Low-Cost Labels, Reliable Choices: Rollout-Calibrated Hyper-Heuristics for Job Shop Scheduling

# Rollout-Calibrated Hyper-Heuristics for Job Shop Scheduling
Source: [https://arxiv.org/html/2605.23957](https://arxiv.org/html/2605.23957)
## Low\-Cost Labels, Reliable Choices: Rollout\-Calibrated Hyper\-Heuristics for Job Shop Scheduling

Yanxiao Li1Yifu Zhao1Zhenhong Peng3Baili Lu1Dexing Yao1Haochen Li1Qinbin He1Sio\-Kei Im4Yapeng Wang1&Xu Yang1 1Faculty of Applied Sciences, Macao Polytechnic University, Macao, 999078, China 2Pazhou Lab \(Huangpu\), Guangzhou, 510555, China 3College of Animal Science and Technology, Zhongkai University of Agriculture and Engineering, Guangzhou, 510225, China 4Macao Polytechnic University, Macao, 999078, China Corresponding author: xuyang@mpu\.edu\.mo

###### Abstract

Learning\-assisted hyper\-heuristics can select among dispatching rules while preserving the feasibility and interpretability of constructive Job Shop Scheduling Problem \(JSSP\) heuristics\. Their main computational cost lies in label generation rather than model fitting, since each supervised label usually requires rolling out candidate rules from a partial schedule\. We study this label\-cost problem together with a reliability problem: a learned selector should not switch away from a strong default rule unless the predicted gain is credible\. The proposed selector uses regret\-normalized rollout labels, a contextual KNN uncertainty estimate, and a gate that acts only when the predicted improvement exceeds an uncertainty\-adjusted margin\. We also vary rollout depth and breadth to measure the cost\-quality trade\-off\. On synthetic JSSP instances, the gated selector achieves the lowest mean RPD among learned selectors, remains close to the best fixed dispatching rule, and reduces Random\-HH mean RPD by more than an order of magnitude\.

## 1Introduction

Selection hyper\-heuristics choose among low\-level rules while a schedule is being built\. For JSSP, this gives a useful middle ground between fixed dispatching rules and end\-to\-end learned constructors: the action space remains small, feasibility is maintained by construction, and the rule choices are directly interpretable\.

This setting falls within the LEAD themes of hyper\-heuristics and operator/algorithm selection\. The learner does not design a new dispatching rule; it decides which existing rule should be trusted at a given partial schedule\. This separation isolates rule selection from the separate problem of inventing new search operators\.

The main bottleneck is supervision\. A common labeler rolls out every candidate rule from a partial schedule and records the terminal makespan\. This gives direct supervision, but a full rollout costsO​\(\|ℋ\|​T\)O\(\|\\mathcal\{H\}\|T\)simulations for a state withTTremaining decisions\. A second issue appears at inference: a learned score can be slightly better than the default rule for accidental reasons, and always following the lowest score can lose to a strong fixed rule\.

Our formulation ties these issues together by storing rollout outcomes as per\-state regret rather than raw or lower\-bound\-normalized makespan, so the target reflects the local ranking of rules\. The selector then uses a variance estimate from a contextual KNN regressor to switch away from the default rule only when the predicted margin is larger than the uncertainty penalty\. To make the supervision cost explicit, we vary rollout depth and breadth and report the resulting cost\-quality trade\-off\.

The scope differs from specialized JSSP solvers, deep\-RL dispatchers such as L2DZhanget al\.\([2020](https://arxiv.org/html/2605.23957#bib.bib8)\), and program search systems such as FunSearchRomera\-Paredeset al\.\([2024](https://arxiv.org/html/2605.23957#bib.bib15)\)and ReEvoYeet al\.\([2024](https://arxiv.org/html/2605.23957#bib.bib16)\)\. The contribution is a reproducible CPU\-only selector and evaluation protocol for studying label cost and conservative switching in learning\-assisted hyper\-heuristics\.

The experiments therefore measure the value of regret labels, the reliability benefit of uncertainty gating, and the amount of rollout depth or breadth that can be removed before the selector loses its advantage\. This evaluation matches a practical deployment choice about how much simulation budget should be spent before a learned selector is used\.

