{"id":"70ef870a-6965-44da-b1b4-5f35a9a04368","arxiv_id":"2608.05585","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":7,"one_line_summary":"A rule-augmented inverse optimization framework jointly learns expert cost preferences and interpretable decision rules, improving out-of-sample route prediction on the Amazon last-mile routing challenge.","lead":"This paper develops an inverse optimization method that learns both an expert's hidden cost preferences and the soft decision rules they follow from observed decisions. It tests the method on synthetic shift and production scheduling examples plus a real Amazon last-mile routing dataset, where adding learned zone-clustering rules improves route prediction.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Feasibility of compressed zone tours is unverified; repeated-zone routes would make the suboptimality loss in Section 3.3 undefined and invalidate the Amazon case study's headline improvements.","rationale":"The reader's weakest_assumption pinpoints exactly the condition on which the central empirical claim depends: the entire suboptimality-based IOP (9) is only meaningful when each observed decision is feasible in the postulated forward model. The Amazon case study is the only real-world demonstration and the source of the headline numbers (0.0632 to 0.0533). The route abstraction in Section 4.3.2 can produce non-Hamiltonian sequences whenever drivers revisit zones, and the paper provides no evidence that this does not occur. Since the constraint set (17) is a Hamiltonian tour, any repeated zone makes x_i not in S_i; the loss being minimized is then not the suboptimality loss, and the algorithm's optimality certificate is invalid. A simple data-level check on the public dataset would settle this. I agree with the reader's conditional verdict: the paper is not acceptable until this feasibility condition is verified or the route abstraction is repaired. The separate issue that the Amazon experiment learns rule rewards and pairwise costs in two stages (Section 4.3.5) rather than jointly is a secondary concern; it weakens the 'jointly' claim but would not by itself invalidate the predictive comparison if the feasibility issue is resolved.","tokens_in":17264,"tokens_out":10730,"duration_ms":106875,"concrete_test":"On the public Amazon Last-Mile Routing dataset, for every train and test route, map each stop to its zone (using the paper's nearest-labeled-stop rule for missing labels) and compress contiguous same-zone runs into a single zone visit. Then check whether the resulting sequence is a Hamiltonian tour: no zone appears more than once and every zone in Z_r appears exactly once. Report the fraction of routes that fail. If the failure rate is above 1%, the feasibility assumption in Section 3.3 is violated for a material share of observations, and the Amazon results must be recomputed using a forward model that permits repeated zones (e.g., a path or TSP with revisits) or by discarding infeasible routes; the reported 6.7% improvement over the no-rules model should be re-evaluated on the feasible subset.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3.3 assumes every observed decision x_i lies in the feasible set S_i of the forward model; otherwise the suboptimality loss Δ_i(θ) is not the degree of suboptimality and the cutting-plane termination test c(u_i;θ)^T x_i ≤ c(u_i;θ)^T x^* can incorrectly declare optimality for infeasible observations. In the Amazon case study (Section 4.3.2), the observed zone tour is obtained by mapping stops to zones and compressing contiguous runs. If a driver leaves a zone and later returns, the compressed sequence contains that zone more than once, so it is not a Hamiltonian tour over Z_r and violates constraints (17b)–(17d). The paper does not report how many of the 6,112 training or 3,072 test routes have repeated zones, nor any feasibility diagnostic. A non-negligible fraction would mean the 0.0632 to 0.0533 Amazon Score improvement is not evidence for the proposed framework, because the loss being minimized in (9) is not well-defined for those observations. This is compounded by Section 4.3.5, where rule rewards and pairwise costs are learned in two stages rather than jointly as in the central formulation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes an inverse mixed-integer optimization framework that jointly learns the cost parameters of a forward optimization problem and the rewards of soft decision rules, expressed as propositional-logic constraints reformulated into mixed-integer form. The method minimizes a suboptimality-based loss via a cutting-plane algorithm. Three case studies are presented: a synthetic shift assignment problem, a synthetic production planning problem, and the Amazon Last Mile Routing Research Challenge. The central empirical claim is that adding