Neural Multi-Objective Combinatorial Optimization for Flexible Job Shop Scheduling Problems
Authors: Igor G. Smit
Yaoxin Wu
Pavel Troubil
Yingqian Zhang
Wim P.M. Nuijten
Venue: ICLR 2026 Poster (2026)
Keywords: Neural Multi-Objective Combinatorial Optimization
Flexible Job Shop Scheduling Problem
Deep Reinforcement Learning
Primary area: optimization

Abstract: Neural combinatorial optimization (NCO) has made significant advances in applying deep learning techniques to efficiently and effectively solve single-objective flexible job shop scheduling problems (FJSPs). However, the more practical multi-objective FJSPs (MOFJSPs) remain underexplored, limiting the applicability of NCO in multi-criteria decision-making scenarios. In this paper, we propose a decomposition-based NCO method to solve MOFJSPs. We present the dual conditional attention network (DCAN), a neural network architecture that takes the objective preferences along with the problem instance, aiming to learn adaptable policies over the preferences. By decomposing an MOFJSP into a set of subproblems with different preferences, the learned DCAN policies generate a set of solutions that reflect the corresponding trade-offs. We customize the Proximal Policy Optimization algorithm based on decomposition to effectively train the policy network for multiple objectives and define the state and reward based on combinations of different objectives. Extensive results showcase that our approach outperforms traditional multi-objective optimization methods and generalizes well across diverse types of problem instances.

Published as a conference paper at ICLR 2026
NEURAL MULTI-OBIECTIVE COMBINATORIAL OPTI-
MIZATION FOR FLEXIBLE JOB SHOP SCHEDULING PROBLEMS
Igor G. Smit Yaoxin Wu Pavel Troubil', Yingqian Zhang Wim P.M. Nuijten
2 DDpartment of Mathematics and Computer Science, Eindhoven University of Technology
Findhoven Artificial Intelligence Systems Institute, Eindhoven University of Technology
'Department of Industrial Engineering & Innovation Sciences, Eindhoven University of Technology
DDelmia R&D, Dassault Systèmes
{i.g smit, wu2, yqzhang, w.w.m.nu.......mmmmmmm.mmmmmmmmm j.m.muuiijtenttteeee n1, pavel ttrutrruubbbi.. com
ABSTRACT
Neural combinatorial optimization (NCO) has made significant advances in apply- ing deep learning techniques to efficiently and effectively solve single-objective flexible job shop scheduling problems (FJSPs). However, the more practical multi-objective FJSPs (MOFISPs) remain underexplored, limiting the applicabil- ity of NCO in multi-criterie decision-making scenarios. In this paper, we propose a decomposittion-based NCO method to solve MOFISPs. We present the dual conditional attention network (DCAN), neural network architecture that takes the objective preferences along with the problem instance, aiming to learn adapt- able policies over the preferences. By decomposing an MOFISP into set of subproblems with different preferences, the learned DCAN policies generate a set of solutions that reflect the corresponding trade-offs. We customize the Proxi- mal Policy Optimization algorithm based on decomposition to effectively train the policy network for multiple objectives and define the state and reward based on combinations of different objectives. Extensive results showcase that our approach outperforms traditional mbjtii-ojeective optimization methods and generalizes well across diverse types of problem instances.
INTRODUCTION
The flexible job shop scheduling problem (FJSP) is one of the most well-studied combinatorial optimization (CO) problems. is complex scheduling task where multiple jobs, each made up of ordered operations, must be processed on machines. Each operation can be performed on several alternative machines with different processing times. The goal create a schedule that optimizes criteria such as minimizing the makespan. FJSP has many practical applications in industries such as semiconductor manufacturing (Tamssaouet et al.) (002)) and aluminum production (Zhang al., 2016). Constraint programming (CP: Baptiste et al. 2001 Col & Teppan 2022), heuristics (Sels) ctt al. 2012), and metahcuristics (Rooyani & Defersha 2019) have made great progress in solving FJSP, focusing mainly single-objective optimization like minimizing makespan. However, real- world scenarios often involve multiple conflicting objectives, such as tardiness, flowtime, and cost.
A straightforward approach multi-objective optimization for FJSP is to form a weighted sum of the objectives and apply single-objective methods. However, this does not provide alternative so- lutions leveraging trade-offs among the objectives. Thus, it is hard to choose appropriate objective weights, as the weights leading to preferred solutions vary across problem instances and scales. Hence, desired solution methods provide Pareto set of solutions with diverse objective trade-off'. To address this issue, one can solve multiple problems with preferences using the same optimization methods. Yet, even single-objective FJSP is NP-hard, rendering such methods too computationally expensive. Instead, more prevalent solution method is to use metaheuristics that generate a set of solutions, particularly multi-objective evolutionary algorithms. However, these metaheuristics re-
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quire extensive effort in manual tuning and specialized operator design to achieve good performance. Moreover, their efficiency and effectiveness tend to deteriorate as problem size increases.
Recently, neural combinatorial optimization (NCO) has attracted increasing attention to solve single- objective FISP. NCO methods aim to learn high-quality solution policies through deep reinforcement learning (DRL), reducing reliance heavily handcrafted strategies and enabling fast inference. NCO methods for single-objective FJSP have made great progress, mainly targeting makespan op- timization (e.g., Song al., 2022 Wang et al. 2023) and extending variants such dynamic or stochastic FJSPs (Zhao el al. 2024 Smit el al.
2025a
The multi-objective FSJP (MOFJSP) has received comparatively little research exploration. Al- though some NCO methods are developed for simple multi-objective CO problems such as multi- objective routing problems (Lin al. [022) Chen al., 2025 Li al. 2021; Zhang al.) 2023d Wang et al. 2024), these approaches are not applicable to the MOFJSP. They depend on episodic rewards and instance-wiss gradients for policy training due to simple problem structures. However, in the context of FJSP, this leads to delayed rewards due to the long decision-making horizon, in- hibiting performance. Moreover, scheduling problems have a substantially different graph structure, which requires distinct problem representations and tailored neural architectures.
We address this gap by proposing a novel decomposition- based neural multi-ohjective combinato- rial optimization (NMOCO) method for the MOFISP, introducing the dual conditional attention net- work (DCAN). DCAN employs a conditional attention mechanism that adapts operation operation-machine attention based on objective preferences, while relying solely a single neural network. Fur- thermore, we tailor the proximal policy optimization (PPO) algorithm (Schulman et al., 2017) for multi-objective optimization by defining the state and reward functions based on different combina- tions of objectives. Experimental results demonstrate that the proposed method outperforms existing multi-objective optimization methods across diverse problem instances and objective combinations. Our main contributions are: (1) a decomposition bbsed PPO framework for multi- multi-objective schedul- ing that is both theoretically grounded and practically applicable; (2) a conditional attention-based network architecture that achieves state-of-the-art performance on flexible job shop scheduling and related variants; (3) new bound-based reward functions and state features for multiple prevalent scheduling objectives, with broad applicability to other nonincreasing or nondecreasing objectives; and (4) extensive experiments showing that our approach consistently outperforms strong meta- heuristic and DRL baselines across variety of objective combinations and problem instances.
2 RELATED WORK
In recent years, many NCO methods have been developed for various scheduling problems, and most use graph neural networks (GNNs) (Smit et al., 2025b). /Zhang et al. (2020) created a constructive DRL approach for the job shop scheduling problem (JSSP), followed by others, who explored dif- ferent network architectures and learning algorithms (e.g., Park et al. 202T] Lei et al. 2022 Kwon
et al. 1021 Corsini et al. 2024: Pirna & Grimm 2024 van Remmerden et al.) 2025a b). For the FISP. Song et al. (2022) first proposed a competitive end-to-end DRL algorithm. They used a het- erogeneous graph and designed heterogeneous GNN using different graph attention layers (GAT; Velīčković et al. 2018) to encode machine and operation nodes. Since then, several network designs and learning structures have been proposed. For instance, Zhang et al. (2023a) integrated DRL and multi-agent RL using multi-agent graph. Others adapted DRL methods to handle various dynamic FJSP variants (Zhao al. 2024| Zhang al. 2023c b). the stochastic FJSP (Smit et al. 2025a), or different FJSP extensions (Zhng et al. [2024] Li et al. 2025). WWang et al. (2023) proposed the cur- rent state-of-the-an -frtthearrt FJSP network architecture, using dual attention network (DAN) that comprises both self- and cross-attention, achieving superior performance over previous approaches.
There are couple of preliminary works on the MOFISP (Luo al., 2021 2022, Wu et al., 2023)
However, they use trivial vector-based state, restrict potential actions to subset of priority dis- patching rules, and limit applicability to a specific variant of the dynamic MOFISP. Moreover, these works train only one policy that optimizes specific trade-off point between objectives. More re- cently, (Su et al.) 2024) proposed a method to learn different policies based on different preference vectors. However, this requires separate actor networks for each preference, resulting in a high com- putational cost. Moreover, their method is restricted to a specific MOFISP with a fixed objective combination, lacking the flexibility to solve other MOFISPs. In this paper, we propose an NCO
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method that uses single neural network solve general MOFISPs with distinct objectives and any combination of them.
Besides learning-based methods, many MOFJSP variants have been addressed in literature using multi-objective evolutionary algorithms, such as particle swarm optimization (Moslehi & Mahnam 2011]. the genetic algorithm NSGA-II (Deng al. 2017 Reijnen al. 2023). and multi-ojjective evolutionary algorithm based on decomposition (MOEAD; Xiao et al. 2024). These approaches
can achieve satisfactory performance for specific problem instances. However, they are highly de- pendent on manual tuning design and their runtimes scale poorly for larger problems.
NMOCO for Routing While the MOFISP has received little attention in NCO research, several works have recently focused on multi-bjeective vehicle routing problems. Li et al. (2021) proposed one of the first approaches in this area, decomposing the multi-objective problem into multiple sub- problems and training separate neural network for each. Zhang et al. (203d)) adopt similar idea within different meta-learning framework. However, these approaches do not scale well and do not allow adapting preference weights at inference time without retraining. [in et al. (2022) par- tially address this limitation using a hypernetwork that maps objective weights to actor parameters, enabling adaptation to different preference vectors during inference, but still requiring a separate actor network per preference, limiting scalability. Subsequent works (Wang et al.) 2024| Chen et al,, 2025 Fan et al. 2025) move single-mi modll approach, conditioning the neural model directly on the preference vector and achieving strong performance. However, these methods are tailored for routing. They build single-objective routing architectures, simple static coordinate based states, and rely on REINFORCE with episodic rewards in environments that are cheap to sample. In contrast, state-of-the-art scheduling methods require stepwise rewards, richer dynamic states, and more elaborate state features, and therefore rely on actor-criic critic methods such as PPO. Moreover, the routing objectives are simple distance-based measures (e.g., Euclidean distances over coordinates), which differ substantially from the practically relevant objectives in scheduling. Consequently, al- though these works demonstrate the promise of preference-conditoone and decomposition based NMOCO methods, their applicability to other problem classes such scheduling is highly limited.
3 BACKGROUND
3.1 MULTTI-OBJECTIVE COMBINATORIAL OPTIMIZATION
where M is the number of objectives and X is the set of feasible solutions. Since objectives are conflicting, there is no single optimal solution for all objectives. Instead, Pareto optimality is introduced.
multi-objective CO (MOCO) problem is defined as min (f) f2()....,
Definition (Pareto dominance). A solution 2 E X' dominates another solution 22 E X' (x1 < x2) if and only if fz)) S fx2), Vi {L..... M) and f((x1) < f((2), 3i {I...., M.. Definition 2 (Pareto optimality). A solution x E X' is Pareto optimal if no other solution x' X dominates it. All Pareto optimal solutions together form the Pareto set P = fxr X X
x' x²) and their objective values form the Pareto front F = (f(x)) P).
The goal of MOCO to find the Pareto set and its front.
Decomposition-Bel -Based Combinatorial Optimization Decomposition is a popular strategy for solving MOCO problems that splits them into multiple subproblems, each being single-objective or multi-objective problem. It provides the basis for, among others. the successful MOEA/D (Zhang
Li 2007) method, which solves the subproblems collaboratively to construct a Pareto set. We consider the most widely used and intuitive weighted sum decomposition method (Ehrgott 2005).
Here, each subproblem minimizes scalarized objective minnexg(=//))
EM A=f( where
E I M is preference vector such that X; 2 0 and ΣM A = 1. The multi-objectivve problem is solved by solving N subproblems that consider N weight vectors.
3.2 MUL.T-OBIEETIVE FLEXIBLE JOB SHOP SCHEDULING
The FISP consists of a pair (J,M) where J is a set of jobs and M a set of machines. A job J: I consists of n; operations O, {On,.... .., Omn, to be performed in order. The total set of
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operations is O U, O, Each operation Oty E O must be processed by single machine, selected from the set of compatible machines My - M.. The processing time of operation Ozg on machine Mr E EMi, is f > and each machine can process one job at a time, A solution to the FISP is schedule, which assigns compatible machine each operation Oz and determines the order of operations on each machine. While the goal of a single-objective FJSP is to find a schedule that optimizes a given objective function, the MOFISP aims to find all schedules in the Pareto set for a given set of objectives. In this paper, we consider the makespan, total tardiness, total earliness, average flowtime, and total costs as objectives. These objectives are among the most commonly occurring in the scheduling literature (see e.g., Xie et al., 2019 Dauzere-Pérés et al. (024) and address a variety of considerations that are relevant in practice. We define the cost of an operation to be inversely related to the processing times, so that faster operations are more expensive and vice versa. Generally, the makespan, tardiness, and flowtime objectives tend to benefit from shorter processing times, while costs and earliness can profit from some slower (but cheaper) choices. Thus, the objectives cover a range of trade-offs.
Definition 3 (Makespan). Given the job completion times {C((,)]J; E J} in schedule x, the makespan is mmakespan (z) = maxJ,ez (z))
Definition 4 (Total tardiness). Given the job completion times (C,())],; E J] in a schedule a and the job deadlines [D.]J; E J]. the total tardiness is ftardiness (a) = Eeer max(C.(z) D,, 0).
Definition 5 (Total earliness). Given the job completion times (C, E J) in schedule x and the job deadlines [Di[J; E J). the total earliness is Seartiness (I) Σnes max(D; - O).
Definition 6 (Average flowtime). Given the job completion times {Ci(x))J, JH and start times [S; E J} in a schedule x, the average flowtime is [ (2) ELETJ (C((z) /13)
Definition 7 (Total costs). Given the processing times (x), E EO} in a schedule x and the maximum potential processing time Pmaz. the total costs are fooets (2) = -o,Eo (Pmaz - Pij (z)).
4 METHODOLOGY
4.1 MARKOV DECISION PROCESS
The scheduling process involves sequential decisions, progressively assigning operations to ma- chines. At each decision moment t, an operation-maching combination (Oij. M&) is chosen to assign operation Oij to machine Mr. In the (multi-objective) Markov Decision Process (MDP), an agent receives the state 8, that represents the partial schedule and selects an action d, = (O M2) A(I) from the feasible actions A()). This set comprises the possible allocations of the first unas- signed operation for each job to compatible machine. The environment then provides reward vector T2 = [rr. [r.M] and new state St+1. The schedule is completed after ( actions.
The relevant operations On(() C O for state 8, exclude those that already have a succes- sor scheduled on the same machine and thus do not directly influence the schedule anymore. The relevant machines Mt( S. M are all machines on which any of the remaining operations can be scheduled. Therefore, the state s; (Ho. HAs HOM) } is defined
State
triplet of operation features Ho = (ho., E [O: E (t)), machine features HM = (hM, E RRMM| JM E M,.()), and operation-machine pair features Hom Wang et al. (2023) and Appendix A for a description of these features. While these features were originally proposed for makespan as single objective, many of them are also relevant across multi- ple objectives. Notably, the lower bound of the completion time C(O.j. st) is particularly relevant, as it matches directly with specific objectives. This feature allows the policy to directly monitor the measures that affect the reward, as noted in the subsequent reward formulation. Therefore, we also include the lower bound feature for each objective in the state. For total tardiness and earliness, we maintain C(Oin,; st) - Dj for each operation. For the average flowtime, we add F(Oin;,&) for each operation. Similarly, a cost lower bound can be included. However, due to the way we define costs, this information is already captured in existing features and there is need to add new feature.
(Mo,,, M+) E Rooa (Oij, M&) A(t)). We refer to
Action Space and State Transition The action space A(t) consists of all compatible operation- machine pairs of the first unscheduled operations per job. By taking action, we process an opera-
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tion machine. The relevant operations (t) and machines (t) are updated and all features are updated correspondingly, giving a new state 8141-
Reward The reward -(ss)) (s for objective is inversely related to the increase in its quality measure H (-). Wang et al. (2023) defined the makespan quality measure using a recursively updated lower bound, which outperforms directly using the objective value because it provides a smoother signal. Concretely, they defined C(Oig. 8t) which equals the scheduled com- pletion time if the operation has been scheduled. Otherwise, it follows the recursion ((Osj. 8,)
C(O 8e) min pf-- The quality measure is H Haakessan (sx) max C(O 8t). We note that defining and maintaining lower bound is possible for any metric that is nondecreasing during the scheduling process. For tardiness and average flowtime, we can also use the completion time lower bounds. Specifically, Htardinesa
-iEET max(C(Oinn: 8t) D., 0) is the qual- ity measure for total tardiness. For the average flowtime, we maintain a lower bound E(Or). which is equal C(Os St) if the first operation has not yet been scheduled. Otherwise, we have E(O s,) C(Oig. St) - Sy. We define H flowtime (3+) = "JEeJ F(Om,,, (t)0m,,sss For costs, the lower bound C(Oir, 8:) is the actual cost if the operation has been scheduled and the lowest possible costs between machines otherwise. Consequently, (s.) = Eo,eoo C(O,j, sx). Sim- ilarly, the reward can be defined for any nondecreasing objective. For carliness, we define an upper bound instead of lowerbound,
this is nonincreasing objective. We use Heartinese Eney max D; C(Qing 8t), 0). Then, we can use the same ry formula to reward decreases in the upper bound. This can be done in similar way for other nonincreasing objectives.
4.2 DECOMPOSITIONS BASED PPO
We propose to solve the MOFISP through a weighted sum decompositiion-base PPO algorithm. We prefer weighted sum decomposition over Tchebycheff decomposition (another commonly used alternative) for two main reasons. Firstly, with weighted sum decomposition. our stepwise rewards converge to the weighted sum episodal reward (cf. Appendix B. In contrast, Tchebycheff scalariza- tion is nonlinear and nonadditive over time, preventing this theoretical alignment. Secondly, despite having a theoretical advantage to find nonconvex fronts, Tehebycheff decomposition is empirically comparable or even inferior to the weighted sum in NCO literature (Chen et al. [2025] Wang et al., 2.24]. Concretely, our goal to find a policy conditioned on the decomposed problem m7 (s, A) that maximizes its expected reward, given the problem instance and preference vector. Formally, given a distribution of problem instances and a distribution of objective preferences A, we aim to find -M
policy тH such that Tे = arg max, (Ex~A.e0<5 | policies. we propose decomposition based PPO algorithm (Algorithm(1) [T]. We base our method on clipped PPO with generalized advantage estimation, incorporating normalized processing times and batch normalization as suggested by (Wang et al. (023). We generate ng problem instances for preference vector À for each instance. In
(== 10|-1 11
Ei=1 Aere, /so. A]). To train such
every NB episodes and for each episode we sample this way, the policy is trained using a wide variety of MOFISP instances and multiple decomposed problems per instance. By sampling frequently and using unique preference vectors per problem instance, we prevent overfitting to specific subproblems. We ensure exploration by probabilistically sampling actions based the output probabilities of the policy.
Algorithm Decomposition baased PPO
Require: Neural network with initialized parameters 0
I: Sample batch of ng instances
2: for nep = 1,2, ..p tep(0 do
3: for b
ng .g do Sample preferences À A 5: for f 0, 1,.,, |0| 1 do
D. In Parallel
4:
Sample action ar, ~ (()),,,))
7:
Perform a,,s and receive $141. and Ttbb Compute T -TTr,b
8:
9:
Collect transition (8t,6, at,8, Tr.b, 8141.8. A A)
10:
Compute generalized advantage estimates
11:
Compute PPO loss (Appendix ( and update
12:
if Rep mod Ne then Sample ng instances
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4.3 NEURAL NETWORK ARCHITECTURE
We put forward two network architectures to learn policies that take the subproblem state and decomposition weights (i.e., preference vectors), based on the DAN architecture (Wang al. 2023). We propose a straightforward yet effective technique for MOFJSP, called WI-DAN: concatenating the preference vector the feature vectors before feeding the resulting vectors to the DAN network. Thus, we set ho., = . || and hag = attention network (DCAN), which uses the dual attention approach of DAN but includes condi- tional attention mechanism that modifies the attention based on the objective preferences. We refer to Wang et al. (2023) for details of DAN. The proposed DCAN consists two attention blocks: the conditional operation message attention block and the conditional machine message attention block. To facilitate understanding, we show visualizations of the network structures in Appendix D
M. ||]. In addition, we propose the dual conditional
Each operation Oij E O. has a feature vector No., and a corresponding preference embedding My, as input to the (1+ 1)-th attention block. Espe- cially, Mo., and h,,, hso are initial linear transformations of the input features of operation Oi and the preference vector X, respectively. Using conditional attention, we update the embeddings:
Conditional Operation Message Attention Block
2+1 No
(1n) Who,,) a( Whis
In the equation, a indicates the attention coefficients after computing the softmax over the scores e, where we compute C(a.b) = LeakyRel.U(a' [ and W are learnable parameters, and a is nonlinearity. Here, the attention mechanism is modified through the preference em- bedding. Intuitively, an artificial node, based on the preference vector, is added the attention mechanism, alongside the operation nodes, thereby affecting the attention between the operations. These artificial node embeddings are in turn also updated in each attention block, such that they modify each block appropriately. This update is similar to the operation embedding update such that Wh',,,) + a(As))))))) Wh's,))
Conditional Machine Message Attention Block For each machine Ms E Ma we have feature vector h'M, and we have hay where hiv and & are different linear transformations of the ma- chine input features and the preference vector. Using conditional machine attention, we compute:
MMall
M, Zhise,) + Bedde Zhd
M
Here, ,3 are the attention coefficients derived from the softmax over u with U(a.b) ((N [ and b, Y, Z. are learnable parameters. The coefficient M()), is an intensity metric between machines MA and M, based on their potential operations (see Ap- pendix E. Since such a metric does not naturally exist between a preference embedding and ma- chine embedding, we take CAM = C(.Asr) C(AM))) as the average across all intensity metrics M,). Analogously, the preference embedding for machine attention is also updated such that 1/42 AM a (oLM ( Zul',, ((ar Zhh Z ). The final output embeddings h and ho.,, are used identically by the actor and critic network as in DAN.
Critic Learning For the critic network, instead of a single scalar output to estimate the weighted sum of objectives. we output one critic value per objective. Thus, the critic is an MLP that takes the aggregated operation and machine embeddings hag = [oo.| Eoseo. h6, ll M.. EMEM... h and outputs a value vector v(s)) E RM To train the critic, we compute the loss over all indi- vidual value estimates Leritic M
10)-1 MM (v.(s)) fr. where fr,; are the general- ized advantage estimates per reward component Te,i- For actor training, the aggregated advantage A - -M X,A,, can still be computed before calculating the PPO loss, while allows the critic to better attribute the losses for each objective. Although theoretically compelling. not find a significant performance improvement over a single-valued critic, presumably indicating that the critic value estimation is considerably less complex than the actor task.
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5 EXPERIMENTS
Baselines We compare with two common multi-objeetive evolutionary algorithms, MOEA/D
(hang & Li 2007) and NSGA-II (Deb et al., 2002). Our implementation based on the operators of Xiao et al. (2024) and Zhang et al. (2011) and the genetic algorithm from Reijnen et al. (2025).
NSGA-II and MOEA/D run for 1000 generations and 80000 evaluations steps, respectively, with
population size of 100, ensuring convergence. The crossover and mutation hyperparameters follow Xiao et al. (222) and Zhang et al. (201T). We also compare with the CP-SAT solver (Perron et al. 2023). We run CP-SAT minute per subproblem (giving total runtime of 101 and 105 minutes per 2- and 3-objective instance, respectively). A 16-core AMD ROME 7H12 machine is used for all baselines. We also implement hypernetwork neural network that closely follows Lin et al. (2022) and Su et al. 2024) combined with our methodology as theirs are not directly applicable.
We use the popular synthetic datasets from Song et al. 2022). Instance sizes 10, 20 10 used for training and testing, and 30x 10 and 40x 10 for testing. Moreover, we evaluate on the mk (Brandimarte] 1993), rdata, edata, and vdata benchmarks (Hurink) al. 1994). For the latter three, processing times across alternative machines are the same for each operation, making costs constant and as such reducing the true number of objectives. From these datasets, the makespan, flowtime, and costs can be calculated directly. We set the deadline of job J/ to D; = 1.5 Eoseo, (minM,eM;; similar to (Wu & Weng 2005
Problem Instances
10x5, 20 15x
Chen & Matis 2013).
Configurations For training, we set Nep 1500, NB 20, nu = 20, and evaluate once every Neval 40 episodes. The hyperparameters for WI-DAN and DCAN match those of DAN (Wang) (et al. 2023]. We use 3 objective combinations: makespan-costs, tardiness-costs, and makespan- flowtimecooss.. For testing, we decompose the 2-objective and 3-objective problems into 101 and 105 uniformly spread subproblems, respectively, which we solve in parallel. In training, we ran- domly sample preference vectors from a flat Dirichlet distribution (Ng al.) 2011). We test 100 problem instances for each synthetic dataset. For inference, we use greedy solution construction and sampling strategy, sampling 10 solutions per subproblem. We measure performance using the nor- malized Hypervolume (HV; ef. Appendix F Guerreiro et al. 2021). reporting the gap- HV
HVcr
to the hypervolume of the CP-SAT approach. Moreover, we report the unique number of solutions in the found Pareto sets, the runtime, and, in Appendix G the IGD+. We present the averages for each instance set. We use NVIDIA A100 GPU and 9-core Intel Xeon Platinum 8360Y CPU. Since the FJSP is a generalized scheduling problem, other problems such as the JSSP and flexible flow shop scheduling problem (FFSP) can also be solved without modifications (see Appendix H
5.1 RESULTS ON SYNTHETIC INSTANCES
Table D shows the performance of our approach for test instances matching the training sizes of the models. These results show that our method, using both WI-DAN and DCAN, learns highly com- petitive policies. We find that for the smallest instances, NSGA-II outperforms the DRL policies for I of the 3 objective combinations. For all other problem sizes and objective combinations, our ap- proach considerably outperforms the metaheuristics, while being much faster. For 20x 10 instances, DRL achieves gap roughly 50% better than MOEA/D. We also observe that the gap to the CP- SAT solutions narrows for larger instances. Although both perform well and also outperform the hypernetwork approach, DCAN consistently outperforms WI-DAN. Especially for the 3-objective problems, DCAN reduces the gap by several percentage points. Moreover, DCAN consistently gen- erates larger Pareto sets, indicating that the conditional attention mechanism improves the network's ability to exploit decomposed subproblems. Sampling further improves HV performance and Pareto set size, the cost of higher runtime. However, the runtime is still very short compared to the baselines. In short, our DRL policies considerably outperform the NSGA-II and MOEA/D base- lines, with DCAN yielding better and larger Pareto sets than the more straightforward WI-DAN. In Appendix we present results for a different synthetic instance set, which offers similar results. In Appendix]] we visualize multiple found Pareto sets. Although our main experiments include and 3-objective problems, our methodology can handle any number of objectives. To illustrate this, we solve the 4-objective problem considering makespan, flowtime, carliness, and costs in Appendix K
Source code http:////ggituh.ccmaaaaaaoaoaaee htwp:/iiiiiiiiiiaaaaaauuuuu/e/ee Nue/eural Multi-Ohjeetive CO for FISP