## 2Related Work

Selection hyper\-heuristics are the closest line of work to our setting because they choose low\-level heuristics for combinatorial optimization problemsBurkeet al\.\([2013](https://arxiv.org/html/2605.23957#bib.bib1)\)\. Online\-learning variants use multi\-armed banditsFialhoet al\.\([2010](https://arxiv.org/html/2605.23957#bib.bib2)\)or reinforcement learningÖzcanet al\.\([2010](https://arxiv.org/html/2605.23957#bib.bib3)\)to control rule choice at each decision point\. Prior work on dispatching rule selection for JSSP includes the imitation\-learning selector of Ingimundardottir and RunarssonIngimundardottir and Runarsson \([2018](https://arxiv.org/html/2605.23957#bib.bib4)\)and RANFORSJunet al\.\([2019](https://arxiv.org/html/2605.23957#bib.bib5)\), where rule priorities are learned from instance features\.

Learning\-based scheduling provides a parallel line of work\. Recent studies treat JSSP as a sequential decision process, as in L2DZhanget al\.\([2020](https://arxiv.org/html/2605.23957#bib.bib8)\)and ScheduleNetParket al\.\([2021](https://arxiv.org/html/2605.23957#bib.bib9)\), often trained with policy gradients\. Related neural combinatorial\-optimization methods such as POMOKwonet al\.\([2020](https://arxiv.org/html/2605.23957#bib.bib10)\)also illustrate the dependence of learned policies on training distributions\. In contrast, our selector uses rollout labels, engineered state features, and a CPU\-only KNN regressor\.

Rollout policies supply the label\-generation mechanism used here\. Bertsekas\-style rollout policiesBertsekas \([2013](https://arxiv.org/html/2605.23957#bib.bib11)\)estimate the value of an action by simulating a base policy\. We borrow this idea for label generation rather than online control, which lets us amortize the simulation cost across many decision points\. Surrogate\-based hyper\-heuristicsHorn and Bischl \([2016](https://arxiv.org/html/2605.23957#bib.bib12)\); Brankeet al\.\([2016](https://arxiv.org/html/2605.23957#bib.bib13)\)similarly trade simulation cost for generalization\. Our Pareto sweep makes the rollout cost visible instead of treating it as a fixed preprocessing step\.

LLM\- and agentic\-driven algorithm design is related but addresses a different part of the pipeline\. LLM\-based systems such as EoHLiuet al\.\([2024](https://arxiv.org/html/2605.23957#bib.bib14)\), FunSearchRomera\-Paredeset al\.\([2024](https://arxiv.org/html/2605.23957#bib.bib15)\), and ReEvoYeet al\.\([2024](https://arxiv.org/html/2605.23957#bib.bib16)\)can generate or refine heuristic code directly\. Our method keeps the rule pool fixed and studies when a compact selector should trust its predictions\.

The work is also connected to metaheuristic design and applications, where LEAD studies often improve search algorithms by designing new movement patterns, update equations, or hybrid prediction pipelines\. Recent examples include TSWOAWeiet al\.\([2025](https://arxiv.org/html/2605.23957#bib.bib18)\)and geometric WOAWeiet al\.\([2026b](https://arxiv.org/html/2605.23957#bib.bib19)\)for engineering design, MRBMOLuet al\.\([2025](https://arxiv.org/html/2605.23957#bib.bib21)\)for numerical optimization, and metaheuristic\-assisted prediction models such as NAWOA\-XGBoostWeiet al\.\([2026a](https://arxiv.org/html/2605.23957#bib.bib20)\)and ASKSSA\-CNN\-BiLSTMLiet al\.\([2026](https://arxiv.org/html/2605.23957#bib.bib22)\)\. These studies modify the optimizer itself\. Our work is complementary: the low\-level rules are fixed, and learning is used only to select among them\.

