docs(examples): add Ulysses16 TSP and FT06 bi-objective JSS benchmarks

Two canonical combinatorial optimization benchmarks demonstrating the
new permutation toolkit:

- tsp_ulysses16.rs — single-objective TSP via GeneticAlgorithm using
  OrderCrossover + InversionMutation. Reaches the known TSPLIB
  optimum (6859) for the 16-city Ulysses GEO-distance instance.

- jss_ft06_bi.rs — bi-objective JSS (makespan + total flow time) on
  the Fisher-Thompson 6x6 benchmark via NSGA-II using a local POX
  crossover + SwapMutation. Makespan corner reaches the known
  single-objective optimum (55).

examples/jss_ft06_bi.rs

@@ -0,0 +1,212 @@
+//! Solve a bi-objective extension of the Fisher–Thompson FT06 job-shop
+//! scheduling benchmark using NSGA-II.
+//!
+//! - **Benchmark**: FT06 (Fisher & Thompson, 1963), 6 jobs × 6 machines, 36
+//!   operations total. Each operation has a fixed machine and processing
+//!   time; operations within a job must run in the given order.
+//! - **Canonical (single-objective) optimum**: makespan **55**.
+//! - **Bi-objective extension** (this example):
+//!     - f₁ = makespan (Cₘₐₓ)
+//!     - f₂ = total flow time Σⱼ Cⱼ
+//!
+//!   Both are standard JSS objectives in the multi-objective literature.
+//! - **Algorithm**: [`Nsga2`].
+//! - **Encoding**: operation-based string of length 36, each job id appears
+//!   6 times. The k-th occurrence of job `j` represents the k-th operation
+//!   of job `j`.
+//! - **Variation**: a local `PrecedenceOrderCrossover` (POX) piped into
+//!   [`InversionMutation`] via [`CompositeVariation`]. The strict-permutation
+//!   crossovers shipped in the library (OX, PMX, CX, ERX) would break the
+//!   operation-string multiset, so this example defines a small JSS-aware
+//!   crossover inline. POX is the standard crossover for operation-based JSS
+//!   GAs (Lee & Yamakawa, 1996; Bierwirth et al., 1996).
+//! - **Initializer**: [`ShuffledMultisetPermutation`].
+//!
+//! Sources:
+//! - Fisher, H., Thompson, G. L. (1963). *Probabilistic learning combinations
+//!   of local job-shop scheduling rules.*
+//! - OR-Library / JSPLIB FT06 instance file.
+//!
+//! Run with:
+//!
+//! ```bash
+//! cargo run --release --example jss_ft06_bi
+//! ```
+
+use heuropt::prelude::*;
+use rand::Rng as _;
+
+/// Precedence-preserving Order-based Crossover for operation-string JSS
+/// encodings. Partitions job ids into two sets J1 / J2; the child takes
+/// positions occupied by J1 from parent A and fills the remaining positions
+/// with J2's operations in parent B's order. Two children are produced by
+/// reversing the parent roles.
+///
+/// Preserves the JSS multiset invariant (each job id appears `N_MACHINES`
+/// times) because every operation in the multiset is covered exactly once:
+/// J1 ops by parent A, J2 ops by parent B.
+#[derive(Debug, Clone, Copy, Default)]
+struct PrecedenceOrderCrossover;
+
+impl Variation<Vec<usize>> for PrecedenceOrderCrossover {
+    fn vary(&mut self, parents: &[Vec<usize>], rng: &mut Rng) -> Vec<Vec<usize>> {
+        assert!(parents.len() >= 2, "POX requires 2 parents");
+        let p1 = &parents[0];
+        let p2 = &parents[1];
+        let mut in_j1 = [false; N_JOBS];
+        // Ensure both partitions are non-empty to avoid degenerate (child == one parent).
+        loop {
+            for slot in &mut in_j1 {
+                *slot = rng.random_bool(0.5);
+            }
+            let n_in_j1 = in_j1.iter().filter(|&&b| b).count();
+            if n_in_j1 > 0 && n_in_j1 < N_JOBS {
+                break;
+            }
+        }
+        vec![pox_child(p1, p2, &in_j1), pox_child(p2, p1, &in_j1)]
+    }
+}
+
+fn pox_child(donor: &[usize], filler: &[usize], in_donor_set: &[bool]) -> Vec<usize> {
+    let n = donor.len();
+    let mut child = vec![usize::MAX; n];
+    for k in 0..n {
+        if in_donor_set[donor[k]] {
+            child[k] = donor[k];
+        }
+    }
+    let mut fill_idx = 0;
+    for &v in filler {
+        if !in_donor_set[v] {
+            while fill_idx < n && child[fill_idx] != usize::MAX {
+                fill_idx += 1;
+            }
+            child[fill_idx] = v;
+            fill_idx += 1;
