and Thiele (2006) when $\frac{l}{l+h}\ge 0.75$, with increasing superiority at higher service-level ratios. Conversely, the advantage of the proposed policy over Bertsimas et al. (2010) lessens with higher service levels and is surpassed by Bertsimas et al. (2010) when $\frac{l}{l+h}\ge 0.88$. These results suggest the optimal RO model choice depends on targeted service levels.

III. Methodology Innovations: The periodic-affine policy introduced in Bandi et al. (2019) is notable for its temporal consistency and computational efficiency. It outperforms the classical affine approximation ordering policy from Ben-Tal et al. (2004) in both performance and tractability. Specifically, the periodic-affine policy achieves an average worst-case profit increase of 14.52% for uncorrelated demands and 24.31% for correlated demands, compared to the affine approximation policy. In terms of computational tractability over 20 periods, the periodic-affine policy shows superior efficiency, completing in just 0.29 seconds, while the affine policy takes 1388.3 seconds.

4.4 Formulations with Setup Cost, Adversarial Problems

Table 6 presents a comparison of formulations incorporating setup costs and adversarial considerations, yielding the following key insights:

I. Computational Tractability: Adversarial considerations significantly increase the computational complexity of multi-period robust inventory models, often rendering them NP-hard due to their MILP nature. To address this, two main strategies are employed. Firstly, algorithmic approaches like Benders’ decomposition have proven effective, particularly with budget and CLT uncertainty sets (Bienstock and Özbay, 2008; Thorsen and Yao, 2017). These methods can solve instances with $T=50$ or $T=100$ within hundreds of seconds on average. Alternatively, model simplifications lead to more manageable formulations. MILP-based approximations (Ardestani-Jaafari and Delage, 2016) convert to LP or SDP, efficiently solvable by standard solvers. The Lagrangian dual approach (Rodrigues et al., 2021) offers a tractable approximation, solving cases with $T=30$ in under 100 seconds.

Setup costs and binary decision variables further complicate inventory management computations. While intermediate-sized instances with binary variables are solvable using optimization solvers (Lim and Wang, 2017; Qiu et al., 2017), larger instances become intractable. For example, Solyalı et al. (2016) reports that the robust model from Ben-Tal and Nemirovski (1998) took 4317.1s to optimally solve only 50% of $T=50$ cases. The model from Bertsimas and Thiele (2006) required 3913.1s for an equivalent percentage, while models from Ben-Tal et al. (2004) and See and Sim (2010) took 7200s without reaching an optimal solution. However, applying the facility-location reformulation technique, originally for lot-sizing problems, enabled efficient solutions of robust inventory models with binary variables, optimally solving all test instances in around 0.8s (Solyalı et al., 2016).

II. Performance Comparisons: In studies by Ardestani-Jaafari and Delage (2016) and Rodrigues et al. (2021), various methods, including approximation approaches and decomposition algorithms from Bienstock and Özbay (2008), are evaluated against robust models without adversarial considerations, like Bertsimas and Thiele (2006):

1.Ardestani-Jaafari and Delage (2016) show that their MILP-based approximation methods yield up to 17.39% lower worst-case cost than Bertsimas and Thiele (2006) for uncertainty budgets between 1 and 10. This is validated using both randomly generated instances and those from Bertsimas and Thiele (2006). With an uncertainty budget below 1.5, the approximation approach outperforms Bertsimas and Thiele (2006) by approximately 8% in average cost and 7.5% in value at risk at the 90% confidence level for $T=10$ scenarios. A similar advantage is observed for $T=100$ scenarios with budgets under 3.

2. Rodrigues et al. (2021) compare the Lagrangian dual method with Bienstock and Özbay (2008) and Bertsimas and Thiele (2006). For $T=30$ instances, the Lagrangian dual method consistently incurs lower costs than Bertsimas and Thiele (2006), with the cost benefit reaching up to 8.69% as setup costs increase from 0 to 150. However, the performance gap between the Lagrangian dual method and Bienstock and Özbay (2008) diminishes with higher setup costs, reducing to under 5%.

III. Methodology Innovations: Departing from traditional worst-case optimization, Lim and Wang (2017) introduces target-attainment decision-making into multi-period robust inventory management. Their TRO model is computationally efficient, allowing decision-makers to balance expected costs and cost variability by adjusting the cost target.

Addressing computational challenges with setup costs, Solyalı et al. (2016) developed a robust inventory formulation using techniques from lot-sizing, facility location, and shortest path problems. This method disaggregates the order quantity for each period, aligning it with specific period demands through targeted scheduling. This strategy shifts uncertainty from constraints to the objective function, resulting in a solvable model in polynomial time. Numerical results in Solyalı et al. (2016) show this model outperforms those in Bertsimas and Thiele (2006); Ben-Tal et al. (2004); Ben-Tal and Nemirovski (1998) in cost efficiency.

