Biparty Multiobjective UAV Path Planning

• BPMO-UAVPP is a framework where two decision-makers separately optimize efficiency and safety objectives, leading to common Pareto solutions.
• It uses discrete waypoint representations with constraints on altitude, turning angles, and risk factors to plan UAV paths in complex urban environments.
• Benchmark results show that algorithms like BPAIMA achieve higher hypervolume metrics compared to NSGA-II, confirming the efficacy of the biparty approach.

Biparty Multiobjective UAV Path Planning (BPMO-UAVPP) is a formulation of UAV path planning in which the optimization problem is not assigned to a single decision-maker, but to two decision-makers (DMs) with distinct and only partially aligned concerns: an efficiency DM and a safety DM. In the formulation introduced in "Evolutionary Biparty Multiobjective UAV Path Planning: Problems and Empirical Comparisons" (Chen et al., 23 Mar 2026), a single fixed-wing or rotary UAV follows a discrete path represented by waypoints, and the final decision is based on solutions that are acceptable from both DM perspectives rather than on a conventional single-party Pareto set. Within the broader literature on multiparty multiobjective optimization, BPMO-UAVPP also functions as a benchmark problem family, including the CEC 2024 competition suite for problems with unknown solutions (Luo et al., 2024).

1. Formal problem statement

In BPMO-UAVPP, the UAV path is discretized into $n+1$ waypoints,

$$P=\{p_0,p_1,\dots,p_n\},\qquad p_i=(x_i,y_i,z_i)\in\mathbb R^3,$$

and each decision vector $X\in\mathbb R^{3(n+1)}$ encodes all waypoint coordinates. The start point $p_0$ and goal point $p_n$ are fixed, while interior waypoints are free (Chen et al., 23 Mar 2026).

The defining structural feature is the presence of two DMs. The efficiency DM chooses two efficiency-related objectives,

$F^{\rm eff}(X)=(f_1(X),f_2(X)),$

and the safety DM chooses two safety-related objectives,

$F^{\rm safe}(X)=(f_3(X),f_4(X)).$

The generic optimization form is

$\min \bigl(F^{\rm eff}(X),F^{\rm safe}(X)\bigr),$

subject to altitude, turning-angle, and slope-angle constraints. This differs from existing work that "considered all efficiency and safety objectives for a single decision-maker (DM) and regarded this as a multiobjective optimization problem (MOP)." The BPMO-UAVPP formulation instead treats urban UAV planning as a biparty decision process in which "there is usually not a single DM but two DMs, i.e., an efficiency DM and a safety DM, and the DMs are only concerned with their respective objectives" (Chen et al., 23 Mar 2026).

A common misconception is that BPMO-UAVPP is merely an ordinary MOP with objectives partitioned into two groups. The modeling choice in BPMO-UAVPP is stronger than that: the two groups are associated with distinct decision-makers, separate nondominated sorts are performed in each DM space, and final solutions are selected from the intersection of both perspectives rather than from a single global dominance relation (Chen et al., 23 Mar 2026).

2. Objectives, path representation, and constraints

Let

$$g_i=\|p_{i+1}-p_i\|,\qquad \Delta z_i=|z_{i+1}-z_i|.$$

The efficiency and safety objectives used in BPMO-UAVPP are drawn from the following set (Chen et al., 23 Mar 2026, Luo et al., 2024).

Category	Objective	Definition
Efficiency	Path length	$f_1(X)=\sum_{i=0}^{n-1}\|g_i\|$
Efficiency	Mission-hover-point distance	$$P=\{p_0,p_1,\dots,p_n\},\qquad p_i=(x_i,y_i,z_i)\in\mathbb R^3,$$0
Safety	Fatality risk	$$P=\{p_0,p_1,\dots,p_n\},\qquad p_i=(x_i,y_i,z_i)\in\mathbb R^3,$$1
Safety	Property risk or noise pollution	$$P=\{p_0,p_1,\dots,p_n\},\qquad p_i=(x_i,y_i,z_i)\in\mathbb R^3,$$2, or $$P=\{p_0,p_1,\dots,p_n\},\qquad p_i=(x_i,y_i,z_i)\in\mathbb R^3,$$3