learned decision rules to pairwise cost learning improves out-of-sample route prediction, with Amazon Score improving from 0.0632 (distance baseline) to 0.0533, about a 15.8% reduction, and from 0.0571 for the best model without decision rules, about a 6.7% improvement (Section 4.3.8). The framework is interpretable and the Amazon case study includes a held-out test set and a structural cluster-crossing diagnostic.","tokens_in":17579,"tokens_out":5668,"duration_ms":64531,"significance":"If the claims hold, the paper makes a useful contribution to interpretable imitation learning by unifying inverse optimization with soft logic-based decision rules. The framework is general, the cutting-plane algorithm is standard and has a finite-convergence argument, and the production-planning experiments demonstrate a clear predictive advantage of rule augmentation in a controlled setting. The Amazon case study is valuable because it is a real operational dataset with an established evaluation metric, and the paper goes beyond aggregate scores by reporting cluster-crossing diagnostics and route-level examples. I also credit the paper for explicitly documenting the suboptimality-loss formulation and its feasibility assumptions, which makes the analysis checkable. The main weaknesses are that the Amazon evaluation relies on an unverified feasibility assumption for the compressed zone tours and on a two-stage heuristic rather than the joint formulation, and that the production-planning results contain a sign inconsistency in the rule-reward interpretation.","major_comments":[{"comment":"The suboptimality loss in Eq. (9) is only well defined if every observed decision x_i lies in the feasible set S_i, as stated in Section 3.3. In the Amazon case study, the observed zone tour is obtained by mapping stops to zones and compressing contiguous same-zone runs. If a driver leaves a zone and later returns, the compressed sequence contains that zone more than once and therefore violates the Hamiltonian-tour constraints (17b)-(17d). The paper does not report how many of the 6,112 training or 3,072 test routes contain repeated zones, nor any feasibility diagnostic. If a non-negligible fraction of routes are infeasible, the reported improvements in Section 4.3.8 (0.0632 to 0.0533) are not computed against a valid loss. Please add statistics on repeated-zone occurrences and either restrict the evaluation to representable routes or extend the forward model to allow zone revisits (e.g., by permitting additional visits through a dummy-copy construction), and re-run the analysis.","section":"Section 3.3 and Section 4.3.2"},{"comment":"The Amazon case study does not solve the joint inverse optimization problem (9)-(10) that is the paper's central formulation. Instead, decision-rule rewards P^d are learned first with theta^d fixed to haversine distances, and then pairwise costs theta^d are refined with P^d fixed. The paper does not quantify the gap between this two-stage heuristic and the joint solution, nor does it justify the claim in the abstract that the approach jointly learns preferences and rules on this real-world example. The comparison in Section 4.3.8 therefore evaluates a heuristic pipeline, not the proposed framework. Please either solve the joint problem on the Amazon data, or explicitly characterize the two-stage procedure as an approximation and report a cross-validated comparison between the joint and two-stage variants, including the difference in Amazon Score.","section":"Section 4.3.5"},{"comment":"There is a sign inconsistency in the production planning case study. In the rule-augmented forward model (16a), the objective is maximized and the term involving m_pj is + sum m_pj z_pj, with z_pj = 1 indicating that the rule 'at most p products' is satisfied. Under this formulation, a positive m_pj rewards satisfaction of the rule, while a negative m_pj discourages setting z_pj = 1. However, Section 4.2.6 reports learned values m_51 = -274.08, m_52 = -289.46, and m_51 = -205.90 and interprets these large negative values as a strong preference for satisfying the at-most-five rule. With the given sign convention, negative values make z_pj = 0 optimal and thus indicate that the rule is not being used. This undermines the interpretability claim of that case study. Please correct the sign convention in the formulation or in the reported values, and verify that the learned rule weights reproduce the ground-truth rule preference under the corrected model.","section":"Section 4.2.4 and Section 4.2.6"},{"comment":"The inverse problem in (8)-(9) defines the observation x_i as the full decision vector x := (x_bar, y, z), including the auxiliary literal variables y and the rule-indicator variables z. The case studies, however, only observe the base decisions (assignments, production quantities, or zone