MOID: Multi-Objective Infeasibility Diagnosis

• Multi-Objective Infeasibility Diagnosis (MOID) is a framework that combines multi-objective optimization with LLM-enabled analysis to diagnose infeasible routing models.
• It treats constraint violations as soft objectives and employs Pareto trade-offs between path cost and violation levels to guide model revisions.
• MOID leverages LLM agents to generate constraint-checking code and solution analyses that provide concrete suggestions for adjusting conflicting routing requirements.

Multi-Objective Infeasibility Diagnosis (MOID) is a framework that combines an automatic routing solver with multi-objective optimization and LLM agents to diagnose routing models whose feasible set is empty (Li et al., 5 Aug 2025). It treats “constraint violation” as a soft objective, optimizes it jointly with the original path-cost objective to generate a set of trade-off solutions, and then uses an LLM-generated solution analysis function to translate those solutions into actionable model modification suggestions (Li et al., 5 Aug 2025). In the routing setting studied in "Multi-Objective Infeasibility Diagnosis for Routing Problems Using LLMs" (Li et al., 5 Aug 2025), MOID is designed for real-world vehicle routing problems (VRPs) in which users often specify conflicting or unreasonable requirements, such as overly tight or contradictory time windows, vehicle capacity limits that are too small relative to demand, route length limits that cannot be respected given geography, pickup-and-delivery precedence constraints, same-vehicle requirements, and priority ordering constraints. The broader constrained multi-objective perspective is informed by the multi-objective-violation landscape developed in "An Instance Space Analysis of Constrained Multi-Objective Optimization Problems" (Alsouly et al., 2022), which formalizes how feasibility structure and objective trade-offs interact.

1. Conceptual basis and problem setting

Infeasibility in routing arises when the imposed constraints produce an empty feasible solution set. In the MOID formulation, this is not treated as a terminal failure state but as a diagnostic object: the system searches for routes that jointly expose the trade-off between path cost and the degree of constraint violation (Li et al., 5 Aug 2025). This distinguishes MOID from LLM-based infeasibility-diagnosis methods that pursue minimal adjustments to restore feasibility and typically ignore the original cost objective, and from ARS (Automatic Routing Solver), which can seek solutions that minimize violation when infeasible but remains single-objective (Li et al., 5 Aug 2025).

The routing model in the paper is defined on a graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$, with depot $0$ and customers $\{1,\ldots,n\}$. Costs $c_{ij}$ are defined on edges $(i,j)$, and binary routing decisions are $x_{ij}\in\{0,1\}$, indicating whether arc $(i,j)$ is used (Li et al., 5 Aug 2025). The objective is

$$\min \sum_{i \in \mathcal{V}} \sum_{j \in \mathcal{V}} c_{ij} x_{ij}, \quad \text{subject to } \mathbf{x} \in C,$$

where $C$ encodes routing constraints such as capacity, distance or length limits, time windows, pickup-and-delivery or precedence, same-vehicle coupling, and priority (Li et al., 5 Aug 2025).

The paper enumerates six constraint families used to build 50 VRP variants (Li et al., 5 Aug 2025):

Constraint family	Abbreviation	Description in the routing model
Vehicle Capacity	C	Capacity limits
Distance Limits	L	Distance/length limits
Time Windows	TW	Time-window requirements
Pickup and Delivery	PD	Pickup-and-delivery/precedence
Same Vehicle	S	Same-vehicle coupling
Priority	P	Priority ordering constraints

These constraints are enforced through LLM-generated “constraint checking and scoring programs” (Li et al., 5 Aug 2025). Incorrect parameterization, such as shrinking time windows, can create an empty feasible set (Li et al., 5 Aug 2025).

2. Multi-objective formulation and infeasibility representation

MOID adopts the standard multi-objective optimization problem (MOP) form

$$\text{minimize}\quad \mathbf{f}(\mathbf{x}) = (f_1(\mathbf{x}), \ldots, f_m(\mathbf{x}))^{\intercal},\quad \text{subject to } \mathbf{x}\in\Omega,$$

and instantiates it as the bi-objective routing vector

$0$0

where $0$1 is the total path cost and $0$2 is a violation score computed by the LLM-generated constraint scoring program (Li et al., 5 Aug 2025). MOID minimizes both objectives simultaneously.

