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Nt1310 Unit 4 Assignment 1

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mu_I} (R_I(\widetilde{S}_I) - T_I^*) \ge V_I(S_I^*)^{{1/\mu_I}} (R_I(S_I^*) - T_I^*) \end{equation*} Since $R_I(S_I^*) \ge T_H^*$, according to Equation (\ref{Equ: DefTi}), we have: $T_I^* = \mu_I T_H^* +(1-\mu_I) R_I(S_I^*)$. After plugging $T_I^*$ in the above inequality, it follows that: \begin{eqnarray}\label{Equ: l2Equ3} &V_I(\widetilde{S}_I) ^{1/\mu_I}& (R_I(\widetilde{S}_I) - \mu_I T_H^* - (1-\mu_I) R_I(S_I^*)) \ge V_I(S_I^*) ^{{1/\mu_I}} (R_I(S_I^*) - \mu_I T_H^* -(1-\mu_I) R_I(S_I^*)) \nonumber \\ \Rightarrow &V_I(\widetilde{S}_I) ^{1/\mu_I}& (R_I(\widetilde{S}_I) -T_H^*) \ge (\mu_I V_I(S_I^*)^{{1/\mu_I}}+ (1-\mu_I) V_I(\widetilde{S}_I)^{1/\mu_I}) (R_I(S_I^*) - T_H^*) \end{eqnarray} Multiply inequality (\ref{Equ: l2Equ3}) by $V_I(\widetilde{S}_I)^{1 …show more content…

Let $x = V_I(S_I^*)^{{1/\mu_I}}$ and $\widetilde{x} = V_I(\widetilde{S}_I)^{1/\mu_I}$, then we get: \begin{equation}\label{Equ: l2Equ5} V_I(\widetilde{S}_I)^{1-1/\mu_I} (\mu_I V_I(S_I^*)^{{1/\mu_I}} +(1-\mu_I) V_I(\widetilde{S}_I)^{1/\mu_I}) \ge V_I(S_I^*) \end{equation} Since $R_I(S_I^*) > T_H^*$, multiply inequality (\ref{Equ: l2Equ5}) by $R_I(S_I^*) - T_H^*$, we have: \begin{equation}\label{Equ: l2Equ6} V_I(\widetilde{S}_I)^{1-1/\mu_I} (\mu_I V_I(S_I^*)^{{1/\mu_I}} +(1-\mu_I)V^{1/\mu_I}) (R_I(S_I^*) - T_H^*) \ge V_I(S_I^*)(R_I(S_I^*) - T_H^*) \end{equation} Thus due to inequality (\ref{Equ: l2Equ4}) and inequality (\ref{Equ: l2Equ6}), we obtain: \begin{equation*} V_I(\widetilde{S}_I) (R_I(\widetilde{S}_I) -T_H^*) \ge V_I(S_I^*) (R_I(S_I^*) - T_H^*) …show more content…

$\square$ \end{proof} One key observation is that: the newly defined local assortment optimization problem at lowest level nests is same as constrained assortment optimization problem under multinomial logit model, which is extensively studied in \cite{rusmevichientong2010dynamic}. By using the polynomial -time algorithm that is proposed in \cite{rusmevichientong2010dynamic}, we can solve this local assortment optimization by only searching $O(N)$ assortments, where $N$ is total number of products within that nest. \section{Candidate Assortment Construction}\label{sec: Construction} Denote total number of products as $n$, and $C = \max C_I$ . \begin{theorem} $\forall I \in V$, we can construct the collection $\mathscr{A}_I \subseteq \Im_I$ such that $|\mathscr{A}_I| \le Cn$, which needs $O(n \log n)$ operations. \end{theorem} \section{Constrained Assortment Optimization Algorithm} \label{sec: Alg} We use $\mathscr{A}_I \subseteq \Im_I$ to denote the collection that contains an optimal solution at node

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