manufacturing schedulers

Abstract: We revisit scheduling with obligatory testing, where each job consists of two operations: a test with an initially known time and an operation whose processing time is revealed upon completion of the job's test. We develop an iterative competitive-ratio-preserving instance-reduction method that enables us to compute exact competitive ratios directly on highly structured instances for various index rules. Our main result is a clean analysis showing that the competitive ratio of the algorithm $1$-SORT is exactly the Golden Ratio $\varphi\approx 1.618$, improving the recent best-known upper bound of $1.861$ for arbitrary testing times. We further leverage this method to obtain near-tight bounds for a simple randomized algorithm on instances with uniform testing times.

Abstract: Dynamic scheduling requires dispatching online arrivals while anticipating future jobs, as poor decisions can compound downstream costs under nonlinear tardiness objectives. We show that a pretrained large language model (LLM) can serve as the exogenous component of a structured world model, forecasting uncertain future job arrivals. Specifically, we exploit an afterstate decomposition, which separates known deterministic shop transitions from uncertain exogenous job arrivals, enabling the LLM to forecast only the latter through a compact belief representation. Building on this, we develop Afterstate World-Model Planning (AWMP), which combines an evolved, belief-conditioned dispatching strategy with short scenario-based rollouts. A surprise-triggered refresh mechanism queries the LLM only when newly observed arrivals deviate sufficiently from previous forecast, and online residual self-calibration further conditions the LLM on its recent forecast errors to handle nonstationary arrivals. Across diverse arrival distributions, AWMP reduces weighted tardiness by 6-21% on dynamic job-shop and 22-46% on flexible job-shop benchmarks, relative to the best of eight classical priority-dispatching rules.

Abstract: Column generation folklore warns that initial columns hurt: a warm start may bias the dual and slow the root of branch-price-and-cut. We show the opposite once the columns are imperfect in the right way. Preloading the support of the optimal root LP, the best columns one could know in advance, fails to finish on half of our hardest instances; a pool of ordinary routes that need not belong to any optimal solution, held outside the master and priced at the current smoothed dual, speeds the root up. Because the columns need not be optimal, producing them becomes a learning task: a neural edge scorer guides a constructive generator that builds the pool in seconds. A second network predicts the initial center of Wentges smoothing from the instance file, and with it the direction of the smoothed pricing queries; the solver's own updates run unchanged. With prefill and center in place, exact pricing dominates root pricing time, so we rebuild bucket-graph labeling around its dominance test for the GPU and certify on the CPU. Inside RouteOpt, the state-of-the-art exact solver for capacitated vehicle routing, the pipeline reaches the same certified root bound in 13.2% of vanilla RouteOpt's root pricing time over 95 instances with 300 to 1000 customers, prefill alone in 26.1%, and 17.5 times faster in the best case. In an exact branch-price-and-cut solver for unrelated parallel machine scheduling, prefill with smoothing and the learned center certifies all 25 benchmark instances, 12 times faster than the solver's own column generation, which certifies 20 of them.

Abstract: The Distributed Flexible Job Shop Scheduling Problem (DFJSP) requires the joint determination of factory assignment, machine selection, and operation sequencing to minimize the makespan. This paper presents an exact SAT-based approach built on an efficient SAT formulation that exploits problem structure through propagation, symmetry breaking, and incremental SAT solving to effectively address these coupled decisions. The proposed method is evaluated on 76 standard benchmark instances from the Fattahi, Brandimarte, and rdata datasets under two- and three-factory configurations. Together with previously established results, the new SAT optimality certificates bring the total number of proven-optimal instances to 72 out of 76, including three newly certified instances; for two of these instances, SAT also improves the previous best-known makespans. Across the benchmark configurations, SAT achieves the best results on four of the five configurations and remains competitive with strong exact approaches such as CP and LBBD1_CPIL, as well as existing metaheuristic methods. These results demonstrate that the proposed SAT framework is an effective exact approach for the DFJSP, combining strong solution quality with rigorous optimality certification and offering a complementary exact method for future DFJSP studies and evaluations on additional benchmark instances.