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Table 1: Results on synthetic instances of the same sizes as the instances used for training
Metahcuristics
Greedy
Sample Hyper
Size
NSGA-II MOEA/D]
Hyper
WI-DAN
DCAN
WI-DAN
DCAN
SCP-SAT
HV
0.7208
0.6616
0,5987
0,6999
0.7104
0.6204
0.7581
0.7647
0.8313
10x5
Gap
13.30%
20.42%
27.98%
15.81%
14.55%
25.37%
8,81%
8.01 %
0.00%
Nr. Sol.
8.14
8.81
6.72
2.91
3.46
7.78
7.19
7.77
13.20
Time (s)
247.85
254.85
0.57
0.57
0.87
2.26
2.37
3,00
Makespan Costs
HV
0,5332
0.4807
0,5083
0,5536
0.5599
0,5268
0.5696
0.5724
0.6095
20x5
Gap
12,53%
21.14%
16.61%
9.17%
8.14%
13.57%
6.56%
6.09%
0,00%
Nr. Sol. Time (s)
11.96
14.67
6.48
3.30
4.04
9.24
5.72
6.82
11.82
652.04
614.81
1.41
1.47
1.99
8.07
8.51
10.43
HV
0,6809
0.5570
0,7039
0.7650
0.7723
0,7388
0.7920
0.8002
0.9243
15x 10
Gap
26.33%
39.73%
23.85%
17.23%
16.44%
20.07%
14.31%
13.42%
0.00%
Nr. Sol.
14.12
16.70
6.95
8.28
9.14
9.17
14.16
15.42
18,16
Time (8)
1694.42
1062.99
2.86
3.07
4.02
21.19
22.72
26.98
HV
0.6289
0,4879
0,7751
0.7966
0.8083
0,7236
0,8073
0.8200
0.8548
20 10
Gap
26.43%
42.93%
9.32%
6.82%
5.44%
15.36%
5.57%
4.07%
0.00%
Nr. Sol. Time (s)
19,21
19.45
16.43
9.44
13.85
7.43
14.14
22.66
14,81
2798.44
1774.73
4.67
5.05
6.57
38.02
41.66
48.32
HV
0,7741
0.7006
0,6335
0.7109
0.7460
0.7103
0.8102
0.8272
0,9144
10x5
Gap
15.35%
23.38%
30.72%
22.25%
18.42%
22.32%
11.40%
9.54%
0.00%
Nr. Sol.
15.41
9.58
6.52
2.66
4.36
11.44
10.28
12,89
25,81
Time (8)
296.60
256.28
0.59
0.59
0.89
2.61
2.66
3.29
HV
0,5331
0,4234
0,5479
0.6372
0.6396
0,5282
0.6645
0.6700
0,7010
Tardiness Costs
20x5
Gap
23.96%
39.60%
21.84%
9.09%
8.76%
24.64%
5.20%
4.42%
0.00%
Nr. Sol.
20.53
17.59
10.23
10.00
10.88
10.87
17.27
19.51
11.26
Time (s)
516.84
619.33
1.55
1.59
2.24
9.47
9.93
12.02
HV
0.6745
0,5131
0.7134
0.7982
0.8094
0,7270
0.8258
0.8338
0,9505
15x 10
Gap
29.04%
46.02%
24.95%
16.03%
14.85%
23.51%
13.12%
12.28
0.00%
Nr. Sol.
22.10
18.96
13.50
15.45
17.04
12.10
26.95
32.39
20.71
Time (8)
1642.20
1048.72
3.12
3.28
4.45
23.90
25.69
30.03
HV
0,5878
0.4229
0,7566
0.8100
0.8112
0,7472
0.8274
0.8306
0.8498
20 10
Gap
30.83%
50.24%
10.97%
4.68%
4.53%
12.08%
2.63%
2.25%
0.00%
Nr. Sol.
24.54
23.23
18.23
17.04
20.03
12.02
29.56
34.01
14,54
Time (8)
2869.48
1745.29
5.51
5.43
7.17
44.66
47.02
54.38
HV
0.6146
0.4969
0.4123
0.4081
0.4647
0.4673
0.4902
0.5130
0.6831
10x5
Gap
10.02% %
27.25%
39.64%
40.26%
31.97%
31.59%
28.24%
24.90%
0.00%
Nr. Sol.
167.40
67.01
23.57
7.33
22.64
48.43
39.50
$2.84
64.98
Makespan Flowtime Costs
Time (s)
229.21
260.88
0.84
0.82
1.14
4.99
4.97
5.66
HV
0.3661
0,2595
0,3800
0,4221
0,4318
0,4021
0,4512
0.4529
0.5210
20x5
Gap
29.73%
50.20%
27.06%
18.97%
17.11%
22.83%
13.39%
13.06%
0.00%
Nr. Sol.
230.89
90,90
31.23
28.15
32.96
71.75
80.70
81.77
59,95
Time (S)
543.45
627.90
2.49
2.52
3.18
18.34
19.39
21.03
HV
0,4654
0.3205
0.4844
0,5583
0.5793
0,4850
0,5782
0.6025
0.7336
15x 10
Gap
36.57%
56.32%
33.97%
23.90%
21.03%
33.89%
21.18%
17.88%
0.00%
Nr. Sol.
86.70
37.91
34.10
29.06
43.96
75.54
82.73
121.23
68.32
Time (s)
1649.08
1020.61
4.36
4.52
5.51
35.56
37.33
41.00
HV 0,3834
0.2547
0,5056
0.5822
0.6142
0.5025
0.6038
0.6314
0.6612
20 10
Gap
42.01%
61.49%
23.54%
11.96%
7.12%
24.00%
8.69%
4.51%
0.00%
Nr. Sol.
106.06
41.38
37.29
43.08
57.83
62.65
114.14
179.74
60.83
Time (s)
2797.74
1674.67
7.35
7.72
9.13
64.14
67.44
74.71
Table 2 shows the generalization to larger problem instances. We policies trained 20 10 in- stances to solve 30x 10 and 40 10 instances. The results show that our policies can generalize fairly well. In fact, they outperform the baselines by even greater margin than in Table 1 Whereas the metaheuristics and CP-SAT deteriorate quickly with larger instances, our approach retains perfor- mance. We find much better hypervolumes and larger Pareto sets for our DRL policies. WI-DAN scales better for MOFJSP with the tardiness and cost objectives, but this pattern is not consistent, as DCAN maintains its advantage for the other objective combinations. In short, our policies can transfer to larger instances, retaining the advantages over the baselines.
5.2 RESULTS ON BENCHMARK INSTANCES
We assess cross- distrriistii performance using the public benchmark datasets in Table Here we present the 3-objective MOFISP, since the 2-objective problems boil down single-objective FJSP
8

