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Errata to: D. Lozovanu and S. Pickl, Optimization of Stochastic Discrete Systems and Control on Complex Networks, Advances in Computational Management Science, DOI 10.1007/978-3-319-11833-8

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Page 124: Equation (2.27)

\( \left\{ \begin{gathered} h_{x} = \mathop {\hbox{min} }\limits_{y \in X\left( x \right)} h_{y} ,\quad \forall x \in X_{C} ; \hfill \\ h_{x} = \sum\limits_{y \in X\left( x \right)} {p_{x,y} h_{x} ,} \quad \forall x \in X_{N} . \hfill \\ \end{gathered} \right. \) (2.27)

\( \left\{ \begin{gathered} h_{x} = \mathop {\hbox{min} }\limits_{y \in X\left( x \right)} h_{y} ,\quad \forall x \in X_{C} ; \hfill \\ h_{x} = \sum\limits_{y \in X\left( x \right)} {p_{x,y} h_{y} ,} \quad \forall x \in X_{N} . \hfill \\ \end{gathered} \right. \) (2.27)

Page 164: Equation (2.75)

\( \omega_{x} = \mathop {\hbox{min} }\limits_{a \in A\left( x \right)} \left\{ {\sum\limits_{y \in X} {p_{x,y}^{a} \omega_{x} } } \right\},\quad \forall x\; \in \;X, \) (2.75)

\( \omega_{x} = \mathop {\hbox{min} }\limits_{a \in A\left( x \right)} \left\{ {\sum\limits_{y \in X} {p_{x,y}^{a} \omega_{y} } } \right.,\quad \forall x\; \in \;X, \) (2.75)

Page 165: Equation

\( {\begin{array}{*{20}c} {\varepsilon_{x} + \omega_{x} \le \mu_{x,a} + \sum\limits_{y \in X} {p_{x,y}^{a} \varepsilon_{y} ,} } & {\forall x\; \in \;X,\quad \forall a\; \in \;A\left( x \right);} \\ {\omega_{x} \le \sum\limits_{y \in X} {p_{x,y}^{a} \omega_{x} ,} } & {\forall x\; \in \;X,\quad \forall a\; \in \;A\left( x \right).} \\ \end{array} } \)

\( {\begin{array}{*{20}c} {\varepsilon_{x} + \omega_{x} \le \mu_{x,a} + \sum\limits_{y \in X} {p_{x,y}^{a} \varepsilon_{y} ,} } & {\forall x\; \in \;X,\quad \forall a\; \in \;A\left( x \right);} \\ {\omega_{x} \le \sum\limits_{y \in X} {p_{x,y}^{a} \omega_{y} ,} } & {\forall x\; \in \;X,\quad \forall a\; \in \;A\left( x \right).} \\ \end{array} } \)

\( \left\{ {\begin{array}{*{20}c} {\varepsilon_{x} + \omega_{x} \le \mu_{x,a} + \sum\limits_{y \in X} {p_{x,y}^{a} \varepsilon_{y} ,} } & {\forall x\; \in \;X,\quad \forall a\; \in \;A\left( x \right);} \\ {\omega_{x} \le \sum\limits_{y \in X} {p_{x,y}^{a} \omega_{x} ,} } & {\forall x\; \in \;X,\quad \forall a\; \in \;A\left( x \right).} \\ \end{array} } \right. \) (2.78)

\( \left\{ {\begin{array}{*{20}c} {\varepsilon_{x} + \omega_{x} \le \mu_{x,a} + \sum\limits_{y \in X} {p_{x,y}^{a} \varepsilon_{y} ,} } & {\forall x\; \in \;X,\quad \forall a\; \in \;A\left( x \right);} \\ {\omega_{x} \le \sum\limits_{y \in X} {p_{x,y}^{a} \omega_{y} ,} } & {\forall x\; \in \;X,\quad \forall a\; \in \;A\left( x \right)} \\ \end{array} } \right. \) (2.78)