Abstract: Designing effective heuristics for diverse combinatorial optimization problems requires substantial expertise and repeated search. Large language models (LLMs) automate heuristic generation and refinement, but heuristic search typically depends on evaluation feedback from the problem being optimized. Generalizing to new problem definitions using only source-task feedback therefore remains a central challenge. We introduce MECo, an LLM-driven multi-task evolutionary framework for zero-shot cross-problem generalization. MECo maintains task-conditioned heuristic populations and uses a transfer gap based on cross-task population performance to guide their interactions. These interactions enable the transfer and recombination of heuristics. A complementary selection criterion then constructs a compact heuristic set by rewarding each member's additional coverage of source combinations. The selected set is applied to target problems without further search or adaptation. Experiments on 32 problem variants across vehicle routing (VRP) and flexible job-shop scheduling (FJSP) show that MECo achieves the lowest mean costs compared with eight automated heuristic design (AHD) baselines under the same budgets. On out-of-domain problems, it outperforms the strongest baseline in each family. Moreover, integrating the framework of MECo with different AHD methods improves their ID and OOD performance in both families, supporting its effectiveness across different methods.

Abstract: Large language models (LLMs) have provided a unified interface for end-to-end combinatorial optimization (CO), but textual serialization alone may obscure spatial and relational structures that are important for generating effective CO solutions. This paper presents a general-purpose vision-language solver that augments textual instance descriptions with input-derived visual representations. A single vision-language model (VLM) is applied across different CO tasks and trained using supervised fine-tuning followed by verifier-guided reinforcement learning. While the visual inputs contain no gold solutions or solution-derived information, our experiments show that the VLM generally improves solution quality over its text-only counterpart, with particularly clear gains on more complex CO problems such as CVRP and JSSP. The advantage of visual information is more pronounced at large problem scales.

Abstract: Scheduling problems arise from repeatedly selecting one item from a set of candidates based on their states. These problems often reduce to assigning priority scores and choosing the highest-ranked item. In this work, we propose a factorized scheduling principle (FSP) framework to learn interpretable and transferable scheduling rules. The FSP framework represents system states as condition distributions and decomposes a global scheduling principle into additive univariate and pairwise components with identifiability constraints. The scheduling principle enables the framework to maintain a simple priority-based structure during deployment. This principle is learned by using a policy-based objective combined with a temporal-difference signal defined on the condition distribution. Experiments on synthetic and realistic scheduling tasks demonstrate the FSP framework's strong performance, interpretability, and zero-shot generalization across different system scales.

Abstract: Solving large-scale linear programs efficiently is an important challenge in many optimization settings. A key technique is column generation, which alternates between solving the master problem over a restricted subset of the variables, and using a pricing subproblem to identify new variables to add. The pricing subproblem is guided by the dual solution of the current restricted master problem, but oscillations in these dual solutions can substantially slow convergence. Dual stabilization methods address this issue. Dual smoothing is a common stabilization method, which guides the pricing subproblem using a combination of the current dual solution and duals from previous iterations. However, while past dual solutions can stabilize the dual trajectory, they do not necessarily guide pricing towards useful new variables. We therefore introduce predictive dual smoothing, which instead combines the current dual solution with a learned prediction of future duals to steer pricing towards variables that are more useful in subsequent iterations. The predictor is trained offline using supervision extracted from standard column generation trajectories and is used only to modify the pricing subproblem's objective function, while exact reduced-cost checks and fallback pricing with the unsmoothed duals preserve correctness. Experiments on cutting stock and generalized assignment problems show that predictive dual smoothing substantially reduces generated columns and wall-clock time relative to standard column generation and existing classical and learned stabilization methods. These gains extend to out-of-distribution instance sizes, and predictive smoothing provides further improvements when combined with strong classical stabilization.