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Published as a conference paper at ICLR 2026
Table 2: Results on large synthetic instances using the policies trained on size 20 10
Mctaheuristics
Greedy
Sample WI-DAN
Size
NSGA-II MOEA/D]
Hyper
WI-DAN
DCAN
Hyper
DCAN
CP-SAT
Makespun Costs
HV
0,5477
0,3848
0.7221
0.7229
0.7437
0.6584
0.7294
0.7493
0.7425
30x 10
Gap
26.24%
48.18%
2.74%
2.64%
-0.17%
11.33
1.77%
-0.91 %
0.00%
Nr. Sol.
29.73
27.80
24.08
11.67
19.46
7.49
18.14
32.62
11.61
Time (s)
5674.45
3580.27
9.54
9.63
12.09
82.49
86.64
100.90
HV
0,4832
0.3128
0.6436
0.6387
0.6629
0,5852
0.6469
0.6688
0.6326
40x 10
Gap
23.61%
50.54%
-1.74%
-0.97%
-4.79%
7.48
-2.28%
-5.33%
0.00%
Nr. Sol. 37
31.28
28.80
12.57
22.31
7.61
22.91
39.86
11.23
Time (s)
10013.26
6233.48
15.69
16.26
20.48
148.92
156.98
181.16
HV
0,4652
0.3082
0.6676
0,7627
0,7464
0,6562
0.7694
0,7601
0.6872
Tardiness Costs
30x 10
Gap
32.31%
55.16%
2.85%
-10.98%
-8.60%
4,51
11.95%
-10.61%
0.00
Nr. Sol.
33.40
28.54
19.07
25.57
24.53
15,53
41.03
41.43
10.18
Time (s)
5664.65
3454.61
10.90
11.58
14.03
98.26
105.23
119.65
HV
0.3751
0.2349
0.5708
0,7040
0.6649
0.5659
0.7084
0.6815
0.5390
40x 10
Gap
30.41%
56.41%
-5,89%
-30611%
-2.35%
.4,99
-31.2%%
26.43%
0.00%
Nr. Sol.
40.00
28.81
15.41
29.35
25.25
14.97
49.78
41.45
7.71
Time (s)
10020.31
5826.27
19.93
19.68
24.12
188.55
194.54
220.47
Flowtime Costs
30x
Gap
HV
41.49%
0.2789
64.13%
0.1709
0,4640
2.64%
-13.33%
0,5401
-19.76%
0,5707
0,4382
8.06
-16.74%
0.5563
-21.93%
0,5811
0,4766
0.00
Makespan
Nr. Sol.
137.67
45.18
44.62
46.62
60.85
50.15
131.78
205.99
46.04
Time (s)
5820.54
3306.71
18.42
16.08
18.58
180.95
150.65
165.46
HV
0,2022
0.1165
0,3993
0,4773
0,5087
0.3724
0.4918
0.5194
0.3043
40x 10
Gap
33.56%
61.73%
-31.22%
-5.84%%
-67.17%
-2.399%
-61.63%
-70.70%
0.00%
Sol. 66.59
44.26
49.97
$1.09
63.53
65.19
151.06
224.75
35.81
Time (s)
10028.12
5705.95
26.53
26.80
30.87
263.06
275.59
298.31
Table 3: Results on public dataset instances for the 3-objective problem using the 15x 10 policies
Metaheuristics
Greedy
Sample
Size
NSGA-II MOEA/D
Hyper
WI-DAN
DCAN
Hyper
WI-DAN
DCAN
CP-SAT
HV
0.3416
0,2517
0.2977
0,2519
0,2743
0,3172
0.2878
0.3047
0,5085
Gap
32.82%
50.49%
41.45%
50.46%
46.06%
37.62%
43.39%
40.07%
0.00%
mk
Nr. Sol.
275.70
94.30
34.30
18.10
27.50
88.40
$1.10
84,50
68.90
Time (s)
1787.04
1122.09
4.59
4.78
5.73
39.05
41.10
44,05
HV
0.6586
0,5652
0.6004
0,5900
0.6060
0.6166
0,6033
0.6367
0.7296
Gap
9.73%
22.53%
17.71%
19.14%
16.94%
15.50%
17.31%
12.73%
0.00%
rdata
Nr. Sol.
8.98
6.98
5.28
6.08
5.03
9.00
9.28
9.48
15.18
Time (s)
2120.14
1164.22
5.34
5.68
6.10
47.79
44.82
47.79
HV
0.6439
0,5773
0,5256
0,5212
0,5272
0,5441
0.5427
0,5689
0,7137
Gap
9.79%
19.11%
26.36%
26.98%
26.14%
23.76%
23.96%
20.30%
0.00%
edata
Nr. Sol.
10.40
6,70
4.23
3.95
3.40
6.23
6.23
7,43
16.00
Time (s)
2106.18
1140.89
5.32
5.66
6.20
47.25
45.20
47.78
HV
0.7180
0,6133
0,6957
0,6799
0.7143
0.7056
0,6800
0.7378
0.7907
Gap
9,20%
22.44%
12.02%
14.01%
9.66%
10.77%
14.00%
6.69%
0.00%
vdata
Nr. Sol.
11.70
6.50
6.70
7.28
6.90
9.18
10.43
10.90
12.03
Time (s)
2236.94
1189.89
5,34
5.65
6.20
47.05
45.85
48.26
for three of the four datasets (results in Appendix [). Table shows NSGA-II outperforms our poli- cies on three out of four benchmark instances, which has several reasons. Firstly, the benchmark datasets contain relatively small instances in which the metaheuristics do not yet deteriorate. Sec- ondly, makespan and flowtime are naturally less conflicting with each other than costs. When costs are constant (in rdata, edata, and vdata), a smaller solution space contains many good solutions. Hence, metaheuristies can more easily find neighboring good solutions via genetic operations. This alleviates the weakness of these algorithms in exploring divergent search spaces. Thirdly, the meta- heuristic runtime is much higher as we run them for many generations. Appendices Mannd N show the results with more comparable runtimes. In those scenarios with shorter runtimes, DCAN out- performs the metaheuristics, underlining its value in scenarios requiring less runtime. In addition, Appendix G shows that the DRL methods mostly outperform NSGA-II with respect the IGD+ metric, indicating their competitiveness. Overall, DCAN and WI-DAN remain competitive, outper- forming MOEA/D and achieving good runtime-performance trade-off compared to NSGA-II.

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Published as a conference paper at ICLR 2026
Table 4: Results of DCAN for varying inference subproblem quantities for the 3-objective problem
N = 10
N 45
N 105
N = 496
N = 1035
Size
Greedy
Sample
Greedy
Sample
Greedy
Sample
Greedy
Greedy
HV
0.4254 37.72%
0,4801
0.4565 33.18%
0.5033
0.4647
0,5130
0.4734
0,4771
Gap
29.72%
26.32%
31.97%
24.90%
30.69%
30.15%
10x5
Nr. Sol.
7.12
24.35
16.44
41.25
22.64
52.84
29.48
31.87
Time (s)
0.74
1.16
0.90
2.70
1.14
5.66
2.93
5.57
HV
0.5792
0.6078
0.6056
0.6248
0.6142
0.6314
0.6269
0.6311
Gap
12.41%
8.08%
8.41%
5.51%
7.12%
4.51%
5.19%
4.55%
20x
Nr. Sol.
9.37
51.01
32.40
123.54
57.83
179.74
141.37
195.75
Time (8)
3.38
8.64
5.23
32.11
9.13
74.71
36.35
71.32
Table 5: Hypervolume results of ablation study. v Vindicates our proposed approach is used
Makespan Flowtime Costs
Makespan Costs
Tardiness Costs
State
Reward
Greedy
Sample
Greedy
Sample
Greedy
Sample
0.5068
0.5385
0.5591
0.5417
0,4165
0.3869
0.5182
0.5383
0.5734
0.5419
0.4475
0.3908
0.8007
0.8131
0.7948
0.8111
0,6036
0.6184
0.8083
0.8200
0.8112
0.8306
0.6142
0.6314
5.3 EFFECT OF SUBPROBLEM QUANTITY
We can decompose problem into different numbers of subproblems to balance computational com- plexity and performance. Table 4 shows that increasing the number of subproblems increases per- formance, albeit with diminishing returns. It also shows that for small instances, sampling strategy with few subproblems outperforms greedy strategy with higher N. The added exploration has a big advantage in these instances. In larger instances the difference fades. Using sampling and increasing the number of subproblems by the same factor of 10 leads to similar results. Thus, tuning the num- ber of subproblems and samples allows for a trade-off between performance and runtime, though the advantage of generating more solutions diminishes as the number of subproblems increases.
5.4 ABLATION STUDY
Table 5 shows the results of the ablation study for the state features and reward formulation. Here, solve the 20x10 instances for the three problems from Table using DCAN. We compare with simple step-wise reward without lower bounds and leaving out the proposed lower bound features. These results highlight the value of our adjustments. Especially our reward formulation crucial to achieve good results. This reward stabilizes the reward signal and achieves better credit assignment. The added features also improve performance, although the effect is smaller than for the rewards.
CONCLUSION
We present a novel NMOCO approach for the MOFJSP, where we use a decomposition-basse PPO algorithm to train conditional policies. These policies take both the FJSP instance and the preference vectors of the decomposed problem to determine the actions. We propose two neural networks based on straightforward preference vector input (WI-DAN) and conditional attention (DCAN). We ex- perimentally show that the proposed approach considerably outperforms the bascline metaheuristic approaches, especially for larger instances, with DCAN outperforming WI-DAN, Our methodology can act as a base for further development of NMOCO techniques for various scheduling variants. Moreover, although we target scheduling problems, we believe components such as decomposition- based PPO, bound-based reward functions, and the conditional attention mechanism, can also be leveraged to develop NCO methods for other CO problems, providing promising research direc- tion for future work. Next to generalizing on optimizing additional and more complex objectives. addition. advanced sampling techniques specifically targeting NMOCO could help better utilize the learned policies.
wider variety of CO problems, future work may focus
10