## 3Method

### 3\.1JSSP and Selection Hyper\-Heuristics

A JSSP instance is a tuple\(𝒥,ℳ,π,p\)\(\\mathcal\{J\},\\mathcal\{M\},\\pi,p\)where each jobj∈𝒥j\\in\\mathcal\{J\}has an ordered operation sequenceπj\\pi\_\{j\}over machinesℳ\\mathcal\{M\}with processing timespp\. A serial schedule generatorGiffler and Thompson \([1960](https://arxiv.org/html/2605.23957#bib.bib17)\)builds a feasible schedule by repeatedly selecting the next operation from the set of ready jobs\. The makespanCmaxC\_\{\\max\}is the largest job completion time\. We use the classical priority dispatching rules covered in standard scheduling\-rule surveys and job\-shop benchmarksPanwalkar and Iskander \([1977](https://arxiv.org/html/2605.23957#bib.bib6)\); Holthaus and Rajendran \([1997](https://arxiv.org/html/2605.23957#bib.bib7)\)\. The rule poolℋ\\mathcal\{H\}containsSPT,LPT,MWKR,LWKR,MOPNR,FIFO, andRandom\. These rules choose by shortest or longest processing time, most or least remaining work, most remaining operations, first\-in\-first\-out order, or a uniform random choice, respectively\. A selection hyper\-heuristic chooses oneh∈ℋh\\in\\mathcal\{H\}at every decision point as a function of the current statess\.

### 3\.2Regret\-Normalized Rollout Labels

Given a partial statess, we evaluate each candidatehhby completing the schedule fromssusinghhfor the next step and then continuing the rollout\. Letm​\(s,h\)m\(s,h\)denote the resulting makespan\. The classical normalized label ism​\(s,h\)/LB​\(I\)m\(s,h\)/\\mathrm\{LB\}\(I\)whereLB\\mathrm\{LB\}is the instance lower bound\. We instead use the per\-state regret

r​\(s,h\)=m​\(s,h\)−minh′∈ℋs⁡m​\(s,h′\)minh′∈ℋs⁡m​\(s,h′\),r\(s,h\)=\\frac\{m\(s,h\)\-\\min\_\{h^\{\\prime\}\\in\\mathcal\{H\}\_\{s\}\}m\(s,h^\{\\prime\}\)\}\{\\min\_\{h^\{\\prime\}\\in\\mathcal\{H\}\_\{s\}\}m\(s,h^\{\\prime\}\)\},\(1\)whereℋs⊆ℋ\\mathcal\{H\}\_\{s\}\\subseteq\\mathcal\{H\}is the candidate subset evaluated atss\. By constructionr​\(s,h\)≥0r\(s,h\)\\geq 0withr​\(s,h\)=0r\(s,h\)=0for the best rule\. The target is therefore a local ranking signal instead of an absolute makespan scale\.

### 3\.3Uncertainty\-Gated Selection

We fit a contextual KNN regressor on\(s,h\)\(s,h\)pairs whose features are described in Section[3\.5](https://arxiv.org/html/2605.23957#S3.SS5)\. For test statesswe compute, for eachh∈ℋh\\in\\mathcal\{H\}:

r^​\(s,h\)\\displaystyle\\hat\{r\}\(s,h\)=∑i∈Nk​\(s,h\)wi​yi∑iwi,\\displaystyle=\\frac\{\\sum\_\{i\\in N\_\{k\}\(s,h\)\}w\_\{i\}\\,y\_\{i\}\}\{\\sum\_\{i\}w\_\{i\}\},σ^​\(s,h\)=stdi∈Nk​\(s,h\)​\(yi\),\\displaystyle\\hat\{\\sigma\}\(s,h\)=\\mathrm\{std\}\_\{i\\in N\_\{k\}\(s,h\)\}\(y\_\{i\}\),\(2\)whereNkN\_\{k\}is thekk\-nearest training neighbor set andwi=1/\(di\+ϵ\)w\_\{i\}=1/\(d\_\{i\}\+\\epsilon\)\. Leth⋆=arg⁡minh⁡r^​\(s,h\)h^\{\\star\}=\\arg\\min\_\{h\}\\hat\{r\}\(s,h\)andh0h\_\{0\}be a fixed default rule \(in our experiments, the best fixed rule on the training set\)\. The*gated*policy appliesh⋆h^\{\\star\}only when the expected improvement overh0h\_\{0\}exceeds a confidence margin:

