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Recipe

Rank designs by goal programming

Use when: you have a stated target for each objective, instead of a preference between the objectives themselves

Goal: every design's total weighted deviation from its stated goals, ranked smallest to largest

Taught in Goal programming

Quick recipe

  1. State each goal and which direction it penalizes
  2. Compute each deviation variable
  3. Weight each deviation that counts, then add them up
  4. Rank designs by total weighted deviation
  1. 1

    For each objective, write down its goal g_i and decide which deviation actually counts against a design.

    • a ceiling goal ("do not exceed") — penalize only the overshoot, d_i^+.
    • a floor goal ("must be at least") — penalize only the shortfall, d_i^-.
    • a goal that must be hit exactly — penalize both d_i^+ and d_i^-.

    Why: goal programming only measures the gap that actually matters; penalizing the wrong direction would punish a design for beating its own target.

  2. 2

    For every objective, compute d_i^+=\max(0,f_i(x)-g_i) and d_i^-=\max(0,g_i-f_i(x)). At most one of the two is ever positive.

    Example

    Goal g=4 (ceiling), candidate scores 6: f(x)-g=2 is an overshoot, so d^+=2 and d^-=0.

    Why: these formulas are exactly the algebraic split of f_i(x)-g_i=d_i^+-d_i^- with both variables required to be nonnegative.

  3. 3

    Multiply each penalized deviation by its own weight (w_i^+ or w_i^-) and sum across every objective for that design.

    Example

    Goal g=10, overshoot three times worse than undershoot (w^+=3, w^-=1). Candidate at 12: d^+=2, penalty =3(2)=6. Candidate at 6: d^-=4, penalty =1(4)=4. The candidate at 6 wins, even though 12 is numerically closer to the goal.

    • w_i^+=w_i^- for a goal penalized both ways — the total reduces to the plain distance |f_i(x)-g_i|.
    • w_i^+\ne w_i^- — a numerically closer candidate can still lose, since the penalty scales unevenly in each direction.

    Why: an asymmetric penalty is often the whole point — overshooting a budget is frequently worse than undershooting it by the same amount.

  4. 4

    Add every objective's weighted deviation together for each design. The smallest total wins; a total of zero means every goal was hit exactly.

    Example

    Goals g_1=4, g_2=4 (both ceilings, weight 1): X=(3,6) totals 2, Y=(5,3) totals 1, Z=(4,4) totals 0. Z wins outright.

    Why: the design with the smallest total deviation is, by construction, the one that comes closest overall to every stated goal at once.

Common mistakes

  • Penalizing the wrong direction — for instance charging d_i^- against a "do not exceed" ceiling goal that only cares about overshoot.
  • Treating an unreachable goal as a modeling failure, rather than a legitimate target that still ranks designs by closeness.
  • Using equal weights when the problem actually states overshoot and undershoot should count differently.
  • Computing both d_i^+ and d_i^- as positive for the same design on the same objective — only one can ever apply.