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Published as a conference paper at ICLR 2026
ACKNOWLEDGMENTS
The LEO (Learning and Explaining Optimization) project co-funded by Holland High Tech TKI HSTM via the PPS allowance scheme for public-private private partnerships. This work used the Dutch national e-infrastructure with the support of the SURF Cooperative using grant no. EINF-15027.
REFERENCES
Philippe Baptiste, Claude Le Pape, and Wim Nuijten. Constraint- scheduling: applying con- straint programming to scheduling problems, volume 39. Springer Science Business Media, 2001. ISBN 0792374088.
Paolo Brandimarte. Routing and scheduling in flexible job shop by tabu search. Annals of Op- erations Research, 41:157-183, 9 1993. ISSN 0254-5330. doi: 10.1007/BF02023073.. URI.
http://lnnk.pringeercom// 10. 1007/B/02223003
Binchao Chen and Timothy I. Matis. A flexible dispatching rule for minimizing tardiness in job shop scheduling. International Journal of Production Economics, 141(1):360-365, 2013. ISSN 0925- 5273. doi: 10. www.::..ppp.p........
URL
50925527312003672 Meta-heuristics for manufacturing scheduling and logistics problems.
Jinbiao Chen, Zhiguang Cao, Jiahai Wang. Yaoxin Wu, Hanzhang Qin, Zizhen Zhang, and Yue-Jiao Gong. Rethinking neural multi-objective combinatorial optimization via neat weight embedding. In The Thirteenth International Conference on Learning Representations, 2025. URL https://
Giacomo Da Col and Erich C. Teppan. Industrial-size job shop scheduling with constraint program- ming. Operations Research Perspectives, 9: 9:100249, 2022. ISSN 221 160. doi: 10. 10. 10.10.6/j.orp. 2022.100249. URL openneviee.neetfoummii--------
Andrea Corsini, Angelo Porrello, Simone Calderara, and Mauro Dell' Amico. Self-labeling the job shop scheduling problem. In A. Globerson, L. Mackey, D. Belgrave, A. Fan, U. Paquet, J. Tomczak, and C. Zhang (eds.), Advances in Neural Information Processing Systems, volume 37, pp. 105528-105551. Curran Associates, Inc., 2024. URL htttss///roceeddigs.eeuuissssee newrips.cc/papprr. Paper- Cowferenceeeceeeeeoooo pdf
Stéphane Dauzère-Pérés, Junwen Ding, Liji Shen, and Karim Tamssaouet. The flexible job shop scheduling problem: A review. European Journal of Operational Research, 314(2):409-432. 2024. ISSN 0377-2217. doi: (0.101669.ejor.2223.00
URL httpss//www..scenccdrree..cot
Sccercc/anccepp///00722222333233
K. Deb, Pratap. S. Agarwal, and T. Meyarivan. fast and elitist multiobjective genetic algorithm: Nsga-ii. IEEE Transactions on Evolutionary Computation, 6(2):182-199) 2002. doi: 10. 10.1 4235.996017.
Qianwang Deng, Guiliang Gong, Xuran Gong, Like Zhang, Wei Liu, and Qinghua Ren. A bee evolutionary guiding nondominated sorting genetic algorithm ii for multiobjective flexible job- shop scheduling. Computational Intelligence and Neuroscience, 2017())5232518, 2017. doi:
URL 10.1155/2011/5232511
Matthias Ehrgott. Multicriteria Optimization. Springer-Verlag Berlin, Heidelberg, 2005. ISBN
3540213988.
Mingfeng Fan, Yaoxin Wu, Zhiguang Cao, Wen Song, Guillaume Sartoretti, Huan Liu, and Guohua Wu. Conditional neural heuristic for multiobjective vehicle routing problems. IEEE Transactions Neural Neworrk and Learning Systems, (663)):67774689 2025. doi: 10.1109/TNNS..000 3371706.
Andreia P. Guerreiro, Carlos M. Fonseca, and Luis Paquete. The hypervolume indicator: Compu- tational problems and algorithms. ACM Comput. Surv,, 54(6), July 2021. ISSN 0360-0300. doi: 10.1145/3453474
11

---

Published as a conference paper at ICLR 2026
Johann Hurink, Bernd Jurisch, and Monika Thole. Tabu search for the job-shop scheduling problem with multi-purpose machines. OR Spektrum, 13:205-211 215, 12 1994. ISSN 0171-6468. doi: 10.1007//F017194511 URL
Yeong-Dae Kwon, Jinho Choo, I1joo Yoon, Minah Park, Duwon Park, and Youngjune Gwon. Ma- trix encoding networks for neural combinatorial optimization. In M. Ranzato, A. Beygelzimer, Y. Dauphin, P.S. Liang, and J. Wortman Vaughan Advances in Neural Information Process- ing Systems, volume 34, pp. 5138-5149 5149. Curran Associates, Inc., 2021. URL https://proceedings Paper.
neurips.cc/paper http:p/prr2222222255333333332212
Kun Lei, Peng Guo, Wenchao Zhao, Yi Wang, Linmao Qian, Xiangyin Meng, and Liansheng Tang. A multi-action deep reinforcement learning framework for flexible job-shop scheduling problem. Expert Systems with Applications, 205: 2022. ISSN 0957-4174. doi: 2022. 11799.. URL wttp:///wwww...ooooooooooooooooa
Jiake Li, Junqing Li, and Ying Xu. Hgnp: A pea-based heterogeneous graph neural network for family distributed flexible job shop. Computers Industrial Engineering, 200:1 10855, 2025. ISSN 83560 doi: 10. 10.1016/j.coo (ttpj.ccee222.118888 URL http::/wwww.ccencccdcccc.oooooos
atticle/pi/003083544007777
Kaiwen Li, Tao Zhang, and Rui Wang. Deep reinforcement learning for multiobjective optimization. IEEE Transactions on Cybernetics, (166)31033-1144 2021.
Xi Lin, Zhiyuan Yang. and Qingfu Zhang. Pareto set learning for neural multi-objeccivee combi- natorial optimization. In International Conference on Learning Representations, 2022. URL
hutps://pennrviie..e//fouuu Vut
Shu Luo, Linxuan Zhang, and Yushun Fan. Dynamic multi-objective scheduling for flexible job shop by deep reinforcement learning. Computers & Industrial Engineering, 159:107489, 2021. ISSN 0360003003333 8352. doi:
URL httpp://wwww.seeeenecccccccccoco
article/pi/00330835222000
Shu Luo, Linxuan Zhang, and Yushun Fan. Real-time scheduling for dynamic partial-no-wait mul- tiobjective flexible job shop by deep reinforcement learning. IEEE Transactions on Automation Science and Engineering, 19(4):3020 300030000-038, 2022. doi: 10. 10.1099AS..22221311417
Ghasem Moslehi and Mehdi Mahnam. pareto approach to multi-objective flexible job-shop scheduling problem using particle swarm optimization and local search. International Journal of Production Economics, 129(1): 201 ISSN 0925-5273, doi: 10. ijpp://.........ccoo 08.004.
URL
Kai Wang Ng, Guo-Liang Tian, and Man-Lai Tang. Dirichlet and related distributions: Theory, methods and applications. John Wiley & Sons, 2011.
Junyoung Park, Jaehyeong Chun, Sang Hun Kim, Youngkook Kim, and Jinkyoo Park. Learning to schedule job-shop problems: representation and policy learning using graph neural network and reinforcement learning. International Journal of Production Research, 59(11):3360-33) -377, 2021. doi: 10. 000100000020753.220000 20002007773.20....11 1870013..
Laurent Perron, Frédéric Didier, and Steven Gay. The cp-sat-lp solver. In Roland H. C. Yap (ed.), 29th International Conference on Principles and Practice of Constraint Programming (CP 2023), volume 280 of Leibniz International Proceedings Informatics (LIPles). pp. 3:1-3:2, Dagstubi, Germany, 2023. Schloss Dagstuhl- Leibniz-Zentrum für Informatik. ISBN 978-3-95977-300.
doi: 10. 10.4230/LIPIcs http:/LIIIc....222..... URL http:://woppppaaaattt....eeeolls
Jonathan Pirnay and Dominik G. Grimm. Self-improvement for neural combinatorial optimization: Sample without replacement, but improvement. Transactions on Machine Learning Rexearch, 2024. ISSN 2835-8856. URL httpss//ppaarevveew.ee///////u/p tion.
Featured Certifica-
Robbert Reijnen, Yingqian Zhang, Zaharah Bukhsh, and Mateusz Guzek. Learning to adapt ge- netic algorithms for multi-objective flexible job shop scheduling problems. In Proceedings of
12

---

Published as a conference paper at ICLR 2026
the Companion Conference Genetic and Evolutionary Computation, GECCO '23 Compan- ion, pp. 315-318, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400701207. doi: 00.114573583113.35907000
Robbert Reijnen, Igor Smit, Hongxiang Zhang, Yaoxin Wu, Zaharah Bukhsh, and Yingqian Zhang. Job shop scheduling benchmark: Environments and instances for learning and non- learning methods, 2025. URL http:///aarrii.oogg////22....2.0
Danial Rooyani and Fantahun M. Defersha. An efficient two-stage genetic algorithm for flexible job-shop scheduling. IFAC- PTAC-PapersOOnlinee 52:2519-2524, 2019. ISSN 24058963. doi: 10. 10 j.ifacol.2019.1 1,585. http:////innngggueeesssccoooores
John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms, 2017. URL http:://arxxi..rrgab//////7....
Veronique Sels, Nele Gheysen, and Mario Vanhoucke. A comparison of priority rules for the job shop scheduling problem under different flow time- and tardiness-related objective functions. International Journal of Production Research, 50:4255-4270, 8 2012. ISSN 0020-7543. doi: 10. 00.10800002005444.220 1..115599 URL.
wwtpc/c/oooooo/.oc/c/oo comm//0//aa//001100000005555555 1080002005030
2011.611539 doi: 10. 0001000000200543.2011..0
Igor G. Smit, Yaoxin Wu, Pavel Troubil, Yingqian Zhang, and Wim P.M. Nuijten. Neural
combinatorial optimization for stochastic flexible job shop scheduling problems. Proceed-
ings of the AAAI Conference on Artificial Intelligence, 39(25))66778-66687 2025a. doi: 10.1609/aaai, v39125.34770. URL http:://....aa...gg//ddepp/////a
Igor G. Smit, Jianan Zhou, Robbert Reijnen, Yaoxin Wu, Jian Chen, Cong Zhang, Zaharah Bukhsh, Yingqian Zhang, and Wim Nuijten. Graph neural networks for job shop scheduling problems: A survey. Computers & Operations Research, 176: 106914, 2025b. ISSN 0305-0548, doi: 1011016 j.cor.2024. 2e0220244.0611,, URL http:///www.eeeccccccccc/c////ea
Wen Song, Xinyang Chen, Qiqiang Li, and Zhiguang Cao. Flexible job-shop scheduling via graph neural network and deep reinforcement learning. IEEE Transactions Industrial Informatics, 19:1600-1610, 2022. ISSN 1551-3203.
Chupeng Su, Cong Zhang, Chuang Wang, Weihong Cen, Gang Chen, and Longhan Xie. Fast pareto set approximation for multi-objective flexible job shop scheduling via parallel preference- conditioned graph reinforcement learning. Swarm and Evolutionary Computation, 88: 18:101 2024. ISSN 20-6-20 doi: 10. 20.1016/..weev.222244
URL https://www.sciencddrreet
E. Taillard. Benchmarks for basic scheduling problems. European Journal of Operational Re- search, (4(2):278-2855 1993. ISSN 0377-2217. doi: 10. 021116007772221709990122222 URL
http::/wwwwwoueeeeee.....cceeees Scheduling.
Project Management anf
Karim Tamssaouet, Stéphane Dauzère- Pérèé, Sebastian Knopp, Abdoul Bitar, and Claude Yugma. Multiobjective optimization for complex flexible job-shop scheduling problems. European Jour- nal of Operational Research, 296:87-100, 2022. ISSN 03772217. doi: 10. 00.10166.ejor22221.00 .03. 069. URL http:://wwwwggooooo.o.oooooeeeep
Jesse van Remmerden, Zaharah Bukhsh, and Yingqian Zhang. Offline reinforcement learn- ing for learning to dispatch for job shop scheduling. Machine Learning. 114:191, 8 2025a. ISSN 0885-6125. doi: 10.10077/1109444055--66663 $10994-0025-08822
URL https://innksprrngerrcom
Jesse van Remmerden, Zaharah Bukhsh, and Yingqian Zhang. Generalizing beyond suboptimality:
Offline reinforcement learning learns effective scheduling through random data, 2025b. URL
httpss//arii.orrr/bb//555....33
Petar Veličković, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Liò, and Yoshua Bengio. Graph attention networks, 2018. URL httpss/arki..oryaas///111.000
13