The paper does not fix a closed-form penalty for violation. Instead, an LLM-generated “constraint scoring program” returns $0$3 by aggregating violations across all constraints specified in natural language, and this scoring acts as a soft objective guiding multi-objective search (Li et al., 5 Aug 2025). Constraints are therefore treated as soft via $0$4, while the metaheuristic uses destroy/repair operators and local search to iteratively reduce both cost and violation (Li et al., 5 Aug 2025).

Pareto dominance and Pareto optimality are used in the standard sense. Dominance is defined by $0$5 iff $0$6 for all $0$7 and there exists $0$8 with $0$9, and $\{1,\ldots,n\}$0 is Pareto-optimal if there is no $\{1,\ldots,n\}$1 such that $\{1,\ldots,n\}$2 dominates $\{1,\ldots,n\}$3 (Li et al., 5 Aug 2025). The output of MOID is therefore an approximation to a Pareto front of cost–violation trade-offs rather than a single repaired model.

The second paper provides a broader CMOP formalization in which infeasibility is represented through aggregate constraint violation (Alsouly et al., 2022). For a CMOP with inequality and equality constraints, it defines

$\{1,\ldots,n\}$4

with

$\{1,\ldots,n\}$5

and $\{1,\ldots,n\}$6 (Alsouly et al., 2022). It also introduces the combined map

$\{1,\ldots,n\}$7

which treats infeasibility and objective quality jointly in a “multi-objective-violation landscape” (Alsouly et al., 2022). This suggests a useful conceptual interpretation of MOID: the routing-specific violation scorer plays the role of a domain-instantiated violation coordinate in a joint objective–violation space.

3. Pipeline and algorithmic components

The MOID pipeline contains three modules (Li et al., 5 Aug 2025). In the generation stage, LLM agents produce constraint-aware heuristic code: a constraint checker and a violation scorer tailored to the natural-language routing description. In the optimization stage, a multi-objective metaheuristic, called the Augmented Multi-objective Solver, searches over routes while minimizing both cost and violation to build a diverse final population approximating the Pareto front. In the analysis stage, an LLM agent generates a “solution analysis function” that processes each trade-off solution and outputs concrete model adjustments to render that solution feasible under a modified model (Li et al., 5 Aug 2025).

The optimization framework uses NSGA-II for non-dominated sorting and diversity via crowding distance, and the paper also compares MOEA/D (Li et al., 5 Aug 2025). The reported settings are population size $\{1,\ldots,n\}$8 and iterations $\{1,\ldots,n\}$9 (Li et al., 5 Aug 2025). Each iteration has three phases.

First, destroy–repair is applied. Destroy operators are selected via roulette wheel among random removal and string removal, and greedy repair reinserts removed customers (Li et al., 5 Aug 2025). The repair operator attempts to restore feasibility where possible or reduce violation when feasibility is impossible (Li et al., 5 Aug 2025).

Second, single-point Pareto local search (SPLS) is executed with 2-OPT, SWAP, and SHIFT (Li et al., 5 Aug 2025). Each operator attempts modifications at each node, producing $c_{ij}$0 candidate solutions (Li et al., 5 Aug 2025). Dominance selection replaces the current solution with a dominating neighbor, and ties are broken using minimum crowding distance to promote diversity (Li et al., 5 Aug 2025). SPLS runs until no improvement or a 1-second timeout (Li et al., 5 Aug 2025).

Third, population update is performed with NSGA-II by non-domination rank and crowding distance (Li et al., 5 Aug 2025). The condensed algorithmic outline is: initialize a population $c_{ij}$1 of size $c_{ij}$2; for $c_{ij}$3, apply destroy–repair and SPLS to each solution; combine improved solutions with the current population; apply NSGA-II to select $c_{ij}$4; and return the final population as an approximate Pareto front, after which each solution is fed to the LLM analysis function to produce suggestions (Li et al., 5 Aug 2025).

