Bi-Objective Traveling Thief Problem (BI-TTP)

• BI-TTP is a bi-objective problem that integrates traveling salesperson and knapsack decisions with explicit interdependence.
• It requires simultaneous routing and packing optimization, where item selection affects travel time by reducing speed along the tour.
• Solution approaches include dynamic programming embedded in evolutionary algorithms, weighted-sum scalarization, and quantum annealing for Pareto front exploration.

The Bi-Objective Traveling Thief Problem (BI-TTP) is a bi-objective variant of the Traveling Thief Problem in which routing and packing must be optimized simultaneously under explicit interdependence between the traveling salesperson problem and the 0-1 knapsack problem. In the cited literature, a solution consists of a tour together with a packing plan, and the Pareto front is the set of non-dominated trade-offs between the chosen objectives (Wu et al., 2018, Chagas et al., 2020, Chagas et al., 2020, Viet et al., 13 Mar 2026). A central structural feature is that knapsack weight accumulated along the route reduces travel speed, so item-selection decisions alter the effective cost of subsequent edges. The literature also shows that BI-TTP is not represented by a single universally fixed objective pair: one line of work uses a reward-versus-weight formulation, whereas others use profit-versus-travel-time formulations. This suggests that BI-TTP is best understood as a family of closely related bi-objective formulations built on the same route-packing coupling.

1. Formal definitions and alternative objective formulations

A standard BI-TTP instance specifies a set of cities, a set of items distributed over those cities, a knapsack capacity, distances, and the maximal and minimal velocities. In one formulation, a feasible solution is a pair $(\pi,z)$ where $\pi=\langle \pi_1,\dots,\pi_n\rangle$ is a permutation of the cities with $\pi_1=1$, and $z\in\{0,1\}^m$ is a binary picking vector satisfying the capacity constraint $\sum_{j=1}^m w_j z_j \le W$. The cumulative knapsack weight after visiting city $\pi_i$ is

$$\omega(i,\pi,z)=\sum_{k=1}^i\sum_{j\in I:\,\mathrm{loc}(j)=\pi_k} w_j z_j,$$

and the total travel time is

$$f_2(\pi,z)=\sum_{i=1}^{n-1}\frac{d(\pi_i,\pi_{i+1})}{v_{\max}-\bigl(v_{\max}-v_{\min}\bigr)\,\omega(i,\pi,z)/W} +\frac{d(\pi_n,\pi_1)}{v_{\max}-\bigl(v_{\max}-v_{\min}\bigr)\,\omega(n,\pi,z)/W}.$$

The corresponding profit objective is

$f_1(\pi,z)=\sum_{j=1}^m p_j z_j.$

This yields the bi-objective optimization problem $\max (f_1(\pi,z),-f_2(\pi,z))$, or equivalently $\pi=\langle \pi_1,\dots,\pi_n\rangle$0 after sign conversion for profit (Chagas et al., 2020).

A closely related formulation uses the same route-packing structure but writes the objectives as total travel time and item profit directly. For a tour $\pi=\langle \pi_1,\dots,\pi_n\rangle$1 and packing $\pi=\langle \pi_1,\dots,\pi_n\rangle$2, the cumulative weight is

$\pi=\langle \pi_1,\dots,\pi_n\rangle$3

the travel speed on leg $\pi=\langle \pi_1,\dots,\pi_n\rangle$4 is

$\pi=\langle \pi_1,\dots,\pi_n\rangle$5

the travel-time objective is

$\pi=\langle \pi_1,\dots,\pi_n\rangle$6

and the profit objective is $\pi=\langle \pi_1,\dots,\pi_n\rangle$7, or in minimization form $\pi=\langle \pi_1,\dots,\pi_n\rangle$8 (Viet et al., 13 Mar 2026).

Wu et al. introduced a different bi-criteria formulation in which the first objective is a net reward

$\pi=\langle \pi_1,\dots,\pi_n\rangle$9

and the second objective is the total picked weight

$\pi_1=1$0

subject to $\pi_1=1$1. In that setting, BI-TTP seeks the Pareto set of all non-dominated $\pi_1=1$2 such that no other feasible solution can improve $\pi_1=1$3 without worsening $\pi_1=1$4, or vice versa (Wu et al., 2018).