\( \left\{ \begin{gathered} \varepsilon_{x} + \omega_{x} = \mu_{x,s\left( x \right)} + \sum\limits_{y \in X} {p_{x,y}^{s\left( x \right)} \varepsilon_{y} ,} \;\;\;\forall x\; \in \;X; \hfill \\ \omega_{x} = \sum\limits_{y \in X} {p_{x,y}^{s\left( x \right)} \omega_{x} ,} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall x\; \in \;X \hfill \\ \end{gathered} \right. \) (2.79)

\( \left\{ \begin{gathered} \varepsilon_{x} + \omega_{x} = \mu_{x,s\left( x \right)} + \sum\limits_{y \in X} {p_{x,y}^{s\left( x \right)} \varepsilon_{y} ,} \;\;\;\forall x\; \in \;X; \hfill \\ \omega_{x} = \sum\limits_{y \in X} {p_{x,y}^{s\left( x \right)} \omega_{y} ,} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall x\; \in \;X \hfill \\ \end{gathered} \right. \) (2.79)

Page 235: Equation

\( \left\{ {\begin{array}{*{20}c} {\omega_{x} = \mathop {\hbox{max} }\limits_{a \in A\left( x \right)} \left\{ {\sum\limits_{y \in X} {p_{x,y}^{a} \omega_{x} } } \right\},\quad \forall_{x} \; \in \;{\rm X}_{1} ;} \\ {\omega_{x} = \mathop {\hbox{min} }\limits_{a \in A\left( x \right)} \left\{ {\sum\limits_{y \in X} {p_{x,y}^{a} \omega_{x} } } \right\},\quad \forall_{x} \; \in \;{\rm X}_{2} ,} \\ \end{array} } \right. \) (3.18)

\( \left\{ {\begin{array}{*{20}c} {\omega_{x} = \mathop {\hbox{max} }\limits_{a \in A\left( x \right)} \left\{ {\sum\limits_{y \in X} {p_{x,y}^{a} \omega_{y} } } \right\},\quad \forall_{x} \; \in \;{\rm X}_{1} ;} \\ {\omega_{x} = \mathop {\hbox{min} }\limits_{a \in A\left( x \right)} \left\{ {\sum\limits_{y \in X} {p_{x,y}^{a} \omega_{y} } } \right\},\quad \forall_{x} \; \in \;{\rm X}_{2} ,} \\ \end{array} } \right. \) (3.18)

\(\begin{gathered} s^{1 * } \left( x \right) \in \left( {\mathop {\arg \hbox{max} }\limits_{a \in A\left( x \right)} \left\{ {\sum\limits_{y \in X} {p_{x,y}^{a} \omega_{x}^{ * } } } \right\}} \right)\bigcap {\left( {\mathop {\arg \hbox{max} }\limits_{a \in A\left( x \right)} \left\{ {\mu_{x,a} + \sum\limits_{y \in X} {p_{x,y}^{a} \varepsilon_{y}^{ * } } } \right\}} \right)} , \hfill \\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad \forall x \in X_{1} \hfill \\ \end{gathered} \) (3.20)

and

\(\begin{gathered} s^{2 * } \left( x \right) \in \left( {\mathop {\arg \hbox{min} }\limits_{a \in A\left( x \right)} \left\{ {\sum\limits_{y \in X} {p_{x,y}^{a} \omega_{x}^{ * } } } \right\}} \right)\bigcap {\left( {\mathop {\arg \hbox{max} }\limits_{a \in A\left( x \right)} \left\{ {\mu_{x,a} + \sum\limits_{y \in X} {p_{x,y}^{a} \varepsilon_{y}^{ * } } } \right\}} \right)} , \hfill \\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad \forall x \in X_{2} \hfill \\ \end{gathered}\) (3.21)

\(\begin{gathered} s^{1 * } \left( x \right) \in \left( {\mathop {\arg \hbox{max} }\limits_{a \in A\left( x \right)} \left\{ {\sum\limits_{y \in X} {p_{x,y}^{a} \omega_{y}^{ * } } } \right\}} \right)\bigcap {\left( {\mathop {\arg \hbox{min} }\limits_{a \in A\left( x \right)} \left\{ {\mu_{x,a} + \sum\limits_{y \in X} {p_{x,y}^{a} \varepsilon_{y}^{ * } } } \right\}} \right)} . \hfill \\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad \forall x \in X_{1} \hfill \\ \end{gathered} \) (3.20)

