Distributionally robust multi-item newsvendor problems with multimodal demand distributions

Abstract

We present a risk-averse multi-dimensional newsvendor model for a class of products whose demands are strongly correlated and subject to fashion trends that are not fully understood at the time when orders are placed. The demand distribution is known to be multimodal in the sense that there are spatially separated clusters of probability mass but otherwise lacks a complete description. We assume that the newsvendor hedges against distributional ambiguity by minimizing the worst-case risk of the order portfolio over all distributions that are compatible with the given modality information. We demonstrate that the resulting distributionally robust optimization problem is $\mathrm{NP}$-hard but admits an efficient numerical solution in quadratic decision rules. This approximation is conservative and computationally tractable. Moreover, it achieves a high level of accuracy in numerical tests. We further demonstrate that disregarding ambiguity or multimodality can lead to unstable solutions that perform poorly in stress test experiments.

Appendix: Extremal distributions

In order to facilitate targeted stress tests for different order portfolios in Sect. 5, we describe here a method to construct extremal distributions for the worst-case CVaR problem (8). The general recipe is to first solve the SDP (18) and to substitute the resulting optimal $\beta ^*$ into (17). An extremal distribution that achieves the supremum in (8) can then be inferred from the dual solution of the SDP (17) by adapting a procedure for generic worst-case expectation problems with piecewise linear objective functions and without support information; see [6, 10]. We extend this procedure so that it can handle support information. Our construction of the extremal distribution will rely on the following corollary of [45, Section 2.1].

Proposition 6.1

Let $\Xi =\{\varvec{\xi }\in \mathbb R^n:[\varvec{\xi }^\intercal \;1]\mathbf W[\varvec{\xi }^\intercal \;1]^\intercal \le 0 \}$ be the ellipsoid specified by the matrix

$$\begin{aligned} \mathbf W= \left[ \begin{array}{c@{\quad }c} \varvec{\Lambda }^{-1} &{} -\varvec{\Lambda }^{-1}\varvec{\nu } \\ -(\varvec{\Lambda }^{-1}\varvec{\nu })^\intercal &{} \varvec{\nu }^\intercal \varvec{\Lambda }^{-1}\varvec{\nu }-\delta ^2 \end{array}\right] \end{aligned}$$

with $\varvec{\Lambda } \in \mathbb S^{n}$, $\varvec{\Lambda }\succ \varvec{0}$, $\varvec{\nu }\in \mathbb R^n$ and $\delta \in \mathbb R$. If

$$\begin{aligned} \left[ \begin{array}{l@{\quad }l} \mathbf Z &{} \varvec{z} \\ \varvec{z}^\intercal &{} 1 \end{array}\right] \succcurlyeq \varvec{0}\quad \text {and}\quad \left<\mathbf W, \left[ \begin{array}{l@{\quad }l} \mathbf Z &{} \varvec{z} \\ \varvec{z}^\intercal &{} 1 \end{array}\right] \right>\le 0\,, \end{aligned}$$

then one can construct a discrete distribution supported on $\Xi$ with mean $\varvec{z}$ and covariance matrix $\mathbf Z-\varvec{z}\varvec{z}^\intercal$.

Proof

This proposition strenghtens a result proved in [45, Section 2.1] in the special case when $\Xi$ is an ellipsoid. The covariance matrix of the discrete distribution constructed in [45] is only upper bounded (in a positive semidefinite sense) by but not necessarily equal to $\mathbf Z-\varvec{z}\varvec{z}^\intercal$.

By using an eigenvalue decomposition, we can factorize $\mathbf Z-\varvec{z}\varvec{z}^\intercal$ as $\sum _{i=1}^n\varvec{w}_i\varvec{w}_i^\intercal$. Thus, we have

$$\begin{aligned} \left\langle \mathbf {W}, \left[ \begin{array}{l@{\quad }l} \mathbf {Z} &{} \varvec{z}\\ \varvec{z}^\intercal &{} 1 \end{array}\right] \right\rangle \le 0 \!\!\iff \!\! \varvec{z}^\intercal \varvec{\Lambda }^{-1}\varvec{z} \!+\! \sum _{i=1}^n\varvec{w}_i^\intercal \varvec{\Lambda }^{-1}\varvec{w}_i \!-\! 2\varvec{\nu }\varvec{\Lambda }^{-1}\varvec{z} \!+\! \varvec{\nu }^\intercal \varvec{\Lambda }^{-1}\varvec{\nu } \!-\! \delta ^2 \le 0\,.\nonumber \\ \end{aligned}$$

