1 Erratum to: Math Meth Oper Res (2014) 79:293–326 DOI 10.1007/s00186-014-0464-y

In the proof of Theorem 1 in the appendix of the original article, the validity of the optimality conditions

$$\begin{aligned} s_{i}^\mathrm{in}\left( \sum _{j:ji\in A(G)}\rho _{ji}-\sum _{\omega \in \varOmega }\sigma _{i}^{\omega }\right) =0 \end{aligned}$$
(114)

for all \((\sigma _{i}^{\omega })^\mathrm{SGSM}\) and \(\rho _{ji}^\mathrm{SGSM}\) is proved by scaling the optimality conditions

$$\begin{aligned} s_{i}^\mathrm{in}\left( \sum _{j:ji\in A(G)}\rho _{ji}-\sigma _{i}\right) =0 \end{aligned}$$
(89)

which are, by assumption, valid for \(\sigma _{i}^\mathrm{GSM}\) and \(\rho _{ji}^\mathrm{GSM}\), by the factor

$$\begin{aligned} \frac{\bar{\alpha }_{i}}{\alpha ^{*}_{i}\bar{n}}\quad \hbox { with } \quad \bar{n} := \sum _{\omega >\omega ^{*}}p^{\omega }>0\quad \hbox { and } \quad \bar{\alpha }_{i} := \sum _{\omega >\omega ^{*}}\alpha ^{\omega }_{i}p^{\omega }>0. \end{aligned}$$

Equation (114) emerges from (89) this way, in general, only if

$$\begin{aligned} \rho _{ji}^{\mathrm{SGSM}}=\frac{\bar{\alpha }_{i}}{\alpha ^{*}_{i}\bar{n}}\rho _{ji}^{\mathrm{GSM}} \qquad \forall ji \in A(G). \end{aligned}$$

The definition of \(\rho _{ji}^{\mathrm{SGSM}}\), however, in the original proof is, up to the reversal of \(i\) and \(j\),

$$\begin{aligned} \rho _{ji}^{\mathrm{SGSM}}:=\frac{\bar{\alpha }_{j}}{\alpha ^{*}_{j}\bar{n}} \rho _{ji}^{\mathrm{GSM}}\qquad \forall ji \in A(G). \end{aligned}$$
(130)

The theorem and the proof in the original article are therefore true, if \(\frac{\bar{\alpha }_{j}}{\alpha ^{*}_{j}\bar{n}}\) does not depend on the node \(j\). This holds, in particular, in the following cases:

  • There is only one node; this is, e.g., the case in the example of Theorem 2 so that the asymmetry between the sets of optimal solutions of the SGSM and the GSM is retained.

  • For each pair of nodes, the ratio between their demands is a constant independent of the scenario; this is, e.g., the case, if the demands for all nodes are identical in each demand scenario.

We conjecture that there are weaker assumptions for which Theorem 1 holds, but they require a modified proof. We apologize for our inadvertent omission.