and

\(\begin{gathered} s^{2 * } \left( x \right) \in \left( {\mathop {\arg \hbox{min} }\limits_{a \in A\left( x \right)} \left\{ {\sum\limits_{y \in X} {p_{x,y}^{a} \omega_{y}^{ * } } } \right\}} \right)\bigcap {\left( {\mathop {\arg \hbox{min} }\limits_{a \in A\left( x \right)} \left\{ {\mu_{x,a} + \sum\limits_{y \in X} {p_{x,y}^{a} \varepsilon_{y}^{ * } } } \right\}} \right)} . \hfill \\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad \forall x \in X_{2} \hfill \\ \end{gathered}\) (3.21)

Page 236: Equation

\( \left\{ {\begin{array}{*{20}c} {\varepsilon_{x} + \omega_{x} \ge \mu_{x,a} + \sum\limits_{y \in X} {p_{x,y}^{a} \varepsilon_{y} ,} } & {\forall x \in X_{1} ,a = \overline{s}^{1} \left( x \right);} \\ {\varepsilon_{x} + \omega_{x} = \mu_{x,a} + \sum\limits_{y \in X} {p_{x,y}^{a} \varepsilon_{y} ,} } & {\forall x \in X_{2} ,a = \overline{s}^{2} \left( x \right);} \\ {\omega_{x} = \sum\limits_{y \in X} {p_{x,y}^{a} \omega_{x} ,} } & {\forall x \in X_{1} ,a = \overline{s}^{1} \left( x \right);} \\ {\omega_{x} = \sum\limits_{y \in X} {p_{x,y}^{a} \omega_{x} ,} } & {\forall x \in X_{2} ,a = \overline{s}^{2} \left( x \right)} \\ \end{array} } \right. \) (3.22)

\( \left\{ {\begin{array}{*{20}c} {\varepsilon_{x} + \omega_{x} = \mu_{x,a} + \sum\limits_{y \in X} {p_{x,y}^{a} \varepsilon_{y} ,} } & {\forall x \in X_{1} ,a = \overline{s}^{1} \left( x \right);} \\ {\varepsilon_{x} + \omega_{x} = \mu_{x,a} + \sum\limits_{y \in X} {p_{x,y}^{a} \varepsilon_{y} ,} } & {\forall x \in X_{2} ,a = \overline{s}^{2} \left( x \right);} \\ {\omega_{x}^{ * } = \sum\limits_{y \in X} {p_{x,y}^{a} \omega_{y} ,} } & {\forall x \in X_{1} ,a = \overline{s}^{1} \left( x \right);} \\ {\omega_{x}^{ * } = \sum\limits_{y \in X} {p_{x,y}^{a} \omega_{y} ,} } & {\forall x \in X_{2} ,a = \overline{s}^{2} \left( x \right)} \\ \end{array} } \right. \) (3.22)

\( \left\{ {\begin{array}{*{20}c} {\varepsilon_{x}^{ * } + \omega_{x}^{ * } \ge \mu_{x,a} + \sum\limits_{y \in X} {p_{x,y}^{a} \varepsilon_{y}^{ * } ,} } & {\forall x \in X_{1} ,a \in A\left( x \right);} \\ {\varepsilon_{x}^{ * } + \omega_{x}^{ * } = \mu_{x,a} + \sum\limits_{y \in X} {p_{x,y}^{a} \varepsilon_{y}^{ * } ,} } & {\forall x \in X_{2} ,a = \overline{s}^{2} \left( x \right);} \\ {\omega_{x}^{ * } \ge \sum\limits_{y \in X} {p_{x,y}^{a} \omega_{x}^{ * } ,} } & {\forall x \in X_{1} ,a \in A\left( x \right);} \\ {\omega_{x}^{ * } = \sum\limits_{y \in X} {p_{x,y}^{a} \omega_{x}^{ * } ,} } & {\forall x \in X_{2} ,a = \overline{s}^{2} \left( x \right)} \\ \end{array} } \right. \)