(33)

Assume first that $\varvec{w}_i=\varvec{0}$ for all $i=1,\dots ,n$. Then, (33) implies that $\varvec{z}\in \Xi$. The Dirac distribution that concentrates unit mass at $\varvec{z}$ is thus supported on $\Xi$, and it has mean $\varvec{z}$ and covariance matrix $\mathbf Z-\varvec{z}\varvec{z}^\intercal =\varvec{0}$. Assume now that $\varvec{w}_{i}\ne \varvec{0}$ for all $i\in \mathcal {N}$, where $\mathcal {N}$ is a nonempty subset of $\{1,\dots ,n\}$. As ${\varvec{\Lambda }}^{-1}\succ \varvec{0}$, we have $\varvec{w}_i^\intercal {\varvec{\Lambda }}^{-1}\varvec{w}_i> 0$ if $i\in {\mathcal {N}}$; $=0$ otherwise. Next, we define the constants

$$\begin{aligned}&s =-(\varvec{z}^\intercal {\varvec{\Lambda }}^{-1}\varvec{z} + \sum _{i\in {\mathcal {N}}}\varvec{w}_i^\intercal {\varvec{\Lambda }}^{-1}\varvec{w}_i - 2\varvec{\nu }{\varvec{\Lambda }}^{-1}\varvec{z} + \varvec{\nu }^\intercal {\varvec{\Lambda }}^{-1}\varvec{\nu } - \delta ^2) \ge 0\,,\\&t =-\sum _{i\in {\mathcal {N}}}\varvec{w}_i^\intercal {\varvec{\Lambda }}^{-1}\varvec{w}_i < 0\,. \end{aligned}$$

For each $i\in {\mathcal {N}}$ we construct the following univariate quadratic function in $\omega$.

$$\begin{aligned} f_i(\omega )&= \left\langle \mathbf {W}, \left[ \begin{array}{c@{\quad }c} (\varvec{z}+\omega \varvec{w}_i)(\varvec{z}+\omega \varvec{w}_i)^\intercal &{} \varvec{z}+\omega \varvec{w}_i \\ (\varvec{z}+\omega \varvec{w}_i)^\intercal &{} 1 \end{array}\right] \right\rangle + s \\&= (\varvec{z} +\omega \varvec{w}_i)^\intercal {\varvec{\Lambda }}^{-1}(\varvec{z} +\omega \varvec{w}_i) - 2\varvec{\nu }^\intercal {\varvec{\Lambda }}^{-1}(\varvec{z}+\omega \varvec{w}_i) + \varvec{\nu }^{\intercal } {\varvec{\Lambda }}^{-1}\varvec{\nu } - \delta ^2 + s\\&= \varvec{w}_i^\intercal {\varvec{\Lambda }}^{-1}\varvec{w}_i\omega ^2+ 2\varvec{w}_i^\intercal {\varvec{\Lambda }}^{-1}(\varvec{z}\!-\!\varvec{\nu })\omega \!+\!\varvec{z}^\intercal {\varvec{\Lambda }}^{-1}\varvec{z} \!-\! 2{\varvec{\nu }}{\varvec{\Lambda }}^{-1}\varvec{z} \!+\!\varvec{\nu }^\intercal {\varvec{\Lambda }}^{-1}\varvec{\nu } - \delta ^2 + s\\&= \varvec{w}_i^\intercal {\varvec{\Lambda }}^{-1}\varvec{w}_i\omega ^2+ 2\varvec{w}_i^\intercal {\varvec{\Lambda }}^{-1}(\varvec{z}-\varvec{\nu })\omega +t \end{aligned}$$

Since $\varvec{w}_i^\intercal {\varvec{\Lambda }}^{-1}\varvec{w}_i>0$ and $t<0$, the quadratic function $f_i(\omega )$ has two real roots $\omega _i>0$ and $\omega _{i+n}<0$. By construction, as $f_i(\omega _i)=f_i(\omega _{i+n})=0$ and $s\ge 0$, the points $\varvec{z}+\omega _i\varvec{w}_i$ and $\varvec{z}+\omega _{i+n}\varvec{w}_i$ are contained in $\Xi$. This allows us to define a discrete probability distribution $\mathbb P$ supported on $\Xi$ with $2|{\mathcal {N}}|$ atoms at $\varvec{z}+\omega _i\varvec{w}_i$ and $\varvec{z}+\omega _{i+n}\varvec{w}_{i}$, $i\in {\mathcal {N}}$, and with