5 Future Research Opportunities and Challenges

Building upon this survey, we identify several potential avenues for future research in robust inventory management, listed as follows:

$•$ Representation of Uncertainty: Through this survey, we find that the CLT uncertainty set and the Wasserstein distance ambiguity set are particularly relevant for inventory management issues in classical RO and DRO approaches, respectively. Mamani et al. (2017) show the significant advantage of robust inventory models using the CLT uncertainty set over those with budget and box uncertainty sets (Bertsimas and Thiele, 2006; Bertsimas et al., 2010), achieving up to 50% and 9% lower simulated costs on average, respectively. Thorsen and Yao (2017) indicates reduced solution conservatism with the CLT uncertainty set, showing a decrease in worst-case cost of up to 10%. The CLT set has also been effectively applied to dual and multiple sources inventory issues (Sun and Van Mieghem, 2019; Xie et al., 2021).

Compared to other ambiguity sets in DRO, such as moment-based and $\varphi$-divergence-based sets, the Wasserstein set offers desirable properties, like finite sample guarantees, asymptotic consistency, and tractability (Mohajerin Esfahani and Kuhn, 2018), while mitigating drawbacks of these sets. DRO with moment-based sets can be overly conservative, while $\varphi$-divergence-based sets risk excluding certain distributions and overlook proximity within the support (Gao and Kleywegt, 2023). Lee et al. (2021) shows that DRO models using the Wasserstein set significantly outperform those with $\varphi$-divergence-based sets, with improvements up to 30% in both average and worst-case costs. Additionally, the Wasserstein set excels in distributionally robust chance-constrained optimization.van der Laan et al. (2022) reveals that DRO models with the Wasserstein set achieve higher service levels in a newsvendor problem, compared to models using mean and $\varphi$-divergence sets.

$•$ Decision Criteria: To date, worst-case objectives and risk measures have been the dominant decision criteria in robust inventory management for newsvendor and multi-period settings. However, the application of target-attainment and min-max regret criteria in these contexts is relatively limited. Target-attainment decision criteria play a crucial role in inventory and supply chain management (Chen et al., 2015; Chen and Tang, 2022). Despite valuable insights from Chen et al. (2015), aspects like practical target formation by inventory managers are yet to be fully explored. It’s essential to determine whether decision-makers can employ heuristic methods or must rely on intuition, a question of significance for both researchers and practitioners. In multi-period settings, Lim and Wang (2017) omit adjustable decision rules for simplicity. Although they demonstrate that a static rule can optimally approximate their TRO model, this approach is not universally applicable (Ben-Tal et al., 2004). Integrating more efficient decision rules into TRO could enhance both objective value and solution robustness.

The min-max regret decision criterion, common in newsvendor settings, is less explored in multi-period robust inventory problems. This criterion often leads to less conservative solutions than worst-case objectives, suggesting potential cost savings in multi-period scenarios. However, the complexity of optimal ordering decisions increases in multi-period inventory management due to carryovers like stock leftovers or shortages from previous periods (Qiu et al., 2017). Overcoming the computational challenges of implementing min-max regret in robust multi-period inventory models remains a significant hurdle.

$•$ Decision Rules: Various adjustable decision rules, such as LR and PLR, have been widely used in robust inventory settings, yet addressing adjustable integer variables remains challenging. In multi-period robust inventory models with setup costs, binary variables for ordering cannot be expressed using continuous rules like LR (Yanıkoğlu et al., 2019). Two main approaches have emerged in response: Firstly, sophisticated functionals adapt integer variables, including the sampling method Bertsimas and Caramanis (2007), piece-wise linear or constant decision rules Bertsimas and Georghiou (2018), and the finite adaptability method Bertsimas and Caramanis (2010); Hanasusanto et al. (2015b). These methods generally yield approximate solutions and are often limited to short planning horizons, like two stages Postek and Hertog (2016). Secondly, segmenting uncertainty into subsets linked to distinct decision variable types, both continuous and integer, as explored in Vayanos et al. (2011) and Postek and Hertog (2016). This approach addresses the complexity in robust inventory management but is confined to specific uncertainty set structures.

Efficiently adapting integer decision variables to realized uncertainties in diverse inventory settings remains a challenge. Recently, Lim et al. (2021) proposed a two-phase method for managing adjustable integer variables, using a static rule for product replenishment (binary variables) in the first phase and fixing these variables to determine continuous decisions via linear rule in the second phase. Additionally, Zhang et al. (2024) introduced a feature-based nonparametric decision rule with a zero suboptimality gap, offering promising directions for future research.

$•$ Combining with Dynamic Programming: The foundational work of Shapiro (2011) shows that DP equations can be effectively formulated for adjustable multi-stage RO, with applications in inventory management. Following this, significant research has explored integrating RO with DP. In DRO, time consistency has been extensively studied in Xin and Goldberg (2021), as applied to inventory management problems. In classic RO, Qiu et al. (2017) introduces a robust dynamic base-stock policy, and Bandi et al. (2019) develops a periodic-affine ordering policy based on DP equations.