tours); the paper never states how y_i and z_i are assigned to each observed decision. This matters because the suboptimality loss c(u_i;theta)^T x_i - min_{x in S_i} c(u_i;theta)^T x and the cut-generation test in Algorithm 1 depend on the chosen z_i and y_i. For example, in the shift assignment and production planning models, z_i can be set to 0 or 1 depending on whether the rule is considered satisfied, and the loss value changes accordingly. Please specify the canonical completion rule used in the experiments (for instance, set y_i from the literal definitions and set z_i to the value that makes the observation feasible while minimizing or maximizing the rule-reward term), and confirm that the same rule is applied in all three case studies.","section":"Section 3.2 and Section 3.3"}],"minor_comments":[{"comment":"There is a typo in the sentence introducing the hypothesized model: 'the analyst knows tat some of the workers' should read 'the analyst knows that some of the workers.'","section":"Section 4.1.3"},{"comment":"The reported rule weights contain a repeated index: 'm_51 = -274.08, m_52 = -289.46, and m_51 = -205.90' should presumably be m_53 = -205.90 for the third facility.","section":"Section 4.2.6"},{"comment":"The distance-based baseline is reported as 0.0632 in the paragraph on decision-rule depth and as 0.0633 in the later comparison against the best no-rules model; please make the numbers consistent.","section":"Section 4.3.8"},{"comment":"The cluster-rule labels in Tables 1-3 use notation such as 'W-x.y' and 'W-x.Z' that is introduced informally; a short sentence defining each component (W, x, y, Z) and the meaning of the dot would improve readability.","section":"Section 4.3.4 and Table 1"},{"comment":"The cutting-plane algorithm is described for the joint formulation, but in the Amazon experiments it is applied to the two-stage procedure; please clarify in the text whether Algorithm 1 is run separately for P^d and theta^d or whether the two-stage approach uses a different termination criterion.","section":"Algorithm 1 and Section 4.3.5"}],"recommendation":"major_revision","confidential_remarks":"The core idea is promising and the framework is presented clearly, but the empirical evaluation needs to address the feasibility of the zone-tour abstraction and the gap between the joint formulation and the two-stage Amazon pipeline. The sign error in the production planning case study should be resolved before acceptance, as it currently weakens the interpretability claims. I would be willing to review a revised version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nI read the Holani et al. paper on inverse mixed-integer optimization for learning interpretable decision models. The central idea — augmenting the forward problem with learnable soft decision rules and recovering both costs and rule rewards from observed decisions — is genuinely new as far as I know. The reformulation via Raman–Grossmann logic constraints and the suboptimality-loss IP are standard but combined cleanly. The synthetic case studies (shift assignment and production planning) are convincing: the recovered rule weights match ground truth, and the comparison with the no-rules model shows the rules add predictive value.\n\nThe real-world Amazon case is where the paper gets shaky. Section 3.3 assumes every observed decision x_i lies in the feasible set S_i of the forward model. In the Amazon setup, that means the compressed zone-level tour must be a Hamiltonian tour over Z_r. The paper compresses contiguous runs of the same zone into a single visit, but if a driver leaves a zone and later returns, the compressed sequence contains that zone more than once, which violates the tour constraints. The paper never reports how many of the 6,112 training or 3,072 test routes exhibit repeated zones, and there is no feasibility diagnostic. If a non-negligible fraction does, the suboptimality loss is not well-defined for those observations, and the 15.8% Amazon Score improvement is not evidence for the framework.\n\nA few smaller issues: the two-stage learning in Section 4.3.5 (first rule rewards, then pairwise costs) is a heuristic that deviates from the joint formulation in Section 3, and the paper does not justify that the two-stage solution is anywhere near the joint optimum. No code or data are provided, and the Amazon results are single runs with no error bars or comparison to prior routing results. These are addressable but currently missing.\n\nDespite these gaps, I think the framework is worth developing. The synthetic results and the clarity of the exposition argue that a serious referee should see this. The feasibility issue is a clear, checkable fix, and if the authors can show the compressed tours are mostly feasible or adjust the model to allow repeated zones, the paper could be reasonably solid.