$$\begin{aligned} \left. \begin{array}{ll} &{}\mathbb P(\{\varvec{z}+\omega _i\varvec{w}_i\}) \!=\! \kappa _i \!=\! \dfrac{\varvec{w}_i^\intercal {\varvec{\Lambda }}^{-1}\varvec{w}_i}{(1-\omega _i/\omega _{i+n})\sum _{i'\in {\mathcal {N}}}\varvec{w}_{i'}^\intercal {\varvec{\Lambda }}^{-1}\varvec{w}_{i'}}\\ &{}\mathbb {P}(\{\varvec{z}+\omega _{i+n}\varvec{w}_{i}\})\!=\! \kappa _{i+n} \!=\!-\kappa _i\frac{\omega _i}{\omega _{i+n}}\!=\! \dfrac{\varvec{w}_i^\intercal {\varvec{\Lambda }}^{-1}\varvec{w}_i}{(1\!-\!\omega _{i+n}/\omega _{i})\sum _{i'\in {\mathcal {N}}}\varvec{w}_{i'}^\intercal {\varvec{\Lambda }}^{-1}\varvec{w}_{i'}} \end{array}\right\} ~\forall i\in {\mathcal {N}}. \end{aligned}$$

Since $\varvec{w}_i^\intercal {\varvec{\Lambda }}^{-1}\varvec{w}_i>0$ and $\omega _i/\omega _{i+n}<0$, we have $\kappa _i>0$ for all $i\in {\mathcal {N}}$. Next, observe that

$$\begin{aligned} \sum \limits _{i\in {\mathcal {N}}} \kappa _i+\kappa _{i+n}=\sum _{i\in {\mathcal {N}}}\frac{[(1-\omega _{i+n}/\omega _{i}) + (1-\omega _i/\omega _{i+n})]\varvec{w}_i^\intercal {\varvec{\Lambda }}^{-1}\varvec{w}_i}{(1-\omega _i/\omega _{i+n})(1-\omega _{i+n}/\omega _{i})\sum _{i'\in {\mathcal {N}}}\varvec{w}_{i'}^\intercal {\varvec{\Lambda }}^{-1}\varvec{w}_{i'}}=1\,. \end{aligned}$$

Thus $\mathbb {P}$ constitutes a valid discrete probability distribution. Furthermore, the first- and second-order moments of $\mathbb {P}$ can be expressed as

$$\begin{aligned} \mathbb {E}_\mathbb {P}[\tilde{\varvec{\xi }}]&= \sum _{i\in {\mathcal {N}}}\kappa _i(\varvec{z}\!+\!\omega _i\varvec{w}_i)+\kappa _{i+n}(\varvec{z}+\omega _{i+n}\varvec{w}_i)=\varvec{z} + \sum _{i\in {\mathcal {N}}}(\kappa _i\omega _i+\kappa _{i+n}\omega _{i+n})\varvec{w}_i\,,\\ \mathbb {E}_\mathbb {P}[\tilde{\varvec{\xi }}\tilde{\varvec{\xi }}^\intercal ]&= \varvec{z} \varvec{z}^\intercal \!+\!\sum _{i\in {\mathcal {N}}}(\kappa _i\omega _i\!+\!\kappa _{i+n}\omega _{i+n})(\varvec{z}\varvec{w}_i^\intercal \!+\!\varvec{w}_i\varvec{z}^\intercal ) \!+\!\sum _{i\!\in \!{\mathcal {N}}} (\kappa _i\omega _i^2\!+\!\kappa _{i+n}\omega _{i+n}^2)\varvec{w}_i\varvec{w}_i^\intercal . \end{aligned}$$

For all $i\in {\mathcal {N}}$ we have

$$\begin{aligned} \kappa _i\omega _i+\kappa _{i+n}\omega _{i+n}=\kappa _i\omega _i-\kappa _i(\omega _i/\omega _{i+n})\omega _{i+n}=0 \end{aligned}$$

and

$$\begin{aligned} \kappa _i\omega _i^2+\kappa _{i+n}\omega _{i+n}^2&= \frac{\varvec{w}_i^\intercal {\varvec{\Lambda }}^{-1} \varvec{w}_i}{\sum _{i'\in {\mathcal {N}}}\varvec{w}_{i'}^\intercal {\varvec{\Lambda }}^{-1} \varvec{w}_{i'}}\left( \frac{\omega _i^2\omega _{i+n}}{\omega _{i+n}-\omega _i}+\frac{\omega _i\omega _{i+n}^2}{\omega _{i}-\omega _{i+n}}\right) \\&= \frac{t}{\sum _{i'\in {\mathcal {N}}}\varvec{w}_{i'}^\intercal {\varvec{\Lambda }}^{-1}\varvec{w}_{i'}}\left( \frac{\omega _i}{\omega _{i+n}-\omega _i}-\frac{\omega _{i+n}}{\omega _{i+n}-\omega _{i}}\right) \\&= \frac{-t}{\sum _{i'\in {\mathcal {N}}}\varvec{w}_{i'}^\intercal {\varvec{\Lambda }}^{-1} \varvec{w}_{i'}}=1\,, \end{aligned}$$