πgated​\(s\)=\{h⋆if​r^​\(s,h0\)−r^​\(s,h⋆\)\>λ​σ^​\(s,h⋆\),h0otherwise\.\\pi\_\{\\mathrm\{gated\}\}\(s\)=\\begin\{cases\}h^\{\\star\}&\\text\{if \}\\hat\{r\}\(s,h\_\{0\}\)\-\\hat\{r\}\(s,h^\{\\star\}\)\>\\lambda\\,\\hat\{\\sigma\}\(s,h^\{\\star\}\),\\\\ h\_\{0\}&\\text\{otherwise\.\}\\end\{cases\}\(3\)The hyperparameterλ\\lambdatrades aggressiveness for reliability:λ=0\\lambda=0recoversarg⁡minh⁡r^\\arg\\min\_\{h\}\\hat\{r\}, while largeλ\\lambdacollapses to the default rule\. We also report an LCB\-style varianthlcb=arg⁡minh⁡r^​\(s,h\)−λ​σ^​\(s,h\)h\_\{\\mathrm\{lcb\}\}=\\arg\\min\_\{h\}\\hat\{r\}\(s,h\)\-\\lambda\\hat\{\\sigma\}\(s,h\)\.

### 3\.4Rollout\-Budget Trade\-off

The cost of label generation is governed by the rollout breadthb=\|ℋs\|b=\|\\mathcal\{H\}\_\{s\}\|, which determines how many candidate rules are evaluated per state, and the rollout depthκ\\kappa, which determines how long the candidate rule is followed before completion resumes with the default ruleh0h\_\{0\}\. The full\-rollout setting withκ=∞\\kappa=\\inftyandb=\|ℋ\|b=\|\\mathcal\{H\}\|is the most expensive case\. By varying\(κ,b\)\(\\kappa,b\)on a small grid, we trace a Pareto curve relating label generation cost, measured by rollout steps or wall\-clock seconds, to the test RPD of the learned selector\.

### 3\.5State Features

The state representation contains 35 instance\- and state\-level features, covering instance size, processing\-time statistics, machine load imbalance, schedule progress, ready\-set processing times, remaining work, and machine and job readiness moments, together with a 7\-dimensional one\-hot encoding of the candidate heuristic\. All features arezz\-normalized at training time and the same normalization is reused at inference\.

### 3\.6Training Pipeline

For each training instance, we sample partial schedules by following uniformly random dispatching rules\. At each sampled state, the labeler evaluates a subset of candidate rules, applies each candidate for the next decision, and completes the remaining schedule according to the rollout depth setting\. Full depth means that the candidate rule is followed to terminal completion; finite depth means that the rollout returns to the default rule afterκ\\kappacandidate\-guided steps\. The recorded makespans are converted to either normalized targets or the regret target in Eq\. \([1](https://arxiv.org/html/2605.23957#S3.E1)\)\.

The default ruleh0h\_\{0\}is chosen on the training split as the fixed rule with the lowest mean makespan\. This makes the gating test conservative: the learned selector must justify switching away from a rule that is already competitive on the deployment distribution\. During inference, the KNN model scores every candidate rule at each decision point and the gate in Eq\. \([3](https://arxiv.org/html/2605.23957#S3.E3)\) decides whether to switch away fromh0h\_\{0\}\.

## 4Experiments

### 4\.1Setup

We evaluate on synthetic JSSP instances at three scales:6×66\\times 6,10×1010\\times 10, and15×1015\\times 10\(jobs×\\timesmachines\)\. Each instance draws an independent random machine permutation per job and processing times uniformly from\[1,99\]\[1,99\]\. We use 150 training instances and 40 held\-out test instances per scale\.

The selector is a CPU\-based contextual KNN regressor withk=7k=7\. Its fitting step normalizes the rollout feature vectors and stores their targets; nearest neighbor distances are computed with NumPy on CPU during inference\. The rollout labels are collected from 25 sampled states per training instance and three exploration trajectories per instance under a uniform\-random rule mixture\. The default training setting evaluates all seven candidate rules at each sampled state and rolls each candidate to terminal completion, with breadthb=7b=7and depthκ=∞\\kappa=\\infty, unless Section[4\.5](https://arxiv.org/html/2605.23957#S4.SS5)explicitly varies the rollout budget\. The gating defaulth0h\_\{0\}is selected automatically as the best fixed rule on the training set, which is typicallyFIFOorMOPNRat our scales\.