While these developments are promising, they open many avenues for future research. The integration of RO with DP, especially in inventory management, presents opportunities for further exploration. Future studies could focus on refining dynamic policies to enhance their applicability and efficiency in diverse and complex inventory scenarios. Current research mainly addresses standard inventory settings like the newsvendor problem (Xin and Goldberg, 2021; Bandi et al., 2019). Future work could expand these methodologies to different inventory contexts, such as perishable goods, healthcare supplies, or high-tech products, where inventory dynamics vary significantly.

To address time consistency in inventory management, future research might consider segmenting the planning horizon into parts, ensuring time-consistent replenishment policies within these segments, rather than across the entire horizon as in Xin and Goldberg (2021). This approach would involve finite re-optimizations at each segment’s breakpoints, achieving an overall time-consistent policy for the entire horizon. Such a segmented strategy could offer a more flexible and practical solution for dynamic inventory management challenges.

$•$ Artificial Intelligence Integration: In the era of big data and Artificial Intelligence (AI), data-driven and AI-related techniques are significantly reshaping operations research and management. Traditional Assumption-driven models, based on specific assumptions about uncertainty/ambiguity sets and the probability distributions and correlations of uncertain parameters, may falter when these assumptions are unmet. In contrast, AI-driven models, with their precise specification of model parameters, adapt effectively across various conditions, gaining relevance in modern operations research and management science (Oroojlooyjadid et al., 2020; Liu et al., 2023; Li et al., 2023). AI techniques in stochastic optimization focus on estimating the conditional expected cost, with methods like Nadaraya-Watson kernel regression (Ban and Rudin, 2019; Srivastava et al., 2021), trees and forests (Bertsimas and Kallus, 2020; Kallus and Mao, 2023), robustness optimization and regularization (Zhu et al., 2022), and smart prediction-then-optimization strategies (Elmachtoub and Grigas, 2022).

In robust inventory management, Zhang et al. (2024) introduces a robust optimal ordering policy for a feature-based newsvendor problem using Lipschitz regularization, marking a significant advancement in applying AI to inventory issues. This approach opens opportunities for applications in more complex scenarios, like robust multi-period or broader inventory management settings. Additionally, the joint estimation and robustness optimization framework in Zhu et al. (2022) offers potential for enhancing inventory management decision-making with AI-driven insights.

6 Conclusion

Inventory management is crucial not only in its own right but also as a foundation for understanding complex supply chain planning challenges. RO has been widely applied in various inventory management problems, accommodating different sources of uncertainty and ambiguity with its flexible modeling frameworks. The field of robust inventory research has made significant strides in addressing model uncertainty, expanding uncertainty modeling, offering practical decision criteria, providing structural results and insights, and improving tractability and computational efficiency. The integration of RO with machine learning and other AI-based techniques is poised to further advance in robust inventory management. This combination can lead to new theories and applications, arising from the interplay between RO and DP, SP, and enhanced solutions’ quality through innovative approximation formulations or tailored efficient heuristics.

Data availability statement

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

Acknowledgments

The authors gratefully acknowledge Prof. Dr. Michael R. Wagner for his generous support and the Editor-in-Chief, Prof. Dr. Yu Ding, and anonymous referees for their contributions in improving the quality of the paper, which substantially contributed to the accomplishment of this work.

Table 1: Comparisons of Analytical Results of Representative Robust Newsvendor Formulations