---

Published as a conference paper at ICLR 2026
Runqing Wang, Gang Wang, Jian Sun, Fang Deng, and Jie Chen. Flexible job shop scheduling via dual attention network-bart based reinforcement learning. JEEE Transactions on Neural Networks and Learning Systems, 2023.
Zhenkun Wang, Shunyu Yao, Genghui Li, and Qingfu Zhang. Multiobjective combinatorial opti- mization using single deep reinforcement learning model. IEEE Transactions Cybernetics, 54(3):1984-199, 2024.
doi: 10.1109/TCYB BLL.///TCYB.2223332224444 2023.3312466.
Zufa Wu, Hongbo Fan, Yimeng Sun, and Manyu Peng. Efficient multi-objectivv optimization on dynamic flexible job shop scheduling using deep reinforcement learning approach. Processes, 11 (7), 2023. ISSN 2227-9717. doi: 10.3390/prl 1072018. URL /ttpps//www..mmpi.cco/////////// 11/7/2018
Zuobao Wu and M.X. Weng. Multiagent scheduling method with earliness and tardiness objectives in flexible job shops. IEEE Transactions Systems, Man, and Cybernetics, Part B (Cybermetics), 35(2):293 2005.
doi: http://SSCCC.2000.8888884
Biao Xiao, Zhengcai Zhao, Yingchen Wu, Xialin Zhu, Shixin Peng, and Honghua Su. An improved moea/d for multi-objective flexible job shop scheduling by considering efficiency and cost. Com- puters & Operations Research, 167:106674, 2024. ISSN 0305-0548. doi: 10. 10.1006/j.cor.2022 106674, wttp://wwwwww.coccoccccccccooooa
Jin Xie, Liang Gao, Kunkun Peng, Xinyu Li, and Haoran Li. Review on flexible job shop scheduling. IET Collaborative Intelligent Manufacturing. 1(3)x67-77, 2019, doi: 10. 10.1049fet-cim2200 2018.0009.
Cong Zhang, Wen Song, Zhiguang Cao, Jie Zhang, Puay Siew Tan, and Xu Chi. Learning to dispatch for job shop scheduling via deep reinforcement learning. In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin (eds.), Advances Neural Information Processing Systems, volume 33, pp. 1621-1632. Curran Associates, Inc., 2020. URL https:/pproeedingss.urrps...
paper
Paper.pppi
Guohui Zhang, Liang Gao, and Yang Shi. An effective genetic algorithm for the flexible job-shop scheduling problem. Expert Systems with Applications, 38(4):353333573 2011. ISSN 0957-
4174. doi: 10.10166j.eswaa www...a0...0000...44..... URL http:://www.eeeecccccccccooo//os 5095741741000953X
Jia-Dong Zhang. Zhixiang He, Wing-Ho Chan, and Chi-Yin Chow. Deepmag: Deep reinforcement learning with multi-agent graphs for flexible job shop scheduling. Knowledge-Basse Systems, 259: 110083, 2023a. ISSN 0950-7051. doi: wwt.////...osss...o 2022. 100083. URL https://www.
Jiae Zhang, Jianjun Yang, and Yong Zhou. Robust scheduling for multi-objective flexible job-shop problems with flexible workdays. Engineering Optimization, 1973-19998, 11 2016. ISSN 0305-215X. doi: 10. 00.108000302215x220166 145216. URL http://wwwwwwwdoooiiionooo////// 10.1000/0302215X. 2016. 1145216
doi: 10. 11w.10000002211X20661115552
Lixiang Zhang, Yan Yan, and Yaoguang Hu. Dynamic flexible scheduling with transportation constraints by multi-agent reinforcement learning. Engineering Applications of Artificial In- telligence, 134: 106699, 2024. ISSN 0952- 1976. doi: 10.1016/f..mm wttp://:cccnapp.222224..44 200.2.09992..,,699,,, 108699. URL
Lu Zhang, Yi Feng, Qinge Xiao, Yunlang Xu, Di Li, Dongsheng Yang, and Zhile Yang. Deep reinforcement learning for dynamic flexible job shop scheduling problem considering vari- able processing times. Journal of Manufacturing Systems, 71:257-273, 2023b. ISSN 0278- 6125. doi: 10. 10.1106/ijjmm jmsy:..2...m.y...m...o URL http:////wwww.cccccccccc..ooocne S027861 1777000122201111
Min Zhang, Liang Wang, Fusheng Qiu, and Xiaorui Liu. Dynamic scheduling for flexible job shop with insufficient transportation resources via graph neural network and deep reinforcement leamn ing. Computers & Industrial Engineering, 186: 109718, 2023c. ISSN 0360- doi: 10. 10. 1016/j.
cie. 109.70... wya.ap/wwwwwstiiiiccccooooeeeeee
14

---

Published as a conference paper at ICLR 2026
Qingfu Zhang and Hui Li. Moea/d: A multiobjective evolutionary algorithm based on de- composition. IEEE Transactions on Evolutionary Computation, 11(6):712-731, 2007. doi:
10.1109/EEV.2200..9222 .9777599
Zizhen Zhang, Zhiyuan Wu, Hang Zhang, and Jiahai Wang. Meta-leaming- deep reinforce- ment learning for multiobjective optimization problems. IEEE Transactions on Neural Networks and Learning Systems, 34(10):7978-799 2023d.
Linlin Zhao, Jiaxin Fan, Chunjiang Zhang, Weiming Shen, and Jing Zhuang. dri-based reactive scheduling policy for flexible job shops with random job arrivals. IEEE Transactions on Automa- tion Science and Engineering, 21(3):2912-2923, 2024. doi: N0.1109/AASE..2233321111..... 266.666......
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A DETAILED DESCRIPTION OF STATE FEATURES
Each state consists of operation features, machine features, and operation machine pair features. The state features ho., for all Oig E O are the following:
Minimum processing time pły among all machines Mr E MMi-
Average processing time płj among all machines Mj MMi--
Span of processing times přj among all machines Mr E Maj-
Proportion of machines that Ozy can be processed on: MMjl/IM
operation Oiş is scheduled, otherwise Number of unscheduled operations in job J
Sum of average processing times of all unscheduled operations in Jr.
Time between when an operation became available for scheduling, and the current schedul- ing time in the system. 0 if the operation is not yet available for scheduling.
Remaining processing time p. of operation Oig at the current scheduling time. the operation is unscheduled.
In addition, the operation feature vector contains the relevant lower bound features, described in Section 4.1 for the objectives that are considered.
For each machine M& Mu. have the following machine features hare
Minimum processing time ply among all operations Oig Ma E MMj.
Average processing time p among all operations Oi : M& E Mej.
Number of unscheduled operations that machine Mr can process.
Number of candidate operations that machine Mg can process.
The moment when machine Mr becomes available.
The time for which machine Mg has been idle at the current scheduling moment.
if Mj is processing operation, otherwise O.
The remaining processing time p. of the current processed operation Oi on machine Mg-
For each considered operation- machine pair (Oij. M&) E A, we use the feature vector h(O
Processing time p.j
Ratio of p.j to max ply
Ratio of pl to the maximum processing time of candidate operations that can be processed by My.
Ratio of pt to the maximum processing time unscheduled operations.
Ratio of př; to the maximum processing time of unscheduled operations that can be pro-
cessed by Mr.
Ratio of ptj to the maximum processing time of the pairs in A.
Ratio of p the remaining workload of job J:-
Sum of waiting times of Oig and Mr.
B THEORETICAL ALIGNMENT OF WEIGHTED SUM REWARD
Proposition The sum stepwise rewards is equal to the negative of the weighted sum of the increase in quality measures H()), given discounting factor = 1:
101-1
M
M Ne (H,())o)) Hi(sv)
Σ
Σ
t=0
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Proof.
- -1
M
101-1
A
Σ
Aprea (7 (9=1)
t=
101-1
Σ
Apra.
Σ
Σ
(Hi(st) Hs((++1))
t=
A; (HH(())))) H1((0))
From Proposition 1 and given that 8p a constant given by the problem instance, it follows that aiming to maximize the expected weighted sum stepwise function aligns with minimizing the in- crease in the weighted sum of the quality measure. Hence, optimizing our reward definition directly corresponds to optimizing our objectives.
C DECOMPOSITION-BASEH PPO LOSS
We use decomposition- -baddd actor-critic clipped PPO algorithm with generalized advantage esti- mation (GAE). The actor loss over given trajectory is defined as:
[
min
(1))))
O
(0) (a< Stc A) Taad (as Stc,
Here, Ar Ed ArA is the aggregated advantage estimate computed from the advantage esti- mates per objective and (0) is the output probability ratio between the current and previous policy for action a, The per-objective generalized advantage estimates follow from:
10 Σ
8c, Ft. + (8((5)1) - (s));, Ar.
(YT)' Se+2.
Here, (s() is the value estimate for objective i of the critic network, and and are hyperparam- eters controlling the bira-variance trade-off of the GAE. As explained before, the critic network updated using the critic loss function:
1 -1 M
Lattee
Σ
Σ
(vi(s)) Pr,a)²
10 M
1=0
Here, fr.; A + U/ (st) the bootstrapped generalized advantage estimate target for objective The final PPO loss consists of the actor loss, critic loss, and an entropy bonus Lentrpy ce 0 H|[To(-ss, A)] that encourages exploration:
LPPO Lasaa + Latttc Lestooy
In this equation, C₁ and c2 coefficients that control the weights of each loss.
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D VISUALIZATION OF NETWORK ARCHITECTURES
To further clarify the proposed WI-DAN and DCAN network architectures, we present several visual overviews this appendix. Figure I offfers a high-level overview of how the preference weights are incorporated in the WI-DAN and DCAN networks. In addition, Figure shows more detailed view of the conditional operation message attention block and the conditional machine message attention block of the DCAN architecture. In this figure, for simplicity, assume single attention head and show a single operation triplet forward pass.
hoy
A
M
2
Concat
Concat
hoa
X
10ul hM
Ox MM
DAN
WI-DAN
2
1
Linear
Linear
hog P
x
1Oul
h
h
OM 0 Qx [M
DCAN
(b) DCAN
Figure 1: Visualization of high-level WI-DAN and DCAN network architectures
h'y
hon-n hoy Nogan
h
h
Mul
Linear
Lincar
Concat
Concat
Concat
Concat
Linear
Linear
Linear
Linear
Softmax
Softmax
Softmax
Softmax
Combine
Combine
Combine
Combine
hoij
h
I+1
h
Mi
I+1
h
1+1
Figure 2: Detailed view of the conditional operational (left) machine (right) message attention blocks in the DCAN network
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E DESCRIPTION OF MACHINE INTENSITY COEFFICIENT
In the conditional machine message attention block, the coefficient ar,) is am intensity metric that measures the competition between machine Mr and Mg. We define Ckg the set of all oper- ations that can be performed
both Mr and Mq. We also define the set of candidate operations Je (Oty BM (Oij. M2a) A} as the set of all operations that appear in at least one potential action (Oij. M&) E A. The intensity metric is then computed using the embeddings ho,, of the operations in Ckyp
Mg)
h045
OcjECeo nJe
Ckyy TJTe is empty, the intensity coefficient values are O.
F HYPERVOLUME INDICATOR
The Hypervolume (HV) widely used metric for assessing performance in multi-objective opti- mization. Given found Pareto front F and reference point R.M, the HV is:
(F, - ( [f(().)) [(x))F
where ye denotes the Lebesgue measure, which indicates the M-dimensional volume, and [f(x), r] (x), x.. x [fas (z), ras] an M-dimensional cube that spans the regions between each point f fz) and the reference point The reference point is a defined point in the objective space that is typically dominated by solutions of interest.
The HV measure is sensitive to the scales of the objectives. Hence, we report the normalized hy- pervolume values. To this end, we first subtract the objective lower bounds, defined by the point from the points on the Pareto front. These lower bounds equal to the objective lower bounds in Section 4.1 at the initial state of the MDP. Then, we compute the hypervolume from these trans- formed points and divide by the product of the ranges between the reference point and lower bound. Thus, we use:
HV. HV, ( (1()) 2,r -,- =] ) -,,) II
(n.
2,)
[(EEEF
To define the reference point r for each problem instance, we initialize 1000 solutions according the initialization procedure of NSGA-II approach, and take the worst value find for each objective in this set solutions.
G IGD+ PERFORMANCE METRIC
IGD+ is performance metric for multi-objective optimization. It is defined as the average distance from each point in given reference Pareto front to the closest point in the solution set found. The distance to the closest solution is computed using a modified Euclidean distance that only accounts for the positive part of the difference in each objective. Formally, given reference set Z and solution set F, IGD+ is calculated as:
IGD (GD*(F,2)
(F,2)=
min d' (z, JCF ZE2
Σ
2
where
M
1/2
a (+, f) max 2;, 0)2
Σ
is the modified distance between the reference point and solution point f. A lower IGD+ value indicates that the approximated front closely follows the reference front both convergence and distribution.
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As noted, the IGD+ requires a reference Pareto set that serves the target. However, in our case, do not have access to the optimal Pareto sets. Hence, to compute the IGD+ we define an alternative reference set. We construct for a specific instance and using the non-dominated set among these. In this way, the resulting IGD+ measures will be influenced by this lack of true optimal reference set, so conclusions should be drawn with care. Nevertheless, the metric still provides valuable insights alongside the hypervolume.
reference sets by taking all the solutions found by all the methods
In Tables 6 7 and 8 present the IGD+ values corresponding to the main experiments. The results largely follow the same pattern observed with the hypervolume and the number of solutions in the Pareto sets. Specifically, our DRL approaches outperform the metaheuristics on the synthetic instances, with the performance gap increasing for larger instance sizes. Moreover, DCAN gen- erally outperforms WI-DAN, However, on instances with tardiness and cost objectives, WI-DAN performs better in terms of IGD+ than in terms of hypervolume, suggesting that it produces more evenly distributed fronts in these specific cases. For the other objectives, DCAN remains superior on this metric. Another notable finding is that for the public dataset instances, our DRL policies achieve better IGD+ scores than the metaheuristics on three out of four instance sets, whereas the hypervolume was worse for all of them. This may indicate that the metaheuristics benefited from a few extreme points on the edges of the objective space, while the DRL policies generated more tightly converged solution sets. Overall, the main conclusions based IGD+ are consistent with those drawn from the hypervolume metric, with a few noteworthy differences that provide additional insight.
H RESULTS ON DIFFERENT SCHEDULING PROBLEMS
We solve JSSP and FFSP problem instances using our approach. We use the same methods and hyperparameters for these problems. For the JSSP, we train and validate on synthetic instances gen- erated using Taillard's method (Tallard 1993). We solve the problem with the makespan and tardi- ness objectives, since each operation has a fixed machine and, thus, fixed costs. Table 9 shows the results for these instances. We find that DCAN performs well for these instances. For the smallest instances, we find again that the metaheuristies are slightly better while DCAN can achieve similar performance. For larger instances, we observe that DCAN performs better again. The metaheuristics start to have trouble with the increased scale, whereas our DRL approach holds good performance while also being considerably faster. We do not outperform CP-SAT on these instances. This is sensible since the JSSP has a smaller search space than the FJSP, which means that CP-SAT will lose performance only at larger instances. We already observe that DCAN gets closer to CP-SAT for the larger sizes, while still being considerably faster.
For the FFSP, we train and validate using synthetic instances that are generated similarly to Kwon let al. (202)) We use two types of instances. One with 15 jobs and 5 stages, where the stages have 3, 2, 3, 2, and 2 machine alternatives, respectively. The other has 20 jobs and stages, where each stage has 3 machine alternatives. We present these results in Table 10 Here, see that, despite the instances being small and therefore advantageous for the metahcuristics, DCAN is competitive or advantageous over the metaheuristics in terms of hypervolume while maintaining its considerable speed advantage. It does not achieve the same hypervolume as CP-SAT. However, small instances more suitable for CP-SAT and the runtime of DCAN is much shorter.
In short, these results confirm that our approach can be applied to other scheduling problems without modifications. We can maintain both efficiency and performance, and thereby our approach is not limited to the FJSP but can be applied to a variety of scheduling problems.
I ADDITIONAL SYNTHETIC INSTANCES
Wang et al. (2023) propose an additional instance set next to the one from Song et al. (2022), which they call SD2. This instance set is less realistic, as each processing time Př, is sampled uniformly from (1, 99). This implies that for the same operation, machine alternatives can be entirely differ- ent. In practice and in the synthetic data that we use, in contrast, the processing times of operations between machines are related to each other. Hence, we work with the our paper. However, for completeness, we also train and test on the SD2 instances using the same method. We present these results in Table 11
realistic instances in
The results similar our main results, with our
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DRL models outperforming the baselines considerably most instances. In terms of hypervol- ume, the difference between WI-DAN and DCAN smaller. This may be caused by the sharper decision boundaries, resulting from the unrelated processing times, that require less sophisticated differentiation between different objective preferences. However, in cases where DCAN is better, the performance improvement over WI-DAN tends to be larger than the other way around. More- over, the DCAN generally generates a Pareto set with more unique solutions. Hence, DCAN remains beneficial over WI-DAN on these instances.
VISUALIZATION OF RESULTS
To better understand the results, we visualize the Pareto fronts of one randomly selected instance per instance set of the synthetic data for the 2-objective problems. Figures 3 and 4 show these fronts. Although these figures are ipseiic--peccffcc and do not represent all solution shapes within each instance set, they do provide an indication of the general patterns.
The plots reflect the overall performance of the different methods, consistent with our numerical evaluation. In general, DRL policies achieve lower objective values than metaheuristics. CP-SAT solutions are highly competitive for the smaller instances, but for larger instances DRL policies tend to find better solutions. We observe that the CP-SAT solutions are generally more diverse and suc- ceed finding more extreme solutions at the edges of the Pareto front, strongly optimizing for one specific objective. The DRL policies, on the other hand, produce slightly more centralized solution sets. This centralization explains a sigmificant part of the advantage CP-SAT has over DCAN and WI-DAN on smaller instances. However, for larger instances, the solutions found by the DRL more diverse and cover a broader range of objective trade-offs.
Overall, DCAN appears to achieve slightly wider spread of solutions than WI-DAN, which may contribute to its better hypervolume performance. All in all, our DRL approach finds well-shaped solution sets that address a meaningful range of trade-offs. Only at the extreme ends of the solution space, where one objective is heavily prioritized, does the DRL approach underperform compared CP-SAT, However, in multi-objectivv optimization, trade-offs that balance the objectives are typ- ically preferred over solutions focusing heavily on single objective, mitigating the impact of this limitation.
K RESULTS ON 4-OBJECTIVE INSTANCES
We solve the 4-objective problem considering, makespan, flowtime, earliness, and costs, using 120 preferences, as presented in Table 12 These results show similar pattern of DCAN outperforming the baselines. The gap to CP-SAT slightly larger, which is mainly due the fact that earliness is non-regular objective. This is more challenging for constructive approaches, and we do not use any post-processing to allow for waiting or other adjustments in our implementation. Despite this, our approach remains superior to the metaheuristics, highlighting its ability to address problems with many objectives differing natures.
L RESULTS ON BENCHMARK INSTANCES FOR 2-OBJECTIVE PROBLEMS
Table 13 shows the results on the benchmark instances for the 2-objective problems. These results show that, since the problems are reduced to singlc-objective problems for rdata, edata, and vdata, only one non-dominated solution is found for those. Hence, these results do not indicate multi- objective performance, but mainly which model is optimized best for makespan tardiness. For the mk dataset, the DRL policies and NSGA-II have similar performance, with NSGA-II having slightly better hypervolume while having much larger runtime.
M SHORTER INFERENCE TIMES FOR BASELINE METHODS
In the main results, the baseline multi-objective optimization algorithms for many genera- tions, leading to a long runtime. For the synthetic data, our DRL approach already outperforms these algorithms with much longer runtimes. For the benchmark datasets, the NSGA-II baseline performs
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slightly better. However, in practice, the available runtime is often limited. This raises the question how the performance compares when the evolutionary algorithms are given less time. Therefore, we the NSGA-II for 50 and 100 generations, and the MOEA/D for 4000 and 8000 evaluations. Table shows the results. We find that our DRI. approach outperforms the baselines for similar runtimes. NSGA-II is only slightly better the edata instances. In other instances, DCAN achieves the best performance. Thus, in these instances where our approach does not outperform the baselines when they have a longer runtime, our approach does have better performance-runting trade-off, making it beneficial in scheduling scenarios with limited runtimes.
N HIGHER NUMBER OF SAMPLES FOR DRL POLICIES
Table 15 shows the results for the benchmark instances using a higher number of samples for DCAN. We find that the performance does improve and the DRL approach becomes more competitive when given the same runtime the NSGA-II. However, it has diminishing returns and does not provide substantial performance boost that makes the DCAN always better than NSGA-II as can be seen in the edata instances. This is because after certain number of samples, more duplicate solutions are produced. More elaborate search strategies can be explored to increase the test time performance of NMOCO methods.
COMBINING WI-DAN AND DCAN
We also explored combining the techniques of WI-DAN with DCAN. These results are shown in Table [6] This shows that combining the methods does not lead to clear performance increase. The conditional attention mechanism already provides strong way to condition the policy the objective preferences, making the additional WI mechanism redundant. Hence, for simplicity, we opted keep them separated.
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Table 6: IGD+ measures for the experiments on synthetic instances, related to Table
Metaheuristics
Greedy
Sample
Size
NSGA-II MOEA/D)
Hyper
WI-DAN
DCAN|
Hyper
WI-DAN
DCAN CP-SAT
10x5
0.0713
0.1031
00.1142
0.0641
0.0574
0.0876
0.0362
0.0324
0.0030
Makespan Costs
20
0.0489
0.0885
0.0526
0.0424
0.0367
0.0391
0.0333
0.0297
0.0063
15x10
0.1434
0.2545
0.0804
0.0478
0.0451
0.0642
0.0348
0.0325
0.0058
20x
0.1414
0.2720
0.0187
0.0160
0.0094
0.0161
0.0121
0.0053
0.0137
10x5
0.0833
0.1293
00.1163
0.0995
0.0748
0.0799
0.0779
0.0551
0.0406
0.0001
Tardiness Costs
20x5
0.1237
0.2244
0.0671
0.0350
0.0277
0.0529
0.0173
0.0127
0.0111
15x 10
0.1796
0.3227
0.0556
0.0267
0.0219
0.0478
0.0180
0.0141
0.0202
20x
0.2049
0.3682
0.0290
0.0114
0.0137
0.0237
0.0053
0.0085
0.0452
10x5
0.1044
0.1695
10.1727
0.1627
0.1390
O. 1409
0.1115
0.1095
0.0005
Makespan
20x5
0.1560
0.2700
0.0877
0.0480
0.0473
0.0708
0.0302
0.0348
0.0109
Flowtime Costs 15x10
0.1506
0.2609
0.0884
0.0310
0.0253
0.0809
0.0238
0.0189
0.0052
20
0.1757
0.2979
0.0609
0.0172
0.0172
0.0530
0.0104
0.0102
0.0225
Table 7: IGD+ measures for the experiments on the large synthetic instances, related to Table 2
Metaheuristics
Greedy
Sample
Size
NSGA-II MOEA/D
Hyper
WI-DAN
DCAN
Hyper
WI-DAN
DCAN CAAANCCP-SSAA SAT
Makespan Costs 30x 10
0.1315
0.2959
0.0096
0.0173
0.0059
0.0081
0.0129
0.0031
0.0259
40x10. 10
0.1092
0.2716
0.0091
0.0188
0.0062
0.0076
0.0129
0.0032
0.0439
Tardiness Costs
30x
0.2114
0.3786
0.0465
0.0058
0.0120
0.0430
0.0024
0.0082
0.0788
40x 10
0.1987
0.3573
0.0605
0.0039
0.0182
0.0575
0.0016
0.0126
0.1087
Makespan 30x 10
0.1859
0.3134
0.0626
0.0102
0.0067
0.0796
0.0058
0.0028
0.0515
Flowtime Costs
40x 10
0.1846
0.2971
0.0531
0.0095
0.0052
0.0510
0.0061
0.0019
0.0969
Table 8: IGD+ measures for the experiments on the public dataset instances for the 3-objective problem, related 3
Metaheuristics
Greedy
Sample
Size NSGA-11 MOEA/D
Hyper
WI-DAN
DCAN
Hyper
WI-DAN
DCAN
CP-SAT
mk 0.2403
0.3009
0.2982
0.3213
0.3072
0.2869
0.2991
0.2917
0.1449
rdata 0.2182
0.3145
0.1240
0.1695
0.1160
0.1117
0.1466
0.0915
0.0310
edata 0.2108
0.2725
0,1841
0.2365
0.1835
0.1693
0.2064
0.1437
0,0304
vdata 0.2146
0.3067
0.0894
0.1405
0.0745
0.0812
0.1251
0.0560
0.0383
Table Results on synthetic JSSP instances of the same sizes as the instances used for training for the makespan and tardiness objectives
Metaheuristics
Greedy
Sample
Size
NSGA-II MOEA/D Hyper
WI-DAN DCAN
Hyper
WI-DAN DCAN
CP-SAT
HV
0.8703
0.8729
0.8354
0.7722
0.8002
0,8450
0.8251
0.8244
0.8778
Gap
0.85%
0.56%
4.83%
12.02%
8.84%
3.73%
6.00%
6.07%
0.00%
6x6
IGD+
0.0057
0,0027
0,0291
0.0642
0.0486
0,0218
0,0322
0,0342
0.0014
Nr. Sol.
4.04
4.03
2.73
1.07
1.56
3.08
2.24
2.22
3.17
Time (s)
998.01
194.50
0.40
0.39
0.61
1.57
1.46
1.82
HV
0.8867
0,8592
0.8767
0.8412
0.8597
0,8844
0.8761
0.8817
0.9203
Gap
3.66%
6.64%
4.74%
8.60%
6.58%
3.91%
4.81%
4.20%
0.00%
10x 10
IGD+
0.0198
0.0421
0,0276
0.0473
0.0371
0.022.5
0.0259
0.0242
0.0001
Nr. Sol.
4.18
4.52
3.48
1.32
2.25
4.23
3.20
4.36
5.35
Time (s)
980.67
662.93
1.66
1.72
2.35
10.91
11.69
13.49
HV
0,8776
0.7955
0,9003
0.8797
0.8935
0.9133
0.9074
0.9104
0.9494
Gap
7.57%
16.21%
5.18%
7.34%
5.89%
3.81%
4.42%
4.11%
0.00%
15x 15
IGD+
0.0459
0.1107
0.0310
0.0412
0.0347
0.0226
0.0245
0.0242
0.0000
Nr. Sol.
4.19
2.85
3.97
1.37
2.98
6.34
4.42
5.92
13.27
Time (s)
4823.55
1694.13
7.57
7.62
9.71
68.58
69.46
82.01
HV
0.8394
0.7791
0.9164
0.9108
0.9145
0.9239
0.9224
0.9245
0.9576
Gap
12.34%
18.65%
4.31%
4.89%
4.50%
3.52%
3.68%
3.46%
0.00%
20x20
IGD+
0.0832
0,1327
0.0263
0.0291
0.0277
0.0217
0.0216
0.0215
0.0000
Nr. Sol.
3,02
2,50
4.33
3,21
4.18
6.60
5.68
6.85
10.50
Time (s)
19926.21
4561.18
28.24
28.29
30.35
267.84
275.62
278.54
23