For fatality risk, the people-related component is

$$P=\{p_0,p_1,\dots,p_n\},\qquad p_i=(x_i,y_i,z_i)\in\mathbb R^3,$$4

with an analogous vehicle term $$P=\{p_0,p_1,\dots,p_n\},\qquad p_i=(x_i,y_i,z_i)\in\mathbb R^3,$$5. In the benchmark formulation, fatality risk is written as

$$P=\{p_0,p_1,\dots,p_n\},\qquad p_i=(x_i,y_i,z_i)\in\mathbb R^3,$$6

where

$$P=\{p_0,p_1,\dots,p_n\},\qquad p_i=(x_i,y_i,z_i)\in\mathbb R^3,$$7

(Luo et al., 2024).

The benchmark suite expands the efficiency side to four candidate components,

$$P=\{p_0,p_1,\dots,p_n\},\qquad p_i=(x_i,y_i,z_i)\in\mathbb R^3,$$8

and the safety side to three candidate components,

$$P=\{p_0,p_1,\dots,p_n\},\qquad p_i=(x_i,y_i,z_i)\in\mathbb R^3,$$9

with scenario-dependent subsets selected for each case (Luo et al., 2024). In the experiments summarized for (Chen et al., 23 Mar 2026), six BPMO cases combine efficiency choices from $X\in\mathbb R^{3(n+1)}$0path-length, length+height, fuel-consumption$X\in\mathbb R^{3(n+1)}$1 versus hover-distance with safety choices $X\in\mathbb R^{3(n+1)}$2fatality vs. property-risk$X\in\mathbb R^{3(n+1)}$3 or $X\in\mathbb R^{3(n+1)}$4fatality vs. noise$X\in\mathbb R^{3(n+1)}$5.

The core constraints are

$X\in\mathbb R^{3(n+1)}$6

$X\in\mathbb R^{3(n+1)}$7

and

$X\in\mathbb R^{3(n+1)}$8

In the benchmark notation, $X\in\mathbb R^{3(n+1)}$9 and $p_0$0 are defined using the horizontal projection $p_0$1 of each segment (Luo et al., 2024). Obstacle avoidance and no-fly-zone constraints may also be added as

$p_0$2

(Chen et al., 23 Mar 2026).

3. Biparty decision-making and the common Pareto front

The operational distinction between BPMO-UAVPP and an ordinary MOP lies in how dominance is computed and how the final solution set is defined. Each DM optimizes its own bi-objective in parallel, and "at each generation, two nondominated sorts are performed—one in $p_0$3 space, one in $p_0$4 space" (Chen et al., 23 Mar 2026).

A solution is considered "common Pareto" if it is not dominated in either DM's space. The final solution set is therefore the common Pareto front: solutions that lie on the front in both DM spaces. Information exchange occurs during selection and clonal allocation, where the ranks obtained from both sorts guide survival and cloning pressure. This creates a coupling between the two decision processes without collapsing them into a single scalar objective (Chen et al., 23 Mar 2026).

The benchmark description for the CEC 2024 competition presents the same problem family from an evaluation standpoint. It emphasizes maintaining two nondominated fronts, one per DM, and combining the resulting performance through Multiparty Hypervolume (MPHV),

$p_0$5

with a common reference point for each DM space (Luo et al., 2024). This suggests that BPMO-UAVPP is not only a modeling construct but also a testbed for assessing whether an algorithm can preserve convergence and diversity simultaneously across two distinct Pareto geometries.

4. Biparty immune-inspired optimization algorithms

The original BPMO-UAVPP study modifies three existing multiobjective immune algorithms—NNIA, HEIA, and AIMA—to produce the biparty variants BPNNIA, BPHEIA, and BPAIMA (Chen et al., 23 Mar 2026). All three follow a multi-party immune-algorithm template. The "only fundamental change from NNIA/HEIA/AIMA is to replace single-sorting by the biparty nondominated sorting operator, $p_0$6."