References	Uncertainty Source	Support Set	Method	Decision Criteria	Model	Key Contribution
Wang et al. (2022)	Demand Volume	Moment-based Ambiguity set	DRO	Worst-case objective	Convex Quadratic Program	Derive a Closed-Form Solution for a Multi-Item, Capacity-Constrained Newsvendor Model
Rahimian et al. (2019)	Demand Volume	Variation-based Ambiguity set	DRO	CVaR Measurement	Minimizing a Convex Combination of CVaR	Derive Closed-Form Expressions for a Single-Item Newsvendor Model Based on Various Confidence Levels of CVaR for Cost
Lee et al. (2021)	Demand Volume	Wasserstein Ambiguity set	DRO	Worst-case objective and CVaR	Convex Optimization Model	Evaluate Risk-Neutral and Risk-Averse Decisions in a Single-Item Newsvendor Model to Derive Closed-Form Solutions
Qiu et al. (2014)	Probability of the Discrete Demand	Box and Ellipsoidal Uncertainty set	RO	Worst-case objective and CVaR	Second-order Cone Program, General Convex Optimization Model	Apply CVaR and Its Extensions to Address the Single-Item Newsvendor Problem Under Both Box and Ellipsoidal Uncertainty
Natarajan et al. (2018)	Demand Volumes and Correlation	Moment-based Ambiguity Set	DRO	Worst-case objective	Semi-definite Program	Derive a Closed-Form Solution for the Single-Item Newsvendor Model and Establishing a Lower Bound for the Multi-Item Newsvendor Model
Das et al. (2021)	Demand Volumes	Moment-based Ambiguity Set	DRO	Worst-case objective	1 Low Service-level: Convex Model 2 High Service-Level: Non-convex Model	Derive a Closed-Form Solution for a Single-Item Newsvendor Model with Low Service-Level; Establish Lower and Upper Bounds for Worst-Case Profit in High-Service Level Scenarios
Hanasusanto et al. (2015a)	Demand Volumes and Correlations	Moment-based Ambiguity Set	DRO	CVaR Measurement	Semi-definite Program of Exponential size (NP-hard)	Consider the Multi-Item Newsvendor Problem Under Multimodal Demand Ambiguity; Propose a DRO Model and Provide an Efficient Approximation for Solvability
Zhang et al. (2024)	Features that Influence Demand Volume	Wasserstein Ambiguity Set	ADRO	Worst-case Objective	Linear Program	Derive an Optimal Policy Mapping Features Related to Demand Ambiguity to Ordering Quantity in a Single-Item Newsvendor Model
Perakis and Roels (2008)	Demand Volume	Moment-based Ambiguity Set	DRO	Min-Max Regret	Semi-infinite Linear Optimization	Derive Various Types of Closed-Form Solutions for a Single-Item Newsvendor Model with Min-Max Regret Objective Based on Different Known Distribution Information
Zhang et al. (2021)	Demand Volumes and Substitution Rate	Budget Uncertainty Set	RO	Worst-case Objective	Mixed Integer Linear Program	Consider a Robust Multi-Item Newsvendor Problem with Substitution; Derive Closed-Form Solutions for Three Special Cases
van der Laan et al. (2022)	Features that Influence Demand Volume	Moment-based, $\varphi -$ divergence and Wasserstein Ambiguity Set	ADRO	Target-attainment	Mixed Integer Linear Program	Employing a distributionally robust chance-constrained optimization approach addresses the newsvendor problem adhering to a specified service-level target.
Ng et al. (2012)	Demand Volumes and Substitution Rates	General Convex Uncertainty Set with Known Means	RO	Target-attainment	Linear Program	Consider a Robust Co-production (Multi-Item) Newsvendor Problem with Downward Substitutions; Implement Target-attainment Criteria to Maximize the Demand Service Level
Yue et al. (2006)	Demand Volumes	Moment-based Ambiguity Set	DRO	Min-Max Regret	General Convex Model	Derive Closed-Form Expressions for Maximum Regret of Any Given Ordering Quantity in a Single-Item Newsvendor Model
Lin and Ng (2011)	Demand Volumes	Box Uncertainty Set	RO	Min-Max Regret	Mixed-integer Linear Program	Propose a Robust Minimax Regret Model to address a Single-item, Multi-market Newsvendor Problem

Table 2: Comparisons of Analytical Results of Representative multi-period Robust Inventory Formulations

References	Uncertainty	Support Set	Method & Rule	Decision Criteria	Model	Key Contribution
Lim and Wang (2017)	Demand Volume	Box Uncertainty Set	RO with Static Decision Rule	Target attainment	Linear Program	Target-Oriented Robust Optimization method for Multi-period Inventory Management; An approximation scheme to establish the optimality of static decision rules.
Bertsimas and Thiele (2006)	Demand Volume	Budget Uncertainty Set	RO with Static Decision Rule	Worst-case objective	Linear or Mixed Integer Linear Program depends on Fixed Cost.	Robust $(s,S)$ policies for both single and network inventory models, considering both capacitated and uncapacitated scenarios.
Mamani et al. (2017)	Demand Volume	CLT uncertainty set	RO with Static, Linear Decision Rules.	Worst-case objective	Linear Program	Closed-form ordering quantities under capacity constraints for both symmetric and asymmetric uncertainty sets, across static and dynamic settings
See and Sim (2010)	Features that Influence Demand Volume	Conic Quadrat ic Uncertainty Set	RO with Piece-wise Linear Decision Rule	Worst-case objective	Second-order Cone Program	A feature-based demand model incorporating trends and seasonality; Introduce an innovative replenishment policy to improve robust inventory formulations, advancing beyond static and linear decision rules.
Xin and Goldberg (2022)	Demand Volumes and their Correlations over time	Moment-based Ambiguity Set	DRO with DP formulation	Worst-case objective	Dynamic Programming	Propose a time-consistent, distributionally robust, multi-period newsvendor model with demand following a martingale process. Closed-form solutions for the min-max optimal ordering policy.
Xin and Goldberg (2021)	Demand Volume	Moment-based Ambiguity Set	DRO with DP Formultion	Worst-case objective	Dynamic Programming	Establish criteria for time consistency in multi-period distributionally robust inventory models. Identify sufficient conditions for both weak and strong time consistency.
Solyalı et al. (2016)	Demand Volumes	Budget Uncertainty Set	RO with Static Decision Rule	Worst-case objective	Mixed-integer Linear Program	Utilize Facility Location Reformulation techniques to make robust inventory models with fixed order costs solvable within polynomial time
Thorsen and Yao (2017)	Demand Volumes and Lead time	Budget and CLT Uncertainty Sets	RO with Static Decision Rule	Worst-case objective	Mixed-integer Linear Program	Explore a multi-period robust inventory model addressing uncertainties in demand and lead time. Consider the adversarial optimization problem and establish two optimal ordering policies: static and base-stock.
Wagner (2018)	Demand Rate Function	Uncertainty set based on strong law of large numbers	RO with Static and Dynamic Decision Rules	Worst-case Objective	Convex optimization model using Calculus	Consider a multi-period, infinite-dimensional robust inventory model with demand driven by a stochastic process. Offer closed-form solutions for the optimal ordering function in the static case and three dynamic variants.
Qiu et al. (2017)	Probability of the Discrete Demand	Box and Ellipsoidal Set	RO with DP formulation	Worst-case objective	Dynamic Program with Binary Variables	Formulate and solve a robust multi-period inventory model through dynamic programming. Derive the optimal $(s,S)$ ordering policy for the model.
Bienstock and Özbay (2008)	Demand Volumes	Box and Budget Uncertainty Sets	RO with static decision rule	Worst-case Objective	Nonconvex Optimization Model	Derive the optimal $(s,S)$ policy for robust multi-period inventory models framed as true min-max optimization problems, avoiding epigraph reformulation.
Bandi et al. (2019)	Demand Volumes and Correlations	Budget Uncertainty Set	RO with periodic linear decision rule	Worst-case objective	Dynamic Programming	Introduce a periodic-affine ordering policy for large-scale, multi-period robust newsvendor models. Show its time consistency, computational tractability, and ability to capture demand correlations across time and locations.
Rodrigues et al. (2021)	Demand Volumes	Budget Uncertainty Set	RO with static and quadratic decision rules	Worst-case objective	Mixed-integer Linear program	Develop a novel, tractable Lagrangian dual model for the inner maximization problem in robust min-max optimization, specifically designed for robust inventory issues.
Ardestani-Jaafari and Delage (2016)	Demand Volumes	Budget, Box and a $L_1$ norm ball Uncertainty Sets	RO with static and linear decision rules	Worst-case objective	A Linear Program or a Semidefinite Program	Introduce a tractable conservative approximation for adversarial problems, and apply it to robust multi-period inventory models.