Abstract: In a landmark paper, Goemans and Rothvoss (2020) established an XP algorithm running in time $\text{enc}(P)^{2^{O(d)}} \cdot \text{enc}(Q)^{O(1)}$ for the \emph{Cone and Polytope Intersection} problem: finding a vector $y \in \text{int.cone}(P \cap \mathbb{Z}^d) \cap Q$ together with a sparse certificate $λ\in \mathbb{Z}_{\ge 0}^{P \cap \mathbb{Z}^d}$ supported on at most $2^{2d+1}$ generators, where $P \subseteq \mathbb{R}^d$ is a bounded rational polyhedron and $Q \subseteq \mathbb{R}^d$ is an arbitrary rational polyhedron. For high-multiplicity bin packing, this gives a running time of ${|I|}^{2^{O(d)}}$, where $|I|$ denotes the encoding length of the input. Recently, Koana and Kumabe (2026) proved that the decision variant of this problem is fixed-parameter tractable (FPT) parameterized by the number of item types $d$ with running time $2^{d^{O(d)}} \cdot {|I|}^{O(1)} = 2^{2^{O(d \log d)}} \cdot {|I|}^{O(1)}$. In this work, we generalize the framework of Koana and Kumabe from standard bin packing to the full Cone and Polytope Intersection Problem of Goemans and Rothvoss, directly encompassing high-multiplicity bin packing, point-in-cone, and scheduling. Secondly, by combining Carathéodory-type integer cone bounds (Eisenbrand and Shmonin, 2006) with active support enumeration, we reduce the running time to: $$2^{2^{O(d)}} \cdot (\text{enc}(P) + \text{enc}(Q))^{O(1)}$$. Under the Exponential Time Hypothesis (ETH), the double-exponential lower bound of Kowalik, Lassota, Majewski, Pilipczuk, and Sokołowski (2024) for point-in-cone and Jansen, Ohnesorge, and Pirotton (2026) for high-multiplicity bin packing implies that this parameter dependence is asymptotically optimal. Finally, we provide an explicit decompression algorithm that extracts a solution with sparse support $|\text{supp}(λ)| \le 2^{2d+1}$ in single-exponential FPT time.

Abstract: We study nonpreemptive scheduling on a single machine with release dates, due dates, positive job weights, and a common processing time. The objective is to minimize total weighted tardiness. Although closely related equal-processing-time problems admit polynomial-time algorithms, the complexity of this problem has remained open in the literature since 2010. We prove that its decision version is strongly NP-complete, even when every job can meet its due date if processed immediately upon release. The reduction is from unweighted MAX-CUT and uses a quadratic number of jobs with polynomially bounded numerical data. Its main ingredient is a constructive normalization theorem that converts every sufficiently inexpensive feasible schedule into a binary choice for each graph vertex; after normalization, total weighted tardiness equals a constant minus a scaled cut value. We also give a deterministic polynomial-time phase-grid assignment algorithm for the shifted objective $Φ=F+p\sum_jw_j$, where $F$ is total weighted tardiness. The algorithm enumerates at most $N$ release-date residues modulo $p$, solves one minimum-cost assignment problem for each residue, and returns the best phase-grid schedule. It runs in $O(N^5)$ arithmetic operations and achieves the tight ratio $3/2-1/(2N)$ for this algorithm. Because the added term $p\sum_jw_j$ is independent of how the jobs are scheduled, the shifted and original objectives have exactly the same optimal schedules. However, the approximation guarantee applies to the shifted objective; for the original objective, the analysis provides an additive bound. Thus, the paper both resolves the long-standing complexity question and provides a complementary worst-case guarantee for the phase-grid assignment algorithm.

Abstract: Bin packing asks whether a collection of items can be packed into at most a given number of bins of a given capacity. We consider the high-multiplicity setting with $d$ distinct item sizes, in which both the item sizes and the number of items of each size are encoded in binary. Goemans and Rothvos (JACM 2020) gave an XP algorithm parameterized by $d$. Whether this problem is fixed-parameter tractable (FPT) in $d$ has remained a central open problem. We resolve this question by giving a deterministic $O^*(2^{d^{O(d)}})$-time algorithm. We formulate bin packing as an integer linear program (ILP) with at most $(d+1)d^d$ variables. A bin configuration records the number of items of each type in one bin. We partition these configurations by their coordinate remainders modulo $d$. For each class, we use one variable for the bin count and $d$ variables for the total item counts. The convex hull of each class has the integer decomposition property, which guarantees that every feasible ILP solution corresponds to a packing.

Abstract: We study the single-machine total-completion-time problem $1||\sum C_j$ when processing times are unknown but each job comes with a reported interval. Jobs are initially processed in nondecreasing order of reported upper bound. If a job is still unfinished after receiving that much service, the reported upper bound has been violated and the policy switches to a resumable geometric fallback. Each interruption of an unfinished job incurs an additive penalty $κ$. For a residual set of $m$ jobs, known size ratio $D=s_1/s_0$, normalized interruption penalty $λ=κ/s_0$, and geometric depth $K$, we derive an explicit worst-case coefficient $Ψ_{m,D,λ}(K)$. When $λ>0$, a finite minimizing depth exists and can be chosen after the trigger from the observed number of unfinished jobs; the optimal depth decreases with the interruption penalty and increases with the residual set size. We also derive the exact worst-case coefficient $R_{n,D}$ for an arbitrary nonpreemptive list when all processing times lie in a known bounded range. These bounds give guarantees for valid intervals, a single interval failure, arbitrary reports, no-trigger outcomes, and random instances. In the single-failure regime, they also give a precise condition under which the proved fallback guarantee is smaller than the bounded-range continuation guarantee.