The learned configurations are*Norm\-Argmin*, which uses lower\-bound\-normalized rollout labels witharg⁡min\\arg\\minselection,*Regret\-Argmin*, and*Regret\-Gated*atλ=1\\lambda=1\. We compare them with the same dispatching\-rule baselinesSPT,LPT,MWKR,LWKR,MOPNR, andFIFOPanwalkar and Iskander \([1977](https://arxiv.org/html/2605.23957#bib.bib6)\); Holthaus and Rajendran \([1997](https://arxiv.org/html/2605.23957#bib.bib7)\); a*Random\-HH*baseline that chooses a rule uniformly at random at each decision and is averaged over five seeds; and the*Oracle\-Fixed*reference, which selects the best fixed rule*per instance, after the fact*\. We report mean relative percentage deviation \(RPD\) against this hindsight reference, the median RPD, and the number of test instances on which a method is best\. Oracle\-Fixed is used only to normalize performance and is not a deployable scheduling policy\. Experiments were run on Ubuntu 20\.04 with an Intel Xeon E5\-2698 v4 CPU \(40 logical cores\), 251 GiB RAM, and Python 3\.13; no GPU was used for label generation, KNN fitting, or inference\.

### 4\.2Example Schedule Visualization

![Refer to caption](https://arxiv.org/html/2605.23957v1/x1.png)Figure 1:Example 6×\\times6 schedules produced by FIFO and Regret\-Gated\.Figure[1](https://arxiv.org/html/2605.23957#S4.F1)gives a schedule\-level view of the policies\. Each bar is one operation, the row is the assigned machine, and the color marks the job\. The visualization is not used as an optimization metric; it checks that the learned selector produces a feasible JSSP schedule\.

### 4\.3Main Results

Table 1:Main test results\. Oracle\-Fixed is a hindsight reference used to compute RPD, not a deployable method\. Arrows indicate metric direction; bold and underlined values mark the best and second\-best non\-oracle entries\.6x610x1015x10Method / referencemean RPD↓\\downarrowwins↑\\uparrowmean RPD↓\\downarrowwins↑\\uparrowmean RPD↓\\downarrowwins↑\\uparrowOracle\-Fixed \(ref\.\)0\.00330\.00320\.0030Regret\-Gated \(ours\)2\.85191\.83230\.8425Regret\-Argmin4\.94163\.15211\.0725Norm\-Argmin5\.30173\.24201\.1024FIFO2\.15\\boldsymbol\{2\.15\}𝟐𝟐\\boldsymbol\{22\}1\.16\\boldsymbol\{1\.16\}𝟐𝟕\\boldsymbol\{27\}0\.78\\boldsymbol\{0\.78\}𝟐𝟕\\boldsymbol\{27\}MOPNR7\.77119\.9979\.043MWKR18\.33122\.59022\.650Random\-HH36\.68047\.42054\.540SPT103\.130197\.560233\.020LPT111\.800224\.920263\.160LWKR165\.370310\.260368\.640![Refer to caption](https://arxiv.org/html/2605.23957v1/x2.png)Figure 2:Mean RPD for learned selectors and key baselines\. Lower is better\.Table[1](https://arxiv.org/html/2605.23957#S4.T1)and Figure[2](https://arxiv.org/html/2605.23957#S4.F2)summarize our main results; the figure focuses on the learned selectors and key baselines\.*Regret\-Gated*has the lowest mean RPD among learned selectors at all three scales\. It remains close to the best fixed rule in the table,FIFO, with gaps of 0\.70, 0\.67, and 0\.06 mean RPD on6×66\\times 6,10×1010\\times 10, and15×1015\\times 10, respectively\. It also improves over the two ungated selectors and reduces Random\-HH mean RPD from 36\.68–54\.54 to 0\.84–2\.85\.

### 4\.4Ablation on Label Targets and Selection Policies

Table 2:Ablation over label targets and selection policies on 10x10\. Arrows indicate metric direction; bold marks the best mean RPD\.![Refer to caption](https://arxiv.org/html/2605.23957v1/x3.png)Figure 3:Ablation over label targets, selection policies, and confidence thresholds\.The ablation in Figure[3](https://arxiv.org/html/2605.23957#S4.F3)varies the label target, selection policy, and confidence parameter across normalized and regret labels, argmin, LCB, and gated selection, andλ∈\{0\.5,1\.0,2\.0\}\\lambda\\in\\\{0\.5,1\.0,2\.0\\\}\. The lowest mean RPD is obtained by regret\-gated selection withλ=1\.0\\lambda=1\.0\. Gating is more stable than always switching: the argmin variants are around 2\.0 mean RPD, whereas the top gated variants are near 1\.1\. LCB performs worse in this setup, suggesting that uncertainty is more useful as a switching test than as a direct score bonus\.