The paper reports that NSGA-II converges to better HV and IGD metrics, whereas MOEA/D exhibits more stable convergence (Li et al., 5 Aug 2025). The broader CMOP literature represented by (Alsouly et al., 2022) provides a related vocabulary for interpreting such behavior, emphasizing the effect of infeasible regions and the interaction between constraints and objectives on search dynamics.

4. Analysis stage, inverse formulations, and suggestion generation

A defining feature of MOID is that it does not stop at producing trade-off routes. It also generates a solution analysis function that takes the original problem description, the constraint-checking code, and a single solution $c_{ij}$5 from the Pareto set, and returns numerical adjustments plus a succinct natural-language suggestion (Li et al., 5 Aug 2025). The function’s purpose is to infer the minimal constraint changes that would make $c_{ij}$6 feasible and, in Strategy 2, optimal for a revised model (Li et al., 5 Aug 2025).

The analysis stage is described using forward and inverse optimization models (Li et al., 5 Aug 2025). The forward model is

$c_{ij}$7

where $c_{ij}$8 encodes the cost vector and $c_{ij}$9 are constraint parameters (Li et al., 5 Aug 2025). In infeasible cases, $(i,j)$0 is empty (Li et al., 5 Aug 2025). The inverse optimization formulation estimates parameters so that a provided decision becomes feasible or optimal:

$(i,j)$1

For linear problems, the paper gives the KKT-based inverse-feasible set used in Strategy 2:

$(i,j)$2

together with a convex estimation formulation

$(i,j)$3

where $(i,j)$4 captures admissible parameters and $(i,j)$5 (Li et al., 5 Aug 2025).

Two design strategies are specified for the solution analysis function (Li et al., 5 Aug 2025). Strategy 1, Direct Constraint Relaxation (DCR), computes for each violated constraint indexed by $(i,j)$6 a change $(i,j)$7 that places $(i,j)$8 on the boundary:

$(i,j)$9

It aggregates across all violations and returns the single most demanding final value per unique constraint (Li et al., 5 Aug 2025). Strategy 2, Estimating Constraint Parameters (ECP), solves the inverse-optimization-based minimal parameter change problem

$x_{ij}\in\{0,1\}$0

The Appendix provides structured prompts for DCR and ECP that instruct the LLM to test the solution against every constraint, compute per-violation relaxations, aggregate to the most demanding final value per constraint, and append recommendations to the original description (Li et al., 5 Aug 2025).

The concrete suggestion types include widening time windows by $x_{ij}\in\{0,1\}$1, increasing vehicle capacity to at least total load on routes in $x_{ij}\in\{0,1\}$2, relaxing per-route maximum length so that each route is within the new bound, adjusting precedence slack or allowing certain relaxations consistent with $x_{ij}\in\{0,1\}$3, and merging or splitting assignments to satisfy same-vehicle requirements or allow exceptions if desired (Li et al., 5 Aug 2025). The paper does not introduce explicit ranking or clustering of suggestions; users instead perceive trade-offs by inspecting different Pareto solutions and their associated suggestions (Li et al., 5 Aug 2025).

5. Empirical evaluation, metrics, and ablations

The experimental evaluation uses 50 types of infeasible VRP variants constructed from the six constraint families C, L, TW, PD, S, and P (Li et al., 5 Aug 2025). Instances are based on Solomon C103 with 25 nodes, and for time windows the start and end times are divided by 10 to induce infeasibility (Li et al., 5 Aug 2025). The reported environment is Intel i7-9700, 32 GB RAM, Windows 10, with DeepSeek V3 as the LLM and 3 runs per experiment (Li et al., 5 Aug 2025).