2. Coupling of routing and packing

The essential difficulty of BI-TTP is the interweaving of its two classical subproblems. If the knapsack is ignored, the problem reduces to a traveling salesperson problem over the cities. If the tour is fixed and travel-time effects are ignored, the problem reduces to a 0-1 knapsack problem over the items. In BI-TTP, however, each stolen item increases the current knapsack weight, which decreases the current speed and therefore increases the travel time of later legs (Chagas et al., 2020).

This dependence is formalized through a velocity function of the current load. One formulation writes

$\pi_1=1$5

so a packing choice does not merely contribute profit; it also modifies the cost structure of the remainder of the route (Chagas et al., 2020). In the profit-versus-time formulations, this coupling is visible in the denominator of each edge-traversal term, because the effective speed depends on the cumulative weight after previously visited cities (Chagas et al., 2020, Viet et al., 13 Mar 2026).

A recurring source of confusion is the assumption that one may optimize the tour and the packing nearly independently and then combine the two results. The cited formulations directly contradict that view. The best tour under constant speed need not remain best once speed becomes load-dependent, and a packing plan that is attractive in a static knapsack model may become inferior once the carrying cost over remaining distance is accounted for. This is why the literature consistently treats BI-TTP as a multi-component optimization problem rather than as a simple juxtaposition of TSP and KP (Wu et al., 2018, Chagas et al., 2020).

3. Exact dynamic programming and indicator-based evolutionary search

For a fixed tour $\pi_1=1$6, Wu et al. consider the Packing-While-Travelling (PWT) subproblem: choose the packing $\pi_1=1$7 to maximize $\pi_1=1$8 subject to the capacity constraint. Building on Neumann et al. (2017), they use an exact dynamic-programming scheme that not only returns one optimal packing but naturally constructs the entire Pareto front of the fixed-tour subproblem (Wu et al., 2018).

With the items ordered along the tour, the dynamic-programming table is

$\pi_1=1$9

The recurrence compares excluding and including the $z\in\{0,1\}^m$0-th item: $z\in\{0,1\}^m$1 where $z\in\{0,1\}^m$2 accounts for the additional slowdown induced by picking item $z\in\{0,1\}^m$3. After each row is computed, dominated cells are discarded by pruning. The naïve version runs in $z\in\{0,1\}^m$4 time and space; with pruning, the practical performance is often much better (Wu et al., 2018).

The final non-dominated pairs $z\in\{0,1\}^m$5 form the DP front $z\in\{0,1\}^m$6 for tour $z\in\{0,1\}^m$7. Wu et al. use these fronts inside an indicator-based evolutionary algorithm in which the individual is the tour $z\in\{0,1\}^m$8, not the full pair $z\in\{0,1\}^m$9. For every tour in the population, the DP subroutine produces a full Pareto front. The union of all fronts is

$\sum_{j=1}^m w_j z_j \le W$0

and its Pareto-optimal surface is $\sum_{j=1}^m w_j z_j \le W$1. Tour quality is then defined by its contribution to that global surface through two indicators: Loss-of-Surface-Contribution (LSC) and Loss-of-Hypervolume (LHV). The general indicator template is

$\sum_{j=1}^m w_j z_j \le W$2

For surface contribution,

$\sum_{j=1}^m w_j z_j \le W$3

and for hypervolume,

$\sum_{j=1}^m w_j z_j \le W$4

Survivor selection repeatedly removes the tour with smallest indicator value, while parent selection biases mating toward tours whose DP fronts make larger contributions to the global surface (Wu et al., 2018).

This architecture is distinctive because the evolutionary layer perturbs only the tour component, while the packing trade-offs for each route are handled exactly by dynamic programming. A plausible implication is that the method avoids one common source of noise in multi-objective search: the need to assess a route using only one sampled packing configuration rather than its full fixed-route Pareto structure.

