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METRICS.md

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Algorithm Sandbox

Metrics

Friend/Enemy Satisfaction

  • Symbols
    • $n =$ number of the students numbered ${1, 2, ..., n}$
    • $m =$ number of the teams numbered ${1, 2, ..., m}$
    • $T_i=$ team $i$
    • $f =$ total number of the friend preferences expressed by all individuals
    • $f_i =$ number of friend preferences expressed by student $i$
    • $fs =$ total number of friend preferences across all individuals that are satisfied
    • $fs_i =$ number of friend preferences that are satisfied for student $i$
    • $e =$ total number of the enemy preferences expressed by all individuals
    • $e_i =$ number of the enemy preferences expressed by student $i$
    • $es =$ total number of enemy preferences across all individuals that are satisfied
      • an enemy preference is satisfied if A indicates B to be an enemy and B is not in A's team
    • $es_i =$ number of enemy preferences that are satisfied for student $i$
  • Metrics
    • Percentage of satisfactions, S
      • Friends $${fs \over f} * 100$$
      • Enemies $${es \over e} * 100$$
    • Average satisfactions, $\bar{S}$
      • Friends $${{\sum\limits_{i=1}^n {fs_i \over f_i} }\over n}$$
      • Enemies $${{\sum\limits_{i=1}^n {es_i \over e_i} }\over n}$$
    • Number of strictly happy people, SHP: A person is considered strictly happy if all their friends are in their team.
      • Friends $${|{i: f_i = fs_i}|}$$
      • Enemies $${|{i: e_i = es_i}|}$$
    • Number of strictly happy teams, SHT: A team is considered strictly happy if all the members of the team are all strictly happy.
      • Fiends $${|{i: \forall j \in T_i : fs_j=f_j}|}$$
      • Enemies $${|{i: \forall j \in T_i : es_j=e_j}|}$$
    • Number of happy people, SP: A person is happy if they have no friend preferences or at least one friend preference is satisfied.
      • Friends $${|{i: f_i = 0 \lor fs_i > 0}|}$$
      • Enemies $${|{i: e_i = 0 \lor es_i > 0}|}$$
    • Number of happy teams (HT) that have ...
      • At least 1 happy person, HT-1P
        • Friends $${|{i: \exists j \in T_i : (f_j = 0 \lor fs_j > 0) }|}$$
        • Enemies $${|{i: \exists j \in T_i : (e_j = 0 \lor es_j > 0) }|}$$
      • At least 1 strictly happy person, HT-1SHP
        • Friends $${|{i: \exists j \in T_i : (fs_j=f_j) }|}$$
        • Enemies $${|{i: \exists j \in T_i : (es_j=e_j) }|}$$
      • Every person is happy, HT-All
        • Friends $${|{i: \forall j \in T_i : (f_j = 0 \lor fs_j > 0) }|}$$
        • Enemies $${|{i: \forall j \in T_i : (e_j = 0 \lor es_j > 0) }|}$$
      • Every person is strictly happy, HT-?
        • Friends $${|{i: \forall j \in T_i : (fs_j=f_j) }|}$$
        • Enemies $${|{i: \forall j \in T_i : (es_j=e_j) }|}$$

Priority Satisfaction

  • Symbols
    • $P_i =$ the $i^{th}$ priority from ${1, 2, ..., k}$
    • $S(P_i, T_j)=1$ if the team $T_j$ has satisfied priority $P_i$, else $S(P_i, T_j)=0$
    • $W_i=$ the weight of the priority $P_i$ (all priorities must have weights associataed)
  • The order of the priorities are defined by weights that are exponentially ordered:
    • Weights are defined as: $$W_i={2^{k-i}\over {2^k-1}}$$
    • Satisfaction Score, PSS-EW $$\sum\limits_{i=1}^k \sum\limits_{j=1}^m W_i * S(P_i, T_j)$$
  • The order of the priorities are defined by weights that are linearly ordered:
    • Weights $$W_i = k+1-i$$
    • Satisfaction Score (same as above) PSS-LW
  • Number of fully satisfied priorities (fully means all teams satisfy that priority)
    • FSS $$\max(x: \forall i\lt x : \forall j\lt m: S(P_i, T_j)=1)$$