---

Published as a conference paper at ICLR 2026
Table 10: Results synthetie FFSP instances of the same sizes the instances used for training for the makespan, flowtime and costs objectives
Metaheuristics NSGA-II MOEA/D
Greedy
Sample
Size
Hyper
WI-DAN
DCAN
Hyper
WI-DAN
DCAN
CP-SAT
HV
0.4492
0.3202
0,3808
0.3898
0.4014
0.4175
0.4336
0.4385
0.5541
15x5
IGD+
Gap
18.92%
0.0892
42.20%
0.1878
31.27%
0,1220
29.66%
0.1185
27.56%
0.1034
24.65%
0.0986
21.74%
0.0903
20.85%
0.0814
0,0013
0.00%
Nr. Sol.
417.96
163.73
30.27
30.94
33.38
77.34
104.36
108.80
76.52
Time (s)
639.38
359.20
1.75
1.73
2.47
13.26
13.14
15.05
HV
0.4039
0.2925
0.4057
0.3082
0,4200
0.4374
0.3647
0.4521
0.5424
20
Gap
25.54%
46.08%
25.20%
43.17%
22.56%
19.35%
32.76%
16.65%
0.00%
IGD+
0.0970
0.1918
0.0553
0.0837
0.0495
0.0420
0.0557
0.0355
0.0098
Nr. Sol. 477.05
207.32
46.69
13.02
50.03
151.83
111.49
177.35
75.71
Time
699.80
389.60
2.13
2.20
2.65
16.84
16.92
17.94
24