4. Scalarization methods: weighted sums and $\sum_{j=1}^m w_j z_j \le W$5-constraints

Another major line of work treats BI-TTP through scalarization. In the weighted-sum method, a parameter $\sum_{j=1}^m w_j z_j \le W$6 controls the balance between profit and travel time: $\sum_{j=1}^m w_j z_j \le W$7 Varying $\sum_{j=1}^m w_j z_j \le W$8 over $\sum_{j=1}^m w_j z_j \le W$9 yields different trade-offs, with $\pi_i$0 emphasizing a pure TSP tour and $\pi_i$1 recovering the KP-only perspective. The method uses randomized versions of existing heuristics: the Chained Lin-Kernighan heuristic for tours, a randomized packing heuristic with item scores

$\pi_i$2

and exploitation through restricted 2-opt moves and probabilistic bit flips (Chagas et al., 2020).

The same paper states the classical limitations of weighted-sum decomposition: an inability to find Pareto points in non-convex regions of the front and a tendency to produce unevenly spaced solutions when the front is curved. Because the BI-TTP subproblems are solved heuristically rather than exactly, the algorithm may still stumble upon some non-convex front points by virtue of heuristic imperfections (Chagas et al., 2020). This is an explicit methodological caveat rather than a contradiction.

A more recent scalarization strategy reformulates BI-TTP by the $\pi_i$3-constraint method. First, the extreme profits $\pi_i$4 and $\pi_i$5 are determined, and the interval $\pi_i$6 is divided into $\pi_i$7 segments

$\pi_i$8

For each band $\pi_i$9, the solver minimizes travel time subject to a profit interval: $$\omega(i,\pi,z)=\sum_{k=1}^i\sum_{j\in I:\,\mathrm{loc}(j)=\pi_k} w_j z_j,$$0 The resulting fractional objective is rewritten with auxiliary variables $$\omega(i,\pi,z)=\sum_{k=1}^i\sum_{j\in I:\,\mathrm{loc}(j)=\pi_k} w_j z_j,$$1, transformed into a Quadratic Unconstrained Binary Optimization model, and solved with quantum annealing, after which a tailored heuristic refinement called LEA (Later and Enough Accumulation) is applied (Viet et al., 13 Mar 2026).

The main algorithmic families in the cited literature can be organized as follows.

Approach	Core mechanism	Reported emphasis
Indicator-based EA + DP	Exact PWT DP front inside a bi-objective EA	Surface contribution and hypervolume guidance
Weighted-sum method	Randomized scalarization over $$\omega(i,\pi,z)=\sum_{k=1}^i\sum_{j\in I:\,\mathrm{loc}(j)=\pi_k} w_j z_j,$$2	Broad heuristic Pareto approximation
NDS-BRKGA	Random-key GA with non-dominated sorting	Elite survival and customized repair
$$\omega(i,\pi,z)=\sum_{k=1}^i\sum_{j\in I:\,\mathrm{loc}(j)=\pi_k} w_j z_j,$$3-constraint + QA	Profit bands, QUBO reformulation, QA, LEA	Broad Pareto front and time efficiency

5. Population-based genetic algorithms and customized representations

The non-dominated sorting based customized random-key genetic algorithm (NDS-BRKGA) represents BI-TTP by a real-valued genotype $$\omega(i,\pi,z)=\sum_{k=1}^i\sum_{j\in I:\,\mathrm{loc}(j)=\pi_k} w_j z_j,$$4, partitioned into tour genes and packing genes. Decoding sorts the tour genes to obtain a permutation of cities $$\omega(i,\pi,z)=\sum_{k=1}^i\sum_{j\in I:\,\mathrm{loc}(j)=\pi_k} w_j z_j,$$5, prepends city $$\omega(i,\pi,z)=\sum_{k=1}^i\sum_{j\in I:\,\mathrm{loc}(j)=\pi_k} w_j z_j,$$6, and converts each packing gene to a binary decision through a threshold at $$\omega(i,\pi,z)=\sum_{k=1}^i\sum_{j\in I:\,\mathrm{loc}(j)=\pi_k} w_j z_j,$$7 (Chagas et al., 2020).