where the second equality follows from the relation $\omega _i\omega _{i+n}=t/(\varvec{w}_i^\intercal {\varvec{\Lambda }}^{-1} \varvec{w}_i)$. Thus, $\mathbb P$ has all the desired properties, that is, $\mathbb E_\mathbb P[\tilde{\varvec{\xi }}]=\varvec{z}$, $\mathbb E_\mathbb P[\tilde{\varvec{\xi }}\tilde{\varvec{\xi }}^\intercal ]=\mathbf Z$ and $\mathbb P(\tilde{\varvec{\xi }}\in \Xi )=1$.

$\square$

To avoid clutter in the subsequent derivation, we introduce the notational shorthands

$$\begin{aligned} \varvec{w}_0 \!=\! \varvec{0},\quad w_0 = 0,\quad \varvec{w}_k \!=\! \varvec{b} - \mathcal {I}_k(\varvec{h}),\quad w_k \!=\! (\varvec{d} \!+\! \mathcal {I}_k(\varvec{h}))^\intercal \varvec{x} - \beta ^* \quad \forall k=1,\dots ,2^n\,. \end{aligned}$$

Using these definitions, we can replace $\max (0,L(\varvec{x}, \varvec{\xi }) - \beta ^*)$ with $\max _{k=0,\dots ,2^n} \varvec{w}_k^\intercal \varvec{\xi } + w_k$, which facilitates a streamlined reformulation of the SDP (17).

$$\begin{aligned} \begin{array}{l@{\quad }l} \inf &{} \displaystyle \sum \limits _{j=1}^mp_j\langle \varvec{\Omega }_j, \mathbf {M}_j \rangle \\ \mathrm{s.t.}&{} \mathbf M_j \in \mathbb {S}^{n+1},\quad \gamma _{{jk}} \in \mathbb {R}_+ \quad \forall j=1,\dots ,m,~k=0,\dots ,2^n \\ &{} \mathbf {M}_j + \gamma _{{jk}} \mathbf {W}_j \succcurlyeq \left[ \begin{array}{c@{\quad }c} \varvec{0} &{} \frac{1}{2}\varvec{w}_k \\ \frac{1}{2}\varvec{w}_k^\intercal &{} w_k \end{array}\right] \;~\begin{array}{l} \forall j=1,\dots ,m\\ \forall k=0,\dots ,2^n \end{array} \end{array} \end{aligned}$$

The dual of the above problem is given by

$$\begin{aligned} \begin{array}{l@{\quad }l} \sup &{} \displaystyle \sum \limits _{j=1}^m\displaystyle \sum \limits _{k=0}^{2^n} \varvec{w}_k^\intercal \varvec{z}_{jk} + w_kz_{jk} \\ \mathrm{s.t.}&{} z_{jk} \in \mathbb {R},\quad \varvec{z}_{jk} \in \mathbb {R}^n, \quad \mathbf {Z}_{jk} \in \mathbb {S}^n\quad \forall j=1,\dots ,m,~ k=0,\dots ,2^n \\ &{} \displaystyle \sum \limits _{k=0}^{2^n} \left[ \begin{array}{c@{\quad }c} \mathbf {Z}_{jk} &{} \varvec{z}_{jk} \\ \varvec{z}_{jk}^\intercal &{} z_{jk} \end{array}\right] = p_j{\varvec{\Omega }}_j \quad \forall j=1,\dots ,m\\ &{} \begin{array}{l@{\quad }l} \left\langle \mathbf {W}_j, \left[ \begin{array}{c@{\quad }c} \mathbf {Z}_{jk} &{} \varvec{z}_{jk} \\ \varvec{z}_{jk}^\intercal &{} z_{jk} \end{array}\right] \right\rangle \le 0,\quad \left[ \begin{array}{c@{\quad }c} \mathbf {Z}_{jk} &{} \varvec{z}_{jk} \\ \varvec{z}_{jk}^\intercal &{} z_{jk} \end{array}\right] \succcurlyeq \mathbf 0 \end{array} \quad \forall j=1,\dots ,m,~ k=0,\dots ,2^n \,. \end{array} \end{aligned}$$