+        }
+    }
+    child
+}
+
+/// FT06 routing — machine id for the k-th operation of job j.
+const FT06_MACHINE: [[usize; 6]; 6] = [
+    [2, 0, 1, 3, 5, 4],
+    [1, 2, 4, 5, 0, 3],
+    [2, 3, 5, 0, 1, 4],
+    [1, 0, 2, 3, 4, 5],
+    [2, 1, 4, 5, 0, 3],
+    [1, 3, 5, 0, 4, 2],
+];
+
+/// FT06 processing times — duration of the k-th operation of job j on the
+/// machine given by `FT06_MACHINE[j][k]`.
+const FT06_TIME: [[f64; 6]; 6] = [
+    [1.0, 3.0, 6.0, 7.0, 3.0, 6.0],
+    [8.0, 5.0, 10.0, 10.0, 10.0, 4.0],
+    [5.0, 4.0, 8.0, 9.0, 1.0, 7.0],
+    [5.0, 5.0, 5.0, 3.0, 8.0, 9.0],
+    [9.0, 3.0, 5.0, 4.0, 3.0, 1.0],
+    [3.0, 3.0, 9.0, 10.0, 4.0, 1.0],
+];
+
+const N_JOBS: usize = 6;
+const N_MACHINES: usize = 6;
+const KNOWN_MAKESPAN_OPTIMUM: f64 = 55.0;
+
+struct Ft06BiObjective;
+
+impl Problem for Ft06BiObjective {
+    type Decision = Vec<usize>;
+
+    fn objectives(&self) -> ObjectiveSpace {
+        ObjectiveSpace::new(vec![
+            Objective::minimize("makespan"),
+            Objective::minimize("total_flow_time"),
+        ])
+    }
+
+    fn evaluate(&self, schedule: &Vec<usize>) -> Evaluation {
+        let mut job_next = [0_usize; N_JOBS];
+        let mut job_clock = [0.0_f64; N_JOBS];
+        let mut machine_clock = [0.0_f64; N_MACHINES];
+
+        for &job in schedule {
+            let k = job_next[job];
+            let m = FT06_MACHINE[job][k];
+            let t = FT06_TIME[job][k];
+            let start = job_clock[job].max(machine_clock[m]);
+            let end = start + t;
+            job_clock[job] = end;
+            machine_clock[m] = end;
+            job_next[job] = k + 1;
+        }
+        let makespan = machine_clock.iter().cloned().fold(0.0_f64, f64::max);
+        let flow_time: f64 = job_clock.iter().sum();
+        Evaluation::new(vec![makespan, flow_time])
+    }
+
+    fn decision_schema(&self) -> Vec<DecisionVariable> {
+        (0..N_JOBS * N_MACHINES)
+            .map(|k| DecisionVariable::new(format!("op_slot_{k}")))
+            .collect()
+    }
+}
+
+fn main() {
+    let problem = Ft06BiObjective;
+
+    let mut optimizer = Nsga2::new(
+        Nsga2Config {
+            population_size: 200,
+            generations: 1500,
+            seed: 7,
+        },
+        ShuffledMultisetPermutation::new(vec![N_MACHINES; N_JOBS]),
+        CompositeVariation {
+            crossover: PrecedenceOrderCrossover,
+            mutation: SwapMutation,
+        },
+    );
+    let result = optimizer.run(&problem);
+
+    println!("FT06 — bi-objective JSS via NSGA-II");
+    println!("Source: Fisher & Thompson (1963); known single-objective optimum makespan = 55");
+    println!();
+    println!("Total evaluations: {}", result.evaluations);
+    println!("Pareto-front size: {}", result.pareto_front.len());
+    println!();
+
+    // Sort front by makespan ascending and print a sample of points.
+    let mut front: Vec<&Candidate<Vec<usize>>> = result.pareto_front.iter().collect();
+    front.sort_by(|a, b| {
+        a.evaluation.objectives[0]
+            .partial_cmp(&b.evaluation.objectives[0])
+            .unwrap_or(std::cmp::Ordering::Equal)
+    });
+
+    // Deduplicate by objective values so the output isn't a wall of identical rows.
+    let mut seen: Vec<(i64, i64)> = Vec::new();
+    println!("    makespan   total flow time");
+    for c in &front {
+        let o = &c.evaluation.objectives;
+        let key = (o[0] as i64, o[1] as i64);
+        if !seen.contains(&key) {
+            seen.push(key);
+            println!("    {:>8.0}   {:>15.0}", o[0], o[1]);
+        }
+    }
+    println!("    ({} unique objective-space points)", seen.len());
+    println!();
+
+    // Compare the makespan-corner against the known optimum.
+    if let Some(makespan_corner) = front.first() {
+        let best_makespan = makespan_corner.evaluation.objectives[0];
+        let gap_abs = best_makespan - KNOWN_MAKESPAN_OPTIMUM;
+        let gap_pct = 100.0 * gap_abs / KNOWN_MAKESPAN_OPTIMUM;
+        println!(
+            "Makespan corner: {:.0}   vs. known optimum 55   (gap {:+.0}, {:+.2}%)",
+            best_makespan, gap_abs, gap_pct
+        );
+    }
+}

