## 6 Pricing Problems
### Discrete Power Control  
In the discrete power control case, we assume that, in each time slot, each network node can transmit with the transmission power level from a discrete set of available power levels that are denoted by \(\mathcal{P}\). Because the transmission power of each node is defined per time slot, we can embed the problem of power assignment into PP, which, in this case, takes the following form:  
\[\max\ \ P=-f+\sum_{s\in\mathcal{S}}\sum_{w\in\mathcal{V}}\sum_{v \in\delta^{+}(w)}\lambda_{s(w,v)}^{*}U_{swv} \tag{8a}\] \[X_{w}\geq Y_{wu},\quad w\in\mathcal{V},u\in\delta^{+}(w)\] (8b) \[X_{w}\leq\sum_{u\in\delta^{+}(w)}Y_{wu},\quad w\in\mathcal{V}\] (8c) \[X_{w}+\sum_{u\in\delta^{-}(w)}Y_{uw}\leq 1,\quad w\in \mathcal{V}\] (8d) \[\sum_{m\in\mathcal{M}}z_{w}^{m}=X_{w},\,w\in\mathcal{V}\] (8e) \[\sum_{p\in\mathcal{P}}h_{w}^{p}=X_{w},\,w\in\mathcal{V}\] (8f) \[\sum_{p\in\mathcal{P}}h_{w}^{p}\mathcal{P}(p)G(w,u)+M(w,u,m)(1-Y _{wu})\geq\] \[\gamma(m)z_{w}^{m}\eta+\gamma(m)\sum_{v\in\mathcal{V}\setminus\{ w,i\}}\sum_{p\in\mathcal{P}}\mathcal{P}(p)G(v,u)Z_{wv}^{mp},\quad(w,u)\in \mathcal{A},m\in\mathcal{M}\] (8g) \[Z_{wv}^{mp}\leq z_{w}^{m},\ Z_{wv}^{mp}\leq h_{v}^{p},\ Z_{wv}^{mp} \geq h_{v}^{p}+z_{w}^{m}-1,\quad m\in\mathcal{M},w\in\mathcal{V},v\in V \setminus\{w\},p\in\mathcal{P}\] (8h) \[U_{swv}\leq g_{sw},\ U_{swv}\leq Y_{wv},\quad s\in\mathcal{S},w\in \mathcal{V},v\in\delta^{+}(v)\] (8i) \[\sum_{s\in\mathcal{S}}g_{sw}\leq\sum_{m\in\mathcal{M}}B(m)F_{w}^{ m},\,w\in\mathcal{V};\ g_{sw}\leq 1,\,s\in\mathcal{S},w\in\mathcal{V}\] (8j) \[F_{w}^{m}\leq|\mathcal{S}|z_{w}^{m},\,F_{w}^{m}\leq f,F_{w}^{m} \geq f-|\mathcal{S}|(1-z_{w}^{m}),\quad m\in\mathcal{M},w\in\mathcal{V}\] (8k) \[Y_{wv}\in\mathbb{B},\,w\in\mathcal{V},v\in\delta^{+}(w);\ X_{w} \in\mathbb{B},\,w\in\mathcal{V}\] (8l) \[z_{w}^{m}\in\mathbb{B},\,m\in\mathcal{M},w\in\mathcal{V};\ h_{v}^ {p}\in\mathbb{B},\,p\in\mathcal{P},v\in\mathcal{V}\] (8m) \[f\in\mathbb{R};\ g_{sw}\in\mathbb{R}_{+},s\in\mathcal{S},w\in \mathcal{V}\ F_{w}^{m}\in\mathbb{R},\,m\in\mathcal{M},w\in\mathcal{V}\] (8n) \[Z_{wv}^{mp}\in\mathbb{B},\,m\in\mathcal{M},p\in\mathcal{P},w\in \mathcal{V},v\in V\setminus\{w\};\ U_{swv}\in\mathbb{R}_{+},s\in\mathcal{S},w \in\mathcal{V},v\in\delta^{+}(w). \tag{8o}\]  
As compared with formulation (7), now a new set of binary variables is used, namely \(h_{w}^{p},p\in\mathcal{P},w\in\mathcal{V}\), where \(h_{w}^{p}\) equals \(1\) if, and only if, node \(w\) transmits with the power level \(p\), and \(0\) otherwise. Subsequently, new constraint (8f) is introduced to ensure that each transmitting node is assigned exactly one transmitting power level from \(\mathcal{P}\). Finally, the SINR constraint is modified in the following way: \(p(w,v)\) is replaced by multiplying the power \(P(p)\) (corresponding to the power level \(p\) assigned to node \(w\) that is identified by variable \(h_{w}^{p}\)) by the path gain denoted by \(G(w,v)\). Finally, binary variables \(Z_{wv}^{mp}\in\mathbb{B},\,m\in\mathcal{M},p\in\mathcal{P},w\in\mathcal{V},v\in V \setminus\{w\}\), together with additional constraints (8h), are introduced to eliminate bi-linearities that otherwise occur because of the multiplication \(h_{v}^{p}z_{w}^{m}\).