Initialization incorporates domain knowledge from both subproblems. The algorithm solves the TSP subproblem with LKH for $$\omega(i,\pi,z)=\sum_{k=1}^i\sum_{j\in I:\,\mathrm{loc}(j)=\pi_k} w_j z_j,$$8 seconds to obtain a near-optimal tour $$\omega(i,\pi,z)=\sum_{k=1}^i\sum_{j\in I:\,\mathrm{loc}(j)=\pi_k} w_j z_j,$$9 and its symmetric reverse, and solves the KP subproblem by a two-stage Greedy+DP heuristic, controlled by a parameter $$f_2(\pi,z)=\sum_{i=1}^{n-1}\frac{d(\pi_i,\pi_{i+1})}{v_{\max}-\bigl(v_{\max}-v_{\min}\bigr)\,\omega(i,\pi,z)/W} +\frac{d(\pi_n,\pi_1)}{v_{\max}-\bigl(v_{\max}-v_{\min}\bigr)\,\omega(n,\pi,z)/W}.$$0, to obtain a packing plan $$f_2(\pi,z)=\sum_{i=1}^{n-1}\frac{d(\pi_i,\pi_{i+1})}{v_{\max}-\bigl(v_{\max}-v_{\min}\bigr)\,\omega(i,\pi,z)/W} +\frac{d(\pi_n,\pi_1)}{v_{\max}-\bigl(v_{\max}-v_{\min}\bigr)\,\omega(n,\pi,z)/W}.$$1. Candidate $$f_2(\pi,z)=\sum_{i=1}^{n-1}\frac{d(\pi_i,\pi_{i+1})}{v_{\max}-\bigl(v_{\max}-v_{\min}\bigr)\,\omega(i,\pi,z)/W} +\frac{d(\pi_n,\pi_1)}{v_{\max}-\bigl(v_{\max}-v_{\min}\bigr)\,\omega(n,\pi,z)/W}.$$2 solutions are built by combining these structures, filtered by non-domination, encoded back into random keys, and mixed with uniformly random individuals (Chagas et al., 2020). The paper explicitly argues that domain-knowledge seeding accelerates convergence.

Capacity feasibility is enforced by a custom repair operator. If a decoded packing is overweight, the operator scans the tour from end to start and removes items whose collection contributes to the overweight, preferentially dropping items collected latest and thereby maximizing early-tour speed. After repair, the corresponding keys are reset to reflect the feasible packing (Chagas et al., 2020).

The evolutionary core follows BRKGA-style biased crossover: $$f_2(\pi,z)=\sum_{i=1}^{n-1}\frac{d(\pi_i,\pi_{i+1})}{v_{\max}-\bigl(v_{\max}-v_{\min}\bigr)\,\omega(i,\pi,z)/W} +\frac{d(\pi_n,\pi_1)}{v_{\max}-\bigl(v_{\max}-v_{\min}\bigr)\,\omega(n,\pi,z)/W}.$$3 where $$f_2(\pi,z)=\sum_{i=1}^{n-1}\frac{d(\pi_i,\pi_{i+1})}{v_{\max}-\bigl(v_{\max}-v_{\min}\bigr)\,\omega(i,\pi,z)/W} +\frac{d(\pi_n,\pi_1)}{v_{\max}-\bigl(v_{\max}-v_{\min}\bigr)\,\omega(n,\pi,z)/W}.$$4 is an elite parent and $$f_2(\pi,z)=\sum_{i=1}^{n-1}\frac{d(\pi_i,\pi_{i+1})}{v_{\max}-\bigl(v_{\max}-v_{\min}\bigr)\,\omega(i,\pi,z)/W} +\frac{d(\pi_n,\pi_1)}{v_{\max}-\bigl(v_{\max}-v_{\min}\bigr)\,\omega(n,\pi,z)/W}.$$5 is drawn from the full population. Diversity is maintained through mutant individuals sampled uniformly in the genotype space. Bi-objective survival uses the NSGA-II mechanism: fast non-dominated sorting, crowding distance in normalized 2-D objective space, and truncation by least-crowded individuals when a front must be cut (Chagas et al., 2020).