### 4\.5Rollout\-Budget Trade\-off

Table 3:Rollout label cost and test quality on 10x10\. Arrows indicate metric direction; bold marks the best RPD\.![Refer to caption](https://arxiv.org/html/2605.23957v1/x4.png)Figure 4:Rollout cost vs\. test RPD on 10×\\times10 JSSP\.Figure[4](https://arxiv.org/html/2605.23957#S4.F4)traces the cost\-quality frontier of rollout\-based label generation\. Full rollouts achieve the lowest RPD, whereas very shallow rollouts withκ=1\\kappa=1yield substantially higher RPD even when all rules are evaluated\. Intermediate depths reduce simulation steps substantially but leave a measurable quality gap, and breadth reduction becomes useful only when the rollout depth is not too small\.

### 4\.6Generalization Across Instance Sizes

We train at10×1010\\times 10and evaluate at15×1015\\times 10as a controlled generalization probe\. Gated selection remains substantially below Random\-HH in RPD and close to the best fixed rule, preserving the ordering observed in the main test\. This experiment is a diagnostic check rather than a full transfer benchmark; it verifies that the selector is not tuned only to one instance size\.

## 5Discussion

The fixed\-rule results clarify the role of the gate\.FIFOis strong on these synthetic distributions, the experiments report it directly, and the gate uses the best training\-set rule as its default\. The gain of gating is therefore evaluated through its ability to avoid both random rule choice and over\-aggressive learned switching, and*Regret\-Gated*is consistently stronger than the learned alternatives\.

The comparison with deep\-RL constructors such as L2DZhanget al\.\([2020](https://arxiv.org/html/2605.23957#bib.bib8)\)and ScheduleNetParket al\.\([2021](https://arxiv.org/html/2605.23957#bib.bib9)\), and with LLM\-driven program search such as FunSearchRomera\-Paredeset al\.\([2024](https://arxiv.org/html/2605.23957#bib.bib15)\)and ReEvoYeet al\.\([2024](https://arxiv.org/html/2605.23957#bib.bib16)\), should be read in this context\. Those systems target a different point in the design space, while our pipeline keeps the model small and the rule pool fixed so that label cost and switching reliability can be isolated cleanly\.

From a deployment perspective, the strongest fixed rule in our synthetic JSSP data is known only after running the comparison\. In a new domain, a practitioner may not know in advance whetherFIFO,MOPNR, or another rule is the right fallback\. The proposed selector does not remove the need for good rules, but it reduces the risk of using a learned selector that switches too often\. This objective is different from pure solver competition and is relevant when interpretability and low deployment cost matter\.

The study should also be read within a bounded scope\. It is a controlled test of rollout\-calibrated rule selection rather than a claim about all scheduling settings\. The instances remain in small and medium regimes; scaling the same labeling procedure to industrial JSSP, including50×2050\\times 20instances and larger, may expose different cost–quality trade\-offs\. The rule pool is also kept fixed to seven dispatching rules, which isolates selector behavior but does not cover rules produced by LLMs or rule\-induction systems\. Because the benchmark assumes static jobs and deterministic processing times, online arrivals, machine breakdowns, or uncertain durations would change the state distribution and would require rollout labels built for those dynamics\.

## 6Conclusion

This paper studied how rollout supervision can be used in learning\-assisted JSSP without making a selector too eager to abandon a strong dispatching rule\. The proposed rollout\-calibrated hyper\-heuristic converts rollout outcomes into per\-state regret labels and uses a KNN uncertainty gate to switch only when the predicted gain is sufficiently reliable\. On the tested synthetic instances, this conservative selector achieves the lowest mean RPD among learned methods, reduces Random\-HH RPD by more than an order of magnitude, and remains close to the best fixed rule\. The rollout\-budget sweep further shows how label cost translates into test quality, which is useful when interpretability, CPU\-only deployment, and simulation budget matter\.

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