\n\nI'd bring this to a reading group, but I would not cite it until the Amazon feasibility question is resolved. Send it to review, with a note asking the referees to pin down the compression issue.","headline":"A genuinely new framework for joint learning of costs and decision rules, but the Amazon case study rests on an unverified feasibility assumption that could undermine its headline numbers.","tokens_in":18049,"tokens_out":3976,"would_cite":false,"duration_ms":42021,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C11","90C27","90B06"],"pacs":[],"model":"deepseek-v4-flash","headline":"A unified inverse-optimization framework learns both an expert's hidden cost preferences and the soft decision rules behind observed choices.","keywords":["inverse optimization","mixed-integer programming","decision rules","interpretable models","expert decision making","imitation learning","last-mile routing","suboptimality loss"],"falsifier":"Take a synthetic expert with known cost weights and known decision rules, generate decisions, then run the proposed inverse algorithm on data where some observed decisions are infeasible in the hypothesized forward model (for example, a driver route that revisits a zone non-contiguously before all its stops are served). If the learned cost weights and rule rewards do not converge to values that reproduce the expert's decisions—or if the model's prediction accuracy is no better than a cost-only baseline—the central claim would be falsified.","tokens_in":17099,"feed_emoji":"🚚","tokens_out":10057,"duration_ms":87388,"temperature":0.7,"pith_summary":"Experts often cannot articulate why they make the choices they do, but the decisions themselves are recorded. This paper proposes an inverse-optimization framework that assumes each observed decision is an approximately optimal solution to an underlying mixed-integer program whose cost coefficients and soft 'decision rules' (logical conditions the expert tends to follow) are unknown. Rather than learning only preferences or only rules, the framework learns both at once: rules are encoded as propositional-logic constraints with learnable rewards, and the inverse problem minimizes the suboptimality of the observed decisions. Across a shift-assignment problem, a production-planning problem, and a real-world Amazon delivery-route dataset, the joint model predicts held-out decisions better than models that learn pairwise costs alone. The routing case is the headline: adding two zone-clustering rules and then refining pairwise costs cuts the Amazon Score from 0.0632 to 0.0533, roughly a 16% improvement over the distance baseline.","feed_headline":"Learned rules plus costs cut Amazon route error 16%","feed_subtitle":"A unified inverse-optimization model recovers not just preferences but the soft decision rules behind expert choices.","key_machinery":"The load-bearing object is the rule-augmented forward MILP: a base mixed-integer linear program with unknown cost coefficients $\\bar{c}(u)$, augmented by a set of soft decision rules $\\Omega_r(\\bar{x},u)$, each represented as a propositional-logic statement reformulated into mixed-integer linear inequalities (via the Raman–Grossmann reformulation) and activated by a binary variable $z_r$ with a learnable reward $d_r(u)$. The inverse problem minimizes the suboptimality loss $\\Delta_i(\\theta)=c(u_i;\\theta)^\\top x_i-\\min_{\\tilde{x}\\in S_i} c(u_i;\\theta)^\\top \\tilde{x}$, and the cutting-plane algorithm of Wang (2009) solves the semi-infinite program by iteratively adding the most violated alternative solutions as cuts. This machinery turns the intractable bilevel formulation into a sequence of linear master problems and MILP forward solves.","core_discovery":"The central claim is that a single inverse-optimization formulation can simultaneously recover the cost parameters of a decision-maker's optimization problem and the soft logical rules that shape their choices, and that doing so yields more accurate, interpretable models than recovering costs alone. The paper develops a rule-augmented forward problem in which each candidate decision rule is a proposition over the decision variables and context, reformulated as mixed-integer linear constraints using the Raman–Grossmann framework, and is activated by a binary variable carrying a learnable reward. The inverse problem then minimizes the suboptimality loss of the observed decisions via a cutting-plane algorithm that alternates between a master problem and forward solves. In the Amazon Last Mile Routing Research Challenge, the learned rules correspond to drivers visiting zones in contiguous hierarchical clusters, and the cluster-crossing diagnostic shows that rules, not pairwise travel costs, reproduce the block structure of human routes. The paper concludes that modeling both preferences and rules is