---

Published as a conference paper at ICLR 2026
Table 11: Results on synthetic instances from instance set SD2 of the same sizes as the instances
used for training
Metahcuristics
Greedy
Sample
Size
NSGA-II MOFA/D]
Hyper
WI-DAN DCAN
Hyper
WI-DAN DCAN
CPPSSAT
HV
0.6245
0,5691
0.6130
0.6292
0,6254
0,6494
0.6740
0.6718
0.6960
Gap
10.27%
18.22%
11.93%
9.59%
10.14%
6.69%
3.16%
3.47%
0.00%
10x5
IGD+
0.0487
0.0815
0,0452
0.0384
0.0395
0.0263
0.0167
0.0175
0.0232
Nr. Sol.
46.12
40.07
20.35
17.96
17.65
46.78
55.96
$2.77
32.69
Time (s)
260.19
254.54
0.57
0.56
0.87
2.33
2.30
2.97
Makespon Costs
HV
0.4554
0,4082
0.4917
0.3237
0.4926
0.5059
0.4732
0.5148
0.4875
Gap
6.59%
16.28%
-0.86%
33.60%
-1.03%
-3.77%
2.95%
-5.59%
0.00%
20x5
IGD+
0.0491
0.0858
0.0203
0.1471
0.0209
0,0122
0,0363
0.0084
0.0259
Nr. Sol.
91.79
61.59
33.04
9.84
26.67
91.68
$6.30
83.89
43.82
Time (s)
542.25
617.20
1.45
1.46
2.13
7.94
8.46
10.62
HV 0,4846
0,3724
0,5966
0,6312
0.6372
0.6177
0.6659
0.6677
0,7402
15x10
Gap
34.54%
49.69%
19.41%
14.72%
13.92%
16.55%
10.03%
9.80%
0.00%
IGD+
0,1501
0,2307
0.0558
0.0413
0.0393
0.0455
0.0263
0.0258
0,0088
Nr. Sol.
96.38
53.51
25.47
22.27
22.39
56.81
64.89
63.71
54.11
Time (s)
1651.52
1033.59
2.92
2.95
3,94
21.12
22.93
26.69
HV 0.4294
0.3183
0.6084
0.6332
0.6374
0.6234
0.6514
0.6540
0.6521
Gap
34.15%
51.19%
6.70%
2.90%
2.25%
4.40%
0.10%
-0.29%
0.00%
20x 10
IGD+
0.1458
0.2291
0.0237
0.0161
0.0149
0.0167
0.0080
0.0072
0.0167
Nr. Sol.
111.64
57.39
33,90
28.39
31.72
95.71
92.10
102.02
49.22
Time
2744.02
1658.01
4.83
5.09
6.20
37.04
41.29
45.56
HV
0.6458
0,5734
0.6210
0.6453
0.6377
0.6578
0.6893
0.6892
0.7614
Gap
15.18%
24.70%
18.44%
15.25%
16.25%
13.60%
9.48%
9.49%
0.00%
10x5
IGD+
0.0656
0.1113
0,0681
0,0573
0,0606
0.0496
0.0345
0.0337
0.0019
Nr. Sol.
83.06
47.34
23.26
21.37
19.48
48.39
62.02
62.61
40.45
Time (8)
235.78
256.09
0.61
0.59
0.91
2.63
2.71
3.34
Tardiness Costs
HV
0.4736
0.3871
0,5541
0.5793
0.5842
0.5760
0.6058
0.6076
0.6226
Gap
23.94%
37.83%
11.00%
6.95%
6.16%
7.48%
2.70%
2.40%
0.00%
20x5
IGD+
0.1039
0.1718
0.0405
0.0275
0.0250
0.0271
0.0125
0.0121
0.0086
Nr. Sol.
100.59
64.46
29.20
29.13
28.64
59.60
67.20
69.97
34.56
Time (s)
531.53
622.17
1,58
1.59
2.24
9.61
10.15
11.95
HV
0.4595
0.3490
0,5922
0.6238
0.6171
0,6126
0.6515
0.6408
0.6990
Gap
34.26%
50.06%
15.27%
10.76%
11.72%
12.36%
6.80%
8.32%
0.00%
15x 10
IGD+
0.1458
0.2252
0.0381
0.0271
0.0307
0.0286
0.0163
0.0212
0.0154
Nr. Sol.
120.96
62.03
33.90
33.76
36.77
80.95
102.46
98.34
44.81
Time (8)
1651.95
1033.17
3.17
3.46
4.46
24.37
26.05
30.11
HV
0.3994
0,2909
0.5916
0.6314
0.6320
0.6083
0.6507
0,6482
0.6337
Gap
36.98%
54.09%
6.65%
0.37%
0.26%
4.01%
-2.68%
-2.28%
0.00%
20x10 10
IGD+
0.1575
0,2361
0.0282
0,0138
0,0140
0.0207
0.0059
0.0071
0.0225
Nr. Sol.
123.98
63.14
38.87
40.15
41.27
93.95
109.92
105.89
37.75
Time (s)
2739.73
1655.64
5.41
5.65
7.01
42.81
46.89
$2.89
HV
0.4531
0,3789
0.3868
0.3594
0,3918
0,4092
0,4123
0.4340
0.5158
Gap
12.16%
26.55%
25.01%
30.32%
24.05%
20.66%
20.07%
15.87%
0.00%
10x5
IGD+
0.0467
0,0822
0,0543
0,0590
0.0507
0.0399
0.0355
0.0322
0,0118
Makespan Flowtime Costs
Nr. Sol.
479.91
193.70
51.90
23.75
49,21
137.85
150.34
183.25
75.62
Time (8)
248.22
260.93
0.83
0.82
1.11
5.05
5.03
5.65
HV 0,2759
0.2212
0.3416
0,3383
0.3608
0.3428
0.3701
0.3827
0.3884
20x5
Gap
28.96%
43.06%
12.04%
12.91%
7.10%
11.74%
4.70%
1.46% %
0.00%
IGD+
0.1182
0,1886
0.0303
0.0287
0.0197
0.0195
0.0160
0.01
0.0173
Nr. Sol.
692,81
228.96
62.85
45.76
64.60
182.79
220.66
242.45
87.55
Time (s)
565.30
626.18
2.51
2.60
3.20
18.76
19.60
21.41
HV 0.2831
0.1992
0.4024
0.4171
0.4174
0.4217
0.4369
0.4354
0.4705
Gap
39.84%
57.67%
14.47%
11.37%
11.29%
10.37%
7.15%
7.47%
0.00%
15x 10
IGD+
0.1217
0.1904
0.0206
0.0191
0.0197
0.0146
0.0122
0.0134
0.0140
Nr. Sol.
360.96
120.93
65.32
46.25
64.42
262.92
277.51
289.54
88.82
Time 1668.57
1009.72
4.40
4.57
5.53
34.83
37.60
41.35
HV 0.2333
0.1597
0.3725
0.3878
0.3917
0.3874
0.4097
0.4133
0.3841
Gap
39.25%
58.43%
3.01%
-0.96%
-1.99%
-0.86%
-6.66%
-7.60%
0.00%
20x 10
IGD+
0.1301
0,2008
0.0250
0,0177
0.0172
0.0196
0.01
0.0114
0.0297
Nr. Sol.
398.97
119.22
67.14
61.14
68.45
275.70
285,84
305.61
83.85
Time (8)
2757.34
1634.10
7.48
7.70
9.07
62.10
66.94
72.69
25

---

Published as a conference paper at ICLR 2026
Table 12: Results on synthetic instances for the 4 objectives makespan, flowtime, earliness, and costs
Metaheuristics
Greedy
Sample
Size
INSGA-II MOEA/D
Hyper
WI-DAN
DCAN
Hyper
WI-DAN DCAN
CP-SAT
HV
0,5628
0.4785
0.3871
0.3618
0,4333
0.4306
0.4508
0.4773
0.653
10x5
Gap
13.82%
26.74%
40.73%
44.60/%
33.65%
34.07%
30.97%
26.92%
0.00%
IGD+
0,0812
0.1106
0.1344
0.1449
0.1076
0.1085
0.0966
0.0855
0.0062
Nr. Sol.
1038.92
147.41
50.71
7.23
47.49
178.60
155.77
183.66
73.54
Time (s)
255.13
267.81
0.92
0.95
1.24
5.89
5.87
6.12
HV
0.3381
0.2561
0.3545
0.3849
0,3895
0,3785
0.4177
0,4147
0.5068
20x5
Gap
33.28%
49.47%
30.05%
24.05%
23.15%
25.32%
17.58%
18.17%
0.00%
IGD+
0,1388
0.2044
0,0808
0.0661
0,0473
0.0620
0.0385
0.0326
0.0094
Nr. Sol.
695.60
111.38
61.37
51.52
51.95
209 96
186.91
165.83
64.14
Time (s)
537.58
629.61
2.87
2.83
3.51
23.07
22.90
23.92
HV
0.4599
0,3283
0.4447
0.4582
0.4611
0.4653
0,4768
0,4826
0.7106
15x10
Gap
35.29%
53,81%
37 37.42%
35.52%
35.12%
34.53%
32.90%
32.09%
0.00%
IGD+
0,1203
0,2069
0,1083
0.0979
0.0999
0.0978
0.0899
0.0902
0.0084
Nr. Sol.
399.58
55.32
63.34
53.10
59.52
250.78
212.46
213.13
90.31
Time (s)
1652.62
1034.72
5.15
5.01
6.05
43.59
43.11
45.12
HV
0.3828
0,2572
0.5325
0,5239
0,5500
0.5436
0,5341
0,5660
0.6485
20x10
Gap
40.97%
60.34%
17.89%
19.22%
15.19%
16.18%
17.65%
12.73%
0.00%
IGD+
0.1521
0,2736
0.0166
0,0156
0,0163
0,0129
0,0119
0.0113
0,0085
Nr. Sol.
185,30
45.69
66.12
52.29
55.78
256.21
168.87
181.97
79.90
Time (8)
2758.37
1697.82
9.36
9.18
10.33
81.66
77.99
82.74
Table 13: Results on public dataset instances for the 2-objective problems using the 15x10 policies
Metaheuristics
Greedy
Sample
Size
INSGA-II MOEA/D
Hyper
WI-DAN
DCAN
Hyper
WI-DAN
DCAN
CP-SAT
HV
0.5575
0.4428
0.4844
0.4784
0.4894
0.5254
0.5231
0.5387
0.6772
Gap
17.67%
34.61%
28.47%
29.35%
27.73%
22.41%
22.76%
20.45%
0.00%
IGD+
0.1709
0,3112
0,3014
0.3086
0.3008
0.2797
0,2829
0.2749
0,1943
Nr. Sol.
30.33
26.60
7.20
6.00
8.10
12.10
13.70
14.80
25.00
Time (s)
1437.42
1148.88
3.09
3.12
4.08
23.81
24.70
29.03
Makespan Costs
HVV
0.8128
0.7795
0.8443
0.8479
0.8473
0.8531
0.8549
0.8553
0.8642
Gap
5.94%
9.79%
2.29%
1.89%
1.95%
1.27%
1.07%
1.03%
0.00%
rdata
IGD+
0,1286
0,1728
0,0504
0.0469
0,0474
0,0416
0,0398
0.0395
0,0306
Nr. Sol.
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
Time (s)
1137.94
1153.30
3.52
3.75
4.85
28.41
30.28
35.03
HV
0.7967
0.7799
0.7843
0.7915
0.7960
0.7943
0.8076
0.8116
0.8329
Gap
4.35%
6.36%
5.83%
4.96%
4.43%
4.63%
3.04%
2.55%
0.00%
edata
IGD+
0.1090
0,1410
0.0808
0.0735
0,0691
0,0639
0.0575
0.0534
0.0322
Nr. Sol.
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
Time (s)
1299.37
1131.10
3.86
3.73
4.75
30.15
30.00
34.68
HV
0.8685
0.8379
0.9093
0.9104
0.9100
0,9112
0.9119
0.9114
0.9101
Gap
4.57%
7.94%
0.09%
-0.03%
0.01%
-0.12%
-0.20%
-0.13%
0.00%
vdata
IGD+
0,0997
0.1413
0.0210
0.0199
0.0202
0.0190
0,0183
0.0189
0.0202
Nr. Sol.
1.00
1.00
1.00
1.00
1.00
1.00
1,00
1.00
1.00
Time (s)
1115.27
1212.02
3.58
3.83
4.93
28.44
31.22
35.92
HV
0.5605
0.4282
0,5313
0.4867
0.5196
0,5559
0.5370
0,5548
0.7454
Gap
24.81%
42.55%
28.72%
34.71%
30.30%
25.42%
27.96%
25.58%
0.00%
mk
IGD+
0.2176
0.3012
0,2713
0.2961
0.2798
0.2593
0.2671
0.2558
0.1682
Nr. Sol.
47.80
30.60
14.90
7.90
11.90
22.60
17.20
20.90
31,60
Time (s)
1758.80
1129.41
3.46
3.46
4.43
28.07
28.08
32.25
Tardiness Costs
HV
0.8335
0.7726
0,8733
0.8896
0.8700
0.8811
0.8952
0.8776
0.9109
Gap
8.49%
15.18%
4.12%
2.34%
4.49%
3.27%
1.72%
3.65%
0.00%
rdata
IGD+
0.2420
0,3381
0.0802
0.0642
0.0838
0.0727
0.0587
0.0762
0,0432
Nr. Sol.
1.00
1,00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
Time (s)
1095.41
1151.52
3.93
4.17
5.33
33.11
34.72
39.60
HV
0.8180
0.7798
0.8262
0.8432
0.8152
0.8409
0.8574
0.8355
0.8915
Gap
8.24%
12.53%
7.33%
5.42%
8.56%
5.68%
3.82%
6.28%
0.00%
edata
IGD+
0.2372
0.3032
0.1117
0.0947
0.1221
0.0973
0.0808
0.1026
0.0477
Nr. Sol.
1.00
1.00
1.00
1.00
1.00
1.00
1,00
1.00
1,00
Time (s)
1164.60
1129.98
3.99
4.20
5.24
32.76
34.81
39.03
HV
0.8848
0,8223
0,9258
0,9346
0.9177
0,9287
0.9376
0.9214
0.9354
Gap
5.41%
12.10%
1.02%
0.09%
1.89%
0.72%
-0.24%
1.50%
0.00%
vdata
IGD+
0.2021
0.2993
0.0435
0.0347
0.0516
0.0406
0.0317
0.0479
0.0340
Nr. Sol.
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
Time (s)
1074.69
1192.19
4.08
4.21
5.21
33.00
34.66
39.50
26

---

Published as a conference paper at ICLR 2026
Table 14: Results public dataset instances for the 3-objective problem using the 15x 10 policies with less generations for the baseline algorithms
Metaheuristics
Greedy
Sample
Size
JNSGA Ilau NSGA-II100
MOEA/D
MOEA/Dauuo
WI-DAN DCAN
|WI-DAN
DCAN
HV
0,2522
0,2839
0.1962
0.2141
0,2519
0.2743
0.2878
0.3047
Gap
50.40%
44.16%
61.41%
57.90%
50.46%
46.06%
43.39%
40.07% %
mk
IGD+
0,2924
0,2766
0.3399
0,3323
0.3213
0.3072
0,2991
0.2917
Nr. Sol.
85.60
123.90
49.00
$7.80
18.10
27.50
51.10
84.50
Time (s)
87.24
199.54
55.92
112.11
4.78
5.73
41.10
44.05
HV
0,5572
0.5904
0,4940
0,5145
0,5900
0,5809
0,6033
0.6052
Gap
23.63%
19.08%
32.29%
29.48%
19.14%
20.39%
17.31%
17.06% %
rdata
IGD+
0,3125
0,2829
0,3703
0.3538
0.1695
0.1160
0,1466
0.0915
Nr. Sol.
7.05
7.50
5.53
5.35
6.08
6.73
9.28
8,90
Time (s)
103.34
205.98
57.21
114.50
5.68
6.86
50.05
54.29
HV
0.5594
0,5861
0.5201
0,5297
0,5212
0.5207
0.5427
0,5463
Gap
21.62%
17.89%
27.12%
25.78%
26.98%
27.04%
23.96%
23.46%
edata
IGD+
0,2856
0,2649
0,3236
0,3194
0,2365
0,1835
0,2064
0,1437
Nr. Sol.
6.73
8.00
5.20
5.48
3.95
4.23
6,23
5.95
Time (s)
103.29
207.07
56.16
112.97
5.66
6.80
49.50
54.21
HV
0,6340
0,6556
0,5436
0,5646
0,6799
0.6701
0.6800
0.6855
Gap
19.82%
17.09%
31.25%
28.59%
14.01%
15.25%
14.00%
13.30%
vdata
IGD+
0,2906
0,2705
0,3666
0,3531
0,1405
0,0745
0.1251
0.0560
Nr. Sol.
7.60
8.78
5.175
5.85
7.28
7.98
10.43
10.85
Time (s)
104.50
208.22
60.10
120.29
5.65
6.81
48.82
54.10
Table 15: Results on public dataset instances for the 3-objective problem using the 15x 10 policies with more inference samples per preference
Metaheuristics
Sample
Size
NSGA-11 MOEA/D
10
100
400
CCPSAT
HV
0,3416
0.2517
0.3047
0.3258
0.3351
0.5085
Gap
32.82%
50.49%
40.07%
35.93%
34.09%
0.00%
mk
IGD+
0.2403
0.3009
0.2917
0.2753
0.2697
0.1449
Sol.
275.70
94.30
84,50
162.30
200.40
68.90
Time (s)
1787.04
1122.09
44.05
411.15
1647.12
HV 0.6586
0.5652
0.6367
0.6487
0.6561
0.7296
Gap
9.73%
22.53%
12.73%
11.10%
10.07%
0.00%
rdata
IGD+
0.2182
0.3145
0.0915
0.0810
0.0754
0.0310
Nr. Sol.
8,98
6.98
9.48
11.72
14.28
15.18
Time (s)
2120.14
1164.22
47.79
485.12
1938.43
HV 0.6439
0.5773
0,5689
0.5811
0,5883
0.7137
Gap
9.79%
19.11%
20.30%
18.58%
17.58%
0.00%
edata
IGD+
0.2108
0.2725
0,1437
0.1326
0.1270
0.0304
Nr. Sol.
10,40
6.70
7.43
9.40
10.55
16.00
Time (8)
2106.18
1140.89
47.78
486.01
1944.45
HV
0.7180
0.6133
0.7378
0.7474
0.7524
0.7907
Gap
9.20%
22.44%
6.69%
5.47%
4.84%
0.00%
ydata
IGD+
0.2146
0.3067
0.0560
0.0482
0.0451
0.0383
Nr. Sol.
11.70
6.50
10.90
12,60
15.15
12.03
Time
2236.94
1189.89
48.26
487.44
1957.78
27

