Discrete-Event Optimization

• Discrete-Event Optimization is a methodological framework for decision-making in systems where random events drive dynamics.
• It encompasses diverse formulations such as supervisory control, queueing-network management, and dynamic combinatorial optimization with a focus on probability-based optimality.
• Algorithmic paradigms like the Omega Median Algorithm and differentiable simulation enhance efficiency by reducing computational complexity in real-world applications.

Discrete-Event Optimization (DEO) addresses decision-making in discrete-event systems (DES) whose dynamics are driven by events such as arrivals, departures, orders, failures, road-condition changes, automation transitions, or service completions. In these systems, the performance of a decision depends on stochastic sample paths generated by random events, while closed-form analysis is typically infeasible; simulation-based optimization is therefore a main tool. The literature also uses DEO to denote language-constrained supervisory synthesis, event-triggered combinatorial re-optimization, queueing-network control, and simulation-driven design or calibration, all of which share the central feature that optimization is coupled to event-driven system evolution (Mao et al., 2016, Matsui et al., 2024, Skackauskas et al., 2023, Che et al., 2024).

1. Formal problem structure

In a canonical stochastic DEO formulation, the feasible solution set is $S$, a random outcome $\omega$ indexes a sample path of the DES, and the performance of decision $s \in S$ on that path is $J(s;\omega)$. The classical stochastic objective is

$$s_E \in \arg\min_{s \in S} \; \mathbb{E}\big[J(s;\omega)\big].$$

Because $\mathbb{E}[J(s;\omega)]$ is rarely available in closed form, the standard computational pattern is repeated simulation and search over $S$. The resulting burden is significant under stochasticity, path dependence, nonstationarity, and the $O(M\cdot I\cdot C)$ complexity of expectation-based simulation optimization, where $M$ is the number of replications per solution, $I$ the number of explored solutions, and $\omega$0 the evaluation cost per trajectory (Mao et al., 2016).

The same DEO label also covers formulations in which the decision object is not a scalar control but a language-constrained supervisory policy. In distributed secret securing, each local agent is modeled by an automaton

$\omega$1

global secrets are tuples of local secret states,

$\omega$2

and the optimization problem is to synthesize protection policies $\omega$3 that make at least one piece of each distributed secret sufficiently protected while minimizing the maximum cost class used (Matsui et al., 2024).

A further formulation appears in event-triggered dynamic combinatorial optimization. There, an event at time $\omega$4 changes objective coefficients, capacities, or constraint coefficients, so the problem is constant on intervals $\omega$5 and changes only at event times. The Dynamic Multidimensional Knapsack Problem is written as a sequence of static states with

$\omega$6

and DEO behavior is assessed across successive state transitions rather than only within one static instance (Skackauskas et al., 2023).

This range of formulations suggests that DEO is best understood as a methodological class rather than a single optimization template: the common structure is that decisions must respect event-driven dynamics, and the objective is evaluated through the induced event evolution.

2. Optimality notions and objective criteria

A distinctive contribution within DEO is the replacement of “optimality in expectation” by “optimality in probability.” In that framework, the preferred decision is not the one with the lowest average cost, but the one whose realized cost is more likely to be no worse than any competitor. The champion solution $\omega$7 is defined by

$\omega$8

The threshold $\omega$9 is fundamental: for continuous sample-path randomness, requiring a strict threshold $s \in S$0 would contradict the champion’s own dominance probability, so $s \in S$1 is the maximal uniform threshold that can hold against all competitors. In discrete settings, ties are natural, and both

$s \in S$2

may exceed $s \in S$3 simultaneously (Mao et al., 2016).

The champion solution privileges winning frequency over winning margin. A solution can be expectation-optimal but not champion, or champion but not expectation-optimal. The two criteria coincide under a non-singularity condition: $s \in S$4 The paper explicitly notes that pathological heavy-tail singularities can break this alignment, and that in dynamic or nonstationary environments with short decision horizons, optimality in probability can be more meaningful because it mitigates expectation bias induced by rare events or regime mixtures that are irrelevant for immediate decisions (Mao et al., 2016).

Across the broader DEO literature, objective criteria vary substantially with the underlying DES and decision semantics.