essential when rules materially shape observed decisions.","pith_inferences":["A fully joint estimation of rule rewards and pairwise costs, instead of the paper's two-stage Amazon pipeline, might trade off rule strength against cost detail differently; comparing the two reveals how much of the 6.7 percent gain is attributable to the staging rather than to the rules themselves.","The feasibility assumption means the framework silently ignores experts whose decisions cannot be represented in the forward model, such as drivers who revisit a zone non-contiguously; extending the loss to penalize infeasible observations is a natural next step.","The cluster-crossing diagnostic (predicted routes crossing clusters more often than observed routes) is a cheap, model-free signal for when a forward model is missing structural rules, and could serve as a stopping rule for rule-set construction in other domains."],"forward_implications":["A model trained on historical decisions can be queried for both the expert's value system and their standing heuristics, making the reasoning auditable for training novices or certifying automated systems.","In routing, the recovered zone-clustering rules predict that drivers serve zones contiguously in a hierarchy; a planning system that ignores this will systematically underestimate the cost of interleaving zones.","Learned rule rewards quantify how strongly each hypothesized heuristic influences decisions, so an analyst can prune or prioritize rules by estimated impact.","The framework carries over to any mixed-integer decision problem where candidate rules can be stated as propositions, not just routing or scheduling."],"supporting_citations":[{"why":"Supplies the reformulation of propositional logic statements into mixed-integer linear constraints, which is how soft decision rules are embedded in the forward model.","marker":"Raman and Grossmann (1994)"},{"why":"Provides the cutting-plane algorithm for inverse mixed-integer linear programs that the paper adapts; its finite-convergence guarantee underlies the solution method.","marker":"Wang (2009)"},{"why":"Introduces the suboptimality-based loss for inverse integer optimization that the paper minimizes to handle imperfect observations.","marker":"Moghaddass and Terekhov (2020, 2021)"},{"why":"Establishes the data-driven inverse-optimization setting with noisy observations that motivates the loss-based formulation.","marker":"Aswani et al. (2018)"},{"why":"Describes the Amazon Last-Mile Routing dataset used in the main real-world case study, including route and zone structure.","marker":"Merchán et al. (2022)"},{"why":"Provides the competition dataset and problem definition that anchors the routing case study.","marker":"Amazon.com Inc (2021a)"}],"fun_headline_variants":["Inverse optimization recovers expert rules and costs, cutting error 16%","Jointly learn expert costs and rules via inverse optimization, cut route error 16%","One inverse-optimization pass learns expert costs and decision rules","Amazon route error drops 16% with new inverse-optimization model"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The framework assumes every observed decision is feasible in the forward model, including the auxiliary decision-rule variables; in the routing application this requires that each driver's stop sequence, after mapping stops to zones and compressing contiguous same-zone runs, forms a Hamiltonian tour over the zones.","fun_headline_variants_meta":{"raw":{"variants":["Inverse optimization recovers expert rules and costs, cutting error 16%","Jointly learn expert costs and rules via inverse optimization, cut route error 16%","One inverse-optimization pass learns expert costs and decision rules","Amazon route error drops 16% with new inverse-optimization model"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001855,"raw_usage":{"total_tokens":7275,"prompt_tokens":927,"completion_tokens":6348,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":543,"completion_tokens_details":{"reasoning_tokens":6267}},"tokens_in":543,"tokens_out":6348,"duration_ms":43187,"temperature":1.0,"reasoning_tokens":6267,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T06:00:43.162716+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a synthetic expert with known cost weights and known decision rules, generate decisions, then run the proposed inverse algorithm on data where some observed decisions are infeasible in the hypothesized forward model (for example, a driver route that revisits a zone non-contiguously before all its stops are served). If the learned cost weights and rule rewards do not converge to values that reproduce the expert's decisions—or if the model's prediction accuracy is no better than a cost-only baseline—the central claim would be falsified.","supporting_citations":[],"review_version":1}