The metrics are Hypervolume (HV), Inverted Generational Distance (IGD), and Analysis Success Rate (ASR) (Li et al., 5 Aug 2025). The paper defines hypervolume as

$x_{ij}\in\{0,1\}$4

with normalization via approximated ideal $x_{ij}\in\{0,1\}$5 and nadir $x_{ij}\in\{0,1\}$6, and reference point $x_{ij}\in\{0,1\}$7 (Li et al., 5 Aug 2025). IGD is

$x_{ij}\in\{0,1\}$8

Against OptiChat and ARS on four VRP variants, the paper reports that MOID yields clear cost–violation trade-offs, whereas ARS tends to prioritize minimal violation ignoring cost (Li et al., 5 Aug 2025). Both converge in approximately 20 iterations, and MOID achieves better HV and IGD (Li et al., 5 Aug 2025).

The quantitative HV and IGD results reported in the paper are as follows (Li et al., 5 Aug 2025):

Variant	HV (OptiChat / ARS / MOID)	IGD (OptiChat / ARS / MOID)
CVRP	$x_{ij}\in\{0,1\}$9 / $(i,j)$0 / $(i,j)$1	$(i,j)$2 / $(i,j)$3 / $(i,j)$4
CVRP-L	$(i,j)$5 / $(i,j)$6 / $(i,j)$7	$(i,j)$8 / $(i,j)$9 / $$\min \sum_{i \in \mathcal{V}} \sum_{j \in \mathcal{V}} c_{ij} x_{ij}, \quad \text{subject to } \mathbf{x} \in C,$$0
DCVRP	$$\min \sum_{i \in \mathcal{V}} \sum_{j \in \mathcal{V}} c_{ij} x_{ij}, \quad \text{subject to } \mathbf{x} \in C,$$1 / $$\min \sum_{i \in \mathcal{V}} \sum_{j \in \mathcal{V}} c_{ij} x_{ij}, \quad \text{subject to } \mathbf{x} \in C,$$2 / $$\min \sum_{i \in \mathcal{V}} \sum_{j \in \mathcal{V}} c_{ij} x_{ij}, \quad \text{subject to } \mathbf{x} \in C,$$3	$$\min \sum_{i \in \mathcal{V}} \sum_{j \in \mathcal{V}} c_{ij} x_{ij}, \quad \text{subject to } \mathbf{x} \in C,$$4 / $$\min \sum_{i \in \mathcal{V}} \sum_{j \in \mathcal{V}} c_{ij} x_{ij}, \quad \text{subject to } \mathbf{x} \in C,$$5 / $$\min \sum_{i \in \mathcal{V}} \sum_{j \in \mathcal{V}} c_{ij} x_{ij}, \quad \text{subject to } \mathbf{x} \in C,$$6
DCVRP-L	$$\min \sum_{i \in \mathcal{V}} \sum_{j \in \mathcal{V}} c_{ij} x_{ij}, \quad \text{subject to } \mathbf{x} \in C,$$7 / $$\min \sum_{i \in \mathcal{V}} \sum_{j \in \mathcal{V}} c_{ij} x_{ij}, \quad \text{subject to } \mathbf{x} \in C,$$8 / $$\min \sum_{i \in \mathcal{V}} \sum_{j \in \mathcal{V}} c_{ij} x_{ij}, \quad \text{subject to } \mathbf{x} \in C,$$9	$C$0 / $C$1 / $C$2

Runtime is reported as similar to ARS, while OptiChat uses a 300 s default Gurobi limit (Li et al., 5 Aug 2025). The case studies on CVRP-L in the Appendix show suggested changes that extend beyond mere feasibility restoration, covering varying degrees of adjustments and associated cost impacts (Li et al., 5 Aug 2025).

The ablation results identify three patterns (Li et al., 5 Aug 2025). First, NSGA-II is more likely to converge to better HV and IGD, while MOEA/D shows smoother, more stable progress. Second, across 100 solution analysis examples, Strategy 1 (DCR) achieves higher ASR than Strategy 2 (ECP) due to simpler computation and coding demands. Third, DeepSeek-V3 consistently generates more correct analysis functions than ChatGPT-4o, and the violation scoring program is stable across runs, with score ranges remaining relatively fixed over 10 repetitions on CVRP and CVRP-L.