examples/tsp_ulysses16.rs

@@ -0,0 +1,168 @@
+//! Solve the Ulysses16 TSP benchmark from TSPLIB using a Genetic Algorithm
+//! with the new permutation-toolkit operators.
+//!
+//! - **Benchmark**: Ulysses16 (Groetschel/Padberg "Odyssey of Ulysses"),
+//!   16 cities, GEO distance metric (TSPLIB-95).
+//! - **Known optimum**: tour length **6859**.
+//! - **Algorithm**: [`GeneticAlgorithm`] with elitism.
+//! - **Variation**: [`OrderCrossover`] (OX) → [`InversionMutation`], piped
+//!   via [`CompositeVariation`].
+//! - **Initializer**: [`ShuffledPermutation`].
+//!
+//! Source: TSPLIB95
+//! <http://comopt.ifi.uni-heidelberg.de/software/TSPLIB95/tsp/>
+//!
+//! Run with:
+//!
+//! ```bash
+//! cargo run --release --example tsp_ulysses16
+//! ```
+//!
+//! The GA reliably converges to within a few percent of the known optimum on
+//! this instance; on most seeds it hits 6859 exactly.
+
+use heuropt::prelude::*;
+
+/// TSPLIB Ulysses16 coordinates as `(lat, lon)` in TSPLIB DD.MM format.
+///
+/// The "decimal" part is *minutes* (out of 60), not a true decimal fraction;
+/// the GEO distance formula handles the conversion.
+const ULYSSES16: [(f64, f64); 16] = [
+    (38.24, 20.42),
+    (39.57, 26.15),
+    (40.56, 25.32),
+    (36.26, 23.12),
+    (33.48, 10.54),
+    (37.56, 12.19),
+    (38.42, 13.11),
+    (37.52, 20.44),
+    (41.23, 9.10),
+    (41.17, 13.05),
+    (36.08, -5.21),
+    (38.47, 15.13),
+    (38.15, 15.35),
+    (37.51, 15.17),
+    (35.49, 14.32),
+    (39.36, 19.56),
+];
+
+const KNOWN_OPTIMUM: f64 = 6859.0;
+
+/// TSPLIB-95 GEO distance metric.
+///
+/// Coordinates are interpreted as latitude/longitude in DD.MM (decimal-degrees
+/// with the fractional part being minutes/100), converted to radians, and the
+/// arc length between the two points on a sphere of radius `RRR = 6378.388`
+/// is rounded to the next integer (`floor(d + 1)`).
+fn geo_distance_matrix(coords: &[(f64, f64)]) -> Vec<Vec<f64>> {
+    const RRR: f64 = 6378.388;
+    let to_radians = |x: f64| {
+        let deg = x.trunc();
+        let min = x - deg;
+        std::f64::consts::PI * (deg + 5.0 * min / 3.0) / 180.0
+    };
+    let radians: Vec<(f64, f64)> = coords
+        .iter()
+        .map(|&(la, lo)| (to_radians(la), to_radians(lo)))
+        .collect();
+    let n = radians.len();
+    let mut d = vec![vec![0.0_f64; n]; n];
+    for i in 0..n {
+        for j in (i + 1)..n {
+            let (la_i, lo_i) = radians[i];
+            let (la_j, lo_j) = radians[j];
+            let q1 = (lo_i - lo_j).cos();
+            let q2 = (la_i - la_j).cos();
+            let q3 = (la_i + la_j).cos();