The generic framework is defined as follows. A population $p_0$7 of $p_0$8 random antibodies is initialized, evaluated under $p_0$9 and $p_n$0, and ranked by $p_n$1. At each generation, the algorithm activates the $p_n$2 best antibodies by the biparty rank, computes clone counts based on biparty-rank and crowding distance, generates offspring by cloning, crossover, and mutation, reevaluates the merged population, re-ranks by $p_n$3, and selects $p_n$4 survivors by layers and crowding. The final multiparty Pareto set is

$p_n$5

(Chen et al., 23 Mar 2026).

The biparty nondominated sorting operator is

$p_n$6

followed by

$p_n$7

that is, a second fast nondominated sort on the pair of layer-ranks from the two DM spaces (Chen et al., 23 Mar 2026).

The three algorithmic variants differ in their variation operators:

Algorithm	Main operators
BPNNIA	SBX crossover + polynomial mutation
BPHEIA	Hybrid DE/SBX crossover (rand/1/bin DE w.p. 0.5, else SBX) + PM
BPAIMA	Adaptive multi-DE strategies (rand/2/bin, rand/1/bin, … w.p. 0.9/0.5/0.1) + PM

Within the broader MPMOP literature, BPMO-UAVPP is also used as an application domain for the Multiparty Immune Algorithm (MPIA) (Chen et al., 29 Mar 2026). MPIA extends the immune-clonal framework through an inter-party guided crossover that uses each individual's nondominated ranks under both DMs and an adaptive activation strategy based on the Multiparty Cover Metric (MCM). In that formulation, MPNDS2 again serves as the core rank-construction mechanism, but activation and operator selection are made adaptive rather than fixed. This places BPMO-UAVPP at the intersection of application-specific UAV planning and general-purpose multiparty evolutionary optimization (Chen et al., 29 Mar 2026).

5. Benchmark instances, parameters, and evaluation protocol

The experimental setup reported for BPMO-UAVPP uses six cases and static urban scenario data (Chen et al., 23 Mar 2026). Building heights are sampled from a lognormal distribution with $p_n$8 and $p_n$9, and population density is generated by a Gaussian radial model representing a dense urban core. Fatality risk maps are precomputed. The benchmark description gives a closely aligned setup: an urban area of size $F^{\rm eff}(X)=(f_1(X),f_2(X)),$0, building heights $F^{\rm eff}(X)=(f_1(X),f_2(X)),$1, population and vehicle densities generated by a radial-basis model, and static precomputed maps and risks with no additional time-varying or stochastic obstacles (Luo et al., 2024).

The constraints and UAV parameters in the BPMO-UAVPP experiments include

$F^{\rm eff}(X)=(f_1(X),f_2(X)),$2

with $F^{\rm eff}(X)=(f_1(X),f_2(X)),$3 and fixed start and end points (Chen et al., 23 Mar 2026). The benchmark specifies

$F^{\rm eff}(X)=(f_1(X),f_2(X)),$4

and preset hover points

$F^{\rm eff}(X)=(f_1(X),f_2(X)),$5

(Luo et al., 2024). In the MPIA study, the decision-variable dimension is reported as

$F^{\rm eff}(X)=(f_1(X),f_2(X)),$6

for the corresponding path encoding (Chen et al., 29 Mar 2026).