6. Theoretical position, limitations, and broader relevance

MOID is a metaheuristic method and provides no global optimality guarantees; it empirically approximates the Pareto front (Li et al., 5 Aug 2025). Its per-iteration computational structure comprises destroy/repair on $C$3 solutions, SPLS generating $C$4 candidates per solution, and NSGA-II sorting across combined candidates. With $C$5, $C$6, and SPLS limited by a 1 s timeout, runtime remains practical according to the reported experiments (Li et al., 5 Aug 2025).

Its practical strengths derive from combining multi-objective generation with post hoc inverse-analysis. Instead of producing a single repaired formulation, it returns multiple representative actionable suggestions from one run (Li et al., 5 Aug 2025). This is significant because infeasible routing models often admit multiple plausible revisions rather than one uniquely preferred modification. A plausible implication is that MOID is especially useful when model revision is a decision-support process involving stakeholder preferences over cost degradation and rule relaxation.

The method also has explicit limitations (Li et al., 5 Aug 2025). It is focused on routing. Its reliability depends on the LLM’s ability to correctly generate constraint-checking and scoring programs and analysis functions. Strategy 2 (ECP) is computationally and programmatically harder, and the quality of the Pareto-front approximation depends on solver hyperparameters, local search richness, and heuristic diversity.

The second paper supplies a broader theoretical context for understanding when infeasibility-aware multi-objective methods are likely to succeed or fail (Alsouly et al., 2022). It concludes that two key characteristics, the isolation of the non-dominated set and the correlation between constraints and objectives evolvability, have the greatest impact on algorithm performance. It quantifies these properties through features such as $C$7, $C$8, $C$9, $$\text{minimize}\quad \mathbf{f}(\mathbf{x}) = (f_1(\mathbf{x}), \ldots, f_m(\mathbf{x}))^{\intercal},\quad \text{subject to } \mathbf{x}\in\Omega,$$0, $$\text{minimize}\quad \mathbf{f}(\mathbf{x}) = (f_1(\mathbf{x}), \ldots, f_m(\mathbf{x}))^{\intercal},\quad \text{subject to } \mathbf{x}\in\Omega,$$1, $$\text{minimize}\quad \mathbf{f}(\mathbf{x}) = (f_1(\mathbf{x}), \ldots, f_m(\mathbf{x}))^{\intercal},\quad \text{subject to } \mathbf{x}\in\Omega,$$2, $$\text{minimize}\quad \mathbf{f}(\mathbf{x}) = (f_1(\mathbf{x}), \ldots, f_m(\mathbf{x}))^{\intercal},\quad \text{subject to } \mathbf{x}\in\Omega,$$3, and $$\text{minimize}\quad \mathbf{f}(\mathbf{x}) = (f_1(\mathbf{x}), \ldots, f_m(\mathbf{x}))^{\intercal},\quad \text{subject to } \mathbf{x}\in\Omega,$$4 (Alsouly et al., 2022). This suggests a principled interpretation of MOID’s performance differences across routing variants: instances with strong isolation of feasible trade-offs or strong conflict between objective progress and violation reduction should be more difficult for any multi-objective diagnostic solver.

The papers also delimit future directions. The routing paper states that the paradigm of multi-objective generation plus LLM inverse-analysis is applicable to other COPs such as scheduling and bin packing, provided that constraint-checking and scoring code can be generated (Li et al., 5 Aug 2025). It also notes that MOID can integrate other MOEAs, decomposition strategies, richer repair operators, and retrieval-augmented prompting for more reliable code generation (Li et al., 5 Aug 2025). The CMOP instance-space paper argues that current benchmarks do not provide enough diversity to fully reveal the efficacy of constrained multi-objective algorithms, and recommends expanding instance generators to vary PF isolation, constraint–objective alignment, and equality-constraint structure (Alsouly et al., 2022). Taken together, these results place MOID at the intersection of infeasibility diagnosis, constrained multi-objective search, and LLM-mediated model revision: a routing-specific framework whose general form is compatible with a wider class of constrained optimization problems (Li et al., 5 Aug 2025, Alsouly et al., 2022).