Table 3: Robust Single-Item Newsvendor Formulations

References	Solution Approach	Data Set	Run Time	Experiment Objectives	Key Findings	Practical Implications
Rahimian et al. (2019)	Optimization Solver	$h=0.5,l=1.0,p=0$ ; Distribution: $\mathcal{F}:2.25+\hspace{0.17em}\log \hspace{0.17em}N(1.303,0.0922,0,10)$ .	Not Reported (Likely Rapid)	Assess the “maximal effective region” and evaluate performance differences between DRO, SP, and RO models.	1.Effective region lies between support set extremes. Ambiguity set size inversely affects effective region size. 2.Function differences vary based on ambiguity set size.	Adjust robustness and risk for optimal results
Lee et al. (2021)	Optimization Solver	$h=1,l=\{1,3,9,19\}$ ; $N\in \{50,500\}$ ; Normal dist. with $\mu =100$ and $\delta =20,40$	Not Reported (Likely Rapid)	Benchmark the Wasserstein model against other divergence models; compare with Scarf’s model; contrast worst-case and CVaR objectives.	1.Wasserstein outperforms KL-divergence and ${\chi }^2$ -distance. 2. Comparable to Scarf’s at $l=1$ ; superior as l increases. 3. CVaR in Wasserstein reduces worst-case cost by 25%..	Emphasize Wasserstein metric benefits for cost-minimization
van der Laan et al. (2022)	Optimization Solver	1.Synthetic Data from Beutel and Minner (2012) 2.Real Data from Fanaee-T and Gama (2014) .	Not Reported (Likely Rapid)	Compare DRO model using various ambiguity set against the Sample-Average Approximation (SAA), Scenario Approximation (SCA), and a stochastic model with known distribution.	Overall, the Wasserstein Distance DRO model demonstrates superior performance across the majority of test cases.	Emphasize Wasserstein metric benefits for ensuring service-levels
Qiu et al. (2014)	Optimization Solver	$p=10,h=2,l=1.5$ ; $d\in \{15,18,20,23,25,28,30,32,35\}$ ; Nominal dist.	Not Reported (Likely Rapid)	Evaluate models focusing on expected profit maximization and CVaR-based profit maximization.	1.Robust model close to predefined probability model. 2.CVaR-based model has larger order quantity. 3.Mean-CVaR model prioritizes higher CVaR.	Balance robustness and risk-aversion.
Das et al. (2021)	Optimization Solver	$\eta =1-c/p\in [0.97,0.9998]$ ; Demand: Exponential, Lognormal, Pareto dist.	Not Reported (Likely Rapid)	Assess DRO model performance using different order moment information; compare with traditional newsvendor model.	1.Higher-order moment information aligns robust order quantity with true optimal. 2.DRO outperforms classical model in contamination scenarios.	Implement in scenarios with heavy-tailed distributions.
Zhang et al. (2024)	Optimization Solver	1.Synthetic Data: $h\in \{0.2,0.5,1\},l=1$ 2.Real Data from Oroojlooyjadid et al. (2020)	1.Synthetic Data: 28.8658s 2.Real Data: 5.8135s	Benchmark the “Shapley Policy” against various methods using both synthetic and real data.	1. Shapley policy excels with small sample size and low holding costs. 2. Shapley policy superior in 8/9 real data cases.	Optimal Adjustable Robust Policy
Perakis and Roels (2008)	Closed-form Solution	$\beta =c/(p+l),\beta \in \{0.1,0.2,\dots ,1.0\}$ ; Seven various types of demand distribution.	Not Reported (Likely Rapid)	Comparison of min-max regret to the newsvendor model’s maximum regret across diverse dist.	Identify the apt probability distribution for the newsvendor model given certain demand distribution details (e.g., mean, range, variance).	Closed-form solution for Min-Max Regret
Yue et al. (2006)	Closed-form Solution	Experiment 1: $l/h$ ranges from 1 to 10. Experiment 2: l = 15.2, h = 10.1 $\mu =900,\delta =122$ .	Not Reported (Likely Rapid)	Comparison of Min-Max Regret model with Scarf’s and other models.	The Min-Max model outperforms others in terms of reduced maximal regret across different parameters.	Min-Max Regret based on Known mean and standard deviation.
Lin and Ng (2011)	Heuristics Bas- -ed on MILP and Local Search	$p\sim U[105,130]$ , $c=100,l=125$ , Markets:10,20,50, Entrance Cost $\sim U[8000,11000]$ ; Demands: various beta dist.	Not Reported (Likely Rapid)	Comparison of multi-merket newsvendor Min-Max Regret model with classic robust and deterministic models.	The Min-Max model surpasses the classic robust alternative by delivering superior mean and variance in simulated profits and exhibits a lower maximal regret compared to others.	Min-Max Regret for multi-market newsvendor.