Setting	Objective criterion	Paper
Stochastic DES optimization	$s \in S$5 or champion dominance probability	(Mao et al., 2016)
Distributed secret securing	Minimize maximum cost class $s \in S$6 subject to secure reachability	(Matsui et al., 2024)
Event-triggered combinatorial DOPs	Minimize average result gap and post-event gap slip	(Skackauskas et al., 2023)
Cockpit HMI design	Reduce cognitive overload, perceptual overload, and EOR while maintaining SA	(Yuskevich et al., 2020)
ED DES calibration	Minimize integrated ECDF discrepancy subject to mean/SD tolerances	(Santis et al., 2021)
Queueing-network control	Minimize $s \in S$7	(Che et al., 2024)

This heterogeneity is not incidental. It reflects the fact that DEO objectives inherit the semantics of the underlying event system: trajectory-wise costs in stochastic DES, language feasibility in SCT, event-to-event degradation and recovery in dynamic combinatorial optimization, and stability-sensitive holding costs in queueing networks.

3. Algorithmic paradigms

For scalar DEO problems with per-scenario unimodality, the Omega Median Algorithm (OMA) provides a direct computational route to the champion solution. For each sampled scenario $s \in S$8, one solves the deterministic $s \in S$9-problem

$J(s;\omega)$0

to obtain an $J(s;\omega)$1-solution

$J(s;\omega)$2

The estimated champion is then the median of the empirical distribution of $J(s;\omega)$3. In the discrete case, one uses

$J(s;\omega)$4

and selects $J(s;\omega)$5 such that $J(s;\omega)$6 and $J(s;\omega)$7. If for every $J(s;\omega)$8, $J(s;\omega)$9 is a scalar unimodal function of $$s_E \in \arg\min_{s \in S} \; \mathbb{E}\big[J(s;\omega)\big].$$0, then the $$s_E \in \arg\min_{s \in S} \; \mathbb{E}\big[J(s;\omega)\big].$$1-median is a champion solution. OMA is consistent by the strong law of large numbers, admits exponential concentration, and reduces complexity from $$s_E \in \arg\min_{s \in S} \; \mathbb{E}\big[J(s;\omega)\big].$$2 to $$s_E \in \arg\min_{s \in S} \; \mathbb{E}\big[J(s;\omega)\big].$$3; in the inventory application, this becomes $$s_E \in \arg\min_{s \in S} \; \mathbb{E}\big[J(s;\omega)\big].$$4 instead of $$s_E \in \arg\min_{s \in S} \; \mathbb{E}\big[J(s;\omega)\big].$$5 (Mao et al., 2016).

A different simulation-based DEO pattern is calibration by Sample Average Approximation and derivative-free constrained optimization. In emergency-department calibration, the decision variables are 46 granular Weibull parameters for triage, medical visit, and exams; the objective is a sum of $$s_E \in \arg\min_{s \in S} \; \mathbb{E}\big[J(s;\omega)\big].$$6 distances between simulated and real ECDFs of Door-to-Doctor Time and Doctor-to-Discharge Time, with relative-deviation constraints on means and standard deviations. The implementation uses $$s_E \in \arg\min_{s \in S} \; \mathbb{E}\big[J(s;\omega)\big].$$7 replications per design point, a derivative-free algorithm for integer black-box constrained problems, and a stopping condition of 3000 function evaluations (Santis et al., 2021).

Recent work extends DEO into differentiable simulation. For queueing-network control, a differentiable DES is obtained by keeping the forward event dynamics exact,

$$s_E \in \arg\min_{s \in S} \; \mathbb{E}\big[J(s;\omega)\big].$$8

while smoothing only the backward pass via straight-through gradients based on softmin. The resulting pathwise policy gradient is computed with reverse-mode autodiff in $$s_E \in \arg\min_{s \in S} \; \mathbb{E}\big[J(s;\omega)\big].$$9 time, and the paper reports that the gradient estimators are several orders of magnitude more accurate than typical REINFORCE-based estimators, with $\mathbb{E}[J(s;\omega)]$0 sample-efficiency gains over state-of-the-art RL methods (Che et al., 2024).

In event-triggered combinatorial DEO, the algorithmic emphasis shifts to rapid post-event adaptation. “ACO with Aphids” keeps a persistent aphid state $\mathbb{E}[J(s;\omega)]$1 across events, relocates it using current-state heuristics,

$\mathbb{E}[J(s;\omega)]$2

seeds pheromone by

$\mathbb{E}[J(s;\omega)]$3

and then updates aphids after search by kill and laydown operators. The additional aphid steps cost $\mathbb{E}[J(s;\omega)]$4 per event, while the ACO core remains $\mathbb{E}[J(s;\omega)]$5 for MKP-like models (Skackauskas et al., 2023).