\( \left\{ {\begin{array}{*{20}c} {\varepsilon_{x}^{ * } + \omega_{x}^{ * } \ge \mu_{x,a} + \sum\limits_{y \in X} {p_{x,y}^{a} \varepsilon_{y}^{ * } ,} } & {\forall x \in X_{1} ,a \in A\left( x \right);} \\ {\varepsilon_{x}^{ * } + \omega_{x}^{ * } = \mu_{x,a} + \sum\limits_{y \in X} {p_{x,y}^{a} \varepsilon_{y}^{ * } ,} } & {\forall x \in X_{2} ,a = \overline{s}^{2} \left( x \right);} \\ {\omega_{x}^{ * } \ge \sum\limits_{y \in X} {p_{x,y}^{a} \omega_{y}^{ * } ,} } & {\forall x \in X_{1} ,a \in A\left( x \right);} \\ {\omega_{x}^{ * } = \sum\limits_{y \in X} {p_{x,y}^{a} \omega_{y}^{ * } ,} } & {\forall x \in X_{2} ,a = \overline{s}^{2} \left( x \right)} \\ \end{array} } \right. \)

\( \left\{ {\begin{array}{*{20}c} {\varepsilon_{x}^{ * } + \omega_{x}^{ * } = \mu_{x,a} + \sum\limits_{y \in X} {p_{x,y}^{a} \varepsilon_{y}^{ * } ,} } & {\forall x \in X_{1} ,a = \overline{s}^{1} \left( x \right);} \\ {\varepsilon_{x}^{ * } + \omega_{x}^{ * } \le \mu_{x,a} + \sum\limits_{y \in X} {p_{x,y}^{a} \varepsilon_{y}^{ * } ,} } & {\forall x \in X_{2} ,a \in A\left( x \right);} \\ {\omega_{x}^{ * } = \sum\limits_{y \in X} {p_{x,y}^{a} \omega_{x}^{ * } ,} } & {\forall x \in X_{1} ,a = \overline{s}^{1} \left( x \right);} \\ {\omega_{x}^{ * } \le \sum\limits_{y \in X} {p_{x,y}^{a} \omega_{x}^{ * } ,} } & {\forall x \in X_{2} ,a \in A\left( x \right)} \\ \end{array} } \right. \)

\( \left\{ {\begin{array}{*{20}c} {\varepsilon_{x}^{ * } + \omega_{x}^{ * } = \mu_{x,a} + \sum\limits_{y \in X} {p_{x,y}^{a} \varepsilon_{y}^{ * } ,} } & {\forall x \in X_{1} ,a = \overline{s}^{1} \left( x \right);} \\ {\varepsilon_{x}^{ * } + \omega_{x}^{ * } \le \mu_{x,a} + \sum\limits_{y \in X} {p_{x,y}^{a} \varepsilon_{y}^{ * } ,} } & {\forall x \in X_{2} ,a \in A\left( x \right);} \\ {\omega_{x}^{ * } = \sum\limits_{y \in X} {p_{x,y}^{a} \omega_{y}^{ * } ,} } & {\forall x \in X_{1} ,a = \overline{s}^{1} \left( x \right);} \\ {\omega_{x}^{ * } \le \sum\limits_{y \in X} {p_{x,y}^{a} \omega_{y}^{ * } ,} } & {\forall x \in X_{2} ,a \in A\left( x \right)} \\ \end{array} } \right. \)

Page 237: Equation

\( \left\{ {\begin{array}{*{20}c} {\varepsilon_{x} + \omega_{x} \ge \mu_{x,a} + \sum\limits_{y \in X} {p_{x,y}^{a} \varepsilon_{y} ,} } & {\forall x \in X_{1} ,a \in A\left( x \right);} \\ {\varepsilon_{x} + \omega_{x} \le \mu_{x,a} + \sum\limits_{y \in X} {p_{x,y}^{a} \varepsilon_{y} ,} } & {\forall x \in X_{2} ,a \in A\left( x \right);} \\ {\omega_{x} \ge \sum\limits_{y \in X} {p_{x,y}^{a} \omega_{x} ,} } & {\forall x \in X_{1} ,a \in A\left( x \right)\left( x \right);} \\ {\omega_{x} \le \sum\limits_{y \in X} {p_{x,y}^{a} \omega_{x} ,} } & {\forall x \in X_{2} ,a \in A\left( x \right)} \\ \end{array} } \right. \)