Table 4: Robust Multi-Item Newsvendor Formulations

References	Solution Approach	Data Set	Run Time	Experiment Objectives	Key Findings	Practical Implications
Natarajan et al. (2018)	Optimization Solver	$N=36,h=l(1-\beta )/\beta ,l=1,\beta \in (0.8,1)$ , Demand moments from actual data	Not Reported (Likely Rapid)	Evaluate robust solutions under asymmetric demand, compare with symmetric models, and sample average method.	DRO model with partitioned stats reduced daily holding costs by 12-28% and shortage costs by 11.72-41.7% compared to other models.	Implement in asymmetric demand
Wang et al. (2022)	Optimization Solver	$N=3,c=[0.2,0.1,0.1],l=[1,2,3],\mu =[4,6,10],\delta =[2,2,3]$ ; Mutually independent nominal dist.	Not Reported (Likely Rapid)	Compare the Moment-based DRO model against stochastic optimization model; Explore the impact of cost and demand parameters.	1.The DRO model matches the stochastic model’s performance when the distribution is known. 2.Increased demand variability leads to higher production with abundant capacity and lower production with tight capacity.	Multi-item newsvendor capacity allocation
Hanasusanto et al. (2015a)	Optimization Solver	$N=3,p=10,c=5,r=1,l=2.5$ ; Mixture of two truncated normal dist.	1.Original Mod el: $\ge$ 6000s for N = 15 2.Partial expa nsion approximation: $\ge$ 6000s for N = 38 3.Quadratic rule approximation: $\le$ 12s for N = 50.	1.Explore the optimality gap and scalability of DRO model 2.Benchmark the DRO model against the stochastic optimization model assuming known demand distribution.	1.The optimality gap for partial expansion and quadratic rule approximation is under 5% and 7%, respectively. 2.The DRO model yields up to 120% more profit than the stochastic model.	Implement in Multimodal demand
Zhang et al. (2021)	1.Branch-and-Cut (BC) algorithm, 2.Conservative Approximation (CA) method	$N=\{10,20\},\overline{d}\sim U[50,100],p\sim U[85,95],c\sim U[40,50],r\sim U[22,30],$ . Substitution rate $\sim U[0,1]$ , Budget of uncertainty:[2,5,8,10]. Various normal dist is used for simulation.	$N=10$ , BC: $\le 17.7s$ , CA: $\le 22.26s$ (Gap $\le 5.29\%$ ); $N=20$ , time limit: 3600s, BC: gap $\le 6.41\%$ , CA: gap $\le 6.14\%$ .	1.Evaluate the performance of branch-and-cut and conservative approximation methods. 2.Investigate the impact of demand correlation on profit; contrast RO model with risk-neutral model	1.Both BC and CA solution approaches perform well for N = 10,20. 2.RO outperforms the risk-neutral model in all testing cases. 3.With a small uncertainty budget, profit falls with a rising correlation; this trend lessens when the budget matches the item count.	Implement in demand Substitution case.
Ng et al. (2012)	Optimization Solver	See Table 1 of the Reference	SAA Model: $\le$ 420s, Probability Maximization Model: $\le$ 14200s, TRO model: $\le$ 460s	Benchmark the TRO model against a SAA model and a probability optimization model.	The TRO model exhibits superior performance, compared to alternative models, both in terms of minimizing shortage points and maximizing the probability of avoiding shortages.	On target Service-levels for co-production newsvendor

