LINMAP - LINear programming technique for Multidimensional Analysis of Preference

1. Substitute z_j=w_j x*_j so every stated preference becomes linear. Minimise total non-negative slack subject to Σw_j=1 and w_j>0.
$min (i, k) \sum s_{ik}; j \sum w_{j} (x_{k j}^{2} - x_{ij}^{2}) - 2 j \sum z_{j} (x_{k j} - x_{ij}) \geq 1 - s_{ik}; z_{j} = w_{j} x_{j}^{*}, j \sum w_{j} = 1, w_{j} > 0, s_{ik} \geq 0$
2. Recover x*_j=z_j/w_j, compute each weighted Euclidean distance, and rank ascending (smaller distance is better).
$D_{i} = j = 1 \sum n w_{j} (x_{ij} - x_{j}^{*})^{2}$

1. Substitute z_j=w_j x*_j so every stated preference becomes linear. Minimise total non-negative slack subject to Σw_j=1 and w_j>0.