The parameter study spans 3072 parameter combinations. The reported best hypervolumes arise with $$f_2(\pi,z)=\sum_{i=1}^{n-1}\frac{d(\pi_i,\pi_{i+1})}{v_{\max}-\bigl(v_{\max}-v_{\min}\bigr)\,\omega(i,\pi,z)/W} +\frac{d(\pi_n,\pi_1)}{v_{\max}-\bigl(v_{\max}-v_{\min}\bigr)\,\omega(n,\pi,z)/W}.$$6, $$f_2(\pi,z)=\sum_{i=1}^{n-1}\frac{d(\pi_i,\pi_{i+1})}{v_{\max}-\bigl(v_{\max}-v_{\min}\bigr)\,\omega(i,\pi,z)/W} +\frac{d(\pi_n,\pi_1)}{v_{\max}-\bigl(v_{\max}-v_{\min}\bigr)\,\omega(n,\pi,z)/W}.$$7, small $$f_2(\pi,z)=\sum_{i=1}^{n-1}\frac{d(\pi_i,\pi_{i+1})}{v_{\max}-\bigl(v_{\max}-v_{\min}\bigr)\,\omega(i,\pi,z)/W} +\frac{d(\pi_n,\pi_1)}{v_{\max}-\bigl(v_{\max}-v_{\min}\bigr)\,\omega(n,\pi,z)/W}.$$8, $$f_2(\pi,z)=\sum_{i=1}^{n-1}\frac{d(\pi_i,\pi_{i+1})}{v_{\max}-\bigl(v_{\max}-v_{\min}\bigr)\,\omega(i,\pi,z)/W} +\frac{d(\pi_n,\pi_1)}{v_{\max}-\bigl(v_{\max}-v_{\min}\bigr)\,\omega(n,\pi,z)/W}.$$9, $f_1(\pi,z)=\sum_{j=1}^m p_j z_j.$0, and regular local search with $f_1(\pi,z)=\sum_{j=1}^m p_j z_j.$1 (Chagas et al., 2020). In competition results, the preliminary “jomar” entry won 1st place on seven of nine instances at EMO-2019, and the refined NDS-BRKGA placed 2nd overall at GECCO-2019 (Chagas et al., 2020).

6. Benchmarks, empirical results, and open directions

The empirical literature uses several benchmark sets. Wu et al. evaluate on the standard TTP benchmark families eil51, eil76, and eil101, each with three item-correlation types: uncorrelated, strongly bounded, and similar weights. Their parameter setting is population size $f_1(\pi,z)=\sum_{j=1}^m p_j z_j.$2, maximum generations $f_1(\pi,z)=\sum_{j=1}^m p_j z_j.$3, five independent runs per setting, and 16 configurations arising from two indicators $f_1(\pi,z)=\sum_{j=1}^m p_j z_j.$4 crossed with eight parent-selection rules. Performance is measured by the global hypervolume of the final union-surface and the total reward $f_1(\pi,z)=\sum_{j=1}^m p_j z_j.$5 of the best single solution, compared with the single-objective state-of-the-art MA2B. Across all 16 settings, harmonic rank-based selection on LHV and best-half on LSC performed best, fitness-proportionate selection was a safe default, the BI-TTP approach consistently found larger hypervolumes than uniform-random selection with $f_1(\pi,z)=\sum_{j=1}^m p_j z_j.$6 under Welch’s $f_1(\pi,z)=\sum_{j=1}^m p_j z_j.$7-test, and the best BI-TTP variants outperformed MA2B on 8 of 9 test instances (Wu et al., 2018).

The weighted-sum study evaluates nine medium and large BI-TTP instances from the EMO-2019 and GECCO-2019 competitions together with 960 single-objective TTP instances for parameter tuning and validation. Its metrics are hypervolume of the Pareto set, number of non-dominated points, single-objective TTP score, runtime, and the curve of hypervolume over time. Against NDSBRKGA on 960 instances, WSM achieved a significantly higher average hypervolume in 82.2% of cases, was worse in 15.0%, and tied in 2.8% under a Wilcoxon test with $f_1(\pi,z)=\sum_{j=1}^m p_j z_j.$8. On the six $f_1(\pi,z)=\sum_{j=1}^m p_j z_j.$9 and $\max (f_1(\pi,z),-f_2(\pi,z))$0 instances, WSM attained the top hypervolume, improving the best known by up to 1.7%, and it matched or exceeded those of 21 leading TTP algorithms on 379 of 960 benchmark instances in the single-objective setting (Chagas et al., 2020).