Table 5: Multi-period Robust Inventory Formulations without Setup Cost and Adversarial Problems

References	Solution Approach	Data Set	Run Time	Experiment Objectives	Key Findings	Practical Implications
Bertsimas and Thiele (2006)	Optimization Solver	$T=20,c=1,h=4,l=6$ $\mu =100,\delta =20,\tilde{d}\in [60,140]$ . Realized Dist: Gamma, Lognormal, and Normal.	Not Reported Likely Rapid	Contrast the RO solution with the DP solution, which is predicated on an assumed distribution, by evaluating the simulated costs based on the realized distribution.	1.The RO solution surpasses the DP’s one by a margin of 10-13%, with its superiority amplifying alongside increasing demand variability. 2.For $h<2.5$ and $l=6$ , DP excels over RO, with relative performance unaffected by c . h influences the service level for both RO and DP.	Balance the robustness and objective value
Mamani et al. (2017)	Closed-form solutions	$T=\{3,10\},c=\{0.1,0.5,1,2\},h=1,l=\{3,5,20,40\}$ ; Multivariate normal dist. with $\mu =5$ , $\delta =\{0.5,1.5,2.5,4,5,7.5,10\}$	Not Reported Likely Rapid	Benchmark the Closed-Form (CF) RO solution against the robust solution of Bertsimas and Thiele (2006) (BT) and the affine policy from Bertsimas et al. (2010) (AP).	CF realizes cost savings exceeding 45% relative to BT, rising with higher $\frac{l}{l+h}$ . Against AP, CF’s cost savings surpass 51% where AP’s optimality is not satisfied, though the margin decreases as $\frac{l}{l+h}$ elevates.	Implement in asymmetric and correlated demand distribution
Wagner (2018)	Closed-form solutions	$T=400,\mu =1,\delta =1/3,h=1,l=2,c=20,\tilde{d}\in [0.5,1.5]$ Demand follows Brownian motion or a seasonal demand $d(t)=\mu +M·\hspace{0.17em}\sin \hspace{0.17em}(2\pi t/T)$ .	Not Reported Likely Rapid	1.Compare two dynamic ordering strategies: C, which utilizes realized demand, and S, which does not. 2.Study the effect of review frequency on the performance of order strategies.	1.Two strategies perform comparably under Brownian motion demand while C policy results in over 15% lower cost under seasonal demand. 2.Increasing review frequency does not necessarily improve robust ordering strategies.	Closed-form robust quantity from the optimal control perspective.
Bandi et al. (2019)	DP-based Heuristics, termed as Periodic Affine (PA) Algorithm	Performance comparisons using real data from a Indian pharmacy company; Computational tractability: synthetic data: $T\in 10,15,20$	PA: $\le$ 0.29s, Affine Policy: $\le 1388.3s$	1.Compare the PA solutions against classic affine (approx) and a myopic policy. 2.Examine the impact of holding and shortage costs on the performance of tested policies.	1.PA excels in the lower profit percentile and under high penalty costs. 2.As penalty costs escalate, PA sustains a reduced inventory compared to the affine approximation and matches its backlog mitigation.	Time-consistent computational tractable order policy

Table 6: Multi-period Robust Inventory Formulations with Setup Cost and Adversarial Problems