At the DES execution level rather than the decision level, compile-time event batching enlarges the scope for optimization by generating contiguous procedures for event sequences of length up to $\mathbb{E}[J(s;\omega)]$6, with batch count

$\mathbb{E}[J(s;\omega)]$7

This allows cross-event inlining, constant propagation, and dead-code elimination that standard event-handler granularity obscures. The observed runtime overhead of the runtime batch selection mechanism is about $\mathbb{E}[J(s;\omega)]$8 when average batch length is 2 (Leinweber et al., 2018).

These paradigms are complementary rather than competitive. OMA exploits per-scenario tractability and unimodality, SAA+DFO targets calibration under black-box noise, differentiable DES uses pathwise gradients for high-dimensional control, event-triggered metaheuristics focus on inter-state adaptation, and simulator-level batching attacks the computational overhead of DES evaluation itself.

4. Representative application domains

The inventory-control example in the champion-solution framework is a periodic-review DES with full backlogging, fixed setup cost $\mathbb{E}[J(s;\omega)]$9, nonstationary demand, and dynamics

$S$0

The finite-horizon rolling objective is

$S$1

with

$S$2

The paper proves that, conditioned on ordering, $S$3 is $S$4-convex in $S$5 and $S$6 is convex in $S$7, so the unimodality condition required by OMA holds. In the reported experiments, $S$8, $S$9, and $O(M\cdot I\cdot C)$0; demands are Poisson with $O(M\cdot I\cdot C)$1 drawn from $O(M\cdot I\cdot C)$2. The empirical cdf from $O(M\cdot I\cdot C)$3 $O(M\cdot I\cdot C)$4-solutions yields a median around 78 in a representative instance, and the median stabilizes within $O(M\cdot I\cdot C)$5 samples. Under stationary demand, the champion estimate was 1.03% worse on average than the exact optimal static $O(M\cdot I\cdot C)$6 policy but beat it in 10 of 20 instances. Under nonstationary demand, the $O(M\cdot I\cdot C)$7-median method with $O(M\cdot I\cdot C)$8 and $O(M\cdot I\cdot C)$9 reduced average cost by 14.52% relative to a per-period $M$0 heuristic and won in all 20 instances, with improvements ranging from 5.54% to 22.20% (Mao et al., 2016).

Human-centered and healthcare DEO provide a different profile. In autonomous-vehicle cockpit design, a SimPy process-interaction DES models road-condition changes, automation transitions, cognitive-function triggers, task start/finish events, resource contention across seven attentional modalities, and metrics for workload, eyes-off-the-road time, and situation awareness. Across 20 replications per design, each of 1000 virtual minutes, the optimized architecture reduced EOR from 11.7% to 10.3%, cognitive overload from 0.87% to 0.57%, and perceptual overload from 0.19% to 0.13%, while SA changed from 92.1% to 92.3% (Yuskevich et al., 2020). In emergency-department calibration, the DEO task is not operational control but parameter identification: the calibrated DES recovers missing service-time parameters for a large Italian ED by matching ECDFs, moments, and occupancy patterns under NHPP arrivals and detailed unit routing (Santis et al., 2021).

Manufacturing and event-triggered combinatorial settings extend DEO beyond conventional DES control. In single-machine machining, DEO integrates order sequencing, machining stability, and learning-based tool-life prediction. The total cost is

$M$1

with $M$2 incorporating processing time and tool changes. In experiments on instances up to $M$3, the integrated cost-only model reduced total cost by 3.73–5.06% versus the baseline Expected Cost Method, while the due-date formulation solved 98 of 100 instances to proven optimality or with gaps no larger than 0.76% on the hardest cases (Ma et al., 2022). In Dynamic Multidimensional Knapsack benchmarks with 55 datasets, 101 states each, and SAM in $M$4, ACO with Aphids achieved an average result gap of 0.519%, compared with 0.733% for Pheromone-Sharing and 1.092% for Full-Restart, corresponding to average gap reductions of 29.2% and 52.5%, respectively (Skackauskas et al., 2023).