Abstract: This paper investigates a two-machine permutation flow shop in which the second machine is unavailable during one fixed interval $[s,t]$. We consider the non-resumable setting: an operation interrupted by the interval must restart from the beginning after the machine becomes available. The objective is to minimize the makespan. We establish three results. First, we give a polynomial-time $10/7$-approximation algorithm. Second, we develop a pseudopolynomial-time exact dynamic program. Third, we prove that the problem does not admit a fully polynomial-time approximation scheme (FPTAS) unless $\mathrm{P}=\mathrm{NP}$, even when the non-availability interval has unit length. Together, these results characterize a distinctive complexity profile: exact optimization is possible in pseudopolynomial time, whereas the usual route from such an algorithm to an FPTAS is impossible unless $\mathrm{P}=\mathrm{NP}$. They also reveal an approximability separation from the corresponding non-resumable problem with the interval on the first machine.

Abstract: We consider the two-machine open-shop scheduling problem in which one machine is unavailable during a fixed interval. We study the resumable setting: an operation interrupted by the unavailability interval may resume, without penalty, when the machine becomes available. The objective is to minimize the makespan. Although the problem is NP-hard and several approximation algorithms are known, whether it admits a fully polynomial-time approximation scheme (FPTAS) has remained open for two decades. We resolve this question affirmatively by giving the first FPTAS, thereby strengthening the previously known polynomial-time approximation scheme (PTAS). As an intermediate result, we develop a new pseudo-polynomial dynamic program with seven state dimensions, improving on the ten-dimensional formulation in the literature.

Abstract: The Travelling Salesman Problem (TSP) remains a key area of research in combinatorial optimization, with applications in logistics, manufacturing, and service delivery. This paper addresses a bi-objective service-oriented TSP in which the clients' ranks in the delivery path matter. Unlike conventional depot-based TSP formulations, the considered problem does not assume a distinguished depot or a fixed tour origin. To address this setting, we adapt the Miller--Tucker--Zemlin (MTZ)-based formulation and derive an original linearization of the resulting model, enabling its solution with off-the-shelf integer linear programming solvers. This adaptation avoids the rigid tour origin imposed by the conventional MTZ formulation, for which fixing the starting node does not affect the tour cost but can affect the objective in a customer-rank-sensitive TSP. To solve this problem, we present a Node-Shift-Encoding (NSE)-based Genetic Algorithm augmented with fuzzy reasoning to update the reference tour throughout the evolutionary process. Experimental evaluation on TSPLIB benchmarks demonstrates that the proposed method achieves improved performance compared with the classical NSE approach.

Abstract: Sequential reinforcement learning (RL) solvers and global diffusion model (DM) solvers for neural combinatorial optimization exhibit complementary failure modes under an optimization-regret view. The former enjoys small marginal regret in the early construction stage, but suffers from horizon-wise compounding errors with super-linear regret growth; the latter avoids horizon compounding but incurs linear or sublinear regret w.r.t. the dimension of the remaining unsolved subspace. We propose Hybrid Neural Solver for Combinatorial Optimization (HyCO), a hybrid inference algorithm that constructs a solution prefix with an RL solver and adaptively switches to a conditional DM to complete the remaining decisions. To characterize why such hybridization helps, when to trigger the handover, and how to realize it in practice, we first develop a unified error-scaling theoretical framework and prove that, under explicit error-scaling assumptions, i) the hybrid structure achieves strictly lower expected regret than either backbone alone, and ii) there exists a unique optimal trigger step that minimizes the hybrid regret. We then design a lightweight adaptive trigger that combines policy entropy and RL-DM disagreement to detect trajectory-level signals of the regime shift as a practical proxy, since the optimal trigger step is defined at the expected-regret level and is not directly computable on individual trajectories. Experimental results on diverse benchmarks demonstrate that HyCO achieves consistent improvements over both backbones and support the empirical effectiveness of adaptive triggering.