9.5 Goal programming

Real decisions often have several objectives that cannot all be met. Goal programming (Charnes and Cooper, 1961) sets a target \(g_k\) for each objective \(f_k(x)\) and introduces deviations \(d_k^-, d_k^+ \geq 0\) with \[ f_k(x) + d_k^- - d_k^+ = g_k , \] the amounts by which the goal is under- and over-achieved. The objective penalises the unwanted deviations: in weighted goal programming, a weighted sum of them; in preemptive (lexicographic) goal programming, the goals are ranked, and each is optimised without worsening those above it, which is approximated by very unequal weights or done exactly by solving a sequence of LPs. The weights encode the decision maker’s trade-offs, and different weights give different compromises (Software 9.6.3).

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