The algorithms compared in (Chen et al., 23 Mar 2026) are NSGA-II, OptMPNDS, OptMPNDS2, BPNNIA, BPHEIA, and BPAIMA, with population size $F^{\rm eff}(X)=(f_1(X),f_2(X)),$7, evaluation limit $F^{\rm eff}(X)=(f_1(X),f_2(X)),$8, SBX parameters $F^{\rm eff}(X)=(f_1(X),f_2(X)),$9 and $F^{\rm safe}(X)=(f_3(X),f_4(X)).$0, PM parameters $F^{\rm safe}(X)=(f_3(X),f_4(X)).$1 and $F^{\rm safe}(X)=(f_3(X),f_4(X)).$2, and DE parameters as described in Sec. III.C of the paper. The principal performance metric is

$F^{\rm safe}(X)=(f_3(X),f_4(X)).$3

where $F^{\rm safe}(X)=(f_3(X),f_4(X)).$4 is the hypervolume with respect to a common reference (Chen et al., 23 Mar 2026). The benchmark paper uses the equivalent multiparty hypervolume perspective and recommends reporting best, median, worst, mean, and standard deviation over 30 runs (Luo et al., 2024).

6. Empirical results, interpretation, and scope

The quantitative comparison in (Chen et al., 23 Mar 2026) reports meanHV $F^{\rm safe}(X)=(f_3(X),f_4(X)).$5 std over 30 runs for all six cases. BPAIMA achieves the highest meanHV in every case:

• Case 1: $F^{\rm safe}(X)=(f_3(X),f_4(X)).$6
• Case 2: $F^{\rm safe}(X)=(f_3(X),f_4(X)).$7
• Case 3: $F^{\rm safe}(X)=(f_3(X),f_4(X)).$8
• Case 4: $F^{\rm safe}(X)=(f_3(X),f_4(X)).$9
• Case 5: $\min \bigl(F^{\rm eff}(X),F^{\rm safe}(X)\bigr),$0
• Case 6: $\min \bigl(F^{\rm eff}(X),F^{\rm safe}(X)\bigr),$1

For comparison, NSGA-II yields substantially lower values, ranging from $\min \bigl(F^{\rm eff}(X),F^{\rm safe}(X)\bigr),$2 to $\min \bigl(F^{\rm eff}(X),F^{\rm safe}(X)\bigr),$3, and OptMPNDS2, BPNNIA, and BPHEIA occupy an intermediate range (Chen et al., 23 Mar 2026). The paper further states that BPAIMA's meanHV exceeds the second-best by 8–20%, and that variances overlap only minimally. Qualitatively, NSGA-II solutions "fluctuate heavily and often violate one DM's optimality," whereas BPAIMA "converges smoothly to a dense common Pareto front in both DM spaces" (Chen et al., 23 Mar 2026).

Subsequent work using BPMO-UAVPP as a real-world testbed reports a similar pattern. In the MPIA study, normal MOEAs such as NSGA-II and NSGA-III are "consistently worst," MPNDS2 and BPAIMA improve performance, and MPIA achieves the highest sumHV across 12 cases, with Wilcoxon tests at $\min \bigl(F^{\rm eff}(X),F^{\rm safe}(X)\bigr),$4 indicating statistical superiority in 11 of 12 cases (Chen et al., 29 Mar 2026). This suggests that the BPMO-UAVPP benchmark is sensitive to algorithmic mechanisms specifically designed to exploit multiparty nondomination information.

The current scope of BPMO-UAVPP is nonetheless limited in clearly identified ways. The original paper lists a single UAV, a static environment, only two DMs, and the absence of explicit no-fly-zone or terrain-shadow constraints in the current testbed as limitations (Chen et al., 23 Mar 2026). The benchmark likewise assumes that "all map and risk parameters are precomputed and static" (Luo et al., 2024). Proposed extensions include multiparty MPMO with a third "cost" or "stealth" department, dynamic replanning under time-varying risks, cooperative multi-UAV BPMO with inter-UAV collision avoidance, and integration of machine-learning-based risk maps from real urban data (Chen et al., 23 Mar 2026).

A plausible implication is that BPMO-UAVPP occupies two roles simultaneously. It is, first, a domain model for urban UAV path planning under departmental conflict between efficiency and safety. It is, second, a canonical benchmark for evaluating multiparty evolutionary algorithms whose objective is not merely to approximate a single Pareto front, but to approach the Pareto front of each DM "as much as possible" (Chen et al., 29 Mar 2026).