References	Solution Approach	Data Set	Run Time	Experiment Objectives	Key Findings	Practical Implications
Lim and Wang (2017)	Optimization Solver	$N=\{1,3,10\},T=\{1,5,10\},f=\{1,10\},c=\{1,2\},h=\{0.1,1,2,3,5\},l=\{2,3,5\}$ Demand: Various Beta Dist or discrete dist.	Not Reported (Likely Rapid)	Benchmark the TRO policy against a DP and a myopic policy assuming known demand distribution.	1.The TRO policy surpasses others across a broad spectrum of cost targets. 2.Adjusting the cost target allows the TRO policy to equilibrate expected costs with cost variance.	Target-oriented decision maker
Qiu et al. (2017)	DP	$p=20,c=10,h=2,l=15,f=100$ , 10 demand scenarios generated from U[100, 200] .	Not Reported (Likely Rapid)	1.Evaluate the effectiveness of the proposed robust DP approach. 2.Explore the impact of uncertainty level on the performance of robust DP.	Robust $(s,S)$ policy parameters closely align with nominal ones across stationary and non-stationary demands, yet heightened uncertainty magnifies the performance disparity.	Incorporating DP with RO
Solyalı et al. (2016)	Optimization Solver	$T=\{20,50\},c=1,f=1000,l=\{5,7,9\}$ and $h=4$ , or $h=\{5,7,9\}$ and $l=4$ . Demand: Uniform or Truncated Normal dist with various correlation levels.	$T=50:\le 0.8s$ , $\le 864.3$ with re-optimization.	Evaluate and rank various multi-period robust inventory management models’ performance.	Performance rank: 1. the cited work’s model with re-optimization, 2. See and Sim (2010) , 3. Ben-Tal et al. (2004) , 4. Bertsimas and Thiele (2006) , 5. the current model, 6. Ben-Tal and Nemirovski (1998) .	Polynomially solvable model incorporating setup costs.
Bienstock and Özbay (2008)	Decomposition Algorithm	1. Static problem: see Table 2 in the reference. 2. Basestock problem: refer to Tables 6 and 7.	Average: Static: $T=500:\le 50.28s$ Basestock: $T=150:\le 37.7s$	1.Assess the static and basestock policy performance in the context of the adversarial problem. 2.Examine the computational efficacy of the proposed decomposition.	The basestock policy exceeds the static policy by as much as 4396% in worst-case cost when the uncertainty set size increases.	Efficient algorithm for adversarial problem .
Thorsen and Yao (2017)	Bender’s Decomposition	Tractability Test: $T\in \{10,20,40\}$ , $c\sim U[0,10]$ $h=10,l\sim U[10,100]$ Monte Carlo: $h=5,l=10,c=1,T=10$ .	Average: Static: $T=40:\le 417.9s$ , Basestock: $T=40:\le 72.4s$	1.Examine the computational efficacy of the proposed decomposition. 2.Benchmark RO method versus SAA method.	RO outperforms SAA in terms of average, maximum, and cost standard deviation in Monte Carlo simulations.	Efficient algorithm for adversarial problem with additional lead time uncertainty
Rodrigues et al. (2021)	Heuristics based on model structure	$T=\{30,100\}$ , $c=1,h=4,l=6$ . $\overline{d}\sim U[0,50]$ , $\widehat{d}\sim U[\mathrm{0.0.2}\overline{d}]$ .	$T=30:$ the Lagrangian Dual model (LD) solves in $\le 80s$ , and with a relaxed uncertainty set (LDS), in $\le 40s$ . $T=100:$ Both LD , LDS $\ge 8h$ .	Contrast the LD model against those in Bertsimas and Thiele (2006) and Bienstock and Özbay (2008) regarding cost and computational time.	$T=30:$ Cost: 1. Bienstock and Özbay (2008) , 2.LD, 3. LDS, 4. Bertsimas and Thiele (2006) . Time: 1. Bertsimas and Thiele (2006) , 2. Bienstock and Özbay (2008) , 3.LDS, 4.LD. $T=100:$ LD-based heuristics offer upper bounds within 600s, Bienstock and Özbay (2008) requires over 8 hours to solve.	Lagrangian Dual approximation for adversarial problem.
Ardestani-Jaafari and Delage (2016)	Optimization Solver	1.Instances from Bertsimas and Thiele (2006) 2.Randomly generated instances: $T=\{10,100\}$ , $c,h,l\sim U[0,10]$ . $\overline{d}\sim U[0,100]$ , $\widehat{d}\sim U[0,\overline{d}]$ .	Not Reported (Likely Rapid)	Contrast MILP-based LP and SDP approximations against Bertsimas and Thiele (2006) and AARC from Gorissen and Den Hertog (2013) .	For Bertsimas and Thiele (2006) instances, the performance ranks: 1. SDP, 2. LP, 3. Gorissen and Den Hertog (2013) , 4. Bertsimas and Thiele (2006) . With $\Gamma \le 2$ , LP and SDP outperform others in random instances.	MILP-based approximation for adversarial problem

Additional information

Notes on contributors

Daoheng Zhang

Daoheng Zhang: Daoheng Zhang is currently a Ph.D. student in Computer Science at the School of Systems and Computing, UNSW Canberra, Australia. He received an MS degree in Management Science and Engineering from Nanjing University in 2017. His research areas are robust optimization and its application to supply chain management.

Hasan Hüseyin Turan

Hasan Hüseyin Turan: Hasan H. Turan is a Senior Lecturer and the Research Lead at Capability Systems Centre, University of New South Wales (UNSW Canberra). He obtained his Ph.D. and master’s degrees both in Industrial and Systems Engineering from Istanbul Technical University and North Carolina State University, respectively. Dr. Turan's research interests revolve around the development and application of data-driven optimization algorithms and simulation models arising in different domains including service and maintenance logistics, defense applications, energy capacity expansion, and telecommunications networks.

Ruhul Sarker

Ruhul Sarker: Ruhul Sarker received his Ph.D. in 1992 from Dalhousie University, Halifax, Canada. He is currently a Professor in the School of Systems and Computing, UNSW Canberra, Australia. His main research interests are Evolutionary Optimization, and Applied Operations Research. He is the lead author of the book Optimization Modelling: A Practical Approach.

Daryl Essam

Daryl Essam: Daryl Essam received his B.Sc. degree in computer science from University of New England, Australia in 1990 and Ph.D. degree from University of New South Wales, Australia, in 2000. Since 1994, he has been with the UNSW Canberra campus, where he is currently a Senior Lecturer and Deputy Head of School (Research) in the School of Systems and Computing. His research interests include genetic algorithms, with a focus on both evolutionary optimisation and large-scale problems.