Queueing-network control shows DEO at larger scale and under stronger stochasticity. The differentiable DES framework handles multi-class queueing networks with non-exponential and non-stationary interarrival and service distributions, using exact forward event selection with smoothed backward sensitivities. The reported empirical outcome is a $M$5 improvement in sample efficiency over PPO- or REINFORCE-based baselines, while maintaining support for realistic nonstationary environments (Che et al., 2024). At the opposite end of abstraction, a grid-based plant–herbivore simulator frames defense, reproduction, and signaling as a computational optimization game in which settings with low energy need and long life under different attack patterns can be analyzed, with proposed applications to sensor-network construction and secure operation (Dietrich et al., 19 Sep 2025).

5. Supervisory-control and language-based DEO

A major DEO strand is formulated directly over DES languages. In distributed secret securing, the optimization problem is to find local protection policies that ensure that for every global secret tuple, at least one local secret state is $M$6-securely reachable with respect to some cost prefix $M$7, while minimizing the overall maximum cost class index $M$8. The paper gives a necessary-and-sufficient solvability condition—Problem DSSP is solvable iff the secret set is secure with respect to $M$9 and $I$0—and a polynomial-time synthesis algorithm, Distributed Reachability Control with Minimum Costs, with complexity

$I$1

The construction relies on supremal controllable sublanguages and a relabelling technique that counts distinct protected transitions without exponential parallel composition (Matsui et al., 2024).

Coordination-control theory adds another optimization layer: the size of the coordinator event set. For a specification language $I$2, one seeks a minimal extension $I$3 such that $I$4 becomes conditionally decomposable,

$I$5

The decision version of this problem, CD MIN EXTENSION, is NP-complete, and the optimization problem of finding a minimum-size coordinator is NP-hard. The paper explicitly states that, unless $I$6, no polynomial algorithm exists to compute the minimal coordinator with respect to the number of events (Komenda et al., 2013). Here, DEO is combinatorial and structural: the optimized object is the communication architecture needed for feasible distributed supervision.

Production planning under SCT illustrates how language feasibility and temporal optimization can be combined without abandoning formal safety guarantees. Safe production sequences are exactly the closed-loop language $I$7, and two heuristics—PMT and HMM—search this space by maximizing parallelism or directly minimizing elapsed time. In the flexible-manufacturing-system case study, the monolithic supervisor has 45,504 states and 200,124 transitions. HMM matches the optimal makespan formula

$I$8

for all tested batch sizes up to $I$9; for $\omega$00, PMT returns a sequence in about 32 seconds and HMM in about 97 seconds, whereas model checking runs out of memory beyond much smaller instances (Alves et al., 2019).

These works make clear that DEO is not confined to simulation-based numerical search. It also includes supervisory synthesis, event-set design, and trajectory selection over closed-loop languages, with optimality defined by secure reachability, nonblockingness, makespan, or coordinator size.

6. Computational limits, assumptions, and research directions

The champion-solution framework is powerful but narrow in its guarantees. Champion existence can fail when there are more than two candidate solutions and unimodality is absent; the paper describes Arrow-style cycles as a failure mode. The guarantees are developed for scalar decisions, whereas multi-dimensional settings would require a coherent median rule and stronger conditions such as coordinate-wise unimodality. OMA also assumes that scenarios are sampled from the current demand model; biased or misspecified scenario generation can make the $\omega$01-median unrepresentative of the true near-term champion. Finally, its offline solver must remain efficient, otherwise the computational gain over expectation-based search diminishes (Mao et al., 2016).

Differentiable DES introduces a different trade-off: smoothing is used only in the backward pass to preserve forward fidelity, but it still introduces bias, and the queueing formulation in the paper assumes that each queue has exactly one compatible server for simplicity. Very long horizons can accumulate small smoothing errors, and autodiff memory can become large without checkpointing (Che et al., 2024). Event-triggered metaheuristics are also limited by representation quality: ACO with Aphids relies on reasonably predictive heuristics, and the tuned parameter set $\omega$02 is problem-specific; the paper notes that weak heuristics or poor tuning can reduce benefits versus Pheromone-Sharing (Skackauskas et al., 2023).

Several application papers make their own restrictions explicit. The cockpit-design DES assumes no concurrent use of the same modality, aggregate cognitive and perceptual workload caps of 10, deterministic task durations and gaze times, and no driver fatigue, learning, or human-subject validation (Yuskevich et al., 2020). The machining models are single-machine, all-orders-released-at-zero formulations with deterministic tool life, fixed $\omega$03, and speed changes only between orders (Ma et al., 2022). The distributed-secret framework minimizes the maximal cost class rather than total