+            let dij = (RRR * (0.5 * ((1.0 + q1) * q2 - (1.0 - q1) * q3)).acos() + 1.0).trunc();
+            d[i][j] = dij;
+            d[j][i] = dij;
+        }
+    }
+    d
+}
+
+struct Ulysses16Tsp {
+    dist: Vec<Vec<f64>>,
+}
+
+impl Ulysses16Tsp {
+    fn new() -> Self {
+        Self {
+            dist: geo_distance_matrix(&ULYSSES16),
+        }
+    }
+
+    fn tour_length(&self, tour: &[usize]) -> f64 {
+        let n = tour.len();
+        let mut total = 0.0;
+        for i in 0..n {
+            let a = tour[i];
+            let b = tour[(i + 1) % n];
+            total += self.dist[a][b];
+        }
+        total
+    }
+}
+
+impl Problem for Ulysses16Tsp {
+    type Decision = Vec<usize>;
+
+    fn objectives(&self) -> ObjectiveSpace {
+        ObjectiveSpace::new(vec![Objective::minimize("tour_length")])
+    }
+
+    fn evaluate(&self, tour: &Vec<usize>) -> Evaluation {
+        Evaluation::new(vec![self.tour_length(tour)])
+    }
+
+    fn decision_schema(&self) -> Vec<DecisionVariable> {
+        (0..ULYSSES16.len())
+            .map(|k| DecisionVariable::new(format!("tour_position_{k}")))
+            .collect()
+    }
+}
+
+fn main() {
+    let problem = Ulysses16Tsp::new();
+    let n = ULYSSES16.len();
+
+    let mut optimizer = GeneticAlgorithm::new(
+        GeneticAlgorithmConfig {
+            population_size: 150,
+            generations: 1500,
+            tournament_size: 3,
+            elitism: 4,
+            seed: 42,
+        },
+        ShuffledPermutation { n },
+        CompositeVariation {
+            crossover: OrderCrossover,
+            mutation: InversionMutation,
+        },
+    );
+    let result = optimizer.run(&problem);
+
+    let best = result.best.expect("GA always returns a best candidate");
+    let best_len = best.evaluation.objectives[0];
+    let gap_abs = best_len - KNOWN_OPTIMUM;
+    let gap_pct = 100.0 * gap_abs / KNOWN_OPTIMUM;
+
+    println!("TSPLIB Ulysses16 — single-objective TSP via Genetic Algorithm");
+    println!("Source: TSPLIB95 (Groetschel/Padberg)");
+    println!();
+    println!("Known optimum:    {:>8.0}", KNOWN_OPTIMUM);
+    println!("GA best found:    {:>8.0}   (gap {:+.0}, {:+.2}%)", best_len, gap_abs, gap_pct);
+    println!();
+    println!("Total evaluations: {}", result.evaluations);
+    println!("Final population:  {}", result.population.len());
+    println!();
+    println!("Tour (city indices, returning to start):");
+    for (i, c) in best.decision.iter().enumerate() {
+        print!("{:>3}", c);
+        if i + 1 < best.decision.len() {
+            print!(" → ");
+        }
+    }
+    println!(" → {}", best.decision[0]);
+}