\( \left\{ {\begin{array}{*{20}c} {\varepsilon_{x} + \omega_{x} \ge \mu_{x,a} + \sum\limits_{y \in X} {p_{x,y}^{a} \varepsilon_{y} ,} } & {\forall x \in X_{1} ,a \in A\left( x \right);} \\ {\varepsilon_{x} + \omega_{x} \le \mu_{x,a} + \sum\limits_{y \in X} {p_{x,y}^{a} \varepsilon_{y} ,} } & {\forall x \in X_{2} ,a \in A\left( x \right);} \\ {\omega_{x} \ge \sum\limits_{y \in X} {p_{x,y}^{a} \omega_{y} ,} } & {\forall x \in X_{1} ,a \in A\left( x \right)\left( x \right);} \\ {\omega_{x} \le \sum\limits_{y \in X} {p_{x,y}^{a} \omega_{y} } ,} & {\forall x \in X_{2} ,a \in A\left( x \right)} \\ \end{array} } \right. \)

Page 238: Equation

\( \begin{aligned} & \quad \,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;s_{k}^{1} \left( x \right) \in \mathop {\arg \hbox{max} }\limits_{a \in A\left( x \right)} \left\{ {\sum\limits_{y \in X} {p_{x,y}^{a} } \omega_{x}^{{s_{k - 1}^{1} }} } \right\},\quad \forall x \in X_{1} ; \\ & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;s_{k}^{2} \left( x \right) \in \mathop {\arg \hbox{max} }\limits_{a \in A\left( x \right)} \left\{ {\sum\limits_{y \in X} {p_{x,y}^{a} } \omega_{x}^{{s_{k - 1}^{1} }} } \right\},\quad \forall x \in X_{2} \\ & {\text{and}}\;{\text{set}}\,s_{k} = s_{k - 1} \;{\text{if}} \\ & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;s_{k - 1}^{1} \left( x \right) \in \mathop {\arg \hbox{max} }\limits_{a \in A\left( x \right)} \left\{ {\sum\limits_{y \in X} {p_{x,y}^{a} } \omega_{x}^{{s_{k - 1}^{1} }} } \right\},\quad \forall x \in X_{1} ; \\ & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;s_{k - 1}^{2} \left( x \right) \in \mathop {\arg \hbox{max} }\limits_{a \in A\left( x \right)} \left\{ {\sum\limits_{y \in X} {p_{x,y}^{a} } \omega_{x}^{{s_{k - 1}^{1} }} } \right\},\quad \forall x \in X_{2} . \\ \end{aligned} \)

\( \begin{aligned} & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;s_{k}^{1} \left( x \right) \in \mathop {\arg \hbox{max} }\limits_{a \in A\left( x \right)} \left\{ {\sum\limits_{y \in X} {p_{x,y}^{a} } \omega_{y}^{{s_{k - 1}^{1} }} } \right\},\quad \forall x \in X_{1} ; \\ & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;s_{k}^{2} \left( x \right) \in \mathop {\arg \hbox{max} }\limits_{a \in A\left( x \right)} \left\{ {\sum\limits_{y \in X} {p_{x,y}^{a} } \omega_{y}^{{s_{k - 1}^{1} }} } \right\},\quad \forall x \in X_{2} \\ & {\text{and}}\;{\text{set}}\,s_{k} = s_{k - 1} \;{\text{if}} \\ & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;s_{k - 1}^{1} \left( x \right) \in \mathop {\arg \hbox{max} }\limits_{a \in A\left( x \right)} \left\{ {\sum\limits_{y \in X} {p_{x,y}^{a} } \omega_{y}^{{s_{k - 1}^{1} }} } \right\},\quad \forall x \in X_{1} ; \\ & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;s_{k - 1}^{2} \left( x \right) \in \mathop {\arg \hbox{max} }\limits_{a \in A\left( x \right)} \left\{ {\sum\limits_{y \in X} {p_{x,y}^{a} } \omega_{y}^{{s_{k - 1}^{1} }} } \right\},\quad \forall x \in X_{2} . \\ \end{aligned} \)