---

Published as a conference paper at ICLR 2026
Table 16: Performance comparison between DCAN and WI-DCAN, the architecture combining the ideas of WI-DAN and DCAN
10x5
20x5
15x10
20x10
|Greedy Sample|Greedy Sample|Greedy Sample Greedy Sample
HV
00.7104
0.7647
0.5599
0.5724
0.7723
0.8002
0.8083
0.8200
DCAN
Nr. Sol.
3.46
7.77
4.04
6.82
9.14
15.42
13.85
22.66
Makespan Costs
WI-DCAN HVV
0.7255
0.7644 4.45
0.5571
0.5716
0.7746
0.8012
0.8122
0.8209
Nr. Sol.
8.45
4.31
6.86
9.60
16.22
15.61
24.72
HV
0,7460
0,8272
0.6396
0.6700
0.8094
0.8338 17.04
0.81 12
0,8306
DCAN
Nr. Sol.
4.36
12.89
10.88
19.51
32.39
20.03
34.01
Tardiness Costs
HV
0.7668
0.8310
0.6359
0.6673
0.7914
0.8211
0.8147
0.8295
WI-DCAN
Nr. Sol.
6.70
16.22
8.08
16.48
14.28
29.48
20.37
34.71
HV
0.4647
0,5130
0,4318
0.4529
0.5793
0.6025
0.6142
0.6314
DCAN
Makespan
Nr. Sol.
22.64
52.84
32.96
81.77
43.96
121.23
57.83
179.74
Flowtime Costs
HV
0.4988
0.5484
0.4028
0.4307
0.5883
0.6061
0.6015
0.6170
WI-DCAN
Nr. Sol
23.6
65.77
33.52
79.74
44.56
112.75
55.03
175.13
28

---

Published as a conference paper at ICLR 2026
NSCA-II
MOEA/D
MOEA/D
CP-SAT
CP-SAT
WI-DAN (greedy)
1075
WI-DAN (greedy)
WI-DAN (sampling)
WI-DAN (sampling)
DCAN (greedy)
1030
DCAN (greedy)
Costs
DCAN (sampling)
Costs
DCAN (sampling)
1025
1000
875
t mhna
P.00
100
110
120
155
140
150
160
170
180
200
220
240
260
280
300
Makespan
Makespan
(a) 10x5
(b) 20x5
2000
1600
MOEA/D
MOEA/D
CP-SAT
1550
WI-DAN (gready)
1900
WI-DAN (gready)
WI-DAN (sampling)
WI-DAN (sampling)
DCAN (grwndy)
DCAN (grendy)
Costs
DCAN (sampling)
Costs
DCAN (sampling)
1450
17000
1400
1350
1600
1300
1500
2:50
1600
350
300
350
400
450
500 530
Makespan
Makespan
(c) 15x10 10
(d) 20x 10
NSCA-II
2000
NSCA-11
MOEA/D
MOEA/D
1530
WI-DAN (grendy)
1900
WI-DAN (grondy)
(sampling)
(sampling)
DCAN (grendy)
DCAN (grondy)
Costs
(sampling)
Costs
(sampling)
1450
1700
1400
1350
1600
2:00 Makespan
3600
350
360
35.
460
450
500
550
Makespan
(e) 30x 10
(1 40x 10
Figure 3: Visualization of solutions of randomly selected instances from different sizes for the makespan and costs
29

---

Published as a conference paper at ICLR 2026
NSCA-II
MOEA/D
MOEA/D
480
CP-SAT
CP-SAT
WI-DAN (greedy)
1030
WI-DAN (greedy)
WI-DAN (sampling)
WI-DAN (sampling)
DCAN (greedy)
DCAN (greedy)
Costs
DCAN (sampling)
Costs
DCAN (sampling)
1000
440
975
950
925
100
100
150
200
250
360
350
1000
1550
2000
2500
3000
3500
Total Tardiness
Total Tardiness
(a) 10x5
(b) 20x5
2000
NSGA-II
MOEADD
MOEA/D
CP-SAT
1550
WI-DAN (gready)
WI-DAN (sampling)
1900
WI-DAN (gready)
WI-DAN (sampling)
1500
DCAN (grwndy)
DCAN (grendy)
Costs
(sampling)
Costs
1800
DCAN (sampling)
1400
1700
1350
1600
1300
500
1000
1500
2000 2500 3000
500
1000
1500
2000 2500 3000 35000 8000 Total Tardiness
Total Tardiness
(c) 15x10
(d) 20
3100
4200
NSGA-I1
NSGA-II
4100
MOFA/D
CP-SAT
CP-SAT
(greedy)
4000
WI-DAN (greedy)
WI-DAN (sampling)
WI-DAN (sampling) (greedy)
2930
(gready)
3900
Costs
DCAN (sampling)
Costs
DCAN (sampling)
3800
2850
3700
2800
3600
3500
2700
3400
2000
6000 80000 106001 20001
5000
20000
25000 30000
Total Tardiness
Total Tardiness
(e) 30
(i 40x 10
Figure 4: Visualization of solutions of randomly selected instances from different sizes for the tardiness and costs
30


Reviews:
[{"review_id": "zn5fGHTvIH", "reviewer": "Reviewer_ck3N", "summary": "The paper proposes to use DRL for the multi objective FJSSP. It compares two different neural architectures and also compares to standard MOCO algorithms such as NSGAII. For large instances the algorithm that samples using the trained neural networks has good results. The conditional attention network outperforms the preference vector input network.", "questions": "How does the results of best network evolve with samples?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Good experimental results for large instances\nComparison of two neural architectures\nDesign of the DCAN architecture\nBetter results than meta-heuristic approaches", "weaknesses": "Simple sampling strategy\nNSGA-II has better result on public dataset instances for the 3-objective problem", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "e2T4FVRpfr", "reviewer": "Reviewer_oBEa", "summary": "This paper proposes a decomposition-based learning method to solve multi-objective flexible job shop scheduling (FJSP), where each subproblem associated with a preference vector. They define two neural architectures based on the DAN architecture from the previous literature for single-objective FSJP. WI-DAN concatenates the preference vector to the feature vector whereas DCAN leverages dual attention with conditional operation message attention block and conditional machine message attention block. The authors evaluate the performance of their proposed method on a variety of FSJP benchmarks, comparing with two commonly used multi-objective evolutionary algorithms, to show effectiveness.", "questions": "See the weaknesses. And further, based on the applicability of multi-objective learning methods for other COPs, the author should consider comparing with those applicable learning methods. Currently, I feel like the number of baselines the author compared is too few. And further, I think it will strengthen the paper if the authors can try to apply their proposed method to more scheduling variants beyond FJSP.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. The paper is easy to read and the proposed method makes sense for multi-objective optimization.\n\n2. Empirical results seem promising for the given method.", "weaknesses": "1. There have been many works on learning for multi-objective combinatorial optimization for other COPs (e.g. routing). This paper does not give sufficient discussion of these related works (e.g. the related work section only mentions a few works for multi-objective FJSP, but not for other COPs at all). Similarly, there’s no discussion on (1) whether the learning method used in this paper has been applied to other COPs, and (2) whether techniques from other COPs can apply to FJSP considered in this paper. The authors should rewrite the related works and discussions to better connect to existing literature. \n\n2. In general, I’m concerned about the novelty of this paper. The decomposition based PPO algorithm seems to be a standard RL algorithm for multi-objective learning, and the neural architecture design seems to be a straightforward extension from the DAN architecture in the previous work. Given this, I’m worried that this paper does not meet the bar for a high quality ML conference like ICLR.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "JhW5P0RxVn", "reviewer": "Reviewer_399M", "summary": "The paper presents two extensions to an existing neural network architecture for the FJSSP, namely the Dual Attention Network, enabling it to address multi-objective combinatorial optimization problems. The authors aim to approximate the standard Pareto front and identify the corresponding Pareto set.\n\nThe first proposed method is a straightforward extension in which the weighted preference vector of the objectives is concatenated with each operation and machine feature. The rest of the network remains unchanged from the single-objective formulation. The second method integrates the preference vector directly into the update process of the operation and machine embeddings during the attention mechanism.\n\nThe authors investigate four optimization objectives: makespan, tardiness, flow time, and cost. The two proposed extensions are evaluated against standard metaheuristic algorithms and a mathematical solver across several benchmark datasets. Additionally, the approach is extended to two related problem types, the classical JSSP and the FFSP (discussed in the appendix).\n\nThe results demonstrate that both neural network extensions achieve significantly faster computation times compared to metaheuristic algorithms and solver-based methods across all instance types. However, the metaheuristics outperform the proposed approaches on certain, smaller problem instances, while the advantage of the proposed methods becomes particularly evident for larger instances that are computationally challenging for traditional optimization techniques. It was also shown that the second proposed approach, in which the preference is integrated into the attention mechanism, performs better than the approach where the preference vector is trivially inserted.\n\nThe authors want to publish code after acceptance.", "questions": "1. Since the study introduces four distinct optimization objectives, why was the method not evaluated on all four objectives simultaneously? The experiments only include comparisons for selected two-objective and three-objective cases. Was this limitation due to the constraints of the metaheuristic algorithms and the solver, or does it reflect a limitation of the proposed model itself?\n2. In Table 3, it would be helpful to see results for additional sampling iterations. Given that the proposed method can generate substantially more samples within the same computational time as the metaheuristic approaches, it is possible that further sampling could enable it to outperform the metaheuristics. Could you clarify why this comparison was not included?\n3. For the ablation studies, why was only NSGA-II considered and not the newer NSGA-III? The latter represents a more state-of-the-art approach for multi-objective optimization.\n4. The focus of current research in multi-objective scheduling is shifting toward real-world applications, where objectives are defined for more practical use cases such as energy consumption and energy cost. Why were such abstract objectives chosen in this study? No citations were provided to justify the selection of the investigated objectives.\n5. There already exist methods that apply deep learning to multi-objective scheduling. Why were these methods ignored? At least some ablation studies, for example on small instances, should have been included as a comparison against the proposed method.\n6. The IGD+ metric was mentioned and used in the Appendix to compare different approaches. Why was it not included in all the tables? Since it represents an alternative to the hypervolume metric, wouldn’t it be reasonable to include both for completeness?\n7. Both methods do not appear to be overly complex, and it seems straightforward to combine them into a single approach—for instance, DCAN with the feature space of WI-CAN. Why was this not investigated?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- Well-motivated problem formulation, clearly highlighting that neural network architectures have rarely been applied to multi-objective variants of the Flexible Job Shop Scheduling Problem (FJSSP).\n\n- Introduction of two new methods that improve solution quality, particularly with respect to inference time: The first method, while not highly innovative, forms a solid baseline. The second method shows clear advancements beyond the baseline.\n\n- Comprehensive evaluation setup: Both methods are tested using greedy and sampling-based inference strategies. This allows a thorough assessment under different inference regimes.\n\n- Novel reward function design: Reward is defined based on the change in the theoretical minimum lower bound before and after an action for each metric. This differs from previous work, which typically uses dense or sparse rewards derived directly from the actual schedule rather than theoretical lower bounds.\n\n- Well-structured and informative appendix: Contains mathematical assumptions, details on key performance metrics, description of the critic network architecture (PPO), and a step-by-step reward calculation.\n\n- Transparent discussion of limitations: Authors acknowledge where their methods underperform on small instances. They provide a convincing explanation why their approach performs better on larger instances.\n\n- Demonstrates understanding of the problem characteristics and adds credibility to the evaluation.", "weaknesses": "- Although the authors acknowledge that other deep learning solutions typically train separate models for individual preference vectors, no such method is included in the experiments. Including at least one representative baseline (e.g., policy-based or value-based DRL approaches from recent literature) would provide meaningful context and allow readers to better assess the relative performance and contribution of the proposed methods.\n\n- The description of how the preference vector is incorporated into the second method (architecture with dual inputs) remains unclear. A visual schematic, for example a processing pipeline similar to those used in DAN literature, would help illustrate how the model processes and uses preference information.\n\n- While the reward formulation (based on changes in theoretical lower bounds) is novel, the paper does not empirically justify why this reward is advantageous over more traditional dense or sparse scheduling rewards. A small ablation experiment (for example in the Appendix) comparing the proposed reward against a common baseline reward would strengthen the motivation and demonstrate its impact on learning behavior and performance.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "K2a7qZi5ok", "reviewer": "Reviewer_XAsV", "summary": "This paper addresses the flexible job shop scheduling problem with multiple objectives. The authors propose a Dual Conditional Attention Network that learns adaptable scheduling policies. They employ a decomposition-based neural combinatorial optimization approach, solving subproblems with different objective preferences, and use PPO for training. The learned policy can generate a set of Pareto-optimal solutions.", "questions": "1. How can practical constraints such as setup times or machine dedication be incorporated into your framework?\n2. How does the model perform when including objectives that are not regular measures (e.g., earliness)?\n3. Many recent studies employ REINFORCE for scheduling. What motivated your choice of PPO?\n4. Using lower bounds as rewards can lead to inconsistent gradients depending on their tightness. How do you handle or mitigate this issue?\n5. Please provide an ablation study to analyze the contribution of each proposed component.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "The paper tackles a practically important and realistic problem, as real-world scheduling problems typically involve multiple conflicting objectives.\n\nThe approach of learning adaptable policies conditioned on objective preferences is promising and could improve generalization.\n\nGenerating a Pareto set of solutions using a single trained model is an attractive feature for practical multi-objective scheduling.", "weaknesses": "The four considered objectives (makespan, total tardiness, average flow time, and total cost) are not well justified. The first three are closely correlated regular measures, while the inclusion of total cost lacks sufficient motivation or explanation regarding its relevance in real-world FJSPs.\n\nThe paper does not analyze the interrelationship among the four objectives, which limits understanding of the trade-offs the model learns.\n\nWhile the paper presents an interesting approach, the overall contribution remains unclear. It would be helpful to explicitly highlight what new insight or technical novelty it brings beyond existing multi-objective scheduling and NCO frameworks.\n\nThe method uses the lower bound of completion time as the state and reward, but this can cause instability in learning depending on the tightness of the bound. This issue should be discussed in detail.\n\nWhile several performance-enhancing techniques are proposed, the paper lacks an ablation study to quantify their individual contributions.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)