The NDS-BRKGA study uses nine benchmark instances from Polyakovskiy et al. (2014), with sizes ranging from $\max (f_1(\pi,z),-f_2(\pi,z))$1 to $\max (f_1(\pi,z),-f_2(\pi,z))$2, and evaluates performance by normalized two-dimensional hypervolume. It reports that LKH needs only about 300 seconds to reach at most a 0.15% gap on the TSP subproblem, and that the GH+DP knapsack solver with $\max (f_1(\pi,z),-f_2(\pi,z))$3 finds optimal KP solutions in acceptable time. Even though BI-TTP does not explicitly maximize the single-objective TTP score, many of the reported bi-objective solutions strictly dominate the best known single-objective solutions on the two smallest instances (Chagas et al., 2020).

The $\max (f_1(\pi,z),-f_2(\pi,z))$4-constraint plus quantum-annealing study uses benchmark instances from Polyakovskiy et al. (2014): $\max (f_1(\pi,z),-f_2(\pi,z))$5 variants with $\max (f_1(\pi,z),-f_2(\pi,z))$6 and $\max (f_1(\pi,z),-f_2(\pi,z))$7 variants with $\max (f_1(\pi,z),-f_2(\pi,z))$8. It compares against NSGA-II, U-NSGA-III, and MOEA/D, using normalized hypervolume and total runtime, including classical CPU time, QPU access, and LEA. The reported results are hypervolume values around $\max (f_1(\pi,z),-f_2(\pi,z))$9 to $\pi=\langle \pi_1,\dots,\pi_n\rangle$00 across all instances in a single hybrid run, approximately $\pi=\langle \pi_1,\dots,\pi_n\rangle$01 to $\pi=\langle \pi_1,\dots,\pi_n\rangle$02 seconds of QPU time for 10 subproblems with less than $\pi=\langle \pi_1,\dots,\pi_n\rangle$03 LEA overhead, overall runtimes such as about $\pi=\langle \pi_1,\dots,\pi_n\rangle$04 seconds for $\pi=\langle \pi_1,\dots,\pi_n\rangle$05 and about $\pi=\langle \pi_1,\dots,\pi_n\rangle$06 seconds for $\pi=\langle \pi_1,\dots,\pi_n\rangle$07, and up to $\pi=\langle \pi_1,\dots,\pi_n\rangle$08 speed-up relative to the cited baselines while obtaining strictly better or comparable hypervolume (Viet et al., 13 Mar 2026).

Across these studies, several limitations and open directions recur. The weighted-sum method inherits the classical difficulty of covering strictly non-convex front regions under exact solution, though heuristic search can mitigate that effect (Chagas et al., 2020). The DP-based hybrid may require more aggressive pruning or approximation when $\pi=\langle \pi_1,\dots,\pi_n\rangle$09 is very large (Wu et al., 2018). The random-key GA notes the computational expense of repeated LKH runs and local search on very large instances, and possible bias from initialization by single-objective subproblem solvers (Chagas et al., 2020). The quantum-annealing formulation is motivated by severe scalability issues in conventional methods and by the complexity of reformulating fractional objectives into QUBO-compatible form (Viet et al., 13 Mar 2026). Future directions explicitly proposed in the literature include richer tour-encoding operators such as edge-assembly crossover, multi-level DP/EAs, per-instance algorithm configuration, alternative scalarization techniques including $\pi=\langle \pi_1,\dots,\pi_n\rangle$10-constraint and Tchebycheff, and extensions to dynamic, multi-depot, and other interwoven multi-component optimization problems (Wu et al., 2018, Chagas et al., 2020, Chagas et al., 2020).

In aggregate, the BI-TTP literature portrays the problem as a stringent testbed for hybrid optimization: exact dynamic programming embedded in evolutionary selection, randomized heuristic scalarization, customized non-dominated sorting genetic algorithms, and quantum-classical $\pi=\langle \pi_1,\dots,\pi_n\rangle$11-constraint pipelines all exploit the same route-packing interdependence but expose different trade-offs in exactness, diversity, scalability, and runtime.