(34)

Strong duality holds as the primal minimization problem is strictly feasible. This implies that the optimal value of (34) is equal to that of the worst-case expectation problem (9). We now denote by $\mathbf Z_{jk}^*$, $\varvec{z}_{jk}^*$, and $z_{jk}^{*}$ for $j=1,\ldots , m$ and $k=0,\ldots , 2^n$ an optimal solution of the SDP (34). Assume first that all $z_{jk}^{*}$ are strictly positive. For any fixed $j$ and $k$ we thus construct the positive semidefinite matrix

$$\begin{aligned} \left[ \begin{array}{c@{\quad }c}\mathbf Z_{jk}^*/z_{jk}^{*} &{} \varvec{z}_{jk}^*/z_{jk}^{*} \\ (\varvec{z}_{jk}^*/z_{jk}^{*})^\intercal &{} 1 \end{array}\right] \succcurlyeq \varvec{0}\,, \end{aligned}$$

which satisfies, together with $\mathbf W_j$, the conditions of Proposition 6.1. Hence, we can construct a discrete distribution $\mathbb P_{jk}^*$ supported in $\Xi _j$ with mean $\varvec{z}_{jk}^*/z_{jk}^{*}$ and covariance matrix $\mathbf Z_{jk}^*/z_{jk}^*-\varvec{z}_{jk}^*/z_{jk}^{*}(\varvec{z}_{jk}^*/z_{jk}^{*})^\intercal$. In the remainder we will argue that $\mathbb P^*=\sum _{j=1}^m\sum _{k=0}^{2^n}z_{jk}^{*} \mathbb P_{jk}^*$ is the sought extremal distribution.

From the equality constraints in (34) it is clear that $\mathbb P^*$ is feasible in (9). Thus,

$$\begin{aligned} \sum _{j=1}^m\sum _{k=0}^{2^n} \varvec{w}_k^\intercal \varvec{z}^*_{jk} + w_kz^*_{jk}&= \sup \limits _{\mathbb P\in {\mathcal {P}}}\mathbb E_{\mathbb P}\left[ \max _{k=0,\dots ,2^n} \varvec{w}_k^\intercal \tilde{\varvec{\xi }} + w_k\right] \\&\ge \mathbb {E}_{\mathbb {P}^*}\left[ \max _{k=0,\dots ,2^n} \varvec{w}_k^\intercal \tilde{\varvec{\xi }} + w_k\right] \\&= \sum \limits _{j=1}^m\sum \limits _{\ell =0}^{2^n} z^*_{j\ell }\mathbb E_{\mathbb P^*_{j\ell }}\left[ \max _{k=0,\dots ,2^n} \varvec{w}_k^\intercal \tilde{\varvec{\xi }} + w_k\right] \\&\ge \sum \limits _{j=1}^m\sum \limits _{k=0}^{2^n} z^*_{jk} \mathbb E_{\mathbb P^*_{jk}}[ \varvec{w}_k^\intercal \tilde{\varvec{\xi }} + w_k] =\sum \limits _{j=1}^m\sum \limits _{k=0}^{2^n} \varvec{w}_k^\intercal \varvec{z}^*_{jk} + w_kz^*_{jk}, \end{aligned}$$

where the first equality follows from strong duality. As all inequalities in the above expression must be binding, $\mathbb P^*$ is indeed extremal in (9).

Assume now that $z_{jk}^{*}=0$ for some $j$ and $k$. The positive semidefiniteness constraint in (34) then implies that $\varvec{z}_{jk}^*=\varvec{0}$, while the trace constraint implies $\langle {\varvec{\Lambda }}_j^{-1},\mathbf {Z}_{jk}\rangle \le 0$, see also the definition of $\mathbf W_j$ in (13). Our assumption ${\varvec{\Lambda }}_j^{-1}\succ \varvec{0}$ thus implies that $\mathbf Z_{jk}^*=\varvec{0}$. This argument suggests that all pairs of indices $j$ and $k$ for which $z_{jk}^{*}=0$ can be ignored altogether. In general, an extremal distribution for the worst-case CVaR problem (8) can thus be constructed as

$$\begin{aligned} \mathbb P^*=\sum \limits _{j=1}^m\sum \limits _{\mathop {z_{jk}^{*} \ne 0}\limits ^{k=0}}^{2^n}z_{jk}^{*} \mathbb P_